Study of a highly integrated approach for lightweight bicycle frame

 ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI) GRADO EN INGENIERÍA ELECTROMECÁNICA
Especialidad Mecánica Study of a highly integrated approach for lightweight bicycle frame design by multi‐objective material selection and CAE tools Author: Lomas Benito, Javier Director: Fabrizio D’Errico Madrid 27/08/2015 2 3 AUTORIZACIÓN PARA LA DIGITALIZACIÓN, DEPÓSITO Y DIVULGACIÓN EN ACCESO ABIERTO (RESTRINGIDO) DE DOCUMENTACIÓN 1º. Declaración de la autoría y acreditación de la misma. El autor D. Javier Lomas Benito, como alumno de la UNIVERSIDAD PONTIFICIA COMILLAS (COMILLAS), DECLARA que es el titular de los derechos de propiedad intelectual, objeto de la presente cesión, en relación con la obra ‘Study of a highly integrated approach for lightweight bicycle frame design by multi‐
objective material selection and CAE tools’ (Proyecto Fin de Grado) 1, que ésta es una obra original, y que ostenta la condición de autor en el sentido que otorga la Ley de Propiedad Intelectual como titular único o cotitular de la obra. En caso de ser cotitular, el autor (firmante) declara asimismo que cuenta con el consentimiento de los restantes titulares para hacer la presente cesión. En caso de previa cesión a terceros de derechos de explotación de la obra, el autor declara que tiene la oportuna autorización de dichos titulares de derechos a los fines de esta cesión o bien que retiene la facultad de ceder estos derechos en la forma prevista en la presente cesión y así lo acredita. 2º. Objeto y fines de la cesión. Con el fin de dar la máxima difusión a la obra citada a través del Repositorio institucional de la Universidad y hacer posible su utilización de forma libre y gratuita ( con las limitaciones que más adelante se detallan) por todos los usuarios del repositorio y del portal e‐ciencia, el autor CEDE a la Universidad Pontificia Comillas de forma gratuita y no exclusiva, por el máximo plazo legal y con ámbito universal, los derechos de digitalización, de archivo, de reproducción, de distribución, de comunicación pública, incluido el derecho de puesta a disposición electrónica, tal y como se describen en la Ley de Propiedad Intelectual. El derecho de transformación se cede a los únicos efectos de lo dispuesto en la letra (a) del apartado siguiente. 3º. Condiciones de la cesión. Sin perjuicio de la titularidad de la obra, que sigue correspondiendo a su autor, la cesión de derechos contemplada en esta licencia, el repositorio institucional podrá: (a) Transformarla para adaptarla a cualquier tecnología susceptible de incorporarla a internet; realizar adaptaciones para hacer posible la utilización de la obra en formatos electrónicos, así como incorporar metadatos para realizar el registro de la obra e incorporar “marcas de agua” o cualquier otro sistema de seguridad o de protección. (b) Reproducirla en un soporte digital para su incorporación a una base de datos electrónica, incluyendo el derecho de reproducir y almacenar la obra en servidores, a los efectos de garantizar su seguridad, conservación y preservar el formato. . 1
Especificar si es una tesis doctoral, proyecto fin de carrera, proyecto fin de Máster o cualquier otro trabajo que deba ser objeto de evaluación académica 4 (c) Comunicarla y ponerla a disposición del público a través de un archivo abierto institucional, accesible de modo libre y gratuito a través de internet.2 (d) Distribuir copias electrónicas de la obra a los usuarios en un soporte digital. 3 4º. Derechos del autor. El autor, en tanto que titular de una obra que cede con carácter no exclusivo a la Universidad por medio de su registro en el Repositorio Institucional tiene derecho a: a) A que la Universidad identifique claramente su nombre como el autor o propietario de los derechos del documento. b) Comunicar y dar publicidad a la obra en la versión que ceda y en otras posteriores a través de cualquier medio. c) Solicitar la retirada de la obra del repositorio por causa justificada. A tal fin deberá ponerse en contacto con el vicerrector/a de investigación ([email protected]). d) Autorizar expresamente a COMILLAS para, en su caso, realizar los trámites necesarios para la obtención del ISBN. d) Recibir notificación fehaciente de cualquier reclamación que puedan formular terceras personas en relación con la obra y, en particular, de reclamaciones relativas a los derechos de propiedad intelectual sobre ella. 5º. Deberes del autor. El autor se compromete a: a) Garantizar que el compromiso que adquiere mediante el presente escrito no infringe ningún derecho de terceros, ya sean de propiedad industrial, intelectual o cualquier otro. b) Garantizar que el contenido de las obras no atenta contra los derechos al honor, a la intimidad y a la imagen de terceros. c) Asumir toda reclamación o responsabilidad, incluyendo las indemnizaciones por daños, que pudieran ejercitarse contra la Universidad por terceros que vieran infringidos sus derechos e intereses a causa de la cesión. d) Asumir la responsabilidad en el caso de que las instituciones fueran condenadas por infracción de derechos derivada de las obras objeto de la cesión. 2
En el supuesto de que el autor opte por el acceso restringido, este apartado quedaría redactado en los siguientes términos: (c) Comunicarla y ponerla a disposición del público a través de un archivo institucional, accesible de modo restringido, en los términos previstos en el Reglamento del Repositorio Institucional 3
En el supuesto de que el autor opte por el acceso restringido, este apartado quedaría eliminado. 7 ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI) GRADO EN INGENIERÍA ELECTROMECÁNICA
Especialidad Mecánica Study of a highly integrated approach for lightweight bicycle frame design by multi‐objective material selection and CAE tools Author: Lomas Benito, Javier Director: Fabrizio D’Errico Madrid 27/08/2015 8 9 ESTUDIO E INTEGRACIÓN DE LA SELECCION DE MATERIALES MULTI‐
OBJETIVO Y HERRAMIENTAS CAD APLICADOS AL DISEÑO DE UN CUADRO DE BICI LIGERO Autor: Lomas Benito, Javier Director: Fabrizio D’Errico Entidad colaboradora: Politecnico di Milano RESUMEN DEL PROYECTO El presente proyecto tiene como objetivo el estudio y diseño de un cuadro de bicicleta ligero, prestando especial atención a la selección de materiales asistida por herramientas CAD. La selección de materiales para una pieza u objeto es una parte más del proyecto de diseño de un producto. Generalmente, los procesos de selección de materiales se basan en métodos derivativos, también conocidos como explícitos; o alternativamente, en métodos no derivativos o implícitos. Los primeros son cuantitativos mientras que los segundos son cualitativos. Una de las innovaciones de este proyecto consiste en el empleo de un método hibrido multi‐
objetivo para la selección de materiales. Este método permite disfrutar ciertas ventajas de los métodos derivativos, como la aplicación de los índices Ashby, junto con la simplicidad y facilidad de manejo que caracteriza a la matriz QFD (Quality Function Deployment). En definitiva, se trata de un método que tiene como objetivo la integración multidisciplinar de equipos. Es decir, que el responsable de marketing, tomando un ejemplo genérico, y el ingeniero de mecánica de fractura pueden aportar su visión/concepción a cerca de las características del producto en una misma herramienta de trabajo. La segunda gran línea innovativa del proyecto es la aplicación de la optimización topológica al diseño o mejora de un producto, en nuestro caso el cuadro de una bicicleta. La optimización topológica es un tipo de optimización estructural que se desarrolló e implementó a nivel de software hace a penas diez años y que, hoy en dia, la están empezando a introducir las grandes empresas para el diseño de nuevos productos o la mejora de productos ya existentes. Partiendo de un espacio de trabajo (“ground structure”), de unas condiciones de contorno y de unas cargas externas; la optimización topológica encuentra la estructura más rigida posible con un límite de masa impuesto por el ingeniero. Ciclo de trabajo de la optimización topológica 10 La tercera parte innovativa y realmente pionera del proyecto es la integración de las dos anteriores: la selección de materiales híbrida multi‐objetivo y la optimización topológica. En este proyecto se ha buscado, a través de la aplicación real a un cuadro de bicicleta, mostrar como se pueden aprovechar los resultados de una optimización topológica para hacer aun más eficiente un proceso de selección de materiales. Para ello, se ha estructurado la tesis en los siguientes capítulos: Capítulo 1. Introducción a la selección de materiales Se da la base teórica con ejemplos prácticos del método derivativo y del método implícito. A continuación, se introduce la matriz QFD (Quality Function Deployment) en la que se basará posteriormente el método hibrido de selección de materiales. Se explica la interpretación de resultados a través de los dos gráficos principales: curvas de utilidad y mapas de burbujas. Finalmente, se aplica el método híbrido QFD4Mat para una selección inicial de materiales para nuestro caso de estudio: marco de una bicicleta. Capítulo 2. Herramientas modernas de diseño asistido por ordenador: optimización topologica El capítulo segundo es completamente teórico. Primero se introduce el concepto de optimización y su aplicación al diseño mecánico. A continuación, se desarrolla la optimización topológica. Se presenta la formulación del problema de forma conceptual y matemática; se explican los fundamentos matemáticos del método de la densidad (distribución de la masa) y, por ultimo, se explica la implementación a nivel computacional. En la segunda parte del capítulo se desarrolla la influencia de los procesos de manufactura en la optimización topológica. Concretamente, se especifica el proceso de fabricación de un cuadro de bicicleta concluyendo las posibles repercusiones del mismo en la optimización del cuadro. Capítulo 3. Definicion de la forma de un cuadro de bicicleta a través de la optimización topológica En primer lugar se efectúa el análisis FEM estructural de un cuadro de bicicleta genérico dimensionado para un ciclista estándar, 80 kg de peso y 1.80 m de altura. Con este primer análisis se busca justificar la elección de los índices Ashby, de la primera matriz QFD4Mat completada en el capítulo uno, y dar una idea de los órdenes de magnitud, para tensiones y desplazamientos, con los que se trabaja en un estudio estático lineal de un cuadro de bici. Posteriormente, se realiza la optimización topológica. Se hace hincapié en la importancia del mallado del FEM, proponiendo tres mallados distintos del espacio de trabajo. Con el último mallado, el más preciso, se imponen las cargas y las condiciones de contorno; y se procede a la ejecución numérica de la optimización, así como a la interpretación de resultados. Se llevan a cabo tres análisis, con objetivos de masa 40,30 y 20 % respectivamente, comentando los resultados para cada uno de ellos. En la tercera parte del capitulo se plantea el rediseño del cuadro de bicicleta considerando los resultados de la optimización topológica 2‐D y del análisis FEM inicial. Se consideran todos los tubos del cuadro como circulares y se varía el espesor y/o el diámetro externo de cada uno en función de los resultados obtenidos. Finalmente, se realiza el análisis FEM estructural del nuevo cuadro optimizado para cada material candidato. Se comprueban, en cada caso, los límites estructurales y de servicio. Se concluye con una tabla comparativa de los resultados más significativos. En dicha tabla aparecen el factor de seguridad, la flecha máxima y la masa total del cuadro, para cada material. 11 Capítulo 4. Redefinición de la matriz QFD4Mat: resultados finales En este capítulo final, se introducen los nuevos tres “key‐factors” definidos al final del capítulo tercero y se deja de considerar el Global Performance Index (GPI). Con la introducción de los nuevos “key‐factors” otros equivalentes se ven sustituidos. La masa total entra como “key‐
factor” de “performance” y como “key‐factor” de costes; en ambos casos con objetivo de minimización. En el caso de “performance” substituye a la resistencia específica, manteniendo las mismas relaciones que esta respecto a los “product requirements”. En el caso del “safety factor” (SF) entra substituyendo al limite elástico (“Yield strength”); con objetivo maximización. De nuevo, se mantienen las correlaciones del límite elástico para el SF respecto a los “product requirements”. La flecha máxima substituye al modulo de Young como índice de rigidez; con objetivo minimización. Una vez más, se mantienen las correlaciones existentes. Finalmente, se analizan los gráficos de burbujas y curvas de valor y se concluye que el material óptimo para la fabricación del cuadro de bicicleta optimizado es el Titanio 3Al‐2.5V (Grade 9), recocido alfa. Para cerrar la tesis, se exponen las conclusiones y se establecen las posibles líneas de mejora. 12 13 STUDY OF A HIGHLY INTEGRATED APPROACH FOR LIGHTWEIGHT BICYCLE FRAME DESIGN BY MULTI‐OBJECTIVE MATERIAL SELECTION AND CAE TOOLS Author: Lomas Benito, Javier Director: Fabrizio D’Errico Collaboration entity: Politecnico di Milano SUMMARY OF THE PROJECT The present thesis scope is the study and design of a lightweight bike frame, employing the material selection assisted by CAD tools. The material selection for a work piece or an object is an important part of the whole product design project. Generally, the material selection processes are based on derivative methods, also known as explicit; alternatively, on non‐derivative or implicit methods. The first ones are quantitative while the second ones are qualitative. One of the innovative ideas of this thesis work is the implementation of the hybrid multi‐
objective method for material selection. This method combines the advantages of derivative methods, like Ashby indices, with the simplicity and user‐friendly interface that characterizes the QFD (Quality Function Deployment) matrix. In short, the hybrid multi‐objective approach has the scope of the interdisciplinary team integration. Actually, the marketing manager, example given, and the fracture mechanics engineer could contribute to the vision of the product characteristics with the same working tool. The second great innovative idea of the thesis work is the implementation of the topological optimization in the conceptual design of a product, in our case of study the bicycle frame. Topological optimization is a kind of structural optimization developed at software level environments nearly ten years ago and, nowadays, it is being introduced by big companies for new design products or the improvement of the ones already launched into the market. Initially defining a design space, boundary conditions and the external loads; topology optimization finds the stiffest structure feasible with the weight spare percentage assessed by the engineer. Topology optimization workflow 14 The third innovative idea and the real pioneer driving force of the thesis work is the integration of the two previous ones: hybrid multi‐objective material selection and topology optimization. The present thesis work aims, by means of the real application to a bike frame, for showing how the results of topological optimization can be exploited for increasing the efficiency of the material selection process. For that goal, the thesis work has been divided into the following chapters: Chapter 1. Introduction to Material Selection strategy by the QFD4Mat approach In this chapter the theory base is given, explained with some practical examples of the derivative method and implicit method. Therefore, the QFD (Quality Function Deployment) matrix is introduced. The targeted hybrid multi‐objective material selection criteria will be based on the named matrix. The main results graphics are explained: value curves and bubble maps. Finally, the QFD4Mat hybrid method is applied for a first‐approach material selection for our case study: bicycle frame. Chapter 2. Modern computer aided approaches for optimal geometry: topology optimization The second chapter is completely theoretical. First of all, optimization concept and his application to mechanical design are introduced. Therefore, topology optimization is explained. The problem formulation is presented in a conceptual and mathematical manner. The density method (Power law) is explained and, thereafter, the computational procedure is illustrated. In the second part of the chapter the influence of the manufacturing processes on topology optimization is explained. Factually, the bike frame’s manufacturing process is illustrated, deciding the possible consequences of it into the frame’s optimization. Chapter 3. Topology optimization for shape definition of a BIKE FRAME Firstly, FEM analysis of a generic bike frame is carried out. The frame is sized for a standard cyclist, 80 kg weight and 1.8 m height. This first analysis aims to justify the Ashby indices selection of the first filled in QFD4Mat matrix in chapter one, and giving an overview about the orders of magnitude typical of a linear static study of a bike frame. Thereafter, topology optimization is carried out. The importance of the mesh is highlighted, proposing three different meshes of the workspace. Taking the last mesh, the most precise, loads and boundary conditions are defined, proceeding afterwards to the numerical execution of the optimization and, finally, the results are commented. In the third part of the chapter the bike frame is redefined considering the results of 2‐D topology optimization and the initial FEM analysis. All the frame tubes are considered as circular, the thickness and diameter of each one are taken as design variables. Finally, a structural FEM analysis of the optimized bike frame is carried out for each candidate material. The Ultimate Limit State (ULS) and the Serviceability Limit State (SLS) are checked and the chapter is concluded with a most significant results table. In that table, the following features appear: Safety Factor (SF), maximum displacement and bike frame total mass, for each material. 15 Chapter 4. Refinement of QFD4Mat analysis: final results and discussions In this final chapter the new key‐factors, defined at the end of the third chapter, are introduced and the GPI (Global Performance Index) is not considered any longer. With the introduction of the new key‐factors, others equivalent are substituted. The total mass is introduced as performance key‐factor and cost key‐factor; in both cases with minimization objective. In the case of performance, it substitutes the specific strength keeping the same correlations respect to the product requirements. The safety factor enters substituting the yield strength, with maximization objective. Once again, the correlations respect the product requirements of the yield strength are kept. The maximum displacement substitutes the Young’s modulus as stiffness index, with minimization objective. One more time, the existent correlations are kept. Finally, the bubble maps graphic and the value curves graphic are analyzed and the Titanium 3Al‐2.5V (Grade 9), alpha annealed, is selected as the optimal material for the manufacturing process of the optimized bike frame. To end the thesis work, the conclusions are shown and possible improvement work lines are pointed out. 16 17 Tableofcontents
1. Introduction to Material Selection strategy by the QFD4Mat approach ..................... 19 1.1. The Material Selection methodologies .............................................................................. 19 1.1.1. Derivative methods. The Ashby approach and “the free search” .............................. 21 1.1.2. Non-derivative methods. Expert survey approach .................................................... 28 1.2. The multi-objective optimization analysis in material selection ....................................... 32 1.3. The QFD4Mat method in preliminary material screening out .......................................... 41 1.4. The assessment of material candidates by Graphic Analysis ............................................ 44 1.4.1. The Value Curve of products .................................................................................... 44 1.4.2. The Bubble Maps grafic tool ..................................................................................... 46 1.5. The scope of work: material optimization by refined QFD4MAT method for study case:
bike frame...................................................................................................................................... 51 1.5.1. Racing bicycle [2] ..................................................................................................... 51 1.5.2. Bike frame [2] ........................................................................................................... 51 1.6. The preliminary QFD4Mat analysis on candidate materials for BIKE FRAME .............. 53 1.7. Limits and constraints of preliminary QFD4Mat analysis ................................................ 61 2. Modern computer aided approaches for optimal geometry: topology optimization .. 63 2.1. Design optimization .......................................................................................................... 63 2.2. Introduction to topology optimization ............................................................................... 65 2.2.1. Problem formulation ................................................................................................. 66 2.2.2. Density Method Approach and mathematical formulation ....................................... 67 2.2.3. Computational procedure .......................................................................................... 68 2.3. The problem of shape constraints by manufacturing approach ......................................... 70 2.3.1. Introduction to manufacturing constraints. Application examples ............................ 70 2.3.2. Bike frame manufacturing process ............................................................................ 71 2.4. 3. Conclusions ....................................................................................................................... 75 Topology optimization for shape definition of a BIKE FRAME .................................. 77 3.1. Starting point: analysis of a generic bike frame ................................................................ 77 3.1.1. Problem formulation ................................................................................................. 77 18 3.1.2. FEM analysis ............................................................................................................. 79 3.1.3. Results ....................................................................................................................... 83 3.2. Topology optimization for shape definition of BIKE FRAME: software simulation ....... 87 3.2.1. Problem formulation ................................................................................................. 87 3.2.2. Topology optimization .............................................................................................. 92 3.2.3. Results ....................................................................................................................... 95 3.3. Final redefined bicycle frame geometry .......................................................................... 102 3.4. FEM analysis of the optimized geometry for each candidate material ........................... 110 3.4.1. AISI 4130 ................................................................................................................ 110 3.4.2. Aluminum 7005-T6 ................................................................................................. 115 3.4.3. Titanium 3Al-2.5V .................................................................................................. 116 3.5. 4. 4.1. Conclusions ..................................................................................................................... 117 Refinement of QFD4Mat analysis: final results and discussions ................................ 119 Translation of Performance Index by FEM results: implementing the Key-Material
Factors …………………………………………………………………………………………..119 4.2. Revising the QFD4Mat matrix ........................................................................................ 119 4.3. Update graphic solutions by Bubble Maps and Material Value Curves.......................... 122 4.4. Conclusions ..................................................................................................................... 125 Discussions, final conclusions and further improvements ................................................... 127 ANNEX A- Material Tables ................................................................................................... 129 ANNEX B- QFD4Mat website program ............................................................................... 131 Bibliography ............................................................................................................................ 135 19 1. Introduction to Material Selection strategy by the QFD4Mat approach 1.1.
The Material Selection methodologies4 Firstly it`s important to said that when we start a material selection process, we talk in terms of “strategy”. We talk about “strategy”, because the scope of material selection is not only related to define a selection method, in reality is enlarged to define plans and actions that will have on a long‐term a substantial impact on the success of a product on the market, and consequently on the success of firms and enterprises. These made material selection a complex and multifaceted challenge. In general aspects, we can say that a material selection strategy is performed having in account tree main consequential tasks:  The translation of customer or user needs (i.e. external analysis) for the product, influenced by material features into technical and non‐technical requirements (i.e. internal analysis)  On the basis of a technical and non‐technical set of requirements, namely the material key‐features inventory, the formulation of performance metrics to measure how well a material matches a set of requirements.  A search procedure, namely a structured material selection method, in order to: - explore a solution‐space - identify materials that meet the constraints and - rank them by their ability to meet the requirements. In fact, a selection strategy works by defining how it is possible to convert a set of inputs or “requirements” of the product, in to a set of outputs. In other words, material selection strategy researches the way of satisfy the costumers and users by accomplishing the requirements and satisfying the expectations. A process that is called in material selection strategy the translation of needs (or external) requirements into technical (or internal) requirements guarantees this phenomenon. It should be the conversion of the customer requirements in to the engineering needs: we pass from the common to the technical domain where we introduce internal metrics and targets in order to rank and screen various candidates. From that list, we will select the one that matches the targets and respects the technical (ex: safe behavior) and non‐technical (ex: price) constrains. It´s important to highlight that we have to make choices in a scarcity environment, because in the vast majority of the cases, we have fixed resources such as the cost budgeted or the manufacturing time, etc… which reduce the possibility of materials. Therefore, we need to seek the optimal solution as the most efficient and powerful compromise between all the aspects we define as important, not just a high performance solution regarding a few of these aspects. The best way to keep in mind all the features product materials should satisfy is to consider them as various aspects of a multifaceted problem. 4
Freely adjusted from the original book “Material Selections by a Hybrid Multi‐Criteria Approach ”,courtesy of author of the book Fabrizio D’Errico, academic tutor at Politecnico di Milano of this thesis work. 20 However, in all the cases we need to work with the same methodologies in order to stablish what aspects are important or not for arrive to the optimal solution which fulfill at the best all the requests. As we just said in the previous paragraph, material selection methodologies in engineering design are devoted to giving support to decisions often made in uncertainty in response to satisfy and manage multiple conflicting criteria. At the same time material selection methodologies are thus constructed to seek out the optimum choice of materials throughout a combination of certain key‐
factors ( controlled independent variables or design parameters) which permit to obtainment a product characterized by a number of desired properties (dependent variables, quality response characteristics, functional requirements). In general, the used planning structure is based into making decisions by Multi Objective Optimization5 (or, MOO) problems. We define a decision‐making problem as multi objective if its solution consists in identifying multiple objectives that somehow have to be combined in order to yield one final solution. Obviously, there is not a single solution which simultaneously maximizes all the technical or non‐
technical objectives. As we said before, in order to realize a competitive product, engineers are generally asked to pursue some macro objectives such as minimizing mass or minimizing volume having less material with a certain number of constraints. At this point, for make the choice of a specific material among varieties that can simultaneously fulfill all the macro objectives and respect all the constraints, we should consider that: -
The properties of different materials are in conflict with each other, and that we cannot simultaneously optimize all of them; Beyond a certain trade‐off, we arrive at the improvement of one feature of interest at the expense of worsening another and; The search for balance is reached by optimizing the resolution of the trade‐offs that exist between features and also by excluding solutions that are not respectful of constraints. The above boundaries are typical for a multi‐objective optimization problem. We discuss in the following paragraphs the two main approaches or methods that are applied to material selection strategy problems, dividing them into two macro groups: a) The derivative, or explicit, methods. b) The non‐derivative or implicit methods. 5
Multi‐objective optimization (also known multi‐criteria optimization) developed in the broad area of Multiple Criteria Decision Making also known as Multiple‐Criteria Decision Analysis (MCDA). MCDA is a sub‐discipline of Operations Research methodology founded in the UK during World War II explicitly to consider multiple criteria in decision‐making high complex environments like the military. Generalizing the Operational Research and derivative methods further developed concerns with mathematical optimization problems involving more than one objective function to be optimized simultaneously, so that decision makers can pursue the best choice.
21 1.1.1. Derivative methods. The Ashby approach and “the free search”6 A derivative method, or explicit method, for material selection starts from translating product requirements and needs into an objective function. An explicit method thus starts from the objective function and expresses itself in terms of material features – e.g. density, elasticity module, etc.– and other parameters which depend on design. Generalizing, such methods consists in writing out the explicit form of the objective function in such a way as to show how it depends on the material variables that can be used to rank the candidate materials. In order to reach a better understanding of the process, let’s consider the following typical classroom case of material screening among candidates. The case consists in selecting the best candidate material to reduce weight onboard a vehicle by substituting panels made of steel (as a baseline scenario), with magnesium alloy or aluminum alloy. The objective function therefore is to select out of the 3 screened materials the one that offers the lightest solution alternative to a real case scenario that is defined by the following design constraints: –
steel pan is the baseline scenario and it accounts for about 30kg (for the sake of simplicity, the case study considers a single large pan, while in reality 30kg is the total weight for around 5 pans) onboard and have length L, width a and thickness b; – the load on the baseline pan is fixed and there is a momentum Mf that bends the pan. Instead we conceive the redesign phase as something that will take advantage of a number of free options (namely, the ‘free‐variables’) left for the new component, such as: –
–
the thickness of the pan b, which is, however, limited to twice the baseline pan; the material, which means the material resistance limit. Two real candidates are therefore considered for substituting steel pan: a wrought magnesium alloy, a commercial ZW30 series, and an aluminum alloy, a commercial AA7075 heat treated alloy. The choice of two material family groups, magnesium and aluminum alloys, are selected because it is known that potentially these two material classes offer advantages in weightsaving strategies because of their density, i.e. mass per volume, lower than steel. As shown in table 1, the density of magnesium and aluminum alloys are respectively 22% and 34% that of steels. On the other hand, 4140 steel has a higher resistance limit7 than the two other candidate materials. This implies that we would expect to use a much greater material volume in order to realize new pan sections in either magnesium or aluminum that could support the same momentum Mf, namely the external load, as supported by the baseline steel pan. 6
Freely adjusted from the original book “Material Selections by a Hybrid Multi‐Criteria Approach “, courtesy of author of the book Fabrizio D’Errico, academic tutor at Politecnico di Milano of this thesis work. 7
In the case study, the resistance limit of a material is considered to be the resistance to bending fatigue, thus assuming the pan is loaded by this load type. This would not be true in a real case scenario, since the pan is not highly stressed by fatigue loading. 22 On the other hand, due to the lower density each kg of steel is expected to be substituted by a smaller mass of light alloy. This key‐concept in automotive sector literature that deals with vehicle weightsaving is known as the substitution factor. The meaning is: how many kg of a baseline scenario material can be substituted by 1 kg of a lighter material. As the reader can guess, it is not correct to calculate the substitution factor as the ratio between the density of the baseline steel and the density of the light material. This procedure does not take into account the lower resistance offered by lighter materials, as commented and shown in table 1. We need to take a further step. As stated, regarding design geometry constraints, the width a of a panel is fixed as is the length L of the panel, while the thickness b of the cross‐section is free. We can reduce the mass by simply reducing the cross‐section, but there is a constraint: the section‐area A must be sufficient to carry the bending moment Mf. We can solve this problem in this way. Considering the state of stress induced by the external momentum, Mf is easily calculated by the formula in : . 1 .2 The design load constraint that requires the pan to resist safely to the bending moment Mf can therefore be expressed by a relationship that states that the strength of material σ_f, the stress to failure, shall be higher than the maximum stress applied to panel, when it is supporting the external load bending momentum Mf. . 3 Combining the previous equations, we can rewrite the last expression as: . 4 . 5 ∙
∙
As our first step consists in calculating the substitution factor for alternative materials, we need to write the ratio between the mass m (obtained multiplying density ρ by section area A ‐ that is expressed by inverting the last equation ‐ by length of panel L, in case of constant thickness) of the alternative panel against the mass of the baseline panel, that is: (eq.6) ∙
/
/
Table 1 gathers the results of the calculation of the mass of alternative panels made in two new materials against baseline steel, considering the constraint of thickness for the alternative light panel that will be limited to twice the baseline steel panel. 23 Table 1. Comparison of weight saved onboard when either conventional
wrought magnesium or wrought aluminum are used to substitute a steel
pan loaded in bending mode to external momentum Mf [1]. So, the material which satisfies the objective function with a minimum value (lightweight design) is the magnesium alloy (type ZW30). The Ashby approach Classified as a derivative method for material selection strategy, the Ashby approach is recognized as the most generalized efficient and relatively user‐friendly method with which to perform quantitative analysis for material selection in engineering cases. In the following, the key‐steps of the Ashby approach are outlined in order to introduce readers with a non‐technical background to its peculiarities. As already illustrated in a previous section, derivative methods carry out quantitative analysis that begins by translating the product requirements and needs into an objective function. More precisely, the aim of the analysis consists in introducing a disciplined approach to a selection problem by identifying its distinguishing key features: what function the component will have, what constraints to take into account, what objectives to target and which free variables to steer through in order to accommodate optimized solutions. Firstly, the analysis process starts from a consideration of the component’s function. Many simple engineering functions can be described in single words or short phrases, unless we need to explain the function in detail. For example, in designing a new lightweight panel for the exterior of a vehicle, the objective function in common language would be defined as: “find me a material and a proper thickness for a panel of length L and width a to support a bending load momentum Mf safely and to make it as light as possible”. In engineering words, this means: as the geometry constraints of the panel, namely the surface dimensions, have been fixed by the design (the pan has primarily to fit the body chassis), the optimal choice of material will be the one that satisfies the major objectives: to resist an external load and keep the pan as light as possible. Seeing as the thickness of pan is a free variable, the screening process can actually count on two types of free‐variables, some pertaining to material features and only one – the thickness a – to geometry. As in any derivative method, here again we have to start firstly from an equation describing our objective for the panel, namely to minimize its mass. We can express the mass as reported here below, which we repeat for the sake of convenience: (eq.7) m ρ∙A∙L 24 It is the equation that targets our objective, so we call it the objective function of the problem. In the previous section we finally expressed eq.7 by design constraints and material free‐variables in the form: (eq.8) ∙
Thus, we start from this point and we express eq.7 as the function of material freevariables, strength σf and density ρ and two assigned design constraints, the momentum Mf and the panel width a. This can be obtained by simply turning eq.8 into: ∙
(eq.9) /
In this way, the optimization function originally written in eq.7 can finally be expressed by combining it with the eq.9. It now looks like this: (eq.10) ∙
∙
∙ 6 ∙
∙
Look at the objective function, in this new form. It is an expression of what was stated above: find a material, that is, act in choosing a specific material that can reduce the mass of the panel while respecting design constraints like the force applied, Mf, and geometry constraints, the width a and length L, by screening materials that have as low a ratio as possible between density ρ and the square root of the strength of the material σf1/2. For Ashby, the objective function can conveniently be written in three parentheses: i.
ii.
iii.
The first parenthesis contains the load constraints for the case, namely the function the component will have in carrying out its work The second parenthesis contains the specified and assigned geometry The last one contains material free‐variables that influence the objective function for the study case. That is: (eq.11) ,
; 25 The width a and the length L of the panel is specified by design, but we are free to choose the cross‐section area since the thickness b is a free variable. The objective is to minimize the mass of the panel, m. We thus write an equation for m (cf. eq. 7), which is the objective function we want to minimize. But there is a constraint: the panel must carry the bending load Mf without yielding in bending. Use this constraint (cf. eq. 3) to eliminate the free variable A and read off the combination of material properties in the last parenthesis to be minimized (cf. eq.10). Ashby calls the term in the final parenthesis, i.e., the one dependent on material properties, as the material index I of the problem that has to be minimized. Anyway, since it is most usual to think of a material index in a form by which when a maximum is sought, an optimum condition is achieved, Ashby prefers to invert the material properties in (eq.10) and define the material index I to maximize as: (eq.12) One enormous advantage introduced by Ashby’s approach is that the three groups of parameters in eq.11 are multiplied together, and can thus be considered separable. This fact implies firstly that the optimum subset of materials can be identified without solving the complete design problem – or even knowing all the details of F and G – and secondly that the overall performance is always maximized by maximizing the material indexes. Ranking materials by material indices Returning to table 1, we worked out a comparison of alternative solutions and baseline scenarios using a very general derivative method to calculate the material substitution factor; thus we recalculated a mass of alternative scenarios against the baseline scenario. Assessing the final weight saved onboard by alternative solutions (refer to last row of table 1), we stated that magnesium is the optimal solution for the case study. We actually ranked different materials to assess which is the best solution for our constraint problem. The Ashby method does the same, but in a more efficient way. It is not necessary to calculate material substitution factors, material by material. As stated in eq.10, or in its general form in eq.11, the optimal solution is the one characterized by the highest material index I calculated for the specific case. Such indices, each associated with maximizing some aspect of performance and, providing criteria that respect the design constrains assigned, permit a quick ranking of materials in terms of their ability to perform well in the given application. Therefore, what we did at this point was to screen candidates that are capable of doing the job, rank them and identify those among them that will function in the best possible way, as shown in Table 2. 26 Table 2. Comparative analysis by material performance index [1]. A wide category of engineering functional problems have been studied by Ashby as mater cases, in order to support the user with the proper index for his study case. The above method is really powerful since it allows for the screening of various candidates using an absolute quantifying approach. The but is, how can we select such candidates? In the study case we illustrated, the specialists, who had sufficient confidence with lightweight design, were aware of specialized literature indications and knew a lot about the basic features of different classes of materials, selected two possible candidates, the magnesium and aluminum alloys. Nevertheless, Ashby wanted his method to be an “open” approach that could be followed by non‐
specialists as well. He thought, such a method would be used both by people who are not experts in materials and also by those who are competent and want to screen among further unexplored solutions. To solve this problem, Ashby introduced the method by using indices of materials on material charts, as described below. The free‐search strategy: the material indeces on charts Figure 1 shows a strength σ plotted against a density ρ, on log scales. Fig. 1. The strength versus density and the mapping out area occupied by each material class [1]. 27 The material indices can be plotted onto the figure using the following procedure. The dotted line is a guideline for plotting lines with slope 2 on the log scale diagram that intercept, as regards each line a subset of materials with a constant value of material performance index. The higher the line with slope 2 is, the higher the material index investigated [1]. The material index = C can be written by taking logarithms of the first and second member: (eq.13) 2
2
Equation 13 is a family of straight parallel lines of slope 2 on the plot of Log (σ) against Log (ρ) and each line corresponds to a certain value of the constant C. Fig.1 shows the 2 slope guideline and the three parallel lines that intercept the three points in the chart identified by the coordinates (125 MPa;1.81 kg/dm3), (220 MPa;2.74 kg/dm3) and (450 MPa;7.85 kg/dm3). They are the three points in the plot corresponding respectively to the magnesium ZW30 type, the aluminum AA7050 type and the AISI 4140 steel. Each line corresponds to three different values of the index. It is now f easy to screen by plotting the subset materials that optimally maximize performance for the case study. All the materials intercepted by a line of constant σ1/2/ρ are of equal performance as a light and safe panel. Those above the line are better, those below, worse, since the former accounts for a higher material performance index and the latter accounts for a lower one (cf. Fig.1). Note the big advantage of this generalized procedure over more common derivative methods. As stated, the procedure to rank out candidate materials is quick and robust. We can refer to the vast number of resolved cases in component optimization (Ashby, 2002), thus applying the proper material performance index to our case. If we have already selected potential candidates, by just comparing the material performance indexes we can rank out the best material for the case study among those selected. But it is possible to widen the selection of candidates starting from the baseline scenario, namely a specific point in the plot and then tracing the line of the proper well‐defined slope to explore which materials are possible further candidates, considering the area above the line. This approach follows what we call a free‐search procedure. Generally such a wide screening out has to be conducted carefully, taking into account other limitations and drawbacks of the candidate materials that are not directly pointed out by the specific chart. For example, from the Fig.1 plot, if we do too free a search, we could end up considering ceramics as the optimal material for producing a thin‐pan for an automobile. Obviously brittleness, cost and manufacturing issues have to be kept in mind to restrict our free‐search to suitable materials, as for example to carbon fiber reinforced polymers or hybrid materials with metallic foams. 28 1.1.2. Non‐derivative methods. Expert survey approach8 A non derivative method or implicit method does not derive from a specific objective function in order to calculate the optimum. In the previous example, the objective remains, but it is not known how the mass of the part can be expressed in terms of the resistance of materials, external load, part geometry. Instead, the non‐derivative methods are also known as “black box methods”. The most recognized is the weight‐sum method. It consists in: 1. identifying the key‐features of a material that can impact on the main objectives. For example, in the case of the lightweight panel previously discussed, the mass of the part and the cost of the material are among the desired keyfeatures. The choice of material can impact on both themes; particularly, the specific strength, i.e., both resistance limit versus density and dollar per kg for raw material procurement should be considered; 2. assigning weight‐factors to each key‐factor by creating a qualitative rating (e.g. weak importance= 1, medium importance =3, high importance =5); in this step a quantifying judgment is made on the relative importance of product keyfeatures for the product in alignment with customer/user requirements; 3. quantifying a numerical value that scales the key‐feature values for candidate materials to a simplified rank (e.g. weak = 1, medium =3, high =5). For example, let us consider the three materials for the automotive panel, the 4140 steel, the 7075 aluminum alloy and the ZW30 magnesium alloy. We deliberate on their specific strength, namely the material limit resistance versus density. A simply calculation leads to 69.1, 80.3 and 57.2 (MPa∙dm3∙Kg‐1) respectively for the magnesium alloy, the aluminum alloy and the steel. Thus, these three values can be scaled to a simplified rank from 0 to 5 in two main different ways: – rate by the largest value: divide the three values by the largest of the pool and obtain the three normalized values 0.86, 1 and 0.71; thus multiply these normalized values for the maximum simple scale value, namely 5. The final quantifying values on a simplified 0 to 5 rank are therefore: 4.30, 5.00 and 3.60; – rate by the average value: calculate for the three values to 69.1, 80.3 and 57.2 MPa∙dm3∙Kg‐1 the mean value 68.8 MPa∙dm3∙Kg‐1. Now consider that on a simply 0 to 5 rank, 3 is the average value. By similitude between average value on a real scale and a simplified scale, the unknown ranked value can be calculated on the simplified scale as: x = 3∙(specific strength)/(average specific strength). Calculation for the three specific strengths of materials leads to 3.01, 3.50 and 2.49. As the three steps above have been completed, the result can be as shown in Table 3. Referring to Table 3, for the three materials P, the results are 2.1, 5.2 and 5 respectively for the magnesium alloy, the aluminum alloy and the steel. The material aluminum alloy AA 7075 with the largest value of the sum P is considered as the best for the application. The alternative ranking “rate by average value” would lead to different results as shown in Table 4. 8
Freely adjusted from the original book “Material Selections by a Hybrid Multi‐Criteria Approach ”,courtesy of author of the book Fabrizio D’Errico, academic tutor at Politecnico di Milano of this thesis work. 29 Table 3 Example of use of weight factors and the “rate by the largest value” calculation for scaling real values to
simplified rank 0 to 5 [1]. Table 4. Example of use of weight factors and the “rate by the average value” calculation for scaling real values to simplified rank 0 to 5 [1]. This method is sometimes considered inherently unstable and sensitive to alternatives (Ashby, 2004). This is clearly shown in Table 5, which gathers data for a light (low density) component that must be strong as well. Data for four possible candidates are shown. The total performance index P is calculated in the third column. Table 5. Example of comparison of “rate by the average value” ranking obtained by omitting and including the CFRP material [1]. 30 The carbon fiber reinforced polymer (CFRP) obviously wins, magnesium is second (minor negative value), aluminum third and finally steel. However, if we remove CFRP from the selection because, for example, it is judged too expensive, the new selection among three candidate materials has reversed the ranking of magnesium and aluminum. On the other hand, one major advantages of such a method is the possibility of taking into account various aspects that are not exclusively related to material properties. For example, it is possible to include the evaluation of production costs, as composed of the cost of materials, machining cost and surface protection. This leads to an extended analysis. An instance of this is given in Table 6 that completes the analysis in Table 3 with the calculation of the global performance index P, which includes the relative influence of product costs. As shown in Table 6, when increase of a feature, like cost, impacts with negative effect, this is accounted by a negative value for the weight‐factor. This feature makes the non‐derivative methods particularly simple and quick to use by a multidisciplinary team, where engineers join with non‐engineers to discuss a strategy for the success of the product on the market. Table 6. Example shown in Table 3 with the P calculation that encloses further non‐technical
features [1]. Expert survey strategy, an implicit method In selecting a material for a product or a component, the primary concern of engineers is to match material properties to the functional requirements of the component. Experts must know what material‐related variables can significantly influence product function, including the type of material, material toughness, hardness, and fatigue resistance. The type of material used for a component, in turn, determines the manufacturing process, as well as all manufacturing process dimensions, such as machinability, formability, weldability, and assemblability. Depending on the specific manufacturing process involved in component fabrication, one or more process variables need to be tested for component and product functionality to be optimal. These variables may include cutting speed and feed, the depth of the cut, the temperature, the presence or absence of lubricants, the duration of machining, the rate of cooling / heating, current density and voltage, and the type and amount of solvent used. 31 The methodology focuses on the development of guidelines based on determining the relationships between product functionality criteria and design and manufacturing variables that are impacted by the choice of material. Questionnaires can be used to guide the selection of materials, and processes are outlined by documenting the ways in which experts do their work. Information gathered in this way is called ‘capture from expert’, and such a method is based on the arrangement of a subset of specific questions in order that the expert can provide specific answers. It is an implicit method since it is not based on a rigorous process that expresses general relationships between the features of materials and their performance, quantifying such relationships using certain values, scoring some candidates on the basis of this relationship and ranking out others. And it is a method that strongly depends on having access to competent experts. Since experts, however, tend not to agree unanimously in their answers, guidelines would help in making sure the job can be done more efficiently. We will focus on an illustration of the general methodology to be adopted. The overall scope of the methodology consists in checking with the experts which key‐features of a material can ensure product functionality by controlling the design and manufacturing variables that greatly impact on the features of the product. Secondly, once such a subset of material key‐
features has been defined, candidate materials can be scored on the basis of suitability. A matrix scoring model is useful in situations where a number of options are available and the very best must be chosen. Table 7 shows how the matrix scoring model works. The scoring scale used in the Table 7 ranges from 1 (poor) to 10 (excellent) and is somewhat arbitrary, since a 5‐point scale, a 7‐point scale, or one with a greater or a smaller number of gradations can be used. A larger scale with more gradations increases the sensitivity of the evaluation process. The weights chosen for different criteria indicate the relative importance of criteria in comparison one to the other (one can use a 10‐point total or a 100‐point total). The final score per each option to be ranked is calculated by the sum of the score per each criteria (5 in the example of Table 7) multiplied by the weight assigned for each criteria category (3 for performance, 4 for reliability, 2 for quality, etc.). The expert fills out the matrix by assigning a weight per category and scoring material options. Thus the final results are strictly dependent on his knowledge of the subject and the quality of the information he is equipped with concerning the response of each option to the assessment criteria. A good general practice used to reduce the non‐objectivity and variability of such an assessment consists in breaking down each of the 5 (or more, if need be) general criteria into sublevels. In order to consider a broad and adequate set of selecting criteria it is useful if the experts’ survey can evaluate candidate materials on the basis of their own relative impact on product functionality, that is on the capability of the product to do something in a safe, reliable, user‐friendly and a high‐quality manner, taking into account product manufacturability and the environment friendliness of product life. 32 Table 7. Example of a matrix score for assessment of options on general criteria relevant for product functionality [1]. 1.2.
The multi‐objective optimization analysis in material selection9 Origin and scope in brief We need to look at how a decision model is usually structured in such cases. The basic assumption is that when a multiple conflicting objective arises decisionmaking is always restrained by the “decision space”: we must make purposeful choices with limited resources. The resources of a material can be, for example, that it might be lighter, but this is limited by the fact it cannot be stronger; as well as that, we look for a material that is both lighter and stronger, but we are limited by the increase in its cost. Economists have developed an operational approach that allows us to assess the public’s choice in multiple conflicting objective problems, depending on their own varying preferences. There are different items among a set of alternatives that conflict with each other: you can usually chose to purchase more goods, but you need to devote more day hours to your work to earn the money to purchase these goods; then, the hours you spend at work diminish the leisure time you have to enjoy what you have bought. Vilfredo Pareto (1848–1923) was probably the first economics researcher whose work might formally be classified as the explicit multi objective decision making approach. As an economist, he was the first (or at least one of the first) to carry out mathematical studies on the aggregation of conflicting criteria into a single composite index in order to quantify decision‐making assessments. A brief overview of Pareto’s basic concepts helps to speed up our comprehension of the explicit method we can apply in multiple conflicting objectives with multiple constraint material selection problems. Utility theory and explicit approach Consumers have varying tastes and preferences for some goods relative to others. And they normally have limited resources – what the economists call a budget ‐ to satisfy their individual needs and desires. Different preferences imply different choices for the total allocation of purchased goods. One person might prefer pizza and coke for his lunch hour, while another prefers salad and lemonade for his healthy break. 9
Freely adjusted from the original book “Material Selections by a Hybrid Multi‐Criteria Approach ”, courtesy of author of the book Fabrizio D’Errico, academic tutor at Politecnico di Milano of this thesis work. 33 And for each of those that like pizza and coke, there is usually a different combination in their weekly consumption of these two things, since one individual thinks that eating “one pizza and drinking three Cokes” per week is better than eating “two pizzas and drinking one Coke”. Economists call this “utility” – the fact individuals prefer one item to another. Generally speaking, if activity A is preferred to alternative B, we then say the utility from A is greater than the utility from B. You might imagine you are in an open‐air market on a Saturday morning and you are deciding how many kilos of bananas and kiwis to buy with your money for your afternoon frozen shake. If you want, the decision would represent the satisfaction level you reach by mixing – in the way you desire – a different combination of bananas and kiwis. Now observe Table 8. It reports the number of grams of bananas listed vertically on the left outside the box, from 0 up to 1,000 grams, while the number of grams of kiwis is listed horizontally below the box, from 0 to 1,000 grams. The entries inside the box show the utility you get from consuming bananas and kiwis together. Thus, the number of boxes at each intersection of a row and a column means the utility number for the consumption and the mixing of a specific combination of bananas and kiwi. For example, if an individual consumes 200 grams of bananas and 200 grams of kiwis, the number in the box states that the utility is 18. Utility is just a numerical indicator of preferences, but it does not matter what units you use to measure – the fact is that economists have no way of measuring utility. The only rule is that the higher the utility, the stronger the preference. For this reason, it is very helpful to explore combinations in the box and rank alternative consumptions. According to Table 8, the fans of banana and kiwi frozen shakes seem to prefer a mixed combination of 800 grams of kiwis and 200 grams of bananas to a combination of 200 grams of bananas and 400 grams of kiwis in that the utility of the former (28) is greater than the utility of the latter (21). Table 8. Example of utility from bananas and kiwis. The numbers inside the box give the utility from consuming the
amounts of bananas and the amounts of kiwi shown outside the box [1]. Thus, by maximizing utility, the consumer is making decisions that lead to the best outcome from his or her point of view. In this way, utility maximization implements the assumption that people make purposeful choices to increase their satisfaction. On the other hand, consumers are limited in how much they can spend when they choose between bananas and kiwis and other goods. For example, suppose the individual allocates a total of $3 per week for his favorite fruit. This limit on total spending is called the budget constraint. Table 9 and Table 10 show expenditures on bananas and kiwis for two different situations. 34 In the boxes all the combinations of bananas and kiwis from Table 8 are shown, but some combinations are outside the $3 budget constraint – these are in the grey shaded area. As a result of the varying price per unit of bananas but a fixed budget, some combinations are outside (i.e. much more expensive than) the budget constraint. Table 9. The numbers inside the box give the total dollar expenditures on
different combinations of kiwis and bananas available for $3 budget
constraint (the beyond‐budget combinations are in grey) at $0.9/kg unitary
price of bananas an $3/kg unitary price of kiwis [1]. Table 10. Similar results obtained for the previous Table 9 but recalculated by an increased $2.7/kg unitary price of bananas and a
fixed $3/kg unitary price kiwis [1]. Economists prefer to express the data in Table 9 by reporting the different value of utility for the varying combinations of possible preference choices in a diagram chart as shown in Fig.2. The axes report the same information about the lines and rows of the matrix in Table 9, namely: x‐axis reports the quantity of kiwis, y‐axis the quantity of bananas you chose to mix for your weekly shakes. 35 Note the chart reports the family curves that interpolate points at the same level of utility (refer once again to the values reported in the cells in Table 9, and thus compare the points shown in the chart in Fig.2). And note the procedure used to pass from Table 9 to the representation in Fig.2: –
–
–
satisfaction for varying combination of bananas and kiwis is summarized by fulfilling Table 9 that reports the non metered values (they are just indicators) of the relative preference for one combination chosen over the other; in economics, such relative preference of choices is called utility; the values in the cells of Table 9 can be transferred to a bi‐dimensional chart; curves are added onto chart that interpolate points at the same level of utility (i.e. the same level of satisfaction). We call these curves indifference curves because all the points on each curve correspond to different combinations of the quantity of bananas and kiwis (look at the coordinate axes). But each combination of bananas and kiwis can be obtained by moving along a specific curve that leaves the consumer indifferent to his choice. For example, the bolded curve in Fig.2 that interpolates points A and B has utility “35”; A and B represent two possible choices for mixing bananas and kiwis that have the same utility for the consumer. Thus, he is indifferent as to the choice of either combination A or combination B since he will be equally satisfied. Let us finally move on to the chart illustrated in Fig.3. It is the same chart we constructed in Fig.2, but with addition of further family lines. They are constructed in this way: (eq.13) $
∙
$
∙
$ and we call it a budget constraint family curve. The meaning is simple: the consumer may buy varying combinations of bananas and kiwis to your own satisfaction, but he is limited by his budget, the $3 of total budget he decided for example to allocate to his weekly banana and kiwi frozen‐shake. Actually he can decide to save money from his budget, for example, spending only $1.1. Thus, he might consider the $1.1 budget line. But this would mean that he is not maximizing his willingness to allocate up to $3 per week to take satisfaction from his frozen shakes. For that reason, the problem of defining the quantity of bananas and kiwis that optimize the budget constraint is solved by selecting any point belonging to the maximum budget line in the chart diagram (see again Fig.3): (eq.14) $
∙
$
∙
3$ The optimized solution among all the points on the budget line is the one that matches the highest utility curve among many. This leads to the definite identification of tangent point P in the chart diagram of Fig.3, a point that corresponds to the quantity of bananas and the quantity of kiwis that satisfy both the budget constraint and the maximum utility objective. 36 Fig. 2. Indifference curves represented on a chart diagram show the utility values of
the example in Table 9 [1]. Fig. 3. When the budget line is tangent to the indifference curve, the consumer cannot do any
better. Compared to the other points of intersection between the indifferent curves and the
budget line “$3”, this point can maximize utility for the consumer: in fact the maximum utility
(35) is obtained among the indifference curves that can match the maximum budget the
consumer can spend. The optimal allocation of bananas and kiwis in respect to quantity is
therefore defined by this intersection point. Note that, if you reduce the consumer’s budget –
e.g. for $1.1 to spend –, the budget line moves on left‐bottom corner and thus the
indifference curve that allow maximum utility changes accordingly [1]. 37 Lightweight door‐car panel Returning to our problem in choosing a material for a lightweight but affordable cost panel; as stated, in the basic optimization theory such a problem is classified as a double conflicting objectives problem – minimizing both the mass and the cost since it is always true the greater the performance of the material, the higher its costs goes. Our scope is to seek out a material among the various candidates that will not necessarily be optimal as regards either of the objectives, but which maximizes our utility. However, in this case, how can we define utility, and what does it mean to pursue ‘‘maximum utility”? Let us proceed step by step. Table 11 reports the panel cost estimated for each material. Table 11. Results to select optimal material for a light, stiff strong panel with further information about the cost of the part made of magnesium, aluminum or steel [1]. Looking at Fig.4; the diagram shows the three alternative solutions plotting mass against cost. Each bubble describes a possible solution as coming from the results summarized in Table 11. Fig. 4. Indifference curves represented on a chart diagram shown for the
utility values of the example in Table 11 [1]. 38 This plot could be sufficient to identify which bubble represents the optimal solution in the case of single‐objective problem: if minimizing the mass is the sole objective, magnesium is the final choice. When cost is the dominant criteria, steel is the best solution. But such a plot is ineffective when we consider the optimization of both criteria – mass and cost. To overcome this obstacle we can now use some of the powerful tools acquired when we were looking through the basics of optimization theory. Assuming we know the utility level for each bubble as illustrated in Fig.5. Note that, unlike goods we purchase for an increase in satisfaction ‐ like the bananas and the kiwis ‐ in this case the increase of utility for any choices is achieved when both mass and cost decrease. This is the meaning of the arrow in Fig.5: it identifies the direction of the increase in utility for any indifference curve that has been plotted. Fig. 5. Indifference curves represented on chart diagram [1].
Therefore, we can conclude that the optimal material would be the aluminum, because it’s the one with the highest utility value. The implicit approach using the Quality Function Deployment Matrix (QFDMatrix) In the late 60s at Mitsubishi’s Kobe Shipyard, Yoji Akao and Shigeru Mizuno were working on the potential relationships between the customer needs and what technical requirements to target in order to match these needs. Originally developed as a simple matrix that put the customer demands on the vertical axis and the methods with which they would be met on the horizontal axis, further improvements were brought to bear in the early ‘70s leading to the definition of a wellstructured tool called in English Quality Function Deployment or QFD, which was recognized almost immediately as a major breakthrough. 39 One very important thing that we need to highlight is that on account of its own structure QFD is an explicit method for designing a product or service which is based on customer demands and that can involve all members of the organization. A QFD methodology flow is depicted in Fig.6. Firstly, we have to comprehend its general scope: the formation and structuring of a creative engineering process that aims effectively to translate the customer’s needs into technical requirements. As such, therefore, QFD is a technique used in order to facilitate the process of converting the customer’s requirements, into a product design. If we are engineers and managers, we are usually so close to our product that our level of expectation and our values are far removed from those of the average customer. What frequently happens is that a high performance product is launched, but it does not meet customer expectations simply because the technical features that characterize the product are not fully perceived by the customer. Or some added features put there to differentiate the product and so to compete in the market are not decisive in targeting the customer’s final choice. A general rule used by QFD is that we should not speak for the public, but that we must listen to the “voice of the customer”, or VOC by gathering the requirements that make their needs more explicit. Fig. 6. Scheme of the QFD matrix. Because of its shape, it is also usually called the “House of Quality” [1]. 40 Explicit or implicit‐ which one is to be preferred in material selection strategy? There are two approaches to the decision making process in several disciplines, and that is also true as for engineering material selection: quantifying methods that point out optimization functions using mathematical relationships, graphs and curves; and non‐quantifying methods that are based on making correlations between what is required, is expected and what it is necessary to put into a product in order to satisfy such requests by doing it better than the competitors. The quantifying methods are precise, effective and based on a verifiable relationship, which, when the objectives and constraints are clear and agreed on, are not at the discretion of any single individual. On the other hand, they are in reality restrained to a limited decision making “space”. Practically speaking, explicit methods of optimization can manage a couple of conflicting objectives using diagram charts. The problem here, and it is by no means a trivial one, is how to define such curves in a solution of three or more space dimensions. Non‐quantifying methods do not do all these things, but complementarily they are suitable for analyzing multiple conflicting aspects without any restraint in their number: a bike saddle case study manages to target 12 requirements from the Voice of Customer and 16 engineering characteristics. Actually, the QFD matrix based method can break the problem of optimization into small pieces, comparing and assessing solutions that have already been developed and then re‐
assembling them all into a final simple number, the Total Weighted Score. As well as that, we have to take into account the investment we need to make in flexibility, in terms of the high level of discretion left to users. QFD is just a tool, sometimes a software tool that is easy‐to‐use, but it does not do all the job for its own. While it forces the Team to set target values for engineering characteristics in order to assure customer satisfaction, and it is able to show how the user can impact with his choices and which leverages appear more advantageous. In other words, it is flexible, but, if it is to be precise, it requires that all the experts that are competent in all the issues relevant to the customers’ requirements and needs should join up and sit around the table it has laid out for them. QFD can act as a common language among all the diversities of a multi‐discipline team that wants to take the product design as a whole. Using it puts people from each company department on equal footing, all able to stimulate the decision making process, no one being sidetracked by partial views. Just like a round table, which, in a most effective metaphor, has no head, and at which everyone who has decided to sit does so with equal status and power for discussion. It is just democratic, and perfectible. And since materials specialists sit at that table with all the others, they can bring some of the results obtained using implicit methods. 41 1.3.
The QFD4Mat method in preliminary material screening out10 Employing QFD in product development processes can help in seeking for a good compromise among the conflicting interests and competing objectives represented by each company department. Summing up, that this is possible because the QFD tool provides an insight which is external to the company and therefore close to the customer: thus exploring product attributes that are vital for customers. One after another, as these attributes get defined, engineering characteristics can be identified which will translate the desired product attributes into technical features that are manageable during the engineering design process and can satisfy the needs and expectations expressed by customers in their own language. We therefore need to define a framework, namely a suitable structure, by which we can organize the key‐requirements of the material so to create a positive impact on product functionality, usefulness and value, three factors that will be essential to its competitiveness. In order to get familiarized with the QFD4Mat matrix, a screenshot has been supplied: Fig. 7. Aspect of the QFD matrix [9].
10
Freely adjusted from the original book “Material Selections by a Hybrid Multi‐Criteria Approach ”, courtesy of author of the book Fabrizio D’Errico, academic tutor at Politecnico di Milano of this thesis work. 42 In the QFD matrix the following fields are present: Product requirements and customer importance value Product requirements are what we call the key‐ attributes desired by a customer. Before we proceed, it is convenient at this point to clarify who our customer actually is. For some products we may only have one or two types of customers, but there will more frequently be a chain of customers, where multiple customers take advantage of using our product. For example, if we design and sell bike‐saddles, the chain of our customers would be: ‐ the bike manufacturer ‐ the bike‐shops ‐ the on‐line after‐market sellers ‐ the user (the biker) Remembering that one major scope of QFD is to capture precious suggestions from any customer we want to target, a complete approach should take into account the needs of each individual part of the whole customer chain. We therefore start to gather from our customer chain all the product key‐requirements that have any part to play in product value creation and that can be controlled by the choice of materials. All such product key‐requirements can be organized into three categories as listed below: ‐ Performance, namely all those material attributes that can directly impact on the functional aspects of the product. In market strategy language these are attributes that are generally desired when pursuing competitiveness by technology‐driven differentiation; ‐ Cost, namely all those economic aspects of a material that can directly impact on the trade‐off between functionality and competitiveness on the marketplace. In market strategy language these are attributes generally researched for pursuing competitiveness by cost‐leadership; ‐ Product attributes receptivity, or simply Receptivity, namely the group or class of product attributes that have a subtle influence on customer product value perception. Fig. 8. Product Requirements and Customer Importance overview
43 The customer importance value is the score that a customer assigns to our company on each of his/her requirements against the best of our competition. Generally a 0 to 5 scale is used to simplify . Technical key factors Technical key factors are the factors related with the properties, functional requirements and manufacturing processes of the material. (i.e: yield strength, weldability, maximum energy storge per unit volume/given velocity [Ashby index]) Any technical key factor is associated with a product requirement by means of: -
Category: P(Performance), C(Cost), R(Receptiveness) Relationship: Direct (▲) , Inverse(▼) Grade of correlation: Strong (●), Medium (○), Weak(˅), Null( ) Fig. 9. Technical Key‐Factors Technical Competitive Comparisons Working on the bottom part of the QFD matrix, namely the box below “Metrics”, we have the “Technical Targets” and “Competitive Comparisons”. The scope of this box is to compare the different materials in any of the key factors individuated previously. For that goal the values, of the key factor for each material, are normalized in a semi‐qualitative 1‐5 scale. The process of normalizing is different depending on the kind of relationship established for that key factor (direct or inverse). The technical targets are expressed by a percentage. Any key factor has one associated. It express the importance of the factor in the hole product according to the VOC (Voice Of the Consumer) and it’s calculated by the grades of correlation and costumer importance values written previously. By these percentages the objective bubble curve will be designed. 44 Fig. 10. Technical Competitive Comparisons 1.4.
The assessment of material candidates by Graphic Analysis Among the several ways of visualizing data, we need something that maintains the high quality of information provided by an image, which can be intuitively comprehensible without requiring any specific skills. Two methods of visualization will therefore be illustrated here that can reply to such needs: the value curves and the bubble maps. 1.4.1. The Value Curve of products11 The first method has its origin in Kim and Mauborgne’s studies, two management strategy scholars. They recently proposed, which they call the Value Curve of products, services and processes (Kim and Mauborgne, 1997 and 2005). Here we focus specifically on products/materials. A Value Curve is a diagram used to compare products over a range of factors by rating them on a qualitative scale, from low to high. Such factors can be product features, product benefits or ways in which a product is distributed or consumed. Here we are interested in the product features related to materials. The combination of these various features defines a product, distinguishing it from the other competitors that may have different Value Curves. Multiple Value Curves can be drawn and superimposed to create a very user‐friendly visual comparison among competitive products and to unearth possible gaps in the market. By investigating the feasibility of these gaps, Kim and Mauborgne warn it may be possible to identify changes to the product that significantly alter the value proposition and enhance the receptiveness of users. In the following, the creation of a product Value Curve is explained. We use the QFD4Mat results taken from the crankshaft case material selection (*View chapter 3 of Material Selections by a Hybrid Multi‐Criteria Approach, Fabrizio D’Errico). In Fig.11, in the next three areas highlighted, we can recognize: 11
Freely adjusted from the original book “Material Selections by a Hybrid Multi‐Criteria Approach ”, courtesy of author of the book Fabrizio D’Errico, academic tutor at Politecnico di Milano of this thesis work. 45 – Area 1 contains the 20 material key‐factors identified in QFD4Mat as influencing the desired product requirements (the VOC features in the left side of matrix); – Area 2 contains the relative weights of 20 material key‐factors, in percentage; – Area 3 contains the assessment by a 0‐5 non‐dimensional scale of the material candidates through the 20 key‐features. Fig. 11. Relevant portions of QFD4Mat that has been constructed for the study case Simply reorganizing the data in Area 1, Area 2 and Area 3 we can draw our ValueCurves for 4 materials. Let us do it. We can proceed as follows: –
–
–
Organize the 20 columns of the key‐features (i.e. Area 1) by horizontally sorting them ascending from lower to higher values in the row highlighted in Area 2; the data in Area 3 will be sorted accordingly. Plot a diagram with the y‐axis ranging between 0 to 5 value and the 20 key‐features on x‐axis. Plot onto such a diagram the data on the 4 materials contained in Area 3.
If we follow the three‐step procedure listed above, the result is that shown in Fig.12 – the Value Curves of the 4 candidate materials along with the 20 key features organized from the least important (on the left) to the higher relative importance (to the right) as also illustrated by the dotted line “Relative Importance”. 46 Fig. 12. The Value Curve visualizing the final results of QFD4Mat in the crank‐shaft case study. 1.4.2. The Bubble Maps grafic tool12 The Value Curves visualizing tool is simple and quick to construct. But it is not the only way to summarize the QFD4Mat results with a high‐quality information content. The second tool presented as follows sets out to be as simple for any user as the Value Curves, and it can overcome one limit of the Value Curves graph. Let us focus firstly on that very limit. With a little insight it could be observed in the final passage of the previous paragraph how we have underlined that the Value Curve can summarize and visualize competition on the key‐
features that are considered relevant in that they impact on product requirements. Specifically in our example, we deal with 20 key‐features that have been selected on top of the QFD4Mat matrix, which aim to answer the 12 product requirements, the left shoulder of the matrix. The Value Curve is finally elaborated considering the “importance mix” resulting from the “causal links” established between technical key‐features (top roof of QFD4Mat matrix) and product requirements (or VOC, the left shoulder of QFD4Mat matrix). Having been constructed by ordering the assessment of technical key‐features for material candidates, the Value Curves embed possible misalignments between VOC and the final user experience of key‐features that try to translate VOC into a tangible product. If we want to reduce such misalignments, we can therefore use another visualizing tool that we call the “Performance‐Cost‐Receptiveness Customer‐ Oriented Diagrams”, or in a more friendly expression “PCR bubbles maps”. These are shown below. Firstly, however, we need to understand the origin of the PCR bubbles maps diagram. 12
Freely adjusted from the original book “Material Selections by a Hybrid Multi‐Criteria Approach ”, courtesy of author of the book Fabrizio D’Errico, academic tutor at Politecnico di Milano of this thesis work. 47 Product character is the result of important features, particularly those that differentiate the product. Around the product scope and its main function, the character of a product can be dissected into a subset of feature categories (Ashby, 2005): –
–
–
–
the context defines the intentions or ‘‘mood’’ by exploring possible answers to the 5 “Wh” questions (Who? Where? When? Why? What?) aiming to define the context and habits of product usage; the materials and processes create the architecture, the hardware of the product; the usability determines the interface with the user; the personality of the product, addressed by the aesthetics, associations and perceptions that the product creates in the user’s emotions (mainly architectural, like shape, surface finishing, opacity or transparency of surface, colors, etc.). Whatever category they belong to, the product features contribute to building up the resulting product character. They can be reorganized into the three main categories, once again set out here: –
–
–
Performance or P, which includes all the features related to technical and functional issues that determine the relative advantage of the product technology, the capability to translate ideas into tangible tasks and actions;
Cost or C, which includes all aspects related to the business and economic aspects of the product;
Receptiveness or R, which groups all those features more related to “Product Psychology”37 (e.g. usability, trialability, complexity, emotions, image, etc.).
Depending on the “side” from which we want to study these product features, namely: a) The side of demand, pertaining to choices and preferences (needs), experience and the final comparison with the expectations (customer satisfaction) of consumers. b) The side of supply, pertaining to research through consumer needs and their translation/codification into tangible products; the three categories P,C and R address respectively: product requirements and product features; the latter aims to supply the best answers to the fully expressed or latent questions in the former. A general scheme for representing the above PCR model is shown in Fig.14. We can gather the non‐dimensional 0 to 5 values obtained for product requirements (demand side) and product features (supply side) into P, C and R parameters. As a last task, we therefore plot onto the same diagram for the supply‐ side and the demand‐side two circles with their centres of coordinates, respectively, P and C, and with their diameters R. Since a drawing is better than words, Fig.13 shows the two resulting bubbles. The ideal situation is obviously when the PCR bubble representing the supply‐side completely matches the demand‐side PCR bubble. 48 Fig. 14. Scheme of model to represent as regards the demand and supply sides all product requirements and
product features developed to answer to requirements expressed by VOC [1]. Fig. 13 .Construction of PCR bubbles for the S‐s and D‐s (left). P and C coordinates are fixed, R parameter increases (right) [1] 49 In the graph in Fig.13 (left), for example, the two circles do not match; in this simplified example this depend on: ‐ Too great a “distance” existing between the “centre coordinates”, namely the fact that the positioning of P and C features evaluated onto the product features supplied by the company is too far away from the P and C requirements on the product supplied by users; ‐ The R parameter is too low. This is the diameter of the circles that graphically explains on a simplified scale: a) how receptive the consumer is to the requirements of the specific product; b) to what extent the designed product features “cover” the product requirements expressed by the consumer. Fig.14 (right) shows how the rise of the R parameter can affect the potential matching of the two bubbles. Instead, Fig. 15 (left) shows the supply‐side PCR bubble that partially covers the demand‐side bubble. A relative portion of the consumers’ bubble schematically (remember that it is a graph constructed in a semi‐qualitative scale) represents consumer's needs expressed in the form of product requirements. This bubble we call the Voice of Customers' (VOC) bubble. When the VOC bubble is partially covered by the material key‐features' bubble, it means the product value that the suppliers propose partially fits in with the consumers' expectations. Fig. 15. The two PCR bubbles match (left). Different of the NPD process (right) [1] Fig.15 (right) summarizes various situations in which the supplying bubbles are positioned outside the demand bubble, but with different results in terms of NPD (New Product Development) process results. 50 Fig. 16. . Benchmarking and market positioning: the market competition visualization by PCR
bubbles [1] Finally Fig. 16 represents a graphic visualization of a benchmarking analysis of market position conducted by the use of PCR bubbles. 51 1.5.
The scope of work: material optimization by refined QFD4MAT method for study case: bike frame 1.5.1. Racing bicycle [2] The case of study of this project is a racing bicycle frame. A racing bicycle, also known as a road bike, is a bicycle designed for competitive road cycling, a sport governed by according to the rules of the Union Cycliste Internationale (UCI). The most important characteristics about a racing bicycle are its weight and stiffness which determine the efficiency at which the power from a rider's pedal strokes can be transferred to the drive‐train and subsequently to its wheels. To this effect racing bicycles may sacrifice comfort for speed. The drop handlebars are positioned lower than the saddle in order to put the rider in a more aerodynamic posture. The front and back wheels are close together so the bicycle has quick handling. The derailleur gear ratios are closely spaced so that the rider can pedal at their optimum cadence. Fig. 17. Racing bicycle (left), cyclist (right)
1.5.2. Bike frame [2] The frame of a racing bicycle must, according to the UCI regulations, be constructed using a "main triangle" with three straight tubular shapes—the top tube, down tube, and seat tube. These three tubes, and other parts of the frame, need not be cylindrical, however, and many racing bicycles feature frames that use alternative shapes. Traditionally, the top tube of a racing bicycle is close to parallel with the ground when the bicycle is in its normal upright position. Some racing bicycles, however, have a top tube that slopes down towards the rear of the bicycle. Frame manufacturers are free to use any material they choose in the frame. For most of the history of road racing, bicycle frames were constructed from steel tubing, and aluminium and titanium alloys were also used successfully in racing bicycles. Racing bicycles in these three materials are still commercially available and are still used by some amateur racing cyclists or in vintage racing classes. However, virtually all professional road racing cyclists now use frames constructed from various carbon fiber composite materials, and a typical modern carbon fiber frame weighs less than 1 kg. 52 Particularly since the introduction of carbon fiber frames, the shape of the tubes that make up the frame has increasingly diverged from the traditional cylinder, either to modify the ride characteristics of the bicycle, reduce weight, or simply achieve styling differentiation. The frame consists of a top tube, down tube, head tube, seat tube, seat stays, and chain stays as seen in Figure 18. The head tube of the frame holds the steerer tube of the fork, which in turn holds the front wheel. The top tube and down tube connect the head tube to the seat tube and bottombracket. The seat tube holds the seat post, which holds the saddle. The bottom bracket holds the cranks, which hold the pedals. The seat stays and chain stays hold the rear dropouts, which connect the rear wheel to the frame. Fig. 18. Tubing diagram of a bike frame 53 1.6.
The preliminary QFD4Mat analysis on candidate materials for BIKE FRAME In this section a preliminary QFD4Mat matrix is developed. The applied excel version is: QFD4Mat‐
customv.03. In the following, the features, the numbers and the grades of relationship, written onto the matrix, are justified: Candidate materials After a market research regarding the most common materials employed for racing bike frames; for our material selection study, we will consider the followings: –
–
–
AISI 4130, normalized at 870ºC, air cooled Aluminum 7005‐T6
Titanium 3Al‐2.5V(Grade 9), alpha annealed
A composite material was wanted to be included as the forth material. However, the lacking disponibility of properties information respect to composites has impeded it. N.B: all the material properties and chemical composition are available in the Annex A. Product Requirements and Explanation of the Customer Importance Value The product requirements (PR) for the bike frame have been broken down into 3 categories: performance, cost, and receptiveness. Below, each of the different requirements are explained and are later related to material key properties in the following sections. Performance 
Plastic deformation is a PR because the frame is not allowed to work out of the elastic region, as it may break from overloading. It should also be a material that will have a safe response against unexpected high stresses from bumps in the road or debris. 
Corrosion resistance is important because the bike frame may experience extreme weather conditions. It has been designated as level 5 importance because it is directly exposed, and it will be the failure root mechanism. 
Fatigue is an important failure mechanism in the bike frame behavior because it is subjected to cyclical moments of bending and axial; and must last for at least a few years. 
Surface fatigue is not quite as important since there are not relative cyclical movements between the frame and the joints. 
Stiffness is given the maximum importance, as it directly relates to the maximum amount of bending allowed on the bike frame. When the cyclist is sprinting, the bike must remain practically undeformed and all the force applied by him must be transmitted to the rear wheel. That is, the physical energy of the athlete must be converted into kinetic energy and nor into deformation energy. Additionally, customers identify the stiffness of the bike frame as its most important contribution. 
Toughness is required to avoid breaking with low impacts. Therefore, it has a fairly medium importance for bike distributors and customers. 54 
Lightness is a main issue for product performance. A significant part of the drive force applied by the athlete is used to beat the inertial force of the bike. So on, the lighter the bike is, the faster the cyclist can go. 
Handling is also an important issue for product performance. The better the handling of the bike, the more efficient the driving will be. 
Global Performance Index – We have included 3 ASHBY indices. After a previous FEM analysis of an standard bike frame (developed in chapter 3) we can concluded that the main stresses of the bars are due to bending and axial forces. So the performance indices selected from the Ashby tables are: Strength-limited design at minimum mass
Function and constraints
Maximize
Tie (tensile strut)
Stiffness, length specified; section area free
f / Stiffness-limited design at minimum mass
Function and constraints
Tie (tensile strut)
Stiffness, length specified; section area free
Maximize
E/p
Beam (loaded in bending)
Stiffness, length, shape specified; section area free
E1/2/
Cost 
Direct variable cost is given a high value as it is the main material cost. 
Restoring cost for non‐conformities is valued as minimally important as there is almost no wasted material in creating the bike frame. 
Cost for shaping processes is at a high‐moderate value as the extrusion and forming (bending) processes are not too complex, but still play a part in the overall expense of the bar. 
Cost of heat treatments is at 2 because, in general, only local heat treatments will be needed (in the welding regions) and the general heat treatments of the materials are included in their raw price. 
Machining costs are stated as medium. After bending the bars in the manufacturing process, they are machined in order to achieve the correct welding angles on the edges. 
The cost of welding is at 5 importance level because it is one of the main manufacturing processes. Furthermore, we have to consider that welding is a high energy‐consuming procedure. So here, we should take into account the cost of electric energy. 55 Receptiveness 
Process cycle complexity is of medium importance 3, as the bars of the frame follow a straightforward and simple process from cutting, to bending, to machining and welding typically. Manufacturing of the part is reviewed more in depth in a later section of this report. 
Delivery time is of medium importance so that the customer can receive their order in a timely fashion. It is of importance to the bike. In fact, it is the core of the bike, so it is rated quite high. Material Key Properties and their correlations with the Product Requirements This section relates to the properties that the project team aims to include in the bike frame material selection process. The properties are explained as they relate to the product requirements. The grades of relationship are justified. ‐Plastic deformation and break by overload: The material needs high toughness to avoid breaking, a high yield strength (YS) to avoid plastic deformation, and a high Ultimate Tensile Strength to avoid necking. ‐Corrosion: The lower the corrosion rate is the better the material reacts against environmental wearing. ‐Fatigue: For higher YS, the persistent slip bands (PSB) phenomenon is constrained. If a material has a higher endurance limit, then the bike frame can be subjected to higher work stresses repeatedly without failing. ‐Surface fatigue: Surface fatigue is initiated by small defects on the surface of a material, where high corrosion rates can motivate the appearance of surface defects. High surface hardness protects against possible micro impacts, which could produce surface defects. This will provide a higher endurance limit, which again lets a material sustain higher stresses over a long period of time. ‐Stiffness: The stiffness is directly proportional to the Young’s modulus because a higher stress is needed to deform elastically at the same rate as a stiffer material. ‐Toughness: The higher the V‐notch impact energy is, the tougher the material is. ‐Lightness: The specific strength is defined as . A material with high specific strength for the same stress and work conditions requires less mass, so it can be lighter. ‐Handling: The handling of the bike is improved if it has a high specific strength and a high young modulus. ‐Direct variable cost: There is a direct and strong relationship between the material cost [€/kg] with the direct variable cost. On the other hand, the productivity, in general, decreases when the costs increase so this last relationship is inverse. 56 ‐Restoring cost for non‐conformities: The cost of unused material, or the wasted material in fabrication processes, is directly related to the material cost [€/kg]. As said before, the productivity decreases when costs are increased. ‐ Cost for shaping processes: A good workability simplifies the process of bending and further manufacturing processes as machining, so these costs are reduced. ‐Cost of heat treatments: Heat treatment and surface treatment cost is directly related. It should be taken into account that the welded regions of the bike frame must be heat treated. ‐Machining costs: The higher the workability of the material is the cheaper the cost of machining. ‐Welding costs: The relationship between weldability and welding costs is strongly inverse. ‐Process cycle complexity: If the material has a good workability, the process cycle complexity is lower because fewer and simpler processes are needed. Moreover, the high complexity of the process cycle decreases the producibility of the anti‐roll bar. ‐Delivery time: If the material has a good workability it will have a simpler manufacturing process, so it could be delivered in less time. It is important to have a reasonable delivery time in order to reach a good productivity rate and this can also be a strong, customer‐driven goal. Conclusions The QFD4MAT excel worksheet has been employed in order to review the candidate materials and select the best material for the bike frame manufacturing and use. In the following screenshots the filled in matrix is shown. Fig. 19. Upper part of the QFD4Mat matrix: performance product requirements 57 Fig. 20. Mid part of the QFD4Mat matrix: cost and receptiveness product requirements As justified before, all the product requirements are related with an specific grade of correlation with the technical‐key factors. Fig. 21. Lower part of the QFD4Mat matrix: technical competitive comparisons The final weighted score provided by the QFDMat excel page is: Fig. 22. Final weighted score for the bike frame analysis
58 Table 12. Global Performance Index sheet Therefore, the global matrix analysis selects the Ti 3Al‐2.5V as the most suitable for the bike‐
frame. Below are two different interpretations of the material selection matrix driven from performance, cost and receptiveness. Both of them are based on graphic analysis. Value curves analysis The obtained value curves are: Fig. 23. Value curves for comparison with key differences circled in yellow 59 N.B: Productivity has been reviewed twice, once for cost and then again for receptiveness. In this graphic we compare individual materials based on the categories in the columns above them. The overall impact of each category increases from left to right and is denoted by the black discontinuos line at the bottom of the graphs. The graph shows value curves where each individual component assessed is shown for each of the candidate materials. The larger the differences between points on each vertical line represent the differences in performance, cost or receptiveness depending on changing the material. The yellow ovals showcase the categories where there is a significant variability between the materials being reviewed. At the performance aspect of surface hardness the titanium has the greates value followed by the steel; on the other hand, aluminum performs with a low value. A similar interpretation could be done for the UTS. In this case the steel performs with the highest value and the aluminum still remains backwards. Talking about endurance stress steel and titanium are considered to perform in the same manner and aluminum performs poorer again. Actually, thinking about the crystal structure of the metals, it is known that the aluminum experiments a decay in the stress limit as long as the number of fatigue cycles increases. Regarding the corrosion rate the titanium has the highest value and the steel the lowest, with a medium value for the aluminum. We have to remember this key factor has been defined inversely and that means the best resistance against corrosion is performed by the titanium. The same analysis is suitable for the material cost. At this point, it is important to point out that the relative importance of the key factors is increasing considerably. Therefore, the same variance between two of the materials is more significative in the global valutation for the last technical factors than for the first on the table. The last and most significant technical factor is the Young’s modulus. The steel gives the highest utility value whereas the titanium and aluminum perform in a poorer manner. In summing‐up the value curves graph does not give a “winner” material but allows the user to understand and compare the individual performing of each material for each of the technical factors. Therefore, the strengths and weaknesses of a candidate material are shown. This could give the user the possibility to change some manufacturing specifications of the material, for example heat treatment, in order to make it more competitive against the others on the most significant specific key factors of the desired product. Bubble maps analysis The center of the bubble is the cost and performance of the individual materials. The diameter of each bubble corresponds to the receptiveness of each material. The target of the bubbles is to match, for the center location and diameter, the VOC (Voice Of Consumer) bubble, represented by the black curve. The aluminum bubble covers the VOC smallest area of the three. That means that some receptiveness requisites seeked by the costumers are not satisfied. However, the cost coordinate of the bubble’s center is the lowest value of the three. Nevertheless, making the comparison with his most immediate competitor, AISI 4130; “loses” against it. The steel has a better performance (vertical coordinate of the bubble center) and covers a bigger area of the seeked receptiveness, with a small higher cost. 60 Fig. 24. Bubble curves graph comparing the candidate materials
Thereafter, titanium and steel are compared: Fig. 25. Titanium and AISI 4130 bubbe curves
61 As seen in the figure 25. The titanium bubble covers quite the same receptiveness of the VOC as the steel bubble. On the other hand the coordinates of the titanium bubble’s center represent a higher performance with a higher cost. The steel is less performing but it allows to put into the market a better cost‐competitive bike frame. Finally, it could be concluded that the best material will be one of the followings: AISI 4130 or Ti 3Al‐2.5V. The chosen one will depend on the profile of the consumer. The titanium frame for a cyclist who is looking for the maximum performance without caring too much about the price or the chromium‐molybdenum alloyed steel for a sportsman who has a limited budget. 1.7.
Limits and constraints of preliminary QFD4Mat analysis We have just given a first approach for the material selection of a racing bike frame. Nevertheless, it is important to point out that the geometry of the frame was not considered. Actually, the global performance index was influenced by three generic Ashby indexes. The chosen Ashby indexes are suitable for any case of beams subjected to bending or ties subjected to tensile. The real challenge of this project is to join the material selection based onto the QFD4Mat matrix and the topology optimization for lightweight design. In the next chapters, an optimal frame design will be targeted. Thereafter, the QFD4Mat matrix will be applied again substituting the generical key factors (Ashby indices) for the specific structural response parameters, given by a FEM analysis, like the maximum displacement or the total weight of the frame. In summary, the previous material selection analysis shows the best material for a generic bike frame. Instead, we are looking for the best material for a specific optimized racing bike frame. 62 63 2. Modern computer aided approaches for optimal geometry: topology optimization An optimization process always has several things in common. It certainly has to have an objective, i.e. to get the maximum in strength or resistance of a structure, it always has constraints, i.e. the weight or the dimensions, and it also must have at least one parameter that can be changed, referred to as a design variable. These design variables are also often subject to constraints or discrete design domains (parts of the structure that are to be optimized). For a numerical simulation all the mentioned parameters obviously have to be described in a mathematical formulation. Since optimization usually is a kind of extremum problem, which means the objective has to be maximized or minimized, there also has to be some kind of sensitivity analysis. The sensitivity of a problem characterizes the change of the objective function due to changes in design variables. This sensitivity analysis has to be implemented by the optimization algorithm. Since there are lots of methods for optimization algorithms available, usually either based on deterministic methods (mathematical programming) or on stochastic methods (i.e. evolutionary algorithms), one has to choose a suitable algorithm for the given problem. Optimization algorithms based on mathematical programming often use gradient‐based methods that involve the calculation of gradients of the objective function and the determination of a search direction in a multidimensional solution space. One such method is the Lagrange Multiplier method, often used in topology optimization codes. 2.1.
Design optimization Design optimization consists on automated modifications of the analysis model parameters to achieve a desire objective while satisfying specified design requirements. The main applications are: 




Structural design improvements: minimize thickness, hence weight Generation of feasible designs from infeasible designs: original model violates stress levels Preliminary Design: candidate designs from topology, topometry, topography optimization Model matching to produce similar structural responses: frequency response, modal test Sensitivity evaluation: identify which regions of the model are most “sensitive” to design changes or imperfections. System parameter identification. The classes of design optimization implemented in the software used for this project (MSC Nastran) are: Fig. 26. Classes of design optimization implemented in MSC Nastran 64 This project work aims the topology optimization of a racing bike frame. Optimization problem statement Any problem of optimization is characterized by the following features: 


Design Variables — Find {X}={X1,X2,…,XN}  E.g. thickness of a panel, area of a stiffener Objective Function — Minimize F(X)  E.g. weight Subject to: — Inequality constraints: 
0
1,2, … ,  Design Criteria and margins — Side constraints: 
1,2, … ,  Gage allowable Fig. 27. Example of a design optimization statement applied to a gusseted tube connection. In any design optimization model there are two models: Fig. 28. Optimization models 65 The workflow followed by MSC Nastran in structural optimization can be schematized as: Fig. 29. Structural optimization workflow by Nastran 2.2.
Introduction to topology optimization Topology optimization determines the optimal shape of a part. The design variables are the effectiveness of each element. Fig. 30. Topology optimization workflow followed by MSCSoftware 66 Topology optimization can be used for: 



Static load path — Multi‐subcase Frequency constraint Frequency response Multidisciplinary — Static+ Modes+ Freq Response+ Sizing The main benefits are: –
–
Used in early design to obtain component designs and shapes Used to redesign existing components 2.2.1. Problem formulation The lay‐out problem that shall be defined in the following combines several features of the traditional problems in structural design optimization. The purpose of topology optimization is to find the optimal lay‐out of a structure within a specified region. The only know quantities in the problem are: –
–
–
–
The applied loads The possible support conditions (BCs) The volume of the structure to be constructed (Work Space) Additional design restrictions such as the location and size of prescribed holes or solid areas. In this problem the physical size and the shape and connectivity of the structure are unknown. Fig. 31. Problem formulation graphic scheme [3]
67 2.2.2. Density Method Approach and mathematical formulation Alternatively called the Power Law approach or Artificial Material approach [3]. It is based on the idea of convexification where an artificial material is used which is homogeneous. 


The density of the artificial material can vary between 0 and 1 The generalized material parameters are simply taken to be proportional to the relative density A power law is used to relate the density with the material property The design variable x is normalized with respect to the nominal density and Young’s modulus: (eq.15) /
(eq.16) ∙
Where ρe describes the density of the element taken into consideration and, ρ0 the density of the artificial material. The same way, Ee represents the stiffness of the element and E0 the stiffness of the artificial material. Each element is a design variable (density) which is “ideally” a discrete variable (0 or 1). Actually, each element in FE model is given an additional property of relative density Xe (eq. 15) which alters the stiffness porperties of the elements. After a based algorithmic iterative process, a set of density values is obtained by optimization, which means the design domain is separated into solid and void regions.This has the effect of redistributing material from regions which don’t require material to those which require. The objective of most common topology optimization problems is to find the minimum compliance c of a structure by a change in the distribution of mass or, in a fixed geometry (volume), the distribution of densities. The objective function can therefore be defined as: ∙ (eq.17)
This compliance is the scalar product of the two vectors and resembles the work done by the force vector along the calculated displacements. Thus the given expression is actually a potential similar to common by the stiffness matrix with the current density distribution. (eq.18)
∙
With some further information the objective function can be written as eq.19 [3]. The compliance here is a linear combination of the compliances of each element. (eq.19)
:
∑
Since it is a normalized value, the design variable can only range between the values 0 (void) and 1 (solid) and therefore has to be restricted. For prevention of possible singularities in the system’s matrices the densities are not restricted by zero but by a lower bound. (eq.20)
0
1
0 Also, since this optimization method is basically a redistribution of material, the mass has to be constrained. 68 (eq.21)
∑
0 ↔ The complete topology optimization problem statement for minimizing compliance therefore reads as [4]: (eq.22)min
|
0,
0,
0,
∊
Although there are many optimization problems that can be solved with Nastran, this problem statement has been customized in Patran and can be easily used on a given geometry. 2.2.3. Computational procedure The direct method of topology design using the material distribution method is based on the numerical calculation of the globally optimal distribution of the density of material ρ which is the design variable. For an interpolation scheme that properly penalizes intermediate densities the resulting 0‐1 design is actually the primary target of our scheme. The optimality criteria method for finding the optimal topology of a structure constructed from a single isotropic material then consists on the following steps: Pre‐processing of geometry and loading: –
–
–
–
Choose a suitable reference domain (the ground structure) that allows for the definition of surface tractions, fixed boundaries, etc. Choose the parts of the reference domain that should be design, and what parts of the ground structure that should be left as solid domains or voids. Construct a finite element mesh for the ground structure. This mesh should be fine enough in order to describe the structure in a reasonable resolution bit‐map representation. Also, the mesh should make it possible to define the a priori given areas of the structure by assigning fixed design variables to such areas. The mesh is unchanged through‐out the design process. Construct finite element spaces for the independent fields of displacements and the design variables. Optimization Compute the optimal distribution over the reference domain of the design variable ρ. The optimization uses a displacement based finite element analysis and the optimality update criteria scheme for the density. The structure of the algorithm is: –
–
–
–
Make initial design, e.g., homogenius distribution of material. The iterative part of the algorithm is then: For this distribution of density, compute by the finite element method the resulting displacements and strains. Compute the compliance of this design. If only marginal improvement (in compliance) over last design, stop the iterations. Else, continue. For detailed studies, stop when necessary conditions of optimality are satisfied. Compute the update of the density variable. This step also consists of an inner iteration loop for finding the value of the Lagrange multiplier Ʌ for the volume constraint. Repeat the iteration loop. 69 For a case where there are parts of the structure which are fixed (as solid and/or void) the updating of the design variables should only be invoked for the areas of the ground structure which are being redesigned (reinforced). Post‐processing of results Interpret the optimal distribution of material as defining a shape, for example in the sense of a CAD representation. Fig. 32. The flow of computations for topology design using the material distribution method and the Method of Moving Asymptotes (MMA) for optimization 70 2.3.
The problem of shape constraints by manufacturing approach 2.3.1. Introduction to manufacturing constraints. Application examples Depending on the manufacturing process the redistribution of mass in the topology optimization is constrained. Actually, topology optimized designs may require major modifications for production or are not producible at all. Some tipycal manufacturing constraints are: Minimum Member Size –
–
To control the size of members in a topology optimal design Enhances simplicity of design, hence its manufacturability Casting Constraint (Draw Direction) –
To prevent hollow profiles Extrusion Constraints –
–
Constant cross‐section along a given direction Essential for designs manufactured by a extrusion process Mirror Simmetry Constraints –
–
Symmetry Constraints force a symmetric design in all cases Support regular or irregular mesh Fig. 33. Influence of manufacturing constraints in a beam topology optimization design The following example [5] shows the influence of manufacturing constraints. Fig. 34. Mount Beam topology optimization formulation
71 The main objective is to minimize the compliance of the front Mount Beam of an airplane engine. Fig. 35. Optimized geometries for no‐costraints case and casting constraints case 2.3.2. Bike frame manufacturing process In the following the bike frame manufacturing process will be explained in order to understand how it could constrained the shape optimization previously announced. Fig. 36. Manufacturing work flow of a bike frame Creation of blank tubes Seamless frame tubes are constructed from solid blocks of steel that are pierced and "drawn" into tubes through several stages. These are usually superior to seamed tubes, which are made by drawing flat steel strip stock, wrapping it into a tube, and welding it together along the length of the tube. Seamless tubes may then be further manipulated to increase their strength and decrease their weight by butting, or altering the thickness of the tube walls. Butting involves increasing the thickness of the walls at the joints, or ends of the tube, where the most stress is delivered, and thinning the walls at the center of the tube, where there is relatively little stress. Butted tubing also improves the resiliency of the frame. 72 Butted tubes may be single‐butted, with one end thicker; double‐butted, with both ends thicker than the center; triple‐butted, with different thicknesses at either end; and quad‐butted, similar to a triple, but with the center thinning towards the middle. Constant thickness tubes, however, are also appropriate for certain bikes. Fig. 37. Seamless tubes manufacturing process [6] Fig. 38. Seamed tubes manufacturing process [6] 73 Hydroforming For hydroforming, the blank tubes with circular cross sections are altered to yield a more complex and stronger geometry. The concept of the hydroforming process is simple, and requires only a steel die and a fluid. The blank tube is placed in the die and the ends of the tube are sealed off. Fluid, usually water, with an anti‐ corrosive additive is then pumped into the tube at high pressure. This forces the tube to expand and thereby conform to the shape of the die. With hydroforming, fluid pressures can often exceed 135 MPa [7]. A diagram of the tube hydroforming process is shown in Figure 39. Fig. 39. Hydroforming a blank tube using a die and high pressure fluid Miter cut tubes In order for tubes to easily and securely connect to one another in the shape of a frame, they must be miter cut. By mitering the end of a tube, two tubes can be joined together in a seamless fashion, making the joint stronger. A machine specifical designed for tube mitering is used most often which produces a clean and accurate cut. Mitering also allows the ends of tubes to sit flush with the sides of the tubes that it will eventually be joined to [22]. An example of a miter cut tube is shown in Figure 40. Fig. 40. Down tube miter cut [10]
Joining the tubes The 5 tubes can be joined into a frame either by hand or machine. Frames may be brazed, welded, or glued, with or without lugs, which are the metal sleeves joining two or more tubes at a joint. Brazing is essentially welding at a temperature of about 1600°F (871°C) or lower. Gas burners are arranged evenly around the lugs which are heated, forming a white flux that melts and cleans the surface, preparing it for brazing. The brazing filler is generally brass (copper‐zinc alloy) or silver, which melt at lower temperatures than the tubes being joined. The filler is applied and as it melts, it flows around the joint, sealing it. For aluminum frames TIG welding is used. It is an arc welding process in which heat is produced between a non‐consumable tungsten electrode and the work metal. TIG welding utilizes the inert gas, argon, to keep the weld area clean which prevents the metal from oxidizing during the welding process. TIG welding is commonly chosen as the welding method for thin tubes and is desirable for the bycicle industry since it provides a high quality finish on the weld surface. Fig. 42. TIG welding Fig. 41. Brazing practice Heat treatment Due to welding the HAZ (Heat Affected Zone) represent the most feasible failure region. In these regions the mechanical properties of the material fall down due to the heated applied during the welding process. Therefore exhibit a lower resistance to fatigue failure and other mechanism failures. To recuperate the properties a heat treatment is needed. Fig. 43. Zones of a TIG weld
75 The heat treatment depends on the frame material (steel, aluminum or titanium) and the kind of welding. As a general rule, steel frames undergo no heat‐treatment post‐manufacture but there are some exceptions. Several tubesets are intentionally heat‐treated before shipping though, and usually described as such. Most of these are company‐specific drawn 4130 steel tubesets. Instead, for aluminum frames the creationg of precipitates in the HAZ, due to the welding process, takes place. The precipitates decrease the hardness and therefore degrade the mechanical properties such as ultimate tensile strength by as much as 34%. For example, with 6061 the tensile strength before welding is 310.3 MPa. After welding, the HAZ has a tensile strength of 186.2 MPa. This drastic loss in strength due to the welding process can seriously affect the life of an aluminum bike frame. Hence a heat treatment is needed. The term heat‐treating describes increasing the strength and hardness of precipitation‐
hardenable wrought alloys. The aluminum selected in this project (Al 7005‐T6) is T6 heat‐
treated. T6 heat‐treating is considered an optimal operation for recovering the mechanical properties of 7005 after TIG welding. The steps for the T6 heat‐treatment are: Fig. 44. T6 Aluminum heat treatment
For titanium frames is not necessary to apply a heat treatment because the HAZ is very thin and localized. 2.4.
Conclusions For a bycicle frame, the manufacturing process does not constraint specially the topology optimization because the variable section of the tubes given by the software could be achieved in the real manufacturing process by a determinate design of the hydroforming die. Therefore, the main parameter to be adjust is the threshold, also known as mass target constraint, in order to build a structure that is realistic: –
–
A continuity should exist in the structure No flying parts should exist 76 77 3. Topology optimization for shape definition of a BIKE FRAME In this section, firstly a FEM analysis of a generic bike frame is carried out. Then the topology optimization of a racing bike frame is conducted, bringing the optimized shape out. This final resulting model of the bike frame will be studied by FEM analysis for each of the candidate materials, with the scope of obtaining the value of the new key‐ factors of the final refined QFD4Mat matrix. 3.1.
Starting point: analysis of a generic bike frame The goal of this section is having a quick overview about the orders of magnitude for the displacements and stresses in a generic bicycle frame, considering an static analysis. As stated in the first QFD4Mat analysis, when the Ashby indixes were chosen, this generic analysis allowed us to know how the frame tubes work (axial, bending and/or torsional loading). Hence the proper Ashby indixes could been chosen. The software MSC Patran with the numerical solver MSC Nastran has been used for all the following FEM analysis and topology optimizations. For the topology optimization the software is based onto the mathematical theory showed in section 2.2. 3.1.1. Problem formulation First of all, the definition of the athlete. A cyclist of 1.80 m height and 80 kg weight is considered. Taking these features into account, the following frame has been chosen: Fig. 45. Frame size analysis [8] 78 Case of study: sitting The case of study will be the sitted cycling position: Fig. 46. Cyclist riding on sitting position
Loads and Boundery Conditions (BCs) For a static analysis only the force due to the weight of the cyclist is studied. Hence, the total downforce applied is: ∙
80 ∙ 10
800 This total force will be distributed onto 3 application points: the saddle, handle bar and pedal (bottom bracket for simplicity). The chosen loads distribution has been taken from the technical paper “Bicycle frame optimization by means of an advanced gradient method algorithm”, L.Maestrelli. Fig. 47. Load conditions on sitting position [12]
The BCs of the model are: 
0

0
79 3.1.2. FEM analysis Taking the dimensions of the selected bike frame (Fig.45) a geometric curve based model is designed: Fig. 48. Bike frame geometry. Edges and and segments numerated. Thereafter, an isotropic material is defined. In this case a generic aluminum has been chosen: 



Elastic modulus: Poisson’s coefficient: Density: Yield strength: 72000
0.3
2.8
3 /
150
N.B: The chosen unit of measures are mm for the the length , MPa for the stresses and N for the forces. The frame will be analized by beam elements. Hence, the beam cross section is defined: Fig. 49. Tubular section defined in Patran The tubular section dimensions: external radio (R1=20mm) and thickness (t=0.7mm) have been chosen regarding generic bike frames and taking and average value. In this case study, the cross section is considered to be costant and the same for all the frame tubes. 80 However, in quite optimized racing frames the section varies in function of the stress distribution in the studied tube. For the same reason, the geometries and external radios vary from one tube to the others. In fact, the thicker ones are the down tube and the head tube because of their higher subjected stresses. Then, the 1‐D property is defined. Fig. 50. 1‐D element properties definition Three element properties are defined. In spite of all the tubes have the same geometry, the normal vector is different in each case. The segments 5,6,7 and 8 are contained in the same plane, so a common normal vector could be defined. Hence, they share the first property, TuBeam_1. The segments 1 and 3 are contained in the same plane and the TubeBeam_2, with the corresponding normal vector, applies to them. Finally TubeBeam_3 property is employed for the segments 2 and 4. The next step is defining the mesh. The Curve Mesher has been used and the Global Edge Length has been settled on Automatic Calculation. Then an Equivalence of nodes has been applied to the joining edges in order to relate the bars. Finally, a total of 78 nodes and 80 elements have been created. Fig. 51. Mesh setting A 3‐D full expand of the mesh geometry is shown in figure 52. 81 Fig. 52. 3‐D FullSpan of the mesh The last step before running the mathematical analysis is settling in the loads and boundary conditions in the software. Fig. 53. . Loads and BCs setting and graphic view
82 Finally the analysis settings are selected and the FEM model is sent to the numerical solver Nastran. Fig. 54. Analysis setting Fig. 55. . Screenshot of the analysis running
In order to understand better how the MSC Software couple the graphical interface Patran and the numerical interface Nastran, the following workflow diagram is supplied: 83 Fig. 56. MSC Sostware FEM analysis workflow 3.1.3. Results Using again Patran as post‐process and results analysis interface the following results were given. Deformed structure Fig. 57. Bike frame deformed structure In the Fig.57 the deformed structure of the bike frame is shown. The displacements are represented by the resultant of the the x and y displacement components. The main displacements are listed in the table below. Point 1 4 5 Displacement [mm] x y δ ‐0,00131 ‐0,09424 0,09425 0,09129 ‐0,08488 0,12465 0,07323 0,02157 0,07634 Table 13. Main displacements
84 Serviceability Limit State (SLS) A SLS of 1 mm is considered taking into account that the frame of racing bycicle must be so stiff as not being deformed to much. Actually, all the physical energy of the cyclist should be converted into kinetic energy, and nor into deformation energy. 1
∀ ∈
⟹ Where N is the set of all the nodes of the structure. Bar stresses Axial bar stresses Fig. 58. Axial bar stresses The maximum traction and compression stresses due to the axial intern force are 6.64 Mpa and 2.41 Mpa, respectively. The maximum compression takes place in the top tube. This result is easily understandable looking at the deformated structure. As shown in table.13, the point 4 (left edge point of the top tube) experiments a higher positive x displacement than the point 5 (the right edge point of the top tube). This difference of same direction “axial” displacements between the two edges of the tube results into compression. On the other hand, the maximum traction is located along the down tube. In the same way, analyzing the displacements of the edges of the tube the axial behavior could be justified. The point 6 is movement‐constrained for the 6 DoFs. Instead, the point 1 (left‐down edge of the down tube) experiments a great downwards displacement into the y direction. The bar is stretched, therefore traction stresses appear. Fig. 59. Tubes cross section. Main points designed
85 Bending bar stresses The post‐processor allows us to graphic the bending bar stresses at the four main points of the section. Looking at the figure.59, two symmetries respect to a generic diameter can be observed. For any bending force, the points C and E will experiment the same stress in absolute value and opposite sign. The same applies for points D and F. The fact that the bending forces will always be same orientated respect to the neutral fiber explains the previous symmetries. Fig. 60. Bending bar stresses at point C Fig. 61. Bending bar stresses at point E As stated before, it could be observed that the upper graphs are equal in absolute values but opposite in sign. In any case, reguarding to the absolute values we can say that the stresses are quite low and they can be neglected respect the ones due to the axial force shown before. Only the maximum values: 0.278 and 0.260 Mpa can be taking into consideration for the total stress. Therefore, it can be said that the bending bar stresses at points C and E are not significant for the principle frame “triangle” set up by the top tube, down tube, seat tube and head tube; and the chain stays. At these regions correspond the dark green colours. 86 Fig. 62. Bending bar stresses at point D Fig. 63. Bending bar stresses at point F Now the bending bar stresses at points D and F are analyzed. In this case no stresses can be neglected in any part of the structure because they are in the same or one lower order of magnitude respect to the stresses due to the axial intern force. Reguarding at the maximum values, 8.53 Mpa and 18 Mpa, we can conclude that the most stressed region of the frame is the head tube. In fact, that region is characterized for supporting a boundary condition and a load at the same time. Also it represents a point of stress concentrations. Maximum combined bar stresses and Ultimate Limite State (ULS) Fig. 64. Maximum combined bar stresses at center
87 Combining the effects of axial internal force and bending internal moment we obtain the fringe diagram shown in figure.64. As said before the most stressed region under compression is the top tube and the most one under traction is the head tube. The maximum values are 2.28 Mpa and 17.6 Mpa, respectively. The less stressed region is the seat tube (intermediate blue colours). Regarding and Ultimate Limite Stress analysis we have: 150
∀ ∈
⟹ Where M is the set of all the elements of the structure. Therefore, an elastic behavior is guaranteed. Ashby indices assessment The chosen ashby indixes for the preliminary QFD4Mat matrix in section 1 are based on the FEM analysis bellow. It has been shown that the tubes of the bike frame work mainly subjected to axial and bending. Hence the Tie(tensile stress) and Beam(bending stress) are actually suitable. Further considerations: weight of the frame Fig. 65. Mass properties display by Patran The mass of the analyzed aluminum frame is 967.6 grams. 3.2.
Topology optimization for shape definition of BIKE FRAME: software simulation 3.2.1. Problem formulation For a topology problem formulation the following features should be defined: –
–
–
–
Topology design region Loads Boundary conditions Additional design restrictions For the topology study, the 2‐D space has been chosen. Unfortunately, the 5,000 nodes limitation of the MSC Student’s version does not allow to work with a desired 3‐D design block. Nevertheless, for a conceptual desing, the 2‐D approach of the problem seems to be enough accurate. 88 The topology design region is the following surface: Fig. 66. Design space. All distances in mm. Then the surface is imported to a patran database. Fig. 67. Design space. Patran overview. Thereafter, an isotropic material is defined. The selected material is the same generic aluminum chosen in section 3.1.1. However the properties of the material are not critical for the topology optimization model since it works with an artificial material and the design variables are the relative densities of the elements. 89 Afther that, the element properties are defined. Since the design space is bidimensional, the element property selected has been a shell: Fig. 68. Shell properties menu The fact of selecting a shell as the element property of the model means that the equations of the structural shell theory are applied to each of the two‐dimensional (quandrangular or triangular) elements. Then, the mesh is defined. This is a quite critical point. An homogeneus and well‐finished mesh is needed to perform an accurate topology analysis. In order to show this three meshes have been designed. In all cases first order elements (Quad4) are selected and the chosen shape is Quad. For hybrid meshing Tria3 elements are also used. Fig. 69. Quad4(left) and Tria3(right) elements
90 One surface. Paver meshing. Paver is best suited for trimmed surfaces, including complex surfaces with more than four sides, such as surfaces with holes or cutouts. Paver is also good for surface requiring “step” mesh transitions, such as going from four to 20 elements across a surface. The following mesh was obtained using paver mesher: Fig. 70. First mesh. One surface. Paver meshing
A total of 4648 nodes and 4505 elements were created. Once again, it is important to remind the student version limitation at 5000 nodes. The mesh seems to be quite dense but it is not very homogeneus. One surface. Hybrid meshing. A hybrid grid contains a mixture of structured portions and unstructured portions. It integrates the structured meshes and the unstructured meshes in an efficient manner. Those parts of the geometry that are regular can have structured grids and those that are complex can have unstructured grids. These grids can be non‐conformal which means that grid lines don’t need to match at block boundaries. The following mesh was obtained using hybrid mesher: Fig. 71. Second mesh. One surface. Hybrid meshing
91 A total of 4469 nodes and 4345 elements were created. This second mesh has a good density and is more homogeneus than the first one. Six surfaces. IsoMesh and Paver meshing. A highly homogeneus and accurate mesh can be designed if the surface is divided into several simpler surfaces. As the matter of fact, the original surface has been divided into six smaller regions. Three of them are quadrilateral, one triangular and the remaining two irregular polygonal. Fig. 72. Six sections broken surface Then, the three quadrangular regions have been meshed with IsoMesh. The remaining three with Paver. The following mesh has been designed: Fig. 73. Third mesh. Six surfaces. IsoMesh and Paver meshing 92 A total of 4657 nodes and 4321 elements were created. This second mesh has a great density and is the most homogeneus of the three meshes. Hence it is the selected mesh for running the topology analysis. NB: Due to the “region by region” meshing, an equivalence action is submitted. Afterwards the loads and boundary conditions are defined. Fig. 74. Loads and BCs applied 3.2.2. Topology optimization The topology optimization analysis is characterized by the following setting menus. Fig. 75. Toptomize general menu
93 Fig. 76. Objectives and constraints (left). Optimization Control Parameters (right) As stated in the fundamentals and mathematical theory of topology optimization, the objective function of the analysis is to minimize the compliance of the structure. The compliance of the structure is the inverse of the rigidity. Hence, the rigidity is maximized. The main constraint is given by the mass target. A selected mass target of 0.4 means that the final structure must have the 40% of the initial design space. Then, a 60% of mas is spared. The maximum design cycles number limits the number of iterations done by the solver program, Nastran. More design cycles may be required to achieve a clear 0/1 material distribution, particularly when manufacturability constraints are used. The penalty factor, also called power factor, has a large influence on the solution of topology optimization problems. A lower value (less than 2.0) often produces a solution that contains large “grey” areas (area with intermediate densities 0.3‐0.7). A higher value (greater than 5.0) produces more distinct black and white (solid and void) designs. However, near singularities often occur when high power value is selected. In our case, an intermediate value of 3 have been chosen. 94 Design domain and load case are selected. Fig. 77. Design domain and subcase select menus
Finally, the model file is sent to the numerical solver Nastran and the optimization takes place. Fig. 78. Nastran running toptimize 95 3.2.3. Results In contrast to sizing and shape optimization for detail design, the layout and load‐path study in the initial, conceptual design stage uses the Topology Optimization function. Topology Optimization is good at dealing with global design responses (such as structural compliance, eigenvalue, and displacements) but not local design responses (such element responses, stress and strain). It is recommended that topology optimization is used to generate a conceptual design proposal with emphasis on global design responses. Therefore the topology optimization results have been taken as an orientation for the re‐defining of the bicycle frame. Three topology analyses have been runned. First one for a weight reduction of 60% of the mass; the second one for a 70% reduction; and the third one for a 80% reduction. It is reminded that these “sparing” percentages are taken respect to the total mass of the design space. First analysis. Mass target=0.4 The following density distribution fringe diagram has been plotted: Fig. 79. Density distribution fringe diagram. Mass target: 40% The most intense coloured regions (red) represent the elements with a relative density nearly 1. That means that they are highly important for the rigidity of the structure. On the other hand, the dark coloured regions (blue) represent the elements with a low relative density, tending to 0. These elements are not significant for the rigidity of the structure. They can be voided. Thereafter some element‐detailed density distribution diagrams are shown. In these diagrams, the elements with a relative density under the selected threshold are eliminated from the design space, remaining a clear figure. They have been posted in increasing order of threshold. 96 97 As much as the threshold increases the internal segment gets thiner and finally disappears. Therefore that structural part is not very significant for the whole structure. In fact, looking at the density distribution diagram (Fig.79) this region is coloured in yellow meaning an intermediate relative density factor. Second analysis. Mass target=0.3 In the second analysis a mass target of 30% has been aimed for. As expected, the obtained design is thiner and the main structure lines are the same as precedence. A little difference about internal sections has been found. Nevertheless, as stated before, these sections do not contribute significantly to the global rigidity and they are voided for thresholds close to one. Firstly, the density distribution fringe diagram has been plotted: 98 Fig. 80. Density distribution fringe diagram. Mass target: 30%
The element‐detailed density distribution diagrams are: 99 100 The two‐crossed sections are not significant for the rigidity of the structure. A yellow colour is associated to them in the density distribution fringe diagram. Hence, they get voided as the threshold increases. Third analysis. Mass target=0.2 In this final topological analysis a mass target of 20% is settled. A thinner geometry is expected, free of internal sections for lower threshold values. The density distribution fringe diagram is: Fig. 81. . Density distribution fringe diagram. Mass target: 20% In this case there is no “seat tube” in the proposed optimized geometry. The element‐detailed density distribution diagrams are: 101 102 As the threshold increases the internal down section gets voided. As shown in the density fringe diagram this section does not contribute to minimize significantly the compliance of the structure. 3.3.
Final redefined bicycle frame geometry Taking into account the results of the sections 3.1 and 3.2 a new frame geometry model is proposed for the posterior FEM analysis of the section 3.4. For simplicity, the geometry of the bicycle frame will be based again on circular tubes. However, in this case we consider as designing variables the thickness and the diameter of the tube. First of all, it is important to justify the fact that the topology optimization results have given a 5‐edge geometry frame with no “seat tube”. For the explanation it is considered the next table: HT TT DT ST SSs CSs Axial stress 5,33E‐03 2,41
6,64
5,33E‐03
2,41 3,02
Bend stress CE 2,70E‐02 2,70E‐02
2,70E‐02
2,70E‐02
2,70E‐01 3,00E‐02
Bend stress DF 18 8,53
7
2,1
2,1 2,1
17,6 2,28
9,6
1,7
2,28 4,34
Max comb stress Table 14. Bar stresses by tubes of the generic bike frame (section 3.1) [MPa] Where the column titles mean, HT(Head Tube), TT(Top Tube), DT(Down Tube), ST(Seat Tube), SSs(Seat Stays) and CSs(Chain Stays); and the row titles are refered to the bar stresses plotted into section 3.1. As shown before the less stressed bar is the seat tube. That is the reason why it is removed from the design space in the topological analysis. The seat tube does not have a high contribution to the global stiffness of the frame. Nevertheless, for a dynamic work case in which the cyclist rides the bicycle vigorously; as an example, an sprint situation, the seat tube hires a great importance in the internal rigidity of the frame. Consequently, the seat tube remains in the redefined structure but it employes an smaller and thinner section. It is reminded that in the first generic bycicle geometry of section 3.1. a constant circular section of 40 mm diameter and 0.7 mm thickness was considered for all the frame tubes. Seat tube redefining A constant section tube is considered with a diameter of 30mm and 0.5mm thick. Fig. 82. New seat tube section. Magnitudes in mm.
103 Head tube redefining As shown by the topology optimization results, the section corresponding to the head tube is the thickest of all sections which forms the optimized structure. That fact points out that the head tube has a big contribution to the global rigidity of the frame. The section 3.1. FEM analysis also justifies it; the highest stresses are localized at the head tube. For that reason, it has been decided to increase the thickness of the tube. The new head tube section has the following dimensions, 40 mm diameter and 0.9 mm thick. Fig. 83. New head tube section. Magnitudes in mm.
Top tube redefining Regarding the topology optimization a smaller section is proposed. In addition the table 14 points out the top tube as a low stressed bar. Therefore a reduction in diameter and thickness are proposed; updating them to 30 mm and 0.6 mm, respectively. Fig. 84. New top tube section. Magnitudes in mm.
104 Chain stays redefining The topology optimization illustrates a section with a similar thickness to the top tube. Nevertheless, it is important to point out that the topology analysis is made in a 2‐D workspace. In the 3‐D reality this rigidity target is assumed by two bars, the two chain stays. Regarding also the bar stresses at the chain stays, it is shown they are not highly forced. Consequently a diameter of 25 mm and a thickness of 0.7 mm are projected. Fig. 85. New chain stay section. Magnitudes in mm.
Seat stays redefining The same interpretation is applied for the seat stays. In this case the bar stresses are quite lower than the ones of the chain stays. In any case, they are lowly stressed. Then, a diameter of 25 mm and a thickness of 0.7 mm are chosen. Fig. 86. New seat stay section. Magnitudes in mm.
105 Down tube redefining Analyzing the topology optimization results and the bar stresses it is concluded that the down tube bar is well defined. No high stresses take place in it and the section shows by the topological analysis illustrates a quite thick section. Hence, the initial diameter of 40 mm and thickness of 0.7 mm are maintained. Bottom bracket design In order to make the new geometry more realistic a bottom bracket is considered. Fig. 87. Bottom bracket detailed picture
The designed bottom bracket tube has the following dimensions: Fig. 88. Bottom bracket design. All magnitudes in mm.
106 Since the bottom bracket has to put up the bearing of the pedal axle, then it is going to be subjected to high torsional and bending loads. For that reason, it is the tube with the highest section thickness (3mm) of all the frame tubes. Finally, the new bike frame geometry is defined in Patran with simple curves. The objective is to model the geometry as simple beams in order to carry out a 1‐D FEM analysis. Fig. 89. Bike frame new geometry model. Main points written down Coordinates [mm] Point x y z 1 0 0 0 2 0 0 40 3 0 0 ‐40 4 ‐398,34 69,5 80 5 ‐398,34 69,5 ‐80 6 ‐59,07 580 0 7 480,93 580 0 8 516 465,24 0 Table 15. New geometry point coordinates
107 Fig. 90. Bike frame new geometry model. Main edges written down Curve 1 2 3 4 5 6 7 8 9 10 Coordinates [mm] Start Point End Point Lenght 1 2 40 1 3 40 2 4 406,33 3 5 406,33 4 6 618,154 5 6 618,154 6 7 540 7 8 120 8 1 694,769 6 1 583 Table 16. New geometry curve attributes Thereafter, the three candidate materials are defined as isotropic materials. The following properties are filled in: AISI 4130 


Young’s modulus= 205 GPa Poisson coefficient= 0.29 Density= 7.85 g/cm^3 Titanium 3Al‐2.5V 


Young’s modulus= 100 GPa Poisson coefficient= 0.30 Density= 4.48 g/cm^3 Aluminum 7005‐T6 


Young’s modulus= 72 GPa Poisson coefficient= 0.33 Density= 2.78 g/cm^3 108 Then the six different circular sections are defined at the beam library of the program. Fig. 91. Beam library menu. New optimized sections created
Thereafter the properties are defined. A different property is defined for each tube, since each tube has a specific section and orientated vector. All the element properties have in common the object, 1‐D, and the type, beam. Fig. 92. Element properties selection. Different property sets created 109 Then the loads and boundary conditions are created. The same loads and BCs of the first FEM analysis are taken. Fig. 93. Loads and BCs applied to the optimized structure
Then the bicycle frame structure is meshed. Fig. 94. Mesh seeds. Element edge length=10mm
110 A total of 417 nodes and 407 elements have been created, before executing the equivalence. The following screenshot shows a 3‐D:fullspan beam display. Fig. 95. 3D:fullspan beam display
3.4.
FEM analysis of the optimized geometry for each candidate material Finally, the finite element model is sent to the numerical solver Nastran. After the numerical analysis, the results file is imported to Patran and the results are displayed. One specific analysis is runned for each material. 3.4.1. AISI 4130 Deformated structure Fig. 96. Deformated bike frame. AISI 4130
111 The displacements shown are plotted as the resultant of the the x and y displacement components. The main displacements are listed in the table below. Displacement [mm] x y δ 0,0027947 ‐0,039367 0,03947 0,036442 ‐0,03761 0,05237 0,026611 0,0079779 0,02778 Point 1 6 7 Table 17. Deformated structure main displacements All the main displacements are of the same order of magnitude, hundreds of millimiter. The maximum displacement corresponds to the point 6 and it has a value of 5,237e‐2 mm. Serviceability Limit State (SLS) A SLS of 1 mm is considered taking into account that the frame of racing bycicle must be so stiff as not being deformed to much. Actually, all the physical energy of the cyclist should be converted into kinetic energy, and nor into deformation energy. 1
∀ ∈
⟹ Where N is the set of all the nodes of the structure. Bar stresses Axial bar stresses Fig. 97. Bar stresses due to axial action [MPa]. The maximum traction stress is located at the down tube with a value of 7,17 MPa and the maximum compression stress is located at the top tube and seat stays with a value of 3,73 MPa.The chain stays are traction stressed with a significant value of 4,26 MPa and, the seat and head tubes are not significantly stressed. 112 It is important to point out that all the stressed members experiment values of the same order of magnitude. That fact brings out the well resizing of the bike frame tubes. Bending bar stresses The post‐processor allows to graphic the bending bar stresses at the four main points of the section. Looking at the section figures related to each tube (view section 3.3.), two symmetries respect to a generic diameter can be observed. For any bending force, the points C and E will experiment the same stress in absolute value and opposite sign. The same applies for points D and F. The fact that the bending forces will always be same orientated respect to the neutral fiber explains the previous symmetries. Fig. 98. Bending bar stresses at point C [MPa]. Fig. 99. Bending bar stresses at point E [MPa]. 113 All the main frame (top tube, head tube, down tube and seat tube), the bottom bracket and the seat stays are low stressed. On the other hand the chain stays exhibit a variable bending. The edges are subjected to opposite sign and nearly equal absolute value stresses. This fact conditions the chain stays to be regions of a high tendency of fatigue failure. Nevertheless, the low absolute values of the stresses give the user a high security margin respect to the endurance limit number of fatigue cycles of the material. It is important to remind, the endurance limit stress is a constant limit for steels, but not for aluminum. Fig. 100. Bending bar stresses at point D [MPa]. Fig. 101. Bending bar stresses at point F [MPa]. 114 A maximum stress of 18.5 MPa is founded at the head tube. The head tube exhibits the maximum compression and traction stresses due to bending at points E and F. Since the head tube suffers a force in his superior edge and it is displacement constraint in the other edge, it concentrates significant stresses in a quite short length. Maximum combined bar stresses and Ultimate Limite State (ULS) Fig. 102. Maximum combined bar stresses [MPa]. Combining the effects of axial internal force and bending internal moment we obtain the fringe diagram shown in figure.102. The Ultimate Limite Stress analysis is done in order to satisfy the structural verifications. Taking into consideration the yield strength of the steel AISI 4130, we have: 435
∀ ∈
⟹ Where M is the set of all the elements of the structure. Therefore, an elastic behavior is guaranteed. Total weight of the frame The total mass of the steel AISI4130 frame brought out by the software is 2160 grams. Table 18. Bike frame mass properties. AISI 4130 115 3.4.2. Aluminum 7005‐T6 Deformated structure Fig. 103. Deformated bicycle frame. Aluminum 7005‐T6. Main displaced
points numbered. The main displacements are listed on the table below. Point 1 6 7 Displacement [mm] x y δ 0,008005 ‐0,11222 0,11251 0,10408 ‐0,10725 0,14945 0,076133 0,022827 0,07948 Table 19. Main displacements. Aluminum 7005‐T6 The largest displacement takes place at point 6 and it has a value of 1,4945e‐1 mm. Serviceability limit state (SLS) A SLS of 1 mm is considered for the same reasons explained in precedence. 1
∀ ∈
⟹ Where N is the set of all the nodes of the structure. Bar stresses Axial bar stresses Since the axial stress only depends on the axial internal action and the area of the cross section: ,
The fringe diagram of axial bar stresses is always the same independently of the material. Therefore the figure 97 applies also for this case. Bending bar stresses The same applies for bending bar stresses. They are only dependent of the bending internal moment and the elastic section modulus. ,
116 The fringe diagram of bending bar stresses is always the same independently of the material. Therefore the figures 98, 99, 100 and 101 apply also for this case. Maximum combined bar stresses and Ultimate Limit State (ULS) For the Ultime Limit State the maximum combined bar stresses are calculated from the axial and bending bar stresses. Hence, the fringe plot is independent of the material. The figure 102 applies again in this case. However, for the ULS verification the material influences since the yield strength is taken into consideration. Given the redefined bicycle frame and targeting Aluminum 7005‐T6 as raw material, we have: 290
∀ ∈
⟹ Where M is the set of all the elements of the structure. Therefore, an elastic behavior is guaranteed. Total weight of the frame The total mass of the Aluminum 7005‐T6 frame brought out by the software is 765 grams. Table 20. Bike frame mass. Aluminum 7005‐T6 3.4.3. Titanium 3Al‐2.5V Deformated structure Fig. 104. Deformated redefined bicycle frame. Titanium 3Al‐2.5V. Main displacement points numbered 117 The main displacements are listed in the table below: Point 1 6 7 Displacement [mm] x y δ 0,0057377 ‐0,080725 0,08093 0,074764 ‐0,07713 0,10742 0,054619 0,016375 0,05702 Table 21. Main displacements for the deformated bike frame. Titanium 3Al‐2.5V The largest displacement takes place at point 6 with a value of 1,074e‐1 mm. Serviceability limit state (SLS) A SLS of 1 mm is considered for the reasons explained in precedence. 1
∀ ∈
⟹ Where N is the set of all the nodes of the structure. Bar stresses As stated before, neither axial bar stresses nor bending bar stresses depend on the material. Hence, the fringe plots of the AISI 4130 stress results applies also for the selected titanium. Nevertheless, the Ultimate Limite State must be verified with the properties of the Ti 3Al‐2.5V. 500
∀ ∈
⟹ Where M is the set of all the elements of the structure. Therefore, an elastic behavior is guaranteed. Total weight of the frame The total mass of the Titanium 3Al‐2.5V frame brought out by the software is 1233 grams. Table 22. Bike frame total mass for Ti 3Al‐2.5V 3.5.
Conclusions Regarding the limit state designs, ULS and SLS, the three materials verify them for the optimized bicycle frame structure. However, three main parameters are taken into consideration in order to compare the three materials. They are: –
–
–
Total frame mass Maximum displacement δ_max Factor of safety for the maximum stress value 118 The values of these parameters for each of the materials are shown in the table below. Material AISI 4130 Al 7005‐T6 Ti 3Al‐2,5V Total Mass [gr] 2160 765 1233 δ_max [mm] 5,237E‐02 1,495E‐01 1,074E‐01 Safety Factor 24 16 27,5 Table 23. Comparison parameters values The stiffer material, the AISI 4130 steel, exhibits a high safety factor giving up lightness. However, the titanium 3Al‐2.5V exhibits a good rigidity (two times lower than the steel rigidity) with the best safety factor and a controlled mass (nearly two times lower than steel frame mass). Finally, the aluminum 7005‐T6 is the lightest (three times lighter than steel) but, on the other hand, it exhibits the lowest safety factor and the highest maximum displacement. All these results will be employed for updating the QFD4Mat matrix in the final chapter. 119 4. Refinement of QFD4Mat analysis: final results and discussions In this last chapter, the initial QFD4Mat matrix of section 1.6. is redefined, substituting some general key‐features and the Globa Performance Index for other specific key‐features related with the results obtained in the FEM analysis of the section 3.4. 4.1.
Translation of Performance Index by FEM results: implementing the Key‐Material Factors As stated in section 1.7, the main limitation of the first QFD4Mat matrix was the generality of the key‐features and the Global Performance Index (GPI). Reminding the calculus of the GPI, it was considered the frame as a generical beam structure, where the components work under axial loading and bending loading. Actually, the chosen AHSBY indeces for a generical tie (tensile strut) and generical beam (loaded in bending) are suitable for a lot of any other applications further than the bicycle frame. Therefore the GPI is not considered anymore in the final matrix. However, new key‐features specifically related with the bicycle frame case are taken into consideration. These new key‐features are: –
–
–
Total frame mass Maximum displacement δ_max Factor of safety for the maximum stress value The software used to enhance the final QFD4Mat analysis has been the QFD4Mat‐customv.03 excel sheet.13 4.2.
Revising the QFD4Mat matrix When introducing the new key‐features, some already considered are substituted. Total frame mass The specific strength (SS) is substitute by the total mass (TM). Since the total mass, as a particular result of the optimized bike frame, is related to lightness and handling product requirements, it should substitute the specific strength that it is, on the contrary, a generic key feature only dependable on the material. The total mass key feature is introduced with inverse relationship respect to the performance product requirements lightness and handling. In fact, the lower the mass the lighter and the better handling the bike frame performs. For the new weight technical coefficients related to the mass, the 0‐5 sheet converter of the excel sheet has been employed, refer to figure 105. The AISI 4130 receives the lowest scaled value, 1.3, since it is the heaviest material. On the other hand, the aluminum 7005‐T6 receives the highest scaled value, 4.3, showing the lightest performance. 13
The QFD4Mat‐customv.03 has been developed by Fabrizio D’Errico (tutor of this thesis work) in his project QFD4Mat. Material choice for product development. It is free software for student purposes. For more information about the project refer to http://qfd4mat.com/ 120 Fig. 105. Total mass 0‐5 scale conversion. Inverse relationship. It is important to point out, the total mass key feature is considered twice, once as performance key‐factor and other as cost key‐factor. It is also considered in costs due to his inverse influence in some of them as the material cost or the heat treatment cost. Actually, if the total mass of the frame is high, more mass material will be employed for the manufacturing process and the heat treatment will be more expensive since more material must be heated. Maximum displacement The Young’s modulus is substituted by the maximum displacement as same reference of frame’s stiffness. Therefore, all the relations and the grades of relationship with the product requirements related to elastic modulus are maintained. However, the maximum displacement key‐feature is defined inverse respect to the product requirements. For example, if the maximum displacement increases the fatigue failure problem tends to be more significant. Therefore an increment of the key‐feature maximum displacement produces a decay in the failure response. As carried on before, the general values of maximum displacement for each bike frame material are converted to 0‐5 scale in order to fill in the respective technical weights of the QFD4Mat matrix. Fig. 106. Maximum displacement 0‐5 scale conversion. Inverse relationship. 121 Safety factor The yield strength key‐feature has been substituted by the safety factor brought out by the FEM analysis of section 3.4. Since the safety factor contains the effect of the yield strength the substitution is justified. Once again, it is passed from a generic key‐feature to a specific key‐
feature of the bike frame performance. Furthermore, all the relations and the grades of relationship are maintained. The safety factor is defined as direct key‐feature as it was defined previously the yield strength. Actually, the safety factor is defined as the yield strength divided by the maximum stress value (a constant value for all the materials). Thereafter the 0‐5 scale conversion table for the safety factor FEM analysis values is fulfilled. Fig. 107. Safety factor 0‐5 scale conversion. Direct relationship. The following screenshots try to visualize the final aspect of the updated QFD4Mat matrix. Fig. 108. Technical key‐factors grouped by performance, cost and receptiveness. First part of the matrix
(performance) shown. 122 Fig. 109. Cost product requirements related to cost key‐features Fig. 110. Receptiveness product requirements related to receptiveness key features. Fig. 111. Technical weight factors and 0‐5 scale values. 4.3.
Update graphic solutions by Bubble Maps and Material Value Curves Finally the new results given by the matrix are interpreted. First of all the final weighted score is displayed. Fig. 112. Final weighted score. Final QFD4Mat matrix.
The final weighted score brings out the titanium as the “winner” of the three candidate materials. It is important to remember this score is made by the sum of all the key‐feature values multiplied each one for his relative weight. Therefore, the inverse relationships are not taken into account in the final weight score. Hence, it is not a very feasible indicator. Consequently, the “winner” material has to be selected by analizing the bubble maps graphic and the material value curves graphic. 123 Value curve analysis Fig. 113. Value curves graphic. Updated QFDFMat matrix. The value curves allow to compare easily the candidate materials for each key‐factor. The technical factors have been ordinated in increasing order of relative weight. That means the key‐factors placed on the right are more significant for the scope of adjusting to the VOC (Voice Of the Consumer) than the ones immediately on the left. The key‐factors in which the biggest differences between materials are present have been pointed out with an orange circle. Looking at surface hardness, the titanium performs great, the steel does it well and the aluminum is the softer of the three. Regarding ultime tensile strength and endurance stress the steel and the titanium perform in a satisfactory manner while aluminum does it in a poor way. This could be explained by the FCC structure of the aluminum. Thereafter, the second orange circle is analized. All the key‐factors in this circle have the same relative weight, 7%. Hence the relative importance curve maintains the slope horizontal. The corrosion rate factor is considered with inverse relationship. That means the titanium has the lowest corrosion rate and the steel performs in the poorest manner against corrosion. The material cost of titanium is very significant; the cost of aluminum quite significant and the steel is the cheapest material. The titanium also performs with the highest toughness, closely followed by the steel while the aluminum liberates a low quantity of energy when it breaks. Significant differences are also presented between the values of the total mass. As expected, the lightest bike frame is made of aluminum and the heaviest made of steel. The total mass as cost key‐factor has even further relative important, 12%. That means that the same difference or gap between the values of a left hand already said key‐factor and the total mass represent very different effects in the global balance of the value curves. In other words, the total mass is a heavy key‐factor where small differences between material values make big differences in the global weight. 124 Finally, the maximum displacement is stated as the most valuable key‐factor. As defined inversely, the steel performs with the lowest maximum displacement and the aluminum frame suffers the largest maximum displacement. Summarising, the value curves do not give us a “winner” material however point out in which key‐factors the materials diverge the most and how significant is this difference in the global comparison between them. Therefore, the R&D product teams are able to recognize what are the key fields in which improvements are needed. Bubble maps analysis The QFD4Mat excel sheet brough out the following bubble curves. Fig. 114. Bubble maps of the updated QFD4Mat matrix First of all, the aluminum bubble (red one) is analized. Its center is low and left side moved respect to the center of the VOC curve. Therefore, the aluminum is cheaper than the expectations of the consumer but it is also less performance respect to the said expectations. In fact, it is only a little bit cheaper but however it is much less performance than expected; the vertical gap between the aluminum bubble center and the VOC bubble center is lower than the horizontal gap. The spare in costs does not compensate the lack in performance. Furthermore, the radius of the aluminum bubble that represents the receptiveness of the product has a lower value than the VOC radius. The receptiveness of the aluminum is the lowest of the three candidate materials. Consequently, aluminum is rejected. Then titanium and the alloy steel are compared. The center of the steel bubble is moved downwards to the left. There is a trade‐off between the reduction in performance and the reduction in cost respect to the VOC bubble. That means, the horizontal distance between such centers is similar to the vertical distance. The radius of the green bubble is quite shorter than the VOC’s one. Hence the receptiveness covered by the steel bubble is lower than the target one by the consumer expectations. On the other hand, the titanium bubble (blue one), 125 is quite less performant (downwards moved) and quite more expensive (rightwards moved) than the VOC’s bubble. Nevertheless, the titanium covers in a satisfactory manner the receptiveness of the consumer. In fact, the blue bubble is shown as the best fit bubble to the consumer’s expectations. 4.4.
Conclusions Finally, taking into account the three valuation procedures given by the QFD4Mat matrix: final weighted score, value curves and bubble curves; the chosen material between the three candidates has been the Titanium 3Al‐2.5V (Grade 9). The bike frame made by the titanium will satisfy greatly the expectations of the athlete in terms of receptiveness and performance but it is a bit expensive refer to the consumer’s budget. However, the bike frame is aimed to be implemented into a racyng bicycle, where a high‐technical and high purchasing power consumer profile is present. 126 127 Discussions, final conclusions and further improvements The present thesis has been developed with a logical and sequential order. The main innovative scope of the thesis was the implementation of the material selection, based onto an hybrid multi‐criteria approach, and the topology optimization assisted by FEM analysis. In the first QFD4Mat material selection analysis no material was clearly selected. There was a breaking even situation between the titanium and the steel. As stated before that previous analysis was based in genereal product requirements of a general bike frame. Instead, thanks to the topology optimization, assisting the re‐designing of the bike frame, new specific key‐
factors could be obtained for the final matrix. In fact, there has been an improvement in the effective application of the matrix; since, finally, it has been supplied with specific geometry and design dependant factors. The first matrix could be used also for a product totally different but based on the general performance of beams subjected to axial and bending forces. On the other hand, once some general values and the general Ashby indices are substituted by specific optimized product geometry factors, the matrix increases his effectivity. Consequently, in the last matrix one material has been pointed out clearly as the most suitable for the optimized design and the other criterias considered. Fig. 115. Influence of the topology optimization in the material selection. Evolutionary workflow. Further improvements Regarding the topology optimization analysis carried out in this thesis it is important to point out the limitation of the student version of the MSCSoftware to 5000 nodes in the creation of FE models. For that reason, a 2‐D topology study has been carried out. Considering that the bike frame geometry is 3‐D, a solid block could be a better design space for the topology optimization. Since topology optimization gives the engineer a conceptual first design, other more‐
detailed kinds of optimization could be combined with it. For example, shape optimization. It is reminded that topology optimization finds an optimal distribution of material given the package space, loads and BCs; and shape optimization allows to vary the shape of the structure to satisfy specific requirements. 128 Regarding the QFD4Mat project, a website platform is been developed in order to create a more user‐friendly product for the material selection14. The results obtained with the beta version of the platform have been the same as the results obtained with the matrix. Some screenshots of the website platform are given in Annex B. 14
For more information about the project refer to http://qfd4mat.com/ ANNEX A‐ Material Tables Designation or trade name C Mn UNI EN 25CrMo4 ‐ AISI 4130 0,28 ‐ 0,33 0,40 ‐ 0,60
Aluminum 7005‐T6 ‐ 0,20‐0,70 Ti 3Al‐2.5V , alpha annealed <=0,050 ‐ Chemical Composition of the selected materials Si Cr Mo V 0,15‐0,30 <=0,35 ‐ 0,80 ‐ 1,10
0,06‐0,20 ‐ 0,15 ‐ 0,25
‐ ‐ ‐ ‐ 2,0‐3,0 Al Fe ‐ 91‐94,7 2,5‐3,5 97‐98,2 <=0,40 <=0,20 Elongation [%] 25,5 13 Kic [Mpa∙m^(1/2)] ‐ ‐ 15 100 Chemical Composition of the selected materials Mg Zn Ti Cu ‐ ‐ ‐ ‐ 1,0‐1,8 4,0‐5,0 0,01‐0,06 <=0,1 ‐ ‐ 92,75‐95,50 ‐ Designation or trade name AISI 4130 Aluminum 7005‐T6 UTS [MPa] 670 350 YS[MPa] 435 290 Hardness HB 197 94 Ti 3Al‐2.5V , alpha annealed 620 500 256 Designation or trade name AISI 4130 Aluminum 7005‐T6 Ti 3Al‐2.5V , alpha annealed Density [kg/m^3] 7850 2780 4480 Mechanical Properties E [GPa] µ[‐] Impact [J] 205 0,29 42 72 0,33 16 100 0,3 86 Other important characteristics Specific Strenght [kN*m/Kg] Price [€/kg] 85,35 ‐ 125,90 ‐ 138,39 ‐ Machinability [%] 70 ‐ ‐ Conversions to 0‐5 scale Mechanical Properties Designation or trade name UTS [MPa] YS[MPa] AISI 4130 Aluminum 7005‐T6 Ti 3Al‐2.5V , alpha annealed 3,7 1,9 3,2 2,1 YS[MPa] (˅) 2,80 3,90 3,4 3,7 2,30 HB E [GPa] 4,3 2,1 4,9 1,7 2,6 1 4 4 Spec.Strenght [kN*m/Kg] 2,2 3,2 5 2,4 5 2 3,6 Impact [J] Kic [Mpa∙m^(1/2)]
Conversions to 0‐5 scale Designation or trade name Price [€/kg] 1 AISI 4130 2 Aluminum 7005‐T6 Ti 3Al‐2.5V , alpha 4 annealed Other important characteristics Machinability [%] Productivity (=cost) 3,1 2,6 2,4 3 3,5 Corrosion Rate (˅) 1,2 3,2 4,6 3,4 ASHBY Indexes E^(1/2)/ρ
Designation or trade name AISI 4130 Aluminum 7005‐T6 Ti 3Al‐2.5V , alpha annealed YS [MPa] 435 290 E [Gpa] 205 72 500 100 ρ[kg/m^3] Numerator 7850 6482669,203
2780 2276839,915
4480 3162277,66 Denominator 7850 2780 Rapport 825,818 819,007 4480 705,866 ASHBY Indexes ρ Numerator 2,05E+11 72000000000 1E+11 ρ Denominator Rapport Numerator Denominator Rapport 7850 26114650 435000000
7850 55414,01 2780 25899281 290000000
2780 104316,5 4480 22321429 500000000
4480 111607,1 Global Performance Index 2,86 3,12 3,02 ANNEX B‐ QFD4Mat website application In the following some screenshots show the aspect of the website QFD4Mat project platform, applied to our case study: bike frame. Fig. 116. Login interface Fig. 117. Project management interface Fig. 118. Material candidates menu 132 Fig. 119. Product requirements menú Fig. 120. Key features menú Fig. 121. Relations grid 133 Fig. 122. Material assesment Fig. 123. Value curves graphic Fig. 124. Bubble maps plot 134 135 Bibliography [1] F. D'Errico, Material Selections by a Hybrid Multi‐Criterial Approach, Springer, 2015. [2] "Wikipedia," [Online]. Available: https://en.wikipedia.org/wiki/Racing_bicycle. [3] M. Bendsoe, Topology Optimization. Theory, methods and applications, Springer, 2003. [4] Kress, G and Keller, D, Structural Optimization, ETH Zürich: Zentrum für Strukturtechnologie, 2007. [5] M. Linari, "Product Shape Optimization‐ MSC Software," 2015. [6] "Explore the World of Piping," http://www.wermac.org/pipes/pipemaking.html. [Online]. Available: [7] L. Lang, ""Hydroforming Highlights: Sheet Hydroforming and Tube Hydroforming"," Journal of Materials Processing Technology, 2004. [8] "Ebicycles," [Online]. sizer/road‐bike. Available: http://www.ebicycles.com/bicycle‐tools/frame‐
[9] "QFD4Mat," [Online]. Available: http://qfd4mat.com/. [10] "Metal Guru. Lessons in bycicle manufacturing," http://metalguruschool.com/?attachment_id=2052. [Online]. Available: [11] "Cube," [Online]. Available: http://www.cube.eu/bikes/road‐race/peloton/cube‐peloton‐
sl‐rednblack‐2015/. [12] Studio Maestrelli, "Bycicle frame optimization by means of an advanced gradient method algorithm". 136