Operational Programme “Education and Lifelong Learning

Operational Programme “Education and Lifelong Learning

. Operational Programme “Education and Lifelong Learning” Continuing Education Programme for updating Knowledge of University Graduates: “Modern Development in Offshore Structures” George D. Manolis, Professor Department of Civil Engineering Aristotle University, Thessaloniki GR‐54124, Greece Tel: (+30 2310) 99 5663, Fax: (+30 2310) 99 5769 E‐mail: [email protected] 1 ‐ SECTION 8.2 ‐ GDM . ΕΠΙΧΕΙΡΗΣΙΑΚΟ ΠΡΟΓΡΑΜΜΑ «ΕΚΠΑΙΔΕΥΣΗ ΚΑΙ ΔΙΑ ΒΙΟΥ ΜΑΘΗΣΗ» Πρόγραμμα Δια Βίου Μάθησης ΑΕΙ για την Επικαιροποίηση Γνώσεων Αποφοίτων ΑΕΙ Χρονική Περίοδος: 2014‐2016 ΠΕΓΑ: ΣΥΓΧΡΟΝΕΣ ΕΞΕΛΙΞΕΙΣ ΣΤΙΣ ΘΑΛΑΣΣΙΕΣ ΚΑΤΑΣΚΕΥΕΣ SECTION 8: ANALYSIS AND DESIGN OF MARINE STRUCTURES 8.1: ΣΥΣΤΗΜΑΤΑ ΘΑΛΑΣΣΙΩΝ ΚΑΤΑΣΚΕΥΩΝ ΣΤΑΘΕΡΟΥ ΠΥΘΜΕΝΑ 8.1.3: Basic Principles of Dynamics (2) Γεώργιος Δ. Μανώλης, καθηγητής Α.Π.Θ. Εργαστήριο Στατικής και Δυναμικής των Κατασκευών Τηλ: +30‐2310‐995663, Fax: +30‐2310‐995769 E‐mail: [email protected] 2 ‐ SECTION 8.2 ‐ GDM . LIST OF SYMBOLS FOR THE SDOF FORMULAS k M c y y y : stiffness : mass : damping : displacement : velocity : acceleration F1 f  t   : external load, with f  t  the dimensionless time variation y0 : initial displacement y 0 : initial velocity ω DLF yst ζ : natural frequency : dynamic load factor : equivalent static displacement : damping ratio   c ccr  c 2M ωd : damped natural frequency d   1   2 Ω : external load frequency 3 ‐ SECTION 8.2 ‐ GDM . TABLE 1: SDOF SYSTEM RESPONSE SYNOPSIS M  
y  k  y  c  y  F1 f  t   Dynamic equilibrium equation
y 
k
 y  0 M
Dynamic equilibrium for free vibration y
y 0
 sin   t   y0 cos   t  
Free vibration solution

y
k
F
y  1 M
M
Dynamic equilibrium for zero damping and constant force y
F1
 1  cos   t   k
Zero damping solution
DLF 
y
y
ky


yst F1 k
F1
4 ‐ SECTION 8.2 ‐ GDM Dynamic load factor definition
. DLF  1  cos   t  y  y0 cos   t  
y 0
 sin   t 

 yst    f (t)  sin    t    d
Dynamic load factor for constant force Duhamel's integral for the SDOF system, zero damping t
0
t

DLF  1  cos   t   1  cos  2     t  t d
T

DLF  cos    t  t d    cos   t 
t  td DLF for a rectangular pulse load in time 
t
 t t 

 cos  2      d    cos  2    
T

 T T 

DLF 
1
 sin   t   sin    t  t d     cos   t   t d 
5 ‐ SECTION 8.2 ‐ GDM DLF for a triangular pulse load in time . DLF for a triangular load in time with rise  sin   t  
 1 

and fall 


0  t  1 td
2
 
  t 
2 
1
DLF    t d  t   2  sin    t  d    sin   t   
t d 
2 

 
 
1 t tt
d
2 d

  t 
2 
DLF 
  2  sin    t  d    sin   t   sin    t  t d   
 t d 
2 
 

td  t
DLF 
2
td
 sin   t  
 t 
t  tr




1
sin    t  t r    sin   t   t  t r DLF  1 
 t r 
1
DLF 
tr
6 ‐ SECTION 8.2 ‐ GDM DLF for a ramp load in time
.  y    y0

y  e t  0
 sin  d  t   y0  cos  d  t  
d

 2 t
 yst    f (t)  e t   sin  d   t    d
d 0
y
F1 



 1  e t   cos   t   sin   t    k 



M  
y  k  y  F1  sin    t  DLF 
1
1   2 2



 sin    t    sin   t   


M  
y  k  y  c  y  F1  sin    t  7 ‐ SECTION 8.2 ‐ GDM Duhamel's integral for the SDOF system
SDOF system solution for a time‐harmonic load SDOF system equation for a time‐harmonic load, zero damping DLF for a SDOF system solution under time‐
harmonic loads SDOF system equation for a time‐harmonic load . y  et  C1  sin  d  t   C2  cos  d  t  

 F1 k  1  2
 DLF max 
2  sin    t   2    2  cos    t   1  2 2   4    2 
2
1
1  
2
2   4    2 
2
2
2
DLFMAX for a SDOF system solution under time‐
harmonic loads 8 ‐ SECTION 8.2 ‐ GDM SDOF system complete solution for a time‐
harmonic load . Figure 1: Dynamic Load Factor (DLF) for a SDOF system to (a) rectangular and (b) triangular pulses 9 ‐ SECTION 8.2 ‐ GDM . Figure 2: Dynamic Load Factor (DLF) for a SDOF system to (a) triangular and (b) ramp loads 10 ‐ SECTION 8.2 ‐ GDM . Figure 3: DLFMAX and corresponding time tMAX values for a SDOF system to (a) rectangular and triangular pulses, (b) triangular load and ((c) ramp load 11 ‐ SECTION 8.2 ‐ GDM . 12 ‐ SECTION 8.2 ‐ GDM . 13 ‐ SECTION 8.2 ‐ GDM . 14 ‐ SECTION 8.2 ‐ GDM . Figure 4: DLFMAX values versus dimensionless frequency ratio for a SDOF system subjected to a sinuisodal load F(t)=F0sin(Ωt) 15 ‐ SECTION 8.2 ‐ GDM . 16 ‐ SECTION 8.2 ‐ GDM . Figure 5: Plasticity index μ and corresponding time tMAX values for an elastoplastic SDOF system subjected to (a) rectangular pulse, (b) triangular pulse, (c) ramp load and (d) triangular load 17 ‐ SECTION 8.2 ‐ GDM . 18 ‐ SECTION 8.2 ‐ GDM . 19 ‐ SECTION 8.2 ‐ GDM . 20 ‐ SECTION 8.2 ‐ GDM . 21 ‐ SECTION 8.2 ‐ GDM . 22 ‐ SECTION 8.2 ‐ GDM . 23 ‐ SECTION 8.2 ‐ GDM . 24 ‐ SECTION 8.2 ‐ GDM . 25 ‐ SECTION 8.2 ‐ GDM