AxisVM 11

Transcription

AxisVM 11

Verification Examples
2012
AxisVM 11 Verification Examples
2
Linear static .............................................................................................................3
Supported bar with concentrated loads. .......................................................................................................................4
Thermally loaded bar structure.....................................................................................................................................5
Continously supported beam with constant distributed load.........................................................................................6
External prestessed beam...........................................................................................................................................9
Periodically supported infinite membrane wall with constant distributed load. ...........................................................11
Clamped beam examination with plane stress elements............................................................................................13
Clamped thin square plate..........................................................................................................................................16
Plate with fixed support and constant distributed load................................................................................................18
Annular plate. .............................................................................................................................................................19
All edges simply supported plate with partial distributed load. ...................................................................................21
Clamped plate with linear distributed load..................................................................................................................23
Hemisphere displacement. .........................................................................................................................................25
Nonlinear static......................................................................................................27
3D beam structure......................................................................................................................................................28
Plate with fixed end and bending moment..................................................................................................................30
Dynamic.................................................................................................................33
Deep simply supported beam.....................................................................................................................................34
Clamped thin rhombic plate........................................................................................................................................37
Cantilevered thin square plate....................................................................................................................................39
Cantilevered tapered membrane. ...............................................................................................................................42
Flat grillages. ..............................................................................................................................................................45
Stability ..................................................................................................................49
Simply supported beam..............................................................................................................................................50
Simply supported beam..............................................................................................................................................52
Design ...................................................................................................................53
N-M interaction curve of cross-section EC2, EN 1992-1-1:2004. ...............................................................................54
RC beam deflection according to EC2, EN 1992-1-1:2004. .......................................................................................55
Required steel reinforcement of RC plate according to EC2, EN 1992-1-1:2004……………………………...………..57
Interaction check of beam under biaxial bending EC3, EN 1993-1-1:2005…………………………...………………….59
Interaction check of beam under normal force, bending and shear force EC3, EN 1993-1-1:2005…………………...61
Buckling resistance of simply supported I beam EC3, EN 1993-1-1:2005…….…………………………………………63
Buckling resistance of simply supported T beam EC3, EN 1993-1-1:2005……………………………………………....65
Buckling of a hollow cross-section beam EC3, EN 1993-1-1:2005…………………………………………………….….67
Lateral torsional buckling of a beam EC3, EN 1993-1-1:2005……………………………………………………………..71
Interaction check of beam in section class 4. EC3, EN 1993-1-1:2005, EN 1993-1-5:2006………………………...…77
Earth-quake design using response-spectrum method.
……………………………………………………..………80
Linear static
3
4
Software Release Number: R3
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: beam1.axs
Thema
Supported bar with concentrated loads.
Analysis
Type
Geometry
Linear analysis.
Side view
Section Area = 1,0 m
2
Loads
Axial direction forces P1 = -200 N, P2 = 100 N, P3 = -40 N
Boundary
Conditions
Material
Properties
Element
types
Mesh
Fix ends, at R1 and R5.
E = 20000 kN / cm
ν = 0,3
Beam element
Target
R1 , R5 support forces
2
Results
Theory
AxisVM
%
R 1 [N]
-22,00
-22,00
0,00
R5 [N]
118,00
118,00
0,00
5
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
Thema
Thermally loaded bar structure.
Analysis
Type
Geometry
Linear analysis.
Side view
Sections:
-4
2
Steel: AS = π x 10 m
-4
2
Copper: AC = π x 10 m
Loads
Boundary
Conditions
Material
Properties
Element
types
Target
P = -12 kN (Point load)
Temperature rise of 10 °C in the structure after assembly.
The upper end of bars are fixed.
Steel:
ES = 20700 kN / cm , ν = 0,3 , αS = 1,2 x 10 °C
2
-5
Copper: EC = 11040 kN / cm , ν = 0,3 , αC = 1,7 x 10 °C
2
-5
-1
-1
Beam element
Smax in the three bars.
Results
Theory
AxisVM
%
Steel Smax [MPa]
23824000
23847900
0,10
Cooper Smax [MPa]
7185300
7198908
0,19
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
Thema
Continously supported beam with point loads.
Analysis
Type
Geometry
Linear analysis.
Side view
(Section width = 1,00 m, height1 = 0,30 m, height2 = 0,60 m)
Loads
P1= -300 kN, P2= -1250 kN, P3= -800 kN, P4= -450 kN
Boundary
Conditions
Elastic supported.
From A to D is Kz = 25000 kN/m/m.
From D to F is Kz = 15000 kN/m/m.
2
E = 3000 kN/cm
ν = 0,3
Three node beam element. Shear deformation is taken into account.
Material
Properties
Element
types
Target
Results
ez, My, Vz, Rz
Diagram ez
Diagram My
Results
6
7
Diagram Vz
Diagram R
Reference
AxisVM
e [%]
eA [m]
0,006
0,006
0,00
eB [m]
0,009
0,009
0,00
eC [m]
0,014
0,014
0,00
eD [m]
0,015
0,015
0,00
eE [m]
0,015
0,015
0,00
eF [m]
0,013
0,013
0,00
Reference
AxisVM
e [%]
0,0
0,2
0,00
MC [KNm]
MD [KNm]
ME [KNm]
88,5
636,2
87,1
630,8
-1,58
-0,85
332,8
164,2
330,1
163,0
-0,81
-0,73
MF [KNm]
0,0
0,4
0,00
MA [KNm]
MB [KNm]
8
Results
VA [KN]
VB [KN]
VC [KN]
VD [KN]
VE [KN]
VF [KN]
Reference
AxisVM
e [%]
0,0
0,1
0,00
112,1
646,8
113,1
647,2
0,89
0,06
335,0
267,8
334,9
267,5
-0,03
-0,11
0,0
-0,1
0,00
Reference
AxisVM
e [%]
2
145,7
154,0
5,70
2
219,5
219,4
-0,05
2
343,8
346,0
0,64
2
386,9
386,4
-0,13
2
224,5
224,7
0,09
2
201,2
200,8
-0,20
RA [KN/m ]
RB [KN/m ]
RC [KN/m ]
RD [KN/m ]
RE [KN/m ]
RF [KN/m ]
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
Thema
External prestessed beam.
Analysis
Type
Geometry
Linear analysis.
Side view
Loads
p = -50 kN /m distributed load
Length change = -6,52E-3 at beam 5-6
Boundary
Conditions
eY = eZ = = 0 at node 1
eX = eY = eZ = 0 at node 4
Material
Properties
E = 2,1E11 N / m
2
4
Beam 1-5, 5-6, 6-4 A = 4,5E-3 m Iz= 0,2E-5 m
2
4
Truss 2-5, 3-6 A = 3,48E-3 m Iz= 0,2E-5 m
2
4
Beam 1-4 A = 1,1516E-2 m Iz= 2,174E-4 m
2
Mesh
Element
types
Three node beam element, 1-5, 5-6, 6-4, 1-4 (shear deformation is taken into account)
Truss element 2-5, 3-6
Target
NX at beam 6-7
My,max at beam 2-3
ez at node 2
9
10
Results
2
3
5
6
2,000
4,000
4
0,600
1
2,000
8,000
Z
X
Diagram ez
ROBOT V6®
AxisVM
%
Nx [kN]
584,56
584,80
0,04
My [kNm]
49,26
-0,5421
49,60
-0,5469
0,68
0,89
ez [mm]
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: plane1.axs
Thema
Periodically supported infinite membrane wall with constant distributed load.
Analysis
Type
Geometry
Linear analysis.
Side view
(thickness = 20,0 cm)
Loads
p = 200 kN / m
Boundary
Conditions
Material
Properties
Element
types
Mesh
vertical support at every 4,0 m support length is 0,4 m
E = 880 kN / cm
ν = 0,16
Parabolic quadrilateral membrane (plane stress)
Target
Sxx at 1-10 nodes (1-5 at middle, 6-10 at support)
2
11
12
Results
Node
Analytical [kN/cm2 ]
AxisVM [kN/cm2 ]
%
1
2
3
4
5
6
7
8
9
10
0,1313
0,0399
-0,0093
-0,0412
-0,1073
-0,9317
0,0401
0,0465
0,0538
0,1249
0,1312
0,0395
-0,0095
-0,0413
-0,1071
-0,9175
0,0426
0,0469
0,0538
0,1247
-0,08
-1,00
2,15
0,24
-0,19
-1,52
6,23
0,86
0,00
-0,16
Reference:
Dr. Bölcskey Elemér – Dr. Orosz Árpád:
Vasbeton szerkezetek Faltartók, Lemezek, Tárolók
13
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: plane2.axs
Thema
Clamped beam examination with plane stress elements.
Analysis
Type
Geometry
Linear analysis.
Side view
Loads
p = -25 kN/m
Boundary
Conditions
Material
Properties
Element
types
Mesh
Both ends built-in.
2
E = 880 kN / cm
ν=0
0,375
1
0,500
Clamped
edge
C
3,000
Z
X
Side view
0,250
Target
14
τxy, max at section C
Results
Diagram
τxy
5,14
791,56
Z
Y
5,28
Diagram
τxy at section C
V = 65,625 kN ( from beam theory )
S y' = 0,0078125 m 3
b = 0,25 m
I y = 0,00260416 m 4
τ xy =
V ⋅ S y'
b⋅Iy
=
AxisVM result
65,625 ⋅ 0,0078125
= 787 ,5 kN / m 2
0,25 ⋅ 0,00260416
τ xy = 791,6 kN / m2
Difference = +0,52 %
AxisVM result V = ∑ n xy = 65,34 kN
Difference = +0,43 %
15
16
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: plate1.axs
Thema
Clamped thin square plate.
Analysis
Type
Geometry
Linear analysis.
Top view
Loads
P = -10 kN (at the middle of the plate)
Boundary
Conditions
Material
Properties
Element
types
Mesh
eX = ez = eZ = fiX = fiY = fiZ = 0 along all edges
2
4,000
E = 20000 kN / cm
ν = 0,3
Plate element (Parabolic quadrilateral, heterosis)
4,000
Y
X
Target
Displacement of middle of the plate
17
Results
-0,001
-0,006
-0,006
-0,012
-0,022
-0,019
-0,012
-0,043
-0,024
-0,043
-0,019
-0,084
-0,065
-0,024
-0,026
-0,065
-0,026
-0,081
-0,024
-0,087
-0,019
-0,006
-0,001
-0,022
-0,337
-0,337
-0,237
-0,084
-0,187
-0,125
-0,065
-0,012
-0,125
-0,237
-0,019
-0,024
-0,337
-0,156
-0,081
-0,006
-0,084
-0,043
-0,012
-0,187
-0,125
-0,065
-0,019
-0,337
-0,257
-0,125
-0,043
-0,006
-0,257
-0,307
-0,257
-0,156
-0,001
-0,022
-0,156
-0,237
-0,168
-0,065
-0,043
-0,043
-0,168
-0,257
-0,081
-0,006
-0,065
-0,168
-0,237
-0,168
-0,012
-0,081
-0,187
-0,156
-0,019
-0,087
-0,156
-0,087
-0,026
-0,012
-0,125
-0,081
-0,026
-0,024
-0,087
-0,125
-0,307
-0,237
-0,307
-0,383
-0,383
-0,337
-0,257
-0,237
-0,156
-0,168
-0,081
-0,087
-0,024
-0,026
-0,383
-0,337
-0,087
-0,168
-0,026
-0,257
-0,383
-0,337
-0,337
-0,156
-0,081
-0,237
-0,307
-0,257
-0,024
-0,168
-0,237
-0,187
-0,087
-0,125
-0,156
-0,065
-0,125
-0,081
-0,019
-0,026
-0,084
-0,065
-0,043
-0,024
-0,043
-0,012
-0,019
-0,022
-0,012
-0,006
-0,006
-0,257
-0,168
-0,087
-0,026
-0,001
Z
X
Y
Displacements
Mode
Mesh
Book1
1
2
3
4
5
2x2
4x4
8x8
12x12
16x16
0,402
0,416
0,394
0,387
0,385
Timoshenko2
AxisVM
0,38
0,420
0,369
0,381
0,383
0,383
Diff1 [%] Diff2 [%]
4,48
-11,30
-3,30
-1,03
-0,52
10,53
-2,89
0,26
0,79
0,79
References:
1.) The Finite Element Method (Fourth Edition) Volume 2.
/O.C. Zienkiewicz and R.L. Taylor/ McGraw-Hill Book Company 1991 London
2.) Result of analytical solution of Timoshenko
Convergency
15,00
10,00
Displacements
5,00
Diff1 [%]
0,00
1
2
3
-5,00
-10,00
-15,00
Mesh density
4
5
Diff2 [%]
18
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: plate2_1.axs
Thema
Plate with fixed support and constant distributed load.
Analysis
Type
Geometry
Linear analysis.
Top view
Loads
Boundary
Conditions
Material
Properties
Element
types
Mesh
Target
Results
2
P = -5 kN / m
eX = eY = eZ = fiX = fiY = fiZ = 0 along all edges
2
E = 990 kN/cm
ν = 0,16
Parabolic triangle plate element
Maximal eZ (found at Node1) and maximal mx (found at Node2)
Component
eZ,max [mm]
mx,max [kNm/m]
Nastran®
AxisVM
%
-1,613
3,060
-1,593
3,059
-1,24
-0,03
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
Thema
Annular plate.
Analysis
Type
Geometry
Linear analysis.
Top view
Loads
Edge load: Q = 100 kN / m
2
Distributed load: q = 100 kN / m
Boundary
Conditions
Material
Properties
Element
types
2
E = 880 kN / cm
ν = 0,3
Plate element (parabolic quadrilateral, heterosis)
19
20
Mesh
3,000
1,000
4,000
Y
X
Target
Smax, emax
Results
Model
Theory
Smax
[kN/cm2]
AxisVM
Smax
[kN/cm2]
%
a.)
b.)
c.)
d.)
e.)
f.)
g.)
h.)
2,82
6,88
14,22
1,33
2,35
9,88
4,79
7,86
2,78
6,76
14,10
1,33
2,25
9,88
4,76
7,86
-1,42
-1,74
-0,84
0,00
-4,26
0,00
-0,63
0,00
Model
Theory
emax
[mm]
AxisVM
emax
[mm]
%
a.)
b.)
c.)
d.)
e.)
f.)
g.)
h.)
77,68
226,76
355,17
23,28
44,26
123,19
112,14
126,83
76,10
220,84
352,89
23,42
44,50
123,17
111,94
126,81
-2,03
-2,61
-0,64
0,60
0,54
-0,02
-0,18
-0,02
Reference:
S. Timoshenko and S. Woinowsky-Krieger: Theory of Plates And Shells
21
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
Thema
All edges simply supported plate with partial distributed load.
Analysis
Type
Geometry
Linear analysis.
Top view
2
Distributed load: q = -10 kN / m (middle of the plate at 2,0 x 2,0 m area)
Boundary
Conditions
Material
Properties
Element
types
Mesh
a.) eX = eY = eZ = 0 along all edges (soft support)
b.) eX = eY = eZ = 0 along all edges ϕ = 0 perpendicular the edges (hard support)
2
E = 880 kN / cm
ν = 0,3
Plate element (Heterosis)
10,000
Loads
5,000
Y
X
Target
mx, max, my, max
Results
a.)
22
Moment
mx, max [kNm/m]
Theory
AxisVM
%
7,24
7,34
1,38
my, max [kNm/m]
5,32
5,39
1,32
Moment
mx, max [kNm/m]
Theory
AxisVM
%
7,24
7,28
0,55
my, max [kNm/m]
5,32
5,35
0,56
b.)
Reference:
23
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
Thema
Clamped plate with linear distributed load.
Analysis
Type
Geometry
Linear analysis.
Top view
2
Loads
Distributed load: q = -10 kN / m
Boundary
Conditions
eX = eY = eZ = fiX = fiY= fiZ = 0 along all edges
Material
Properties
E = 880 kN / cm
ν = 0,3
Element
types
Mesh
Plate element (Heterosis)
2
3
1
4
10,000
Y
X
2
10,000
q
Target
24
mx, my
Results
Results
mx, 1 [kNm/m]
my, 1 [kNm/m]
mx, 2 [kNm/m]
mx, 3 [kNm/m]
my, 4 [kNm/m]
Theory
AxisVM
%
11,50
11,50
33,40
17,90
25,70
11,48
11,48
33,23
17,83
25,53
-0,17
-0,17
-0,51
-0,39
-0,66
Reference:
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: hemisphere.axs
Thema
Hemisphere displacement.
Analysis
Type
Geometry
Linear analysis.
Hemisphere (Axonometric view)
t = 0,04 m
Loads
Point load P = 2,0 kN
C
2,0 kN
2,0 kN
A
B
Z
X
Y
25
Boundary
Conditions
eX = eY = eZ = 0 at A
eX = eY = eZ = 0 at B
Material
Properties
E = 6825 kN / cm
ν = 0,3
Element
types
Shell element
1.) guadrilateral parabolic
2.) triangle parabolic
ex at point A
Target
26
2
Results
e x [m]
Theory
0,185
AxisVM quadrilateral
AxisVM triangle
0,185
0,182
e [%]
0,00
-1,62
Nonlinear static
27
28
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: nonlin1.axs
3D beam structure.
Analysis
Type
Geometry
Geometrical nonlinear analysis.
Fy =-300,00 kN
Fz =-600,00 kN
3,000 m
1,732 m
Thema
1,732 m
Fy =-300,00 kN
Fz =-600,00 kN
Node1
Beam1
Y
1,732 m
3,000 m
X
D
Fz =-600,00 kN
A
C
Z
Z
Y
Loads
Boundary
Conditions
Material
Properties
CrossSection
Properties
Element
types
Target
4,000 m
B
X
X
Py = -300 kN
Pz = -600 kN
eX = eY = eZ = 0 at A, B, C and D
S 275
2
E = 21000 kN / cm
ν = 0,3
HEA 300
2
4
4
4
Ax = 112.56 cm ; Ix = 85.3 cm ; Iy = 18268.0 cm ; Iz = 6309.6 cm
Beam
eX, eY, eZ, at Node1
Nx, Vy, Vz, Tx, My, Mz of Beam1 at Node1
1,732 m
Results
29
Comparison with the results obtained using Nastran V4
®
Component
Nastran
AxisVM
%
eX [mm]
eY [mm]
eZ [mm]
Nx [kN]
Vy [kN]
Vx [kN]
Tx [kNm]
My [kNm]
Mz [kNm]
17,898
-75,702
-42,623
-283,15
-28,09
-106,57
-4,57
-519,00
148,94
17,881
-75,663
-42,597
-283,25
-28,10
-106,48
-4,57
-518,74
148,91
-0,09
-0,05
-0,06
0,04
0,04
-0,08
0,00
-0,05
-0,02
30
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: nonlin2.axs
Thema
Plate with fixed end and bending moment.
Analysis
Type
Geometry
Geometrical nonlinear analysis.
1,0 m
Edge1
Edge2
12,0 m
Z
Y
X
Loads
Boundary
Conditions
Material
Properties
Cross
Section
Properties
Element
types
Mz = 2600 kNm (2x1300 Nm) acting on Edge2
eX = eY = eZ = fiX = fiY = fiZ = 0 along Edge1
2
E = 20000 N / mm
ν=0
Plate thickness: 150 mm
Rib on Edge2: circular D = 500 mm (for distributing load to the mid-side-node)
Parabolic quadrilateral shell (heterosis)
Rib on Edge2 for distributing load to the mid-side-node
Target
31
ϕZ at Edge2
Results
5,5502 rad
1,0 m
Edge1
Edge2
12,0 m
Z
Y
X
Theoretical results based on the differential equation of the flexible beam:
M

M l plate
I plate E plate  → ϕ z =
I plate E plate
ϕ z = κ ⋅ l plate 
a b 3 1 ⋅ 0.15 3
I plate =
=
= 2.8125 ⋅10 −4
12
12
E plate = 2 ⋅1010 N m 2
κ =
l plate = 12 m
M = 2.6 ⋅10 6 Nm
ϕz =
2.6 ⋅10 6 ⋅12
= 5.5467 rad
2.8125 ⋅10 −4 ⋅ 2 ⋅ 1010
Comparison the AxisVM result with the theoretical one:
Component
fiZ [rad]
Theory
AxisVM
%
5,5467
5,5502
0,06
BLANK
32
Dynamic
33
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: dynam1.axs
Thema
Deep simply supported beam.
Analysis
Type
Geometry
Dynamic analysis.
Beam (Axonometric view)
Cross section (square 2,0 m x 2,0 m)
Loads
Self-weight
Boundary
Conditions
eX = eY = eZ = fiX = 0 at A
eY = eZ = 0 at B
Material
Properties
E = 20000 kN / cm
ν = 0,3
3
ρ = 8000 kg / m
Element
types
Target
Three node beam element (shear deformation is taken into account)
2
First 7 mode shapes
34
35
Results
Mode 1: f = 43,16 Hz
Mode 2: f = 43,16 Hz
Mode 3: f = 124,01 Hz
Mode 4: f = 152,50 Hz
Mode 5: f = 152,50 Hz
Mode 6: f = 293,55 Hz
Mode 7: f = 293,55 Hz
Results
36
Comparison with NAFEMS example
Mode
1
2
3
4
5
6
7
NAFEMS (Hz)
AxisVM (Hz)
%
42,65
42,65
125,00
148,31
148,31
284,55
284,55
43,16
43,16
124,01
152,50
152,50
293,55
293,55
-1,20
-1,20
0,79
-2,83
-2,83
-3,16
-3,16
37
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
Thema
Clamped thin rhombic plate.
Analysis
Type
Geometry
Dynamic analysis.
Top view of plane
Loads
Self-weight
Boundary
Conditions
eX = eY = fiZ = 0 at all nodes (ie: eX, eY, fiZ constained at all nodes)
eZ = fiX = fiY = 0 along the 4 edges
Material
Properties
E = 20000 kN / cm
ν = 0,3
3
ρ = 8000 kg / m
Element
types
Mesh
Parabolic quadrilateral shell element (heterosis)
10
,0
00
2
10,000
Y
X
Target
38
First 6 mode shapes
Results
Mode 1: f = 8,02 Hz
eR
eR
0,506
0,470
0,433
0,397
0,361
0,325
0,289
0,253
0,217
0,181
0,144
0,108
0,072
0,036
0
0,463
0,429
0,396
0,363
0,330
0,297
0,264
0,231
0,198
0,165
0,132
0,099
0,066
0,033
0
Mode 2: f = 13,02 Hz
eR
eR
0,520
0,483
0,446
0,409
0,486
0,451
0,416
0,382
0,347
0,312
0,278
0,243
0,208
0,174
0,139
0,104
0,069
0,035
0
Mode 3: f = 18,41 Hz
eR
0,372
0,335
0,297
0,260
0,223
0,186
0,149
0,112
0,074
0,037
0
Mode 4: f = 19,33 Hz
0,498
0,462
0,427
0,391
0,356
0,320
0,284
0,249
0,213
0,178
0,142
0,107
0,071
0,036
0
Mode 5: f = 24,62 Hz
Results
0,449
0,417
0,385
0,353
0,321
0,289
0,257
0,225
0,192
0,160
0,128
0,096
0,064
0,032
0
Mode 6: f = 28,24 Hz
Mode
1
2
3
4
5
6
eR
NAFEMS (Hz)
AxisVM (Hz)
%
7,94
12,84
17,94
19,13
24,01
27,92
8,02
13,02
18,41
19,33
24,62
28,24
1,01
1,40
2,62
1,05
2,54
1,15
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
Thema
Cantilevered thin square plate.
Analysis
Type
Geometry
Dynamic analysis.
Top view
Loads
Self-weight
Boundary
Conditions
Material
Properties
eX = eY = eZ = fiX = fiY = fiZ = 0 along y-axis
Element
types
Mesh
E = 20000 kN / cm
ν = 0,3
3
ρ = 8000 kg / m
2
Parabolic quadrilateral shell element (heterosis).
39
Target
First 5 mode shapes
Results
Mode 1: f = 0,42 Hz
Mode 3: f = 2,53 Hz
Mode 5: f = 3,68 Hz
40
41
Mode 2: f = 1,02 Hz
Mode 4: f = 3,22 Hz
Mode
1
2
3
4
5
NAFEMS (Hz)
AxisVM (Hz)
%
0,421
1,029
2,580
3,310
3,750
0,420
1,020
2,530
3,220
3,680
-0,24
-0,87
-1,94
-2,72
-1,87
42
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
Thema
Cantilevered tapered membrane.
Analysis
Type
Geometry
Dynamic analysis.
Side view
Loads
Self-weight
Boundary
Conditions
eZ = 0 at all nodes (ie: eZ constained at all nodes)
eX = eY = 0 along y-axis
Material
Properties
E = 20000 kN / cm
ν = 0,3
3
ρ = 8000 kg / m
Element
types
Mesh
1,000
5,000
2
10,000
Y
X
Target
43
First 4 mode shapes
Results
1,000
5,000
10,000
Y
X
1,000
5,000
Mode 1: f = 44,33 Hz
10,000
Y
X
Mode 2: f = 128,36 Hz
1,000
5,000
44
10,000
Y
X
1,000
5,000
Mode 3: f = 162,48 Hz
10,000
Y
X
Mode 4: f = 241,22 Hz
Results
Mode
1
2
3
4
NAFEMS (Hz)
AxisVM (Hz)
%
44,62
130,03
162,70
246,05
44,33
128,36
162,48
241,22
-0,65
-1,28
-0,14
-1,96
45
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
Thema
Flat grillages.
Analysis
Type
Geometry
Dynamic analysis.
Top view
Loads
Self-weight
Boundary
Conditions
Material
Properties
eX = eY = eZ = 0 at the ends (simple supported beams)
2
A = 0,004 m
4
Ix = 2,5E-5 m
4
Iy = Iz = 1,25E-5 m
Three node beam element (shear deformation is taken into account)
1,000
0,500
4,500
1,000
Element
types
Mesh
2
2,000
Cross
Section
E = 20000 kN / cm
2
G = 7690 kN / cm
ν = 0,3
3
ρ = 7860 kg / m
1,500
Y
X
1,500
1,500
1,000
0,500
Target
46
First 3 mode shapes
1,679
1,879
1,605
1,638
1,586
1,035
1,241
1,114
Results
ZY
X
1,938
2,254
0,856
-1,837
-2,065
-1,813
2,040
Mode 1: f = 16,90 Hz
Z Y
X
ZY
X
Mode 3: f = 51,76 Hz
-1,845
-1,992
2,040
1,585
-1,130
1,721
-1,620
-1,581
-1,667
Mode 2: f = 20,64 Hz
Mode
1
2
3
47
Reference
AxisVM (Hz)
%
16,85
20,21
53,30
16,90
20,64
51,76
0,30
2,13
-2,89
Reference:
C.T.F. ROSS: Finite Element Methods In Engineering Science
BLANK
48
Stability
49
50
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: buckling1.axs
Thema
Simply supported beam.
Analysis
Type
Geometry
Buckling analysis.
Front view
2
2
7
3
20,0
6
S 1
1
8
5
9
z
y
4
1,0
G
10,0
4
4
6
Cross section (Iz =168,3 cm , It =12,18 cm , Iw =16667 cm )
Loads
Boundary
Conditions
Material
Properties
Element
types
Mesh
Bending moment at both ends of beam
MA = 1,0 kNm, MB = -1,0 kNm
eX = eY = eZ = 0 at B
kz = kw = 1
2
E = 20600 kN / cm
ν = 0,3
2
G = 7923 kg / m
Parabolic quadrilateral shell element (heterosis)
Target
51
Mcr = ? (for lateral torsional buckling)
Results
Analytical solution
M cr =
M cr =
π 2 ⋅ E ⋅ IZ
L2
IW
L2 ⋅ G ⋅ I t
+ 2
IZ
π ⋅ E ⋅ IZ
π 2 ⋅ 20600 ⋅ 168,3 16667
2002
AxisVM result
Mcr = 125,3 kNm
Difference
+0,6%
168,3
+
2002 ⋅ 7923 ⋅ 12,18
= 12451 kNcm = 124,51 kNm
π 2 ⋅ 20600 ⋅ 168,3
52
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: buckling2.axs
Thema
Analysis
Type
Geometry
Buckling analysis.
Front view (L = 1,0 m)
1
2
5
S 1
2
G
10,0
5
S 1
2
10,0
2
G
3
3
4
4
12,0
30,0
z
y
z
y
Section A1
Section A2
Cross-sections
Loads
P = -1,0 kN at point B.
Boundary
Conditions
eY = eZ = 0 at B
Material
Properties
E = 20000 kN / cm
ν = 0,3
Element
types
Target
Beam element
2
Pcr = ? (for inplane buckling)
Results
P cr [kN]
Theory
AxisVM
e [%]
3,340
3,337
-0,09
Design
53
54
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: RC column1.axs
Thema
N-M interaction curve of cross-section (EN 1992-1-1:2004).
Analysis
Type
Geometry
Linear static analysis+design.
2φ20
3φ28
Section: 300x400 mm
Covering: 40 mm
Loads
Boundary
Conditions
Material
Properties
Target
Results
Concrete:
2
fcd=14,2 N/mm
ec1=0,002 ecu=0,0035 (parabola-constans σ-ε diagram)
Steel:
2
fsd=348 N/mm
esu=0,015
Compare the program results with with hand calculation at keypoints of M-N interaction
curve.
N
1
2
6
5
3
4
1
2
3
4
5
6
N [kN]
-2561
-1221
0
+861
0
-362
M [kNm]
+61
+211
+70
-61
-190
-211
M(N) AxisVM
+61,4
+209,7
+70,5
-61,4
-191,2
-209,7
Reference: Dr. Kollár L. P., Vasbetonszerkezetek I. Műegyetemi kiadó
e%
+0,7
-0,6
+0,7
+0,7
+0,6
-0,6
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
Thema
RC beam deflection according to EC2, EN 1992-1-1:2004.
Analysis
Type
Geometry
Material nonlinear analysis.
q = 17 kN/m
L = 5,60 m
Side view
2φ20
35 cm
covering = 3 cm
β = 0,5
4φ20
25 cm
Section
Loads
q = 17 kN /m distributed load
Boundary
Conditions
Material
Properties
Element
types
Target
Concrete: C25/30, ϕ = 2,1
Steel: B500B
Parabolic quadrilateral plate element (Heterosis)
ez, max
55
56
Z
X
Diagram ez
Aproximate calculation:
e = ζ ⋅ e II + (1 − ζ ) ⋅ e I = 20,06 _ mm
where,
eI is the deflection which was calculated with the uncracked inertia moment
eII is the deflection which was calculated with the cracked inertia moment
σ
ζ = 1 − β ⋅  sr
 σs



2
Calculation with integral of κ:
e = 19,82 mm
Calculation with AxisVM:
e = 19,03 mm (different -4,0%)
-0,002
-5,239
-10,101
-14,242
-17,393
-19,360
-20,029
-19,360
-17,393
-14,242
-10,101
-5,239
-0,002
Results
57
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
Thema
Required steel reinforcement of RC plate according to EC2, EN 1992-1-1:2004.
Analysis
Type
Geometry
Linear analysis.
Szabvány : Eurocode
Eset
: ST1
50 kN
4,0
Y
X
Side view
Cross-section
Loads
Pz = -50 kN point load
Boundary
Conditions
Material
Properties
Element
types
Mesh
Clamped cantilever plate.
Concrete: C25/30
Steel: B500A
Parabolic quadrilateral plate element (heterosis)
Eset
: ST1
1,0
4,0
Clamped
edge
Y
X
Top view
Target
58
AXT steel reinforcement along x direction at the top of the support
Results
Lineáris számítás
Eset
: ST1
E (W)
: 1,09E-11
E (P)
: 1,09E-11
E (ER)
: 8,49E-13
Komp.
: axf [mm2 /m]
1,0
4,0
Clamped
edge
ST1, axf: 2093 mm2 /m
Z
Y
X
Diagram AXT
Calculation according to EC2:
500
= 435 N / mm 2
1,15
f cd =
25
= 16,6 N / mm2
1,5
ξc 0 =
c ⋅ ε cu ⋅ ES
0,85 ⋅ 0,0035 ⋅ 20000
=
= 0,54
ε cu ⋅ ES + f yd 0,0035 ⋅ 20000 + 435
f yd =
d = 300 – 53 = 247 mm
x 

M sd = M Rd = b ⋅ xc ⋅ f cd  d − c  = 200 kNm
2

439 > h
xc = 

 55 
ξc =
xc
55
=
= 0,22 < ξ c 0 = 0,54 Steel reinforcement is yielding
d 247
AS =
b ⋅ xc ⋅ f cd 55 ⋅1000 ⋅16,6
=
= 2099 mm 2
f yd
435
Calculation with AxisVM:
AXT = 2093 mm2 / m
Different = -0,3 %
59
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: 3_10 Plastic biaxial bending interaction.axs
Thema
Interaction check of simply supported beam under biaxial bending (EN 1993-1-1).
Analysis
Type
Geometry
Steel Design
h = 270 mm
b = 135 mm
tf = 10 mm
tw = 7 mm
l = 6000 mm
2
A = 45,95 cm
3
W y,pl = 484,1 cm
3
W z,pl = 97 cm
IPE270 cross section
Loads
qy = 1,5 kN/m
qz = 20,4 kN/m
Boundary
Conditions
ex = ey = ez = 0 at A
ey = ez = 0 at B
Material
Properties
S 235
2
E = 21000 kN/cm
ν = 0,3
Element
types
Target
Beam element
Results
Analytical solution in the following book:
60
Interaction check taking into account plastic resistances
Dunai, L., Horváth, L., Kovács, N., Verőci, B., Vigh, L. G.: “Acélszerkezetek méretezése az
Eurocode 3 alapján, Gyakorlati útmutató” (Design of steel structures according to
Eurocode 3, ) Magyar Mérnök Kamara Tartószerkezeti tagozata, Budapest, 2009.
Exercise 3.10., page 28.
Analitical
solution
AxisVM
e[%]
My,Ed [kNm]
91,8
91,8
-
Mz,Ed [kNm]
6,75
6,75
-
Mpl,y,Rd [kNm]
113,74
113,76
+0,02
Mpl,z,Rd [kNm]
22,78
22,79
+0,04
α
2
2
-
β
1
1
-
capacity ratio [-]
0,948
0,947
-0,11
61
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: 3_12 _MNV_Interaction.axs
Thema
Analysis
Type
Geometry
Interaction check of simply supported beam under normal force, bending and shear force.
(EN 1993-1-1, EN 1993-1-5)
Steel Design
h = 200 mm
b = 200 mm
tf = 15 mm
tw = 9 mm
l = 1400 mm
2
A = 78,1 cm
2
Av = 24,83 cm
3
Iy = 5696 cm
3
W y,pl = 643 cm
IPE270 cross section
Loads
Fz = 300 kN at thirds of beam
N = 500 kN at B
Boundary
Conditions
ey = ez = 0 at B
Material
Properties
S 235
2
E = 21000 kN/cm
ν = 0,3
Element
types
Beam element
Target
Interaction check of axial force, shear force and bending moment.
Results
62
Exercise 3.12., page 31-33.
Analytical
solution
AxisVM
results
e[%]
NEd [kN]
500
500
-
Vz,Ed [kN]
300
300
-
My,Ed [kNm]
140
140
-
Npl,Rd [kN]
2148
2148
-
capacity ratio [-]
0,233
0,233
-
Vpl,z,Rd [kN]
394,2
394,5
+0,08
capacity ratio [-]
0,761
0,761
-
Mpl,y,Rd [kNm]
176,8
176,7
-0,06
capacity ratio [-]
0,792
0,792
-
Ρ
0,273
0,271
-0,73
MV,Rd [kNm]
163,96
163,93
-0,02
N
0,233
0,233
-
A
0,232
0,232
-
MNV,Rd [kNm]
142,2
142,2
-
capacity ratio [-]
0,985
0,984
-0,10
Pure compression
Pure shear
Pure bending
Interaction check
63
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: 3_15 Központosan nyomott rúd - I szelvény.axs
Thema
Buckling resistance of simply supported beam (EN 1993-1-1).
Analysis
Type
Geometry
Steel Design
h = 300 mm
b = 250 mm
tf = 14 mm
tw = 8 mm
l = 4500 mm
2
A = 94 cm
4
Iy = 19065,8cm
4
Iz = 3647,1 cm
iy = 14,1 cm
iz = 6,2 cm
“I” cross section, symmetric about y and z axis
Loads
Boundary
Conditions
Material
Properties
Element
types
Target
Normal force at point A
NA= -1,0 kN
ey = 0 at A
ex = ey = ez = φx = φz = 0 at B
kz = kw = 1
S 235
2
E = 21000 kN / cm
ν = 0,3
Beam element
Buckling resistance Nb,Rd = ?
Results
64
Exercise 3.15., P. 37-39.
Analytical
solution
AxisVM
e[%]
[-] *
0,673
0,673
-
[-]
0,771
0,769
-0,26
Χy [-] *
0,8004
0,7989
-0,19
Χz [-]
0,6810
0,6815
+0,07
Nb,Rd [kN]
1504,3
1505,3
+0,07
λy
λz
65
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: 3_21 Központosan nyomott rúd - T szelvény.axs
Thema
Buckling resistance of simply supported beam (EN 1993-1-1).
Analysis
Type
Geometry
Steel Design
h = 180 mm
b = 250 mm
tf = 16 mm
tw = 16 mm
l = 3000 mm
2
A = 68,8 cm
4
Iy = 2394,25cm
4
Iz = 2089,48 cm
4
Ics= 58,71 cm
6
Iw = 1108,0 cm
iy = 5,90 cm
iz = 5,51 cm
Loads
Boundary
Conditions
Material
Properties
Element
types
Target
Welded “T” section, symmetric to z but not y
Normal force at point A
NA= -1,0 kN
ey = 0 at A
ex = ey = ez = φx = 0 at B
kz = kw = 1
S 235
2
E = 21000 kN/cm
ν = 0,3
Beam element
Buckling resistance Nb,Rd = ?
Results
66
Exercise 3.21., P. 47-49.
Analitical
solution
AxisVM
e[%]
zs [cm]
49,0
49,0
-
zw [cm]
4,10
4,04
-1,46
iw [cm] *
9,05
9,03
-0,22
[-]
0,542
0,542
-
Χy [-]
0,8204
0,8195
-0,11
Nb,Rd,1 [kN]
1326,4
1325,0
-0,11
[-] *
0,667
0,667
-
Χz [-] *
0,7432
0,7446
+0,19
Nb,Rd,2 [kN] *
1201,6
1203,9
+0,19
λy
λz
* hidden partial results, Axis does not show them among the steel desing results
67
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: Külpontosan nyomott rúd - RHS szelvény.axs
Topic
Buckling of a hollow cross-section beam (EN 1993-1-1).
Analysis
Type
Geometry
Steel Design
h = 150 mm
b = 100 mm
tf = 10 mm
tw = 10 mm
L = 4,000 m
2
A = 43,41 cm
4
Iy = 1209,8 cm
4
Iz = 635,7 cm
iy = 52,8 mm
iz = 38,3 mm
3
W el,y = 161,3 cm
3
W el,z = 127,1 cm
3
W pl,y = 205,6 cm
3
W pl,z = 154,6 cm
RHS 150x100x10,0 cross section (hot rolled)
Loads
Boundary
Conditions
Material
Properties
Element
types
Steel
Design
Parameters
Target
Bending moment at both ends of beam and axial force
NEd,C = 200 kN
MEd,A = MEd,B = 20 kNm
ex = ey = ez = 0, warping free at A
ey = ez = 0, warping free at B
S 275
2
E = 21000 kN / cm
ν = 0,3
Beam element
Buckling length:
Ly = L
Lz = L
Lw = L
Check for interaction of compression and bending.
Results
68
Analytical solution:
Section class: 1.
Compression – flexural buckling
2
2
π E Iy
π 21000 ⋅ 1209,8
N cr, y =
=
= 1567,2 kN
2
Ky L
400
2
2
π E I z π 21000 ⋅ 635,7
N cr,z =
=
= 823,5 kN
2
Kz L
400
N pl,Rd = A ⋅ f y = 43,41 ⋅ 27,5 = 1193,8 kN
λy =
λz =
N pl
N cry
N pl
N crz
=
=
1193,8
1567,16
1193,8
823,48
= 0,8728
= 1,2040
imperfection factor based on buckling curve “a” (hot rolled RHS section):
α y = α z = 0,21
φ=
1 + α ⋅ (λ - 0.2) + λ 2
χ :=
2
1
φ + φ 2 - λ2
χ y = 0,7516
χ z = 0,5275
N b,Rd =
χ y A fy
γ1
=
2
2
0,5275 ⋅ 43,41 cm ⋅ 27,5 kN/cm
1,0
= 629,72 kN > N Ed, x = 200 kN
Bending – lateral torsional buckling
Wpl,y f y 205,6 cm 3 ⋅ 27,5 kN/cm 2
M pl,Rd, y =
=
= 56,54 kNm > M Ed = 10 kNm
γ1
1,0
C1 = 1,000
k = k w =1
z
2
π E Iz
M cr = C1
2
(kL)
 kz

 kw

2
 I w (kL) 2 G I t

+
=
2
 Iz
π
E
I
z

kN
2
4
π 21000
⋅ 635,7cm
cm 2
M cr = 1,0 ⋅
2
(400 cm)
M cr = 977,41 kNm
kN
2
4
(400 cm) ⋅ 8077
⋅ 1436,2 cm
766 cm
cm 2
+
4
kN
2
4
635,7 cm
π ⋅ 21000
⋅ 635,7 cm
cm 2
6
Wy f y
λ LT =
M cr
λLT > 0,2
3
2
205,6 cm ⋅ 27,5 kN/cm
=
977,41 kNm
69
= 0,2405
torsional buckling may occur
α LT = 0,76
φ=
1+α
(λ
- 0.2) + λ
LT
2
LT
2
χ LT :=
M
LT
b, Rd
1
2
φ + φ - λ LT
2
= 0,5443
= 0,9684
= χ LT ⋅ M
= 0,9684 ⋅ 56,54kNm = 54,76kNm
pl , Rd , y
Interaction of bending and buckling
2
2
N Rk = A ⋅ f y = 43,41 cm ⋅ 27,5 kN/cm = 1193,8 kN
M y,Rk = M pl,Rd, y = 56,54kNm
Equivalent uniform moment factors according to EN 1993-1-1 Annex B, Table B.3.:
φ = 1,0
C my = 0,6 + 0,4φ = 1,0 > 0,4
For members susceptible to torsional deformations the interaction factors may be
calculated according to EN 1993-1-1 Annex B, Table B.2.:



N Ed

 < C my 1 + 0,8
χ y N Rk /γ M1 



200
200




k yy = 1,0 1 + (0,87 - 0,2) ⋅
 < 1,0  1 + 0,8 ⋅

0,7531 ⋅ 1193,78 /1,0 
0,7531 ⋅ 1193,78 /1,0 




k yy = C my 1 + (λ LT - 0,2)
N Ed
k yy = min (1,149 ; 1,178) = 1,149


k zy = 1 −


0,1 ⋅ λ
N Ed,x
N Ed,x
 

0,1
z
⋅
≥ 1 −
⋅


C
− 0,25 χ z N /γ
C
− 0,25 χ z N /γ
Rk M1 
Rk M1 
 

mLT
mLT
k zy = 1 −
200
200
0,1 ⋅ 1,2040
 
0,1

⋅
≥1−
⋅


1,0 − 0, 25 0,5275 ⋅ 1193,78 /1,0  
1,0 − 0, 25 0,5275 ⋅ 1193,78 /1,0 
k zy = max (0,9490 ; 0,9577) = 0,9577
N Ed
χ y ⋅N Rk /γ M1
=
200
0,7516 ⋅ 1193,78
N Ed
200
0,5275 ⋅ 1193,78
M y,Ed
χ y ⋅ M y,Rk /γ M1
+ 1,149 ⋅
+ k zy
χ z ⋅ N Rk /γ M1
=
+ k yy
20
0,9684 ⋅ 56,54
M y,Ed
M y,Rk /γ M1
+ 0,9577 ⋅
70
=
= 0,6426
=
20
0,9684 ⋅ 56,54
= 0,6674
Analytical solution
AxisVM
e [%]
NRk = Npl,Rd [kN]
1193,8
1193,9
-
λ y [-]
0,873
0,870
-0,3
λz
[-]
1,204
1,201
-0,2
Χy [-]
0,7516
0,7516
-
Χz [-]
0,5275
0,5274
-
Nb,Rd [kN]
629,7
629,7
-
Mc,Rd = Mpl,Rd [kNm]
56,54
56,54
-
C1
1,000
1,000
-
Mcr [kNm]
977,41
977,40
-
λ LT [-]
0,2405
0,2405
-
ΧLT [-]
0,9684
0,9684
-
Mb,Rd [kNm]
54,76
54,57
-0,3
Cmy [-]
1,0
1,0
-
kyy [-]
1,149
1,150
-
kzy [-]
0,9577
0,9577
-
Interaction capacity ratio 1 [-]
0,643
0,643
-
Interaction capacity ratio 2 [-]
0,667
0,667
-
71
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: 3_26 Külpontosan nyomott rúd - I szelvény.axs
Thema
Lateral torsional buckling of a beam (EN 1993-1-1).
Analysis
Type
Geometry
Steel Design
h = 171 mm
b = 180 mm
tf = 6 mm
tw = 9,5 mm
L = 4,000 m
2
A = 45,26 cm
4
Iy = 2510,7 cm
4
Iz = 924,6 cm
iy = 74 mm
iz = 45 mm
3
W el,y = 293,7 cm
3
W el,z = 102,7 cm
3
W pl,y = 324,9 cm
3
W pl,z = 156,5 cm
Iw = 58932 cm
4
It = 15 cm
HEA180
Loads
Boundary
Conditions
Material
Properties
Element
types
Axial force at B: Nx = -280 kN
Point load in y direction at the thirds of the beam: Fy = 5 kN
Distributed load in z direction: qz = 4,5 kNm
ex = ey = ez = 0, warping free at A
ey = ez = 0, warping free at B
S 235
2
E = 21000 kN / cm
ν = 0,3
Beam element
6
Steel
Design
Parameters
72
The elastic critical load factor is:
αcr = 4,28
As αcr = 4,28 < 15
II. order analysis is required.
For this, the beam element needs to be meshed. Divison of the beam element into 4.
Buckling length:
Ly = L
Lz = L
LT buckling length:
Lw = L
Target
Buckling check for interaction of axial force and bi-axial bending.
Results
Internal forces from the second order analysis
NEd,x = 280 kN
MEd,y = 9,84 kNm MEd,z = 8,81 kNm
VEd,y = 6,50 kN VEd,z = 9,61 kN
73
Analytical solution:
Section class: 1.
Normal force
2
N cr,y =
N cr,z =
π
E Iy
=
Ky L
π
2
E Iz
2
21000 ⋅ 2510,7
400
π 2 21000 ⋅ 924,6
=
Kz L
π
400
= 3252,3 kN
= 1197,7 kN
N pl,Rd = A ⋅ f y = 45,26 ⋅ 23,5 = 1063,6 kN
λ =
y
λ =
z
N pl
N cry
N pl
N crz
=
1063,6
=
1063,6
3252,3
1197,7
= 0,5719
= 0,9424
based on buckling curve “b” in y direction and “c” in z direction:
χ y = 0,8508
χ z = 0,5741
χ y A fy
N b,Rd,1 =
N b,Rd,2 =
γ1
χz A fy
γ1
=
=
2
2
0,8508 ⋅ 45,26cm ⋅ 23,5kN/cm
= 904,92 kN > N Ed,x = 280 kN
1,0
2
2
0,5741 ⋅ 45,26cm ⋅ 23,5kN/cm
1,0
= 610,62 kN > N Ed,x = 280 kN
Bending
M pl,Rd, y =
M pl,Rd,z =
Wpl,y f y
γ1
Wpl,z f y
γ1
=
=
3
2
324,9 cm ⋅ 23,5 kN/cm
1,0
3
2
156,5 cm ⋅ 23,5 kN/cm
1,0
= 76,35 kNm > M Ed, y = 9,84 kNm
= 36,78 kNm > M Ed,z = 8,81 kNm
Calculation of the critical moment:
C1 = 1,132
(due to the My moment diagram)
2
π E Iz
M cr = C1
2
(kL)
M cr = 1,132
2
 k z  I w (kL) 2 G I t


+
=
2
 k w  Iz
π E Iz


2
2
4
π 21000 kN/cm ⋅ 924,6 cm
2
(400 cm)
M cr = 174,1 kNm
58932 cm
924,6 cm
6
4
+
2
2
4
(400 cm) ⋅ 8077 kN/cm ⋅ 15 cm
2
2
4
π ⋅ 21000 kN/cm ⋅ 924,6 cm
74
For rolled section, the following procedure may be used to determine the reduction
factor (EN 1993-1-1,Paragraph 6.3.2.3.):
Wy f y
λ LT =
φ=
M cr
1+α
(λ
LT
174,10 kNm
- 0.4) + 0.75 ⋅ λ
b, Rd
1
2
φ + φ - 0.75 ⋅ λ LT
= 0,6622
2
LT
2
χ LT :=
M
LT
3
2
324,9 cm ⋅ 23,5 kN/cm
=
= 0,7090
= 0,8881
2
= χ LT ⋅ M
= 0,8881 ⋅ 76,35kNm = 67,81kNm
pl , Rd , y
Interaction of axial force and bi-axial bending
N Rk = N pl,Rd = 1063,6 kN
M y,Rk = M pl,Rd, y = 76,35 kNm
M z, Rk = M pl,Rd, z = 36,78 kNm
Equivalent uniform moment factors according to EN 1993-1-1 Annex B, Table B.3.:
ψ = 0, α = 0 in both directions
C
my
=C
mLT
= 0,95 + 0,05α = 0,95
C mz = 0,90 + 0,10α = 0,90


k yy = C my 1 + (λ - 0,2)
y


(distributed load)
(concentrated load)
N Ed,x



 ≤ C my 1 + 0,8




N Ed,x
k yy = 0,95 ⋅ 1 + (0,5719 - 0,2) ⋅
280



 ≤ 0,95 ⋅  1 + 0,8 ⋅

0,8508 ⋅ 1063,6 /1,0 
0,8508 ⋅ 1063,6 /1,0 

280
k yy = min (1,0593 ; 1,1851) = 1,0593


k zy = 1 −


0,1 ⋅ λ
N Ed,x
N Ed,x
 

0,1
z
⋅
≥ 1 −
⋅


C
− 0,25 χ z N /γ
C
− 0, 25 χ z N /γ
Rk M1 
Rk M1 
mLT
 
mLT

k zy = 1 −
280
280
0,1 ⋅ 0,9424
 
0,1

⋅
≥1−
⋅


0,95 − 0,25 0,5741 ⋅ 1063,6 /1,0  
0,95 − 0, 25 0,5741 ⋅ 1063,6 /1,0 
k zy = max (0,9383 ; 0,9345) = 0,9383
k zz = C
75
N Ed,x


1 + (2 ⋅ λ - 0,6)

 ≤ C mz
mz 
z
χ z N Rk /γ M1 




k zz = 0,90 1 + (2 ⋅ 0,9424 - 0,6)
N Ed,x


1 + 1,4

χ z N Rk /γ M1 


280



 ≤ 0,90 1 + 1,4

0,5741 ⋅ 1063,6 /1,0 
0,5741 ⋅ 1063,6 /1,0 

280
k zz = min (1,4303 ; 1,478) = 1,4303
k yz = 0,6 k
zz
= 0,8582
N Ed,x
+ k yy
χ y ⋅N Rk /γ M1
=
280
0,8508 ⋅ 1063,6
N Ed,x
χ z ⋅ N Rk /γ M1
=
0,5741 ⋅ 1063,6
χ
⋅ M y,Rk /γ M1
LT
+ 1,0593 ⋅
+ k zy
280
M y,Ed
9,84
0,8881 ⋅ 76,35
+ 0,8582 ⋅
M y,Ed
χ
⋅ M y,Rk /γ M1
LT
+ 0,9383 ⋅
9,84
0,8881 ⋅ 76,35
+ k yz
k zz
M z,Ed
M z,Rk /γ M1
8,81
36,78
= 0,3094 + 0,1537 + 0,2056 = 0,6687
M z,Ed
M z,Rk /γ M1
+ 1,4303 ⋅
8,81
36,78
=
=
= 0, 4586 + 0,1362 + 0,3426 = 0,9374
*
76
Analytical solution
AxisVM
e [%]
Npl,Rd [kN]
1063,6
1063,6
-
Ncr,y [kN]
3252,3
3252,4
-
Ncr,z [kN]
1197,7
1197,7
-
λy, rel [-]
0,5719
0,5719
-
λz, rel [-]
0,9424
0,9424
-
Χy [-]
0,8508
0,8509
-
Χz [-]
0,5741
0,5741
-
Mpl,Rd,y [kNm]
76,35
76,36
-
Mpl,Rd,z [kNm]
36,78
36,78
-
C1 [-]
1,132
1,125
-0,6*
Mcr [kNm]
174,1
173,0
-0,63
λLT, rel [-]
0,6622
0,6644
+0,3
ΧLT [-]
0,8881
0,8887
+0,1
Mb,Rd [kNm]
67,81
67,73
-0,1
Cmy = CmLt [-]
0,95
0,95
-
Cmz [-]
0,90
0,95
+5,5**
kyy
1,0593
1,0593
-
kzz
1,4303
1,5096
+5,5***
kyz
0,8582
0,9058
+5,5***
kzy
0,9383
0,9383
-
Interaction capacity ratio 1
0,6687
0,6801
+1,7***
Interaction capacity ratio 2
0,9374
0,9564
+2,0***
AxisVM calculates this factor using a closed form expression, while in the hand
calculation C1 was derived from a table. The effect of this on the final result
-4
(efficiency) is 10 , thus on the safe side.
** See EC3 Annex B, Table B.3: the difference is due to the fact, that AxisVM calculates
the equivalent uniform moment factor (Cmy, Cmz, CmLT) for both uniform load and
concentrated load, and then takes the higher value. The effect on the final result
(efficiency) is +1~2%.
*** the difference is due to the different Cmz value
77
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: Double-symmetric I - Class 4.axs
Thema
Interaction check of beam in section class 4 (EN 1993-1-1, EN 1993-1-5)
Analysis
Type
Geometry
Steel Design
h = 1124 mm
tw = 8 mm
b = 320 mm
tf = 12 mm
L = 8,000 m
2
A = 164,8 cm
4
Iy = 326159,4 cm
3
W el,y = 5803,6 cm
Double-symmetric welded I shape
Loads
Boundary
Conditions
Material
Properties
Element
types
Target
Axial force at B: N Ed,C = 700 kN
Distributed load in z direction: qz = 162,5 kNm
The internal forces in the mid-section: MEd,y = 1300 kNm, NEd,x = - 700 kN
ey = ez = 0 at B
S 355
2
E = 21000 kN / cm
ε=0,81
ν = 0,3
Beam element
Check the strength capacity ratios for axial force, bending and interaction.
Results
78
Exercise 3.4., P. 14-16.
Exercise 3.6., P. 19-21.
Exercise 3.13., P. 34.
Analytical
solution
AxisVM
e [%]
0,43
0,43
-
0,831
0,858
+3,1
0,931
0,910
-2,3
140,0
142,0
+1,4
4
4
-
2,957
2,975
+0,6
0,313
0,311
-0,6
340,8
342,4
+0,5
99,98
97,46
-2,6
3549
3460
+2,6
0,2
0,2
-
0,43
0,43
-
0,831
0,858
+3,1
0,931
0,910
-2,3
139,95
142,0
+1,4
-0,969
-0,959
+1,0
23,09
22,84
-1,1
1,231
1,245
+1,1
0,739
0,731
-1,1
408,6
410,4
+0,4
5131
4976
-3,1
1821,5
1766,5
-3,1
0,71
0,74
+4,1
0,91
0,94
+3,3
Uniform compression
Uniform bending
Small differences occur because AxisVM does not take into account welding when
calculating the effective section sizes.
79
80
Date: 17. 10. 2012.
Tested by: InterCAD
Page number:
File name: Earthquake-01-EC.axs
Thema
Earth-quake design using response-spectrum method.
Analysis
Type
Geometry
Linear frequency analysis with 5 modes.
Linear static analysis.
C ode : Euroco de
C ase : FR +
5,0
00
90,0°
5,1
96
6,000
90,0°
30,0°
7,000
Y
X
Top view
4,000
3,500
C ode : Eurocode
C ase : F R +
Z
X
Front view
8,0
00
81
Code : Eurocode
Case : ST1
All nodal masses are Mx=My=Mz =100000 kg
All beams 60x40 cm
Inertia about vertical axis is multiplied by 1000.
Node D
All columns 60x40 cm
Column B
Column A
Support C
All supports are constrained in all directions.
eX=eY=eZ=fiX=fiY=fiZ=0
Z
Y
X
Perspective view
Section beams: 60x40 cm
Ax=2400 cm2 Ay=2000 cm2 Az=2000 cm2
Ix=751200 cm4 Iy=720000 cm2 Iz=320000000 cm4
Section columns: 60x40 cm
Ax=2400 cm2 Ay=2000 cm2 Az=2000 cm2
Ix=751200 cm4 Iy=720000 cm2 Iz=320000 cm4
Loads
Nodal masses on eight nodes. Mx=My=Mz=100000 kg
Model self-weight is excluded.
Spectrum for X and Y direction of seismic action:
T[s]
Sd
1
2
3
4
5
0
0,2000
0,6000
1,3000
3,0000
1,150
2,156
2,156
0,995
0,300
6
4,0000
...
0,300
...
S d [m/s 2 ]
2,156
1,150
0,709
0,300
2,0000
Boundary
Conditions
Nodes at the columns bottom ends are constrained in all directions.
eX=eY=eZ=fiX=fiY=fiZ=0
Material
Properties
C25/30
E=3050 kN/cm2 ν =0,2 ρ = 0
T[s]
Element
types
Target
Results
82
Three node straight prismatic beam element. Shear deformation is taken into account.
Compare the model results with SAP2000 v6.13 results.
The results are combined for all modes and all direction of spectral acceleration.
CQC combination are used for modes in each direction of acceleration.
SRSS combination are used for combination of directions.
Period times of first 5 modes
Mode
T[s] SAP2000
1
0,7450
2
0,7099
3
0,3601
4
0,2314
5
0,2054
T[s] AxisVM
0,7450
0,7099
0,3601
0,2314
0,2054
Modal participating mass ratios in X and Y directions
Mode
Difference
εX
εX
%
SAP2000
AxisVM
1
0,5719
0,5719
0
2
0,3650
0,3650
0
3
0
0
0
4
0,0460
0,0460
0
5
0,0170
0,0170
0
Summ
1,0000
1,0000
0
Difference [%]
0
0
0
0
0
εY
SAP2000
0,3153
0,4761
0,1261
0,0131
0,0562
0,9868
Internal forces at the bottom end of Column A and Column B
Column A Column A Difference Column B
SAP2000
AxisVM
%
SAP2000
Nx [kN]
315,11
315,15
+0,01
557,26
Vy [kN]
280,34
280,34
0
232,88
Vz [kN]
253,49
253,49
0
412,04
Tx [kNm]
34,42
34,41
-0,03
34,47
My [kNm]
625,13
625,12
-0,002
1038,74
Mz [kNm]
612,31
612,31
0
553,41
εY
AxisVM
0,3154
0,4760
0,1261
0,0131
0,0562
0,9868
Difference
%
+0,03
-0,02
0
0
0
0
Column B
AxisVM
557,29
232,88
412,04
34,46
1038,70
553,41
Difference
%
+0,005
0
0
-0,03
-0,004
0
Support forces of Support C
Support C
SAP2000
Rx [kN]
280,34
Ry [kN]
253,49
Rz [kN]
315,11
Rxx [kNm]
625,13
Ryy [kNm]
612,31
Rzz [kNm]
34,42
Support C
AxisVM
280,34
253,49
315,15
625,12
612,31
34,41
Difference
%
0
0
+0,01
-0,002
0
-0,03
Displacements of Node D
Node D
SAP2000
eX [mm]
33,521
eY [mm]
19,944
eZ [mm]
0,229
0,00133
ϕX [rad]
0,00106
ϕY [rad]
0,00257
ϕZ [rad]
Node D
AxisVM
33,521
19,945
0,229
0,00133
0,00106
0,00257
Difference
%
0
+0,005
0
0
0
0
Normal forces:
83
Bending moments:
84
85
Displacements:
86

AxisVM 11

Transcription

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