Verification Examples AxisVM-v11

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


Thermally loaded bar structure.


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

AxisVM 11 Verification Examples Software Release Number: R3 Date: 17. 10. 2012. Tested by: InterCAD Page number: File name: beam3.axs Thema

Continously supported beam with point loads.


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 ]


External prestessed beam.


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]

AxisVM 11 Verification Examples Software Release Number: R3 Date: 17. 10. 2012. Tested by: InterCAD Page number: File name: plane1.axs Thema

Periodically supported infinite membrane wall with constant distributed load.


Linear analysis.

Side view (thickness = 20,0 cm) Loads

p = 200 kN / m


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

Software Release Number: R3 Date: 17. 10. 2012. Tested by: InterCAD Page number: File name: plane2.axs Thema

Clamped beam examination with plane stress elements.


Linear analysis.

Side view Loads

p = -25 kN/m


Both ends built-in. 2

E = 880 kN / cm ν=0 Parabolic quadrilateral membrane (plane stress)

0,375

1

0,500

Clamped edge

C

3,000

Z

X

Side view

0,250

AxisVM 11 Verification Examples Target

14

τxy, max at section C

Results

Diagram

τxy

5,14

791,56

Z

Y

5,28

Diagram

τxy at section C

AxisVM 11 Verification Examples 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


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Software Release Number: R3 Date: 17. 10. 2012. Tested by: InterCAD Page number: File name: plate1.axs Thema

Clamped thin square plate.


Linear analysis.

Top view (thickness = 5,0 cm) Loads

P = -10 kN (at the middle of the plate)


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

Software Release Number: R3 Date: 17. 10. 2012. Tested by: InterCAD Page number: File name: plate2_1.axs Thema

Plate with fixed support and constant distributed load.


Linear analysis.

Top view

Loads Boundary Conditions Material Properties Element types Mesh

Target Results

(thickness = 15,0 cm) 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

AxisVM 11 Verification Examples Software Release Number: R3 Date: 17. 10. 2012. Tested by: InterCAD Page number: File name: plate3.axs Thema

Annular plate.


Linear analysis.


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


All edges simply supported plate with partial distributed load.


Linear analysis.

Top view (thickness = 22,0 cm) 2

Distributed load: q = -10 kN / m (middle of the plate at 2,0 x 2,0 m area)


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


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.)



23


Clamped plate with linear distributed load.


Linear analysis.

Top view (thickness = 22,0 cm) 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


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


AxisVM 11 Verification Examples Software Release Number: R3 Date: 17. 10. 2012. Tested by: InterCAD Page number: File name: hemisphere.axs Thema

Hemisphere displacement.


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

AxisVM 11 Verification Examples 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

Software Release Number: R3 Date: 17. 10. 2012. Tested by: InterCAD Page number: File name: nonlin1.axs 3D beam structure.


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

AxisVM 11 Verification Examples 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

Software Release Number: R3 Date: 17. 10. 2012. Tested by: InterCAD Page number: File name: nonlin2.axs Thema

Plate with fixed end and bending moment.


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


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

AxisVM 11 Verification Examples Software Release Number: R3 Date: 17. 10. 2012. Tested by: InterCAD Page number: File name: dynam1.axs Thema

Deep simply supported beam.


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


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


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

Software Release Number: R3 Date: 17. 10. 2012. Tested by: InterCAD Page number: File name: dynam2.axs Thema

Clamped thin rhombic plate.


Dynamic analysis.

Top view of plane (thickness = 5,0 cm) 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


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

AxisVM 11 Verification Examples Software Release Number: R3 Date: 17. 10. 2012. Tested by: InterCAD Page number: File name: dynam3.axs Thema

Cantilevered thin square plate.


Dynamic analysis.


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


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


Cantilevered tapered membrane.


Dynamic analysis.

Side view (thickness = 10,0 cm) 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

Parabolic quadrilateral membrane (plane stress)

1,000

5,000

2

10,000 Y

X


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


Flat grillages.


Dynamic analysis.

Top view Loads

Self-weight


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


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

AxisVM 11 Verification Examples 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

Software Release Number: R3 Date: 17. 10. 2012. Tested by: InterCAD Page number: File name: buckling1.axs Thema

Simply supported beam.


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 A 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)


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

Software Release Number: R3 Date: 17. 10. 2012. Tested by: InterCAD Page number: File name: buckling2.axs Thema



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

eX = eY = eZ = 0 at A eY = eZ = 0 at B

Material Properties

E = 20000 kN / cm ν = 0,3


Beam element

2

Pcr = ? (for inplane buckling)

Results

P cr [kN]

Theory

AxisVM

e [%]

3,340

3,337

-0,09


Design

53


54

Software Release Number: R3 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).


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


RC beam deflection according to EC2, EN 1992-1-1:2004.


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




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


Required steel reinforcement of RC plate according to EC2, EN 1992-1-1:2004.


Linear analysis. Szabvány : Eurocode Eset : ST1

50 kN

4,0 Y

X

Side view

Cross-section Loads

Pz = -50 kN point load


Clamped cantilever plate. Concrete: C25/30 Steel: B500A Parabolic quadrilateral plate element (heterosis) Szabvány : Eurocode Eset : ST1

1,0

4,0

Clamped edge

Y

X

Top view


58

AXT steel reinforcement along x direction at the top of the support

Results Lineáris számítás Szabvány : Eurocode 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

Software Release Number: R3 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).


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

AxisVM 11 Verification Examples 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

Software Release Number: R3 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


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

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

Beam element

Target

Interaction check of axial force, shear force and bending moment.


62

Analytical solution in the following book: 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 3.12., 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

Software Release Number: R3 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).


Steel Design


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 = ?


64

Analytical solution in the following book: 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 3.15., 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

Software Release Number: R3 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).


Steel Design


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 = ?


66

Analytical solution in the following book: 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 3.21., 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

Software Release Number: R3 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).


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.


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

AxisVM 11 Verification Examples 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  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

AxisVM 11 Verification Examples 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

Software Release Number: R3 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).


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

AxisVM 11 Verification Examples 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  χ y N Rk /γ M1  χ y N Rk /γ M1     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

AxisVM 11 Verification Examples 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

Software Release Number: R3 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)


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



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 ex = ey = ez = 0 at A 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.


78

Analytical solution in the following book: 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 3.4., 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

Software Release Number: R3 Date: 17. 10. 2012. Tested by: InterCAD Page number: File name: Earthquake-01-EC.axs Thema

Earth-quake design using response-spectrum method.


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]

AxisVM 11 Verification Examples 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

AxisVM 11 Verification Examples Normal forces:

83

AxisVM 11 Verification Examples Bending moments:

84


85

AxisVM 11 Verification Examples Displacements:

86

Verification Examples AxisVM-v11

Verification Examples AxisVM-v11

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