A REVOLUTIONARY TOOLKIT FOR OPENSEES ...- Midas Gen 2020, MIDAS Information Technology Co., Ltd. 1.2...

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A REVOLUTIONARY TOOLKIT FOR OPENSEES

VERIFICATION TESTS – STKO 2020 v. 1.1 and OpenSees 3.2.0

July, 2020

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1 INTRODUCTION .................................................................................................... 3

1.1 Overview ............................................................................................................ 3

1.2 Consistent units .................................................................................................. 3

1.3 Element tests ...................................................................................................... 3

1.4 Material tests ...................................................................................................... 4

1.5 Nonlinear analysis tests ........................................................................................ 4

2 LINEAR ELASTIC TESTS USING THEORETICAL SOLUTIONS................................... 5

2.1 ST1: Static analysis of overhanging beam .............................................................. 5

2.2 ST2: Symmetric frame structure subjected to rotational forces ................................. 7

2.3 ST3: Beam with elastic supports and an internal hinge ............................................. 9

2.4 ST4: Analysis of cantilever beam with in-plane vertical load at free end ................... 12

2.5 ST5: Stress concentration around a hole in a square plate ...................................... 15

2.6 ST6: Static analysis of simply supported square plate under a uniform pressure load . 18

2.7 ST7: Thin cylindrical shell under two point loads .................................................... 21

2.8 ST8: Static analysis of tapered plate (beam) under static load ................................ 24

2.9 ST9: Twisted solid cantilever beam ...................................................................... 26

2.10 ST10 Shell Elements: Twisted cantilever beam ...................................................... 29

2.11 ST11: Static analysis of a circular slab subjected to a pressure load......................... 34

3 LINEAR ELASTIC TESTS: NAFEMS BENCHMARKS ................................................ 37

3.1 LE1: Elliptic membrane - Plane stress elements ..................................................... 37

3.2 LE3: Hemispherical shell with point loads ............................................................. 41

3.3 LE5: Z-section cantilever .................................................................................... 44

3.4 LE6: Skew plate under normal pressure ............................................................... 47

3.5 LE10: Thick plate under pressure ........................................................................ 50

4 FREE VIBRATION TESTS USING THEORETICAL SOLUTIONS ................................ 53

4.1 FVT1, Truss element: two springs and two lumped masses ..................................... 53

4.2 FVB1, Beam element: beams on springs ............................................................... 54

4.3 FVB2, Beam element: analysis of a shaft with three disks ....................................... 55

4.4 FVB3, Beam element: pyramid ............................................................................ 56

4.5 FVB4, Beam and shell element: cantilever model ................................................... 59

4.6 FVS1, Cantilever shell model ............................................................................... 61

4.7 FVS2, Skewed cantilever plate ............................................................................ 64

5 FREE VIBRATION TESTS: NAFEMS BENCHMARKS ................................................ 67

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5.1 FV2: Pin-ended double cross: in-plane vibration Elements tested ............................ 67

5.2 FV4: Cantilever with off-center point masses ........................................................ 70

5.3 FV12: Free thin square plate ............................................................................... 72

5.4 FV15: Fixed thin rhombic plate ............................................................................ 77

5.5 FV16: Cantilevered thin square plate ................................................................... 81

5.6 FV22: Clamped thick rhombic plate...................................................................... 88

5.7 FV32: Cantilevered tapered membrane ................................................................ 91

5.8 FV52: Simply supported “solid” square plate ......................................................... 94

6 NONLINEAR GEOMETRY TESTS ........................................................................... 97

6.1 NLG1. Corotational Truss Element ....................................................................... 97

6.2 NLG2: Snap through .......................................................................................... 99

6.3 NLG3: Static large displacement analysis of a tower cable ..................................... 101

6.4 NLG4: Cantiliver subjected to bending moment .................................................... 104

7 NONLINEAR GEOMETRY TESTS: NAFEMS BENCHMARKS ................................... 107

7.1 3DNLG-1: Elastic large deflection response of a cantilever under an end load ........... 107

7.2 3DNLG-7: Elastic large deflection response of a hinged spherical shell under pressure

loading ................................................................................................................... 110

8 NONLINEAR MATERIAL TESTS .......................................................................... 113

8.1 NLM1: Plane Strain Plasticity .............................................................................. 113

8.2 NLM2: 3D Plasticity ........................................................................................... 115

9 NONLINEAR ANALYSIS TESTS .......................................................................... 118

9.1 TH1: Dynamic modal response for 2-D rigid frame................................................ 118

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1 INTRODUCTION

1.1 OVERVIEW

This is the verification manual of OpenSees solver 3.2.0 prepared with STKO 2020

v.1.1 (hereafter STKO).

STKO verification manual includes:

Code verification. The code results are compared to analytical solutions

Calculation verification. Verification of the errors in the code output due to

discretization.

The manual contains NAFEMS test problems and standard problems, the code output

is compared to analytical solutions and other commercial software output. The

programs used to verify the manual are:

- SIMULIA Abaqus Unified FEA, release Abaqus 2018, Dessault Systems

- Midas Gen 2020, MIDAS Information Technology Co., Ltd.

1.2 CONSISTENT UNITS

STKO is unitless, consistent units must be used.

Consistent Units

Length m mm ft in

Time s s s s

Mass kg ton lbf × s2/ft lbf × s2/in

Force N N lbf lbf

Temperature C C F F

Velocity m/s mm/s ft/s in/s

Acceleration m/s2 mm/s2 ft/s2 in/ s2

Angular velocity rad/s rad/s rad/s rad/s

Density kg/m3 ton/mm3 slug/ft3 lbf × s2/in4

Moment N × m N × mm ft × lbf in × lbf

Stress Pa Mpa psf Psi

Energy J mJ ft × lbf in × lbf

g-Gravity Constant

9.81E+00 9.81E+03 3.22E+01 3.86E+02

Steel Density 7.83E+03 7.83E-03 1.52E+01 7.33E-04

Steel Modulus 2.07E+11 2.07E+05 4.32E+09 3.00E+07

1.3 ELEMENT TESTS

Quadrilateral Elements

Quad Element Plain Stress and Plain Strain

ShellMITC4 Plain Stress, nonlinear geometry

ShellMITC9 Plain Stress

https://opensees.berkeley.edu/wiki/index.php/Quad_Element

https://opensees.berkeley.edu/wiki/index.php/Shell_Element


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ShellDKGQ Plain Stress

ASDShellQ4 Plain Stress, nonlinear geometry

SSPquad Element Plain Stress and Plain Strain

ShellNLDKGQ Plain Stress, nonlinear geometry

Bbar Quadrilateral Plain Strain

Enhanced Quadrilateral Plain Strain

Triangular Elements

Tri31 Element Plain Stress and Plain Strain

ShellDKGT Plain Stress

ShellNLDKGT Plain Stress, nonlinear geometry

Brick Elements

Brick Elements

Standard Brick Element

Bbar Brick Element

Twenty Node Brick Element

Twenty Seven Node Brick Element

SSPbrick Element

Tetrahedron Elements

FourNodeTetrahedron

Truss Elements

Truss Element

Corotational Truss Element

Beam-Column Elements

ElasticBeamColumn

DispBeamColumnElement

ForceBeamColumnElement

Elastic Forced Based

1.4 MATERIAL TESTS

Plane Strain Plasticity

3D Plasticity

1.5 NONLINEAR ANALYSIS TESTS

Dynamic modal response

https://opensees.berkeley.edu/wiki/index.php/ShellDKGQ

https://opensees.berkeley.edu/wiki/index.php/SSPquad_Element

https://opensees.berkeley.edu/wiki/index.php/ShellNLDKGQ

https://opensees.berkeley.edu/wiki/index.php/Bbar_Plane_Strain_Quadrilateral_Element

https://opensees.berkeley.edu/wiki/index.php/Enhanced_Strain_Quadrilateral_Element

https://opensees.berkeley.edu/wiki/index.php/Tri31_Element

https://opensees.berkeley.edu/wiki/index.php/ShellDKGT

https://opensees.berkeley.edu/wiki/index.php/ShellNLDKGT

https://opensees.berkeley.edu/wiki/index.php/Standard_Brick_Element

https://opensees.berkeley.edu/wiki/index.php/Bbar_Brick_Element

http://opensees.berkeley.edu/OpenSees/manuals/usermanual/734.htm

https://opensees.berkeley.edu/wiki/index.php?title=Twenty_Seven_Node_Brick_Element&action=edit&redlink=1

https://opensees.berkeley.edu/wiki/index.php/SSPbrick_Element

https://opensees.berkeley.edu/wiki/index.php/FourNodeTetrahedron

https://opensees.berkeley.edu/wiki/index.php/Truss_Element

https://opensees.berkeley.edu/wiki/index.php/Corotational_Truss_Element

https://opensees.berkeley.edu/wiki/index.php/Elastic_Beam_Column_Element

https://opensees.berkeley.edu/wiki/index.php/Displacement-Based_Beam-Column_Element

https://opensees.berkeley.edu/wiki/index.php/Force-Based_Beam-Column_Element


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2 LINEAR ELASTIC TESTS USING THEORETICAL SOLUTIONS

2.1 ST1: STATIC ANALYSIS OF OVERHANGING BEAM

Elements tested

Elastic Beam Column

Benchmark description

Figure 1: model description, adapted from Midas Figure 2: STKO FE Model

• Problem description. Static analysis of overhanging beam loaded on the

overhangs with uniformly distributed loads.

Dimension: L1= 3.048 m, L2= 3.048 m, Ltot = 6.096 m.

Section Property: Iyy = 3.28E-3 m4.

Only a half model may be analyzed due to symmetry.

• Material. Linear elastic, E = 2.07 E+11 N/m2.

• Boundary conditions. ux= Ry =0 on node 1, ux= Ryz =0 on node 2.

• Loading. Uniform distributed load of 1.46 E+05 N/m is applied on the overhangs

in the –Z direction.

Test Results

Reference solution: maximum deflection δmax= 4.6228 E-03 m.

Values obtained and percentage differences with respect to the reference solution.

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Figure 3: Z deflection, ElasticBeamColumn

Table 1. Maximum Deflection, δmax (m)

Analytical STKO

Value Value Diff.

m m %

Maximum Deflection. 4.62E-03 4.62E-03 0

References

• Static 03: Midas verification examples. MIDAS Information Technology Co.

• Timoshenko, S., “Strength of Materials, Part I, Elementary Theory and Problems”,

3rd Ed.

Input files

ST1_ElasticBeamColumn.scd

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2.2 ST2: SYMMETRIC FRAME STRUCTURE SUBJECTED TO ROTATIONAL

FORCES

Elements tested

Elastic Beam Column



• Problem description. 2D Static analysis (x-y plane) of symmetric frame

structure subjected to rotational forces.

Dimension: L1 = 0.254 m, L (node 3-9) = 1.016 m, H1 = 0.254 m, H (node 6-

12) = 1.016 m.

Section Property: Iyy = 3.47 E-08 m4.


• Boundary conditions. Constrain all DOFs on node 1.

• Loading. Concentrated load of 44.38 N is applied to the node 3 in the -Y

direction, to node 9 in the Y direction, to the node 6 in the X direction and to the

node 12 in -X direction.

Test Results


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Figure 6: Y-displacement, ElasticBeamColumn

Table 2. Y Displacement, δy at 3 (m)

SAP2000 MIDAS STKO

m m m

Node 3 4.52E-04 4.52E-04 4.52E-04

References


Input files


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2.3 ST3: BEAM WITH ELASTIC SUPPORTS AND AN INTERNAL HINGE

Elements tested

Elastic Beam Column


Figure 7: model description, adapted from Midas

Figure 8: STKO FE Model

• Problem description. 2D Static analysis (x-y plane) of beam with elastic

supports and an internal hinge.

Dimension: L1= 7.32 m; L2 = 9.15 m; H = 3.66 m

Section Property Elements 1, 2: A = 1.16 E-02 m2, Iyy = 2.27 E-03 m4

Section Property Elements 3: A = 1.63 E-02 m2, Iyy = 1.67 E-03 m4


• Boundary conditions. Constrain all DOFs on node 2, spring constant (Z

direction), K= 1.75 E+07 N/m on node 1 and 4, release Ry of the node 3 of the

element 3 in the element local coordinates.

• Loading. Concentrated load P1 of 2.22 E+04 N is applied on node named 5 in

the X direction, a concentrated load P2 of 6.67 E+04 N is applied on node named

6 in the –Z direction.

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Test Results

Reference solution are shown in the following for nodes 1, 3 and 4:

Displacements at node 1: δx = 3.29 E-04 m, δy = 5.45 E-04 m, θy = -9.9 E-05 rad.

Displacements at node 3: δx = 3.29 E-04 m, δy = -5.49 E-05 m, θy = 4.44 E-04 rad.

Displacements at node 4: δx = 3.29 E-04 m, δy = -1.47 E-03 m, θy = -3.62 E-04 rad.


Figure 9: X-displacement, ElasticBeamColumn

Figure 10: Y-displacement, ElasticBeamColumn

Table 3. X Displacement, δx (m)

THEOR. SAP200 MIDAS -Gen STKO

Value Value Diff. Value Diff. Value Diff.

m m % m % m %

Node 1 3.29E-04 3.29E-04 0 3.29E-04 0 3.29E-04 0

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Node 3 3.29E-04 3.29E-04 0 3.29E-04 0 3.29E-04 0

Node 4 3.29E-04 3.29E-04 0 3.29E-04 0 3.29E-04 0

Table 4. Y Displacement, δy (m)



m m % m % m %

Node 1 5.45E-04 5.45E-04 0 5.45E-04 0 5.45E-04 0

Node 3 -5.49E-05 -5.49E-05 0 -5.49E-05 0 -5.50E-05 0

Node 4 -1.47E-03 -1.47E-03 0 -1.47E-03 0 -1.47E-03 0

Table 5. Z Rotation, θy (rad)



m m % m % m %

Node 1 -9.90E-05 -9.90E-05 0 -9.90E-05 0.00 -9.92E-05 0

Node 3 4.44E-04 4.44E-04 0 4.44E-04 0.00 4.44E-04 0

Node 4 -3.62E-04 -3.61E-04 0 -3.62E-04 0.00 -3.61E-04 0

References


Input files


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2.4 ST4: ANALYSIS OF CANTILEVER BEAM WITH IN-PLANE VERTICAL

LOAD AT FREE END

Elements tested

ASDShellQ4

ShellANdeS

ShellMITC4

ShellMITC9

ShellDKGQ

ShellDKGT



• Problem description. 2D Static analysis (x-y plane) of cantilever beam

subjected to an in-plane vertical load at the free end.

Dimension: L= 7.62 E-02 m; H = 1.5 E-02 m;

Thickness: t = 2.54 E-03 m


• Mesh. Element mesh is shown in Figure 11 with the following dimensions:

L1 = 0.0381 m, H1= 0.00254 m.

• Boundary conditions. ux = uy = 0 on nodes 1, 3, 5, 6, 7, 8, 9.




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• Loading. Concentrated load P1 of 4.45 E+01 N is applied to node named 2 and

4 in the -Y direction, a concentrated load P2 of 8.90 E+01 N is applied to node

named 17, 18, 19, 20 and 21 in the –y direction.

Test Results


Figure 13: Y-displacement, ASDShellQ4

Table 6. Y Displacement, δy at 2 (m)

Node 2 – Y Displacement

m

MSC-NASTRAN -1.33E-03

STAAD-PRO -1.38E-03

MIDAS-GEN -1.33E-03

STKO

ASDShellQ4 -1.33E-03

shellANdes -1.35E-03

shellMITC4 -4.25E-04

shellMITC9 -1.43E-03

ShellDKGQ -1.37E-03

ShellDKGT -7.71E-04

References




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Input files

ST4_ASDShellQ4.scd

ST4_ShellANdeS.scd

ST4_ShellDKGQ.scd

ST4_ShellDKGT.scd

ST4_ShellMITC4

ST4_ShellMITC9

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2.5 ST5: STRESS CONCENTRATION AROUND A HOLE IN A SQUARE

PLATE

Elements tested

ASDShellQ4

Quad Element

ShellMITC4

ShellDKGQ

SSPquad Element



• Problem description. 2D Static analysis of square plate due to effects of a

circular hole at the center under an in-plane uniform line load.

Only a quarter model may be analyzed due to symmetry.

Dimension: LTOT= 4.06 E-01 m, LMODEL= 2.03 E-01 m.

Radius of the hole: R = 1.27 E-02 m. Thickness: t = 2.54 E-02 m.

• Material. Linear elastic, E = 6.89 E+03 N/m2, = 0.1.

• Mesh. Element mesh is shown in Figure 14.



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• Boundary conditions. uy = Rx = Ry = Rz = 0 on nodes named 1, 2, 3, 4, 5, 6,

7, 8, 9 and ux= Rx = Ry = Rz = 0 on nodes named 64, 65, 66, 67, 67, 68, 69, 70,

71, 72.

• Loading. Uniform in-plane compression in the -X direction of outward pressure

of 1.75E+02 N/m is applied on the right edge.

Test Results


Figure 16: S11 stress, SSPquad

Table 7. x Stress, S11 (N/m2) - absolute value –other software

THEOR. SAP2000 MIDAS -Gen

Value Val. Diff. Val. Diff.

N/m2 N/m2 % N/m2 %

El 49 1.61E+04 1.65E+04 2.38 1.63E+04 1.61

El 50 1.05E+04 1.08E+04 3.06 1.07E+04 1.99

El 51 8.17E+03 8.26E+03 1.16 8.21E+03 0.56

El 52 7.25E+03 7.31E+03 0.88 7.29E+03 0.58

El 53 7.09E+03 7.00E+03 -1.23 7.01E+03 -1.11

El 54 6.94E+03 6.93E+03 -0.22 6.91E+03 -0.47

El 55 6.92E+03 6.88E+03 -0.59 6.90E+03 -0.37

El 56 6.91E+03 6.82E+03 -1.34 6.84E+03 -0.97

Table 8. x Stress, S11 (N/m2) - absolute value - STKO

ASDSHELLQ4 QUAD

ELEMENT SSPQUAD SHELLMITC4 SHELLDKGQ

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Val. Diff. Val. Diff. Val. Diff. Val. Diff. Val. Diff.

N/m2 % N/m2 % N/m2 % N/m2 % N/m2 %

El 49 1.632E+04 1.54 1.630E+04 1.40 1.630E+04 1.40 1.641E+04 2.07 1.399E+04 -12.99

El 50 1.075E+04 2.68 1.070E+04 2.20 1.070E+04 2.20 1.071E+04 2.27 1.156E+04 10.45

El 51 8.242E+03 0.93 8.260E+03 1.16 8.240E+03 0.91 8.268E+03 1.26 1.177E+04 44.16

El 52 7.305E+03 0.82 7.320E+03 1.04 7.290E+03 0.62 7.315E+03 0.97 4.163E+03 -42.54

El 53 7.006E+03 -1.12 7.020E+03 -0.92 7.010E+03 -1.06 7.000E+03 -1.20 7.242E+03 2.21

El 54 6.920E+03 -0.35 6.940E+03 -0.06 6.950E+03 0.08 6.921E+03 -0.34 6.717E+03 -3.27

El 55 6.914E+03 -0.11 6.900E+03 -0.31 6.910E+03 -0.17 6.907E+03 -0.21 8.175E+03 18.11

El 56 6.876E+03 -0.51 6.850E+03 -0.89 6.847E+03 -0.93 6.887E+03 -0.36 6.045E+03 -12.53

References


• Timoshenko, S. and Goodier, J.N. “Theory of Elasticity”, McGraw-Hill, New York,

1951, pp 78-80.

Input files

ST5_ASDShellQ4.scd

ST5_QuadElement.scd

ST5_shellDKGQ.scd

ST5_shellMITC4.scd

ST5_SSPquad.scd

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2.6 ST6: STATIC ANALYSIS OF SIMPLY SUPPORTED SQUARE PLATE

UNDER A UNIFORM PRESSURE LOAD

Elements tested

ShellANdeS

ShellDKGT



• Problem description. 2D Static analysis of simply supported square plate under

a uniform pressure load.

Only a quarter model may be analyzed due to symmetry.

Dimension: LTOT= 8 E-01 m, LMODEL= 4 E-01 m. Thickness: t = 8 E-03 m.



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Triangular base x height = 0.1x0.05 m.

• Boundary conditions. uy = Rx = 0 on nodes named 1, 2, 3, 4, 5, ux = Ry = 0 on

nodes named 1, 10, 19, 26, 37, uz = Ry = 0 on nodes named 37, 38, 39, 40, 41

and uz = Rx = 0 on nodes 5, 14, 23, 32 and 41.

• Loading. Uniform pressure load P of 1.00 E+03 N/m2 is applied in the -Z

direction.

Test Results

Reference solution: δz at 1 is 1.689 E-04 m.


Figure 19: Z displacement, ShellANDeS

Table 9. z Displacement, δz (m)

THEOR. MIDAS-GEN

STKO

ShellANdeS ShellDKGT

Value Val. Diff. Val. Diff. Val. Diff.

N/m2 N/m2 % N/m2 % N/m2 %

El 49 1.69E-04 1.68E-04 -0.77 1.67E-04 -1.12 1.68E-04 -0.53


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References


• Timoshenko, S.P., and Woinowsky-Krieger, S “Theory of Plates and Shells”, 2nd

Edition, McGraw-Hill, 1959.

Input files

ST6_ShellANdeS.scd

ST6_ShellDKGT.scd

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2.7 ST7: THIN CYLINDRICAL SHELL UNDER TWO POINT LOADS

Elements tested

ASDShellQ4

ShellANdeS

ShellDKGQ

ShellDKGT

ShellMITC4

ShellMITC9



• Problem description. 3D Static analysis of a thin cylindrical shell. A pair of equal

and opposite point loads act on a thin cylindrical shell transverse to the cylindrical

axis. simply supported square plate under a uniform pressure load.

Only a quarter of a half model may be analyzed due to symmetry.

Dimension: LTOT= 1.524 E+04 mm, LMODEL= 7.62 E+03 mm.

Radius: R = 7.62 E+03 mm. Thickness: t = 7.62 E+01 mm.




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• Material. Linear elastic, E = 2.07 E+04 N/mm2, = 0.3.


Base x height = [7.62 E+03/12mm] x [7.62 E+03/4/12mm].

• Boundary conditions. uz = Rx = Ry = 0 on nodes from 157 to 169, ux = Ry = Rz

= 0 on nodes from 1 to 157, uy = Rx = Rz = 0 on nodes from 13 to 169.

• Loading. Force P is equal to 4.448 N.

In the quarter of a half model analyzed F157,y = 1.112 N.

Test Results

Reference solution: δy at 157 is 1.15 E-02 mm.


Figure 22: Y displacement, ASDShellQ4

Table 10. y Displacement, δy (m) –other software

THEOR. MIDAS -Gen

Value Value Value Diff.

mm mm mm %

Node 157 1.15E-02 1.18E-02 1.17E-02 2.27

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Table 11. y Displacement, δy (m) –STKO

Node 157 – y Displacement

mm %

ASDShellQ4 1.16E-02 1.22

shellANdes 1.17E-02 1.67

shellMITC4 1.16E-02 1.14

shellMITC9 1.22E-02 6.02

ShellDKGQ 1.17E-02 2.08

ShellDKGT 1.18E-02 2.44

References


• Timoshenko, S.P., and Woinowsky-Krieger, S “Theory of Plates and Shells”, 2nd

Edition, McGraw-Hill, 1959.

Input files

ST7_ASDShellQ4.scd

ST7_ShellANdeS.scd

ST7_ShellDKGQ.scd

ST7_ShellDKGT.scd

ST7_ShellMITC4.scd

ST7_ShellMITC9.scd



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2.8 ST8: STATIC ANALYSIS OF TAPERED PLATE (BEAM) UNDER STATIC

LOAD

Elements tested

ShellANdeS

ShellDKGT




• Problem description. 3D Static analysis of a tapered cantilever plate of

rectangular cross-section subjected to a vertical load at its tip.

Dimension: L1= 7.62 E+01 mm, L2= 5.08 E+02 mm.

Thickness: t = 1.27 E+01 mm.



• Boundary conditions. Constrain all DOFs on nodes from 1, 2, 3.

• Loading. FTIP,z = -4.48 E+01 N.

Test Results

Reference solution: δz at tip is 1.08 mm.



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Figure 25: Y displacement, ShellANDeS

Table 12. z Displacement, δz (m)

THEOR. MIDAS -Gen

STKO

shellANdes

STKO

shellDKGT


m m % m % m %

Node at Tip. -1.08E+00 -1.08E+00 0.01 -1.08E+00 0.00 -1.08E+00 0.00

References


Input files

ST8_ShellANdeS.scd

ST8_ShellDKGT.scd

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2.9 ST9: TWISTED SOLID CANTILEVER BEAM

Elements tested

SSPbrick


Bbar Brick Element


FourNodeTetrahedron



• Problem description. 3D Static analysis of a twisted solid cantilever beam of

rectangular cross-section subjected to in-plane and out of plane shear forces.

Dimension: L= 1.20 E+01 m.

Rectangular cross-section base x height: 1.1 m x 3.2 E-01 m.



• Boundary conditions. ux = uz = 0 on nodes 1, 3, 4, 6, ux = uy = uz = 0 on nodes

2 and 5.

• Loading. 2 load cases are analysed.

CASE 1 – In-Plane Shear Force: F1,z = -1 E+01 N.

CASE 2 – Out-of-Plane Shear Force: F2,y = -1 E+01 N.

Test Results

CASE 1 – In-Plane Shear Force: δz = -5.424 E-03 m.

CASE 2 – Out-of-Plane Shear Force: δy = -1.754 E-03 m.





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Figure 28: CASE 1 - Z displacement, SSPbrick

Figure 29: CASE 2 - Y displacement, SSPbrick

Table 13. Case 1 and 2 displacement (m) – other software

THEOR. SAP200 MIDAS -Gen

Value Value Diff. Value Diff.

m m % m %

CASE 1: Disp. δz -5.42E-03 -5.40E-03 -0.53 -5.40E-03 -0.53

CASE 2: Disp. δy -1.75E-03 -1.73E-03 -1.14 -1.73E-03 -1.14

Table 14. Case 1 and 2 displacement (m) – STKO

SSPBRICK STDBRICK BBARBRICK 20NODEBRICK 4NODETETR.

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Val. Diff. Val. Diff. Val. Diff. Val. Diff. Val. Diff.

N/m2 % N/m2 % N/m2 % N/m2 % N/m2 %

CASE 1 -5.42E-03 -0.04 -1.12E-03 -79.35 -1.26E-03 -76.77 -5.63E-03 3.80 -0.00168 -69.03

CASE 2 -1.74E-03 -0.74 -5.85E-04 -66.65 -6.47E-04 -63.11 -1.78E-03 1.48 -0.00069 -60.55

References


Input files

ST9_20nodeBrick.scd

ST9_BbarBrick.scd

ST9_SSPbrick.scd

ST9_standardBrick.scd

ST9_4nodeTetrahedron.scd

29

S T K O

2.10 ST10 SHELL ELEMENTS: TWISTED CANTILEVER BEAM

Previous benchmark ST9 is performed with shell elements.

Elements tested

ASDShellQ4

ShellMITC4

ShellMITC9

ShellDKGQ

ShellDKGT



• Problem description. 3D Static analysis of a twisted cantilever beam of

rectangular cross-section subjected to in-plane and out of plane shear forces.

Dimension: L= 1.20 E+01 m.

Rectangular cross-section base x height: 1.1 m x 3.2 E-01 m.


• Mesh. A coarse, a fine mesh and a very fine mesh are tested for each element.

Mesh type are shown in the next paragraph.

• Boundary conditions. ux = uz = 0 on nodes 1, 3, 4, 6, ux = uy = uz = 0 on nodes

2 and 5.

• Loading. 2 load cases are analysed.

CASE 1 – In-Plane Shear Force: F1,z = -1 E+01 N.

CASE 2 – Out-of-Plane Shear Force: F2,y = -1 E+01 N.




30

S T K O

Mesh type

Figure 32: coarse mesh

Figure 33: fine mesh

Figure 34: very fine mesh

Test Results

CASE 1 – In-Plane Shear Force: δz = -5.424 E-03 m.

CASE 2 – Out-of-Plane Shear Force: δy = -1.754 E-03 m.


31

S T K O

Figure 35: CASE 1 - Z displacement, ASDShellQ4 – coarse mesh

Figure 36: CASE 2 - Y displacement, ASDShellQ4 – coarse mesh

CASE 1: Values obtained and percentage differences with respect to the reference

solution.

Table 15. CASE 1: z Displacement, δz (m)

Element

Coarse Mesh Fine Mesh Very Fine Mesh

Value Diff. Value Diff. Value Diff.

N/m2 % N/m2 % N/m2 %

ASDShellQ4 -5.41E-03 -0.29 -5.41E-03 -0.19 -5.42E-03 -0.12

ShellMITC4 -1.39E-03 -74.35 -2.94E-03 -45.86 -4.68E-03 -13.63

ShellMITC9 -1.51E-03 -72.10 -3.14E-03 -42.08 -4.68E-03 -13.69

ShellDKGQ -2.05E-03 -62.27 -3.67E-03 -32.38 -8.19E-03 50.94

ShellDKGT -5.35E-03 -1.28 -5.39E-03 -0.62 -5.40E-03 -0.49



32

S T K O

Figure 37: z Displacement δz as a function of mesh type

CASE 2: Values obtained and percentage differences with respect to the reference

solution.

Table 16. CASE 1: z Displacement, δy (m)

Element



N/m2 % N/m2 % N/m2 %

ASDShellQ4 -1.76E-03 0.16 -1.75E-03 -0.03 -1.75E-03 -0.05

ShellMITC4 -7.38E-04 -57.94 -1.10E-03 -37.55 -1.56E-03 -11.29

ShellMITC9 -6.56E-04 -62.59 -1.09E-03 -37.74 -1.54E-03 -12.16

ShellDKGQ -9.08E-04 -48.25 -1.45E-03 -17.43 -3.98E-03 126.74

ShellDKGT -1.67E-03 -4.53 -1.73E-03 -1.61 -1.74E-03 -0.54

1.30E-03

2.30E-03

3.30E-03

4.30E-03

5.30E-03

6.30E-03

7.30E-03

8.30E-03

12 32 52

CA

SE 1

: z D

isp

lace

men

t, δ

zat

en

d (

m)

-ab

solu

te

valu

e

Mesh global seed (-)

THEOR. RESULT

shellMITC4

shellMITC9

shellDKGQ

shellDKGT

ASDShellQ4



33

S T K O

Figure 38: z Displacement δy as a function of mesh type

References


Input files

ST10_ASDShellQ4_coarse_mesh.scd

ST10_ASDShellQ4_fine_mesh.scd

ST10_ASDShellQ4_very_fine_mesh.scd

ST10_ShellMITC4_coarse_mesh.scd

ST10_ShellMITC4_fine_mesh.scd

ST10_ShellMITC4_very_fine_mesh.scd

ST10_ShellMITC9_coarse_mesh.scd

ST10_ShellMITC9_fine_mesh.scd

ST10_ShellMITC9_very_fine_mesh.scd

ST10_ShellDKGQ_coarse_mesh.scd

ST10_ShellDKGQ_fine_mesh.scd

ST10_ShellDKGQ_very_fine_mesh.scd

ST10_ShellDKGT_coarse_mesh.scd

ST10_ShellDKGT_fine_mesh.scd

ST10_ShellDKGT_very_fine_mesh.scd

5.00E-04

1.00E-03

1.50E-03

2.00E-03

2.50E-03

3.00E-03

3.50E-03

4.00E-03

12 32 52

CA

SE 1

: z D

isp

lace

men

t, δ

yat

en

d (

m)

-ab

solu

te v

alu

e


THEOR. RESULT

shellMITC4

shellMITC9

shellDKGQ

shellDKGT

ASDShellQ4

34

S T K O

2.11 ST11: STATIC ANALYSIS OF A CIRCULAR SLAB SUBJECTED TO A

PRESSURE LOAD

Elements tested

ASDShellQ4

ShellMITC4

ShellMITC9

ShellDKGQ

ShellDKGT


Figure 39: model description, adapted from

Midas


• Problem description. 3D Static analysis of a circular slab subjected to a

pressure load.

Dimension: R= 1000 mm.

Thickness: t= 5 mm.



• Boundary conditions. Constrain all DOFs on nodes A, B, C and D.

• Loading. Uniform pressure load P of 1 E-03 MPa is applied in -Z direction.

Test Results

Reference solution: displacement δz= 6.5 mm.





35

S T K O

Figure 41: Figure 42: Z displacement, ASDShellQ4

Table 17. Displacement, δz (m) – OTHER SOFTWARE

THEOR. MIDAS -Gen

Value Value Diff.

mm mm %

Displacement δz -6.50 -6.50 0.00

Table 18. y Displacement, δz (m) –STKO

Displacement δz

Value (mm) Diff.

(%)

ASDShellQ4 -6.48E+00 -0.24

shellMITC4 -6.48E+00 -0.24

shellMITC9 -6.64E+00 2.20

ShellDKGQ -6.56E+00 0.87

ShellDKGT -6.56E+00 0.87

References

• Static 41. Midas verification examples. MIDAS Information Technology Co.



36

S T K O

Input files

ST11_ASDShellQ4.scd

ST11_ShellMITC4.scd

ST11_ShellMITC9.scd

ST11_ShellDKGQ.scd

ST11_ShellDKGT.scd



37

S T K O

3 LINEAR ELASTIC TESTS: NAFEMS BENCHMARKS

3.1 LE1: ELLIPTIC MEMBRANE - PLANE STRESS ELEMENTS

Elements tested

ASDShellQ4

Tri31 Element

Quad Element

SSPquad Element

ShellMITC4

ShellMITC9

ShellDKGQ

ShellDKGT


Figure 43: model description Figure 44: STKO FE Model

• Problem description. Plane stress problem of thickness 0.1m subjected to a

uniform pressure, functions defining the curves AB and CD are:

AB curve: (𝑥

3.25)

2+ (

𝑦

2.75)

2= 1

CD curve: (𝑥

2)

2+ 𝑦2 = 1

Thickness = 0.1 m

• Material. Linear elastic, E = 2.1E+11 N/m2, = 0.3, = 7.8E+03 kg/m3.



• Boundary conditions. uy=0 along edge DA, ux=0 along edge CB.

• Loading. Uniform outward pressure of 1.00E+07 N/m2 (10 MPa) is applied on

the outer face.







38

S T K O

Mesh type

Figure 45: coarse mesh Figure 46: fine mesh Figure 47: very fine mesh

Test Results

Reference solution: yy at D is 9.27E+07 N/m2 (92.7 MPa).


Table 19. Direct stress, yy at D (Pa)

Element



N/m2 % N/m2 % N/m2 %

ASDShellQ4 6.12E+07 -33.98 8.41E+07 -9.28 9.21E+07 -0.65

Tri31Element 4.81E+07 -48.11 8.18E+07 -11.76 8.95E+07 -3.45

QuadElement 7.41E+07 -20.06 8.94E+07 -3.56 9.34E+07 0.76

SSPquadEl. 4.51E+07 -51.35 6.88E+07 -25.78 8.43E+07 -9.06

ShellMITC4 7.41E+07 -20.06 8.94E+07 -3.56 9.34E+07 0.76

ShellMITC9 9.59E+07 3.45 9.29E+07 0.22 9.27E+07 0.00

ShellDKGQ 7.07E+07 -23.73 9.47E+07 2.16 8.49E+07 -8.41

ShellDKGT 5.96E+07 -35.71 8.90E+07 -3.99 9.75E+07 5.18



39

S T K O

Figure 48: Direct stress at the point P as a function of mesh type

References

• NAFEMS Ltd, The Standard NAFEMS BENCHMARKS TNSB Rev. 3, NAFEMS Ltd,

Scottish Enterprise Technology Park, Whitworth Building, East Kilbride, Glasgow,

United Kingdom,1990.

Input files

Le1_ASDShellQ4_coarse_mesh.scd

Le1_ASDShellQ4_fine_mesh.scd

Le1_ASDShellQ4_very_fine_mesh.scd

Le1_Tri31Element_coarse_mesh.scd

Le1_Tri31Element_fine_mesh.scd

Le1_Tri31Element_very_fine_mesh.scd

Le1_QuadElement_coarse_mesh.scd

Le1_QuadElement_fine_mesh.scd

Le1_QuadElement_very_fine_mesh.scd

Le1_SSPquad Element_coarse_mesh.scd

Le1_SSPquad Element_fine_mesh.scd

Le1_SSPquad Element_very_fine_mesh.scd

Le1_ShellMITC4_coarse_mesh.scd

Le1_ShellMITC4_fine_mesh.scd

Le1_ShellMITC4_very_fine_mesh.scd




4.00E+07

5.00E+07

6.00E+07

7.00E+07

8.00E+07

9.00E+07

1.00E+08

4 14 24

Dir

ect

stre

ss, σ

yyat

D (

Pa)


THEOR. RESULT

ASDShellQ4

TRI31 Element

quad Element

SSPQuad Element

shellMITC4

shellMITC9

shellDKGQ

shellDKGT

40

S T K O

Le1_ShellDKGQ_coarse_mesh.scd

Le1_ShellDKGQ_fine_mesh.scd

Le1_ShellDKGQ_very_fine_mesh.scd

Le1_ShellDKGT_coarse_mesh.scd

Le1_ShellDKGT_fine_mesh.scd

Le1_ShellDKGT_very_fine_mesh.scd

41

S T K O

3.2 LE3: HEMISPHERICAL SHELL WITH POINT LOADS

Elements tested

ASDShellQ4

ShellMITC4

ShellMITC9

ShellDKGQ

ShellDKGT



• Problem description. Hemispherical shell with Point Load.

Radius: r = 10 m,

Thickness: t = 0.04 m.

Function defining the hemispherical surface: 𝑥2 + 𝑦2+𝑧2 = 100

• Material. Linear elastic, E = 6.825E+10 N/m2, = 0.3.



• Boundary conditions. ux= uy=uz=0 at E. Along edge AE, uy= x= z= 0. Along

edge BE, ux= y= z= 0.

• Loading: FB,y = -2.0E+03 N, FA,x = 2.0E+03 N.





42

S T K O

Mesh type


Test Results

Reference solution: ux= 0.185 m at point A.

The values enclosed in parentheses are percentage differences with respect to the

reference solution.

Table 20. Displacement ux at A (m)

Element



N/m2 % N/m2 % N/m2 %

ASDShellQ4 1.84E-01 -0.80 1.84E-01 -0.30 1.85E-01 -0.02

ShellMITC4 2.94E-02 -84.09 1.22E-01 -34.25 1.76E-01 -5.05

ShellMITC9 2.59E-02 -85.99 1.36E-01 -26.47 1.81E-01 -2.19

ShellDKGQ 3.97E-02 -78.56 1.40E-01 -24.48 1.81E-01 -2.18

ShellDKGT 4.54E-02 -75.46 1.81E-01 -2.05 1.85E-01 -0.12





43

S T K O

Figure 54: Displacement ux at point A as a function of mesh type

Reference




Input files
















0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

4 24 44 64

x D

isp

lace

men

t, u

xat

A (

m)


THEOR. RESULT

shellMITC4

shellMITC9

shellDKGQ

shellDKGT

ASDShellQ4










44

S T K O

3.3 LE5: Z-SECTION CANTILEVER

Elements tested

ASDShellQ4

ShellMITC4

ShellMITC9

ShellDKGQ

ShellDKGT



• Problem description. Z-section cantilever under torsional loading. All units in

the figures are in meters.




• Boundary conditions. All displacements are zero along the edge at x= 0.

• Loading. The torque is applied by two uniformly point loads, S = 6.0E+05 N as

shown in Figure 55.





45

S T K O

Mesh type


Test Results

Reference solution: Axial stress = -1.08E+08 N/m2 (108 MPa) at midsurface, point A.

Table 21. Axial stress, xx at A (Pa)

Element



N/m2 % N/m2 % N/m2 %

ASDShellQ4 -1.21E+08 12.04 -1.15E+08 6.48 -1.13E+08 4.63

ShellMITC4 -1.08E+08 0.00 -1.14E+08 5.56 -1.13E+08 4.63

ShellMITC9 -1.32E+08 22.22 -1.20E+08 11.11 -1.16E+08 7.41

ShellDKGQ -1.27E+08 17.59 -1.18E+08 9.26 -1.14E+08 5.56

ShellDKGT -1.08E+08 0.00 -1.11E+08 2.78 -1.12E+08 3.70

Figure 60: Axial stress xx at point A as a function of mesh type

-1.30E+08

-1.25E+08

-1.20E+08

-1.15E+08

-1.10E+08

-1.05E+08

6 11 16 21

Dir

ect

stre

ss, σ

xxat

D (

Pa)


THEOR. RESULT

ASDShellQ4

sheLLMITC4

shellMITC9

shellDKGQ

shellDKGT





46

S T K O

References




Input files

Le5_ASDShellQ4_coarse.scd



Le5_ShellMITC4_coarse.scd



Le5_ShellMITC9_coarse.scd



Le5_ShellDKGQ_coarse.scd



Le5_ShellDKGT_coarse.scd












47

S T K O

3.4 LE6: SKEW PLATE UNDER NORMAL PRESSURE

Elements tested

ASDShellQ4

ShellMITC4

ShellMITC9

ShellDKGQ

ShellDKGT



• Problem description. Skew plate under normal pressure, plate thickness = 0.01

m, l =1 m.

• Material. Linear elastic, E = 2.1E+11 N/m2, Poisson's ratio = 0.3, density = 7800

kg/m3.

• Boundary conditions. uz = 0 along edges AD, DC, BC, and AB. ux = uy = 0 at

point A and uy = 0 at point D.

• Loading. Uniform pressure pz= –7.0E+02 N/m2 (out of plane).

• Mesh. A coarse (2 × 2) and a fine (4 × 4) mesh are tested for each element.

Mesh type

Figure 63: coarse mesh Figure 64: fine mesh

Test Results

Target solution: Maximum principal stress = 8.02E+05 N/m2 (0.802 MPa) on the lower

surface at point E.





48

S T K O

Table 22. Maximum principal stress, max at E (Pa)

Element

Coarse Mesh (2x2) Fine Mesh (4x4)

Value Diff. Value Diff.

N/m2 % N/m2 %

ASDShellQ4 3.49E+05 -56.48 6.92E+05 -13.72

ShellMITC4 3.49E+05 -56.48 6.92E+05 -13.72

ShellMITC9 2.99E+05 -62.72 6.26E+05 -21.95

ShellDKGQ 4.69E+05 -41.52 9.56E+05 19.20

ShellDKGT 1.18E+06 47.13 9.91E+05 23.57

Figure 65: Maximum principal stress max at point E as a function of mesh type

References




Input files







0.00E+00

2.00E+05

4.00E+05

6.00E+05

8.00E+05

1.00E+06

1.20E+06

1.40E+06

2 3 3 4 4Max

imu

m P

rin

cip

al s

tres

s, σ

max

at E

(Pa

)


THEOR. RESULT

shellMITC4

shellMITC9

shellDKGQ

shellDKGT

ASDShellQ4











49

S T K O





50

S T K O

3.5 LE10: THICK PLATE UNDER PRESSURE

Elements tested


SSPbrick Element

Bbar Brick Element


FourNodeTetrahedron



• Problem description. Thick plate under uniform pressure, thickness = 0.6 m

(Units: m, N), and the functions defining the curves CD and AB are:

CD curve: (𝑥

3.25)

2+ (

𝑦

2.75)

2= 1

AB curve: (𝑥

2)

2+ 𝑦2 = 1


• Mesh. A coarse and a fine mesh are tested. Mesh type are shown in the next

paragraph.

• Boundary conditions. uy=0 on face BCB’C’, ux= uy=0 on face CDC’D’, and

ux=0 on face ADA’D’. uz=0 along edge EE’.

• Loading. Uniform normal pressure of 1.0E+06 N/m2 on the upper surface of

the plate.





51

S T K O

Mesh type

Figure 68: coarse mesh Figure 69: fine mesh

Test Results

Reference solution: yy at B is 5.38E+06 N/m2 (5.38MPa).


Table 23. Direct stress, yy at D (MPa)

Coarse Mesh Fine Mesh


Element N/m2 % N/m2 %


5.07E+06 -5.76 5.61E+06 4.28

SSPBrick Element 2.06E+06 -61.71 4.23E+06 -21.38

Bbar Brick Element

3.68E+06 -31.60 4.87E+06 -9.48


5.42E+06 0.74 5.35E+06 -0.56

FourNodeTetrahedron

3.09E+06 -42.57 4.80E+06 -10.78

References




Input files

Le10_StandardBrickElement_coarse_mesh.scd

Le10_StandardBrickElement_fine_mesh.scd

Le10_SSPbrick_coarse_mesh.scd

Le10_SSPbrick_fine_mesh.scd

Le10_BbarBrickElement_coarse_mesh.scd

Le10_BbarBrickElement_fine_mesh.scd









52

S T K O

Le10_20nodeBrickElement_coarse_mesh.scd

Le10_20nodeBrickElement_fine_mesh.scd

Le10_4nodeTetrahedron_coarse_mesh.scd

Le10_4nodeTetrahedron_fine_mesh.scd

53

S T K O

4 FREE VIBRATION TESTS USING THEORETICAL SOLUTIONS

4.1 FVT1, TRUSS ELEMENT: TWO SPRINGS AND TWO LUMPED MASSES

Elements tested

Truss Element



• Problem description. Simple frictionless two DOF system is constructed with

two springs and two lumped masses. l1= l2= 0.254 m, Area=6.4516E-05 m2 , k=

EA/l

• Material. Linear elastic, E = 6.8948E+08 N/m2, m1= 35.403 kg, m2= 8.851 kg

• Boundary conditions. ux=uy=0 at node 3, uy=0 at nodes 1 and 2.

Test Results


Table 24. Angular velocity, ω (rad/sec)

1st mode 2nd mode


Theoretical, ω 10.83 46.18

Truss Element, ω 10.83 0% 46.18 0%

Theoretical Eingenvalue 1 0.531

Truss Element Eingenvalue 1 0% 0.531 0%

References

• Eigen 01: Midas verification examples. MIDAS Information Technology Co.

Input files

FVT1.scd

https://opensees.berkeley.edu/wiki/index.php/Truss_Element

54

S T K O

4.2 FVB1, BEAM ELEMENT: BEAMS ON SPRINGS

Elements tested

ElasticBeamColumn

2 nodes link



• Problem description. Beams on springs and a lumped mass. l1=2.134 m, l2=

0.914 m, k= 52538.05 N/m, Moment of inertia Iyy =4.162314256E-07 m4

• Material. Linear elastic, E = 2.06843E+11 N/m2, m2= 453.67 kg

• Boundary conditions. As shown in figure above. In addition ux = 0 at nodes 1

and 3.

Test Results



MIDAS GEN STKO

rad/sec rad/sec

Mode 1, ω 11.7833 11.7833

References


Input files

FVB1.scd


55

S T K O

4.3 FVB2, BEAM ELEMENT: ANALYSIS OF A SHAFT WITH THREE DISKS

Elements tested

ElasticBeamColumn



• Problem description. l1= l2= l3=254 mm, rotational mass moment of inertia = Im1

= Im2 = Im3 = 1.13 kg m2, torsional stiffness: IXX = 4.162 E-07

• Material. Linear elastic, E = 7.171E+010 N/m2 , = 0.3

• Boundary conditions. ux = uy = uz = x = y = z = 0 at node 4. ux = uy = uz = y

= z = 0 at nodes 1, 2, and 3.

Test Results



Eigen 02: STKO

ω

Value Value Diff.

rad/sec rad/sec %

1st mode 89.00 89.00 0.00

2nd mode 249.40 249.40 0.00

3rd mode 360.40 360.40 0.00

References


Input files

FVB2.scd


56

S T K O

4.4 FVB3, BEAM ELEMENT: PYRAMID

Elements tested

Elastic Beam Column Element


Figure 76: model description, adapted

from Midas Figure 77: STKO FE Model

Figure 18.178: elevation of the structure Figure 18.2: plan of the structure

• Problem description. l1= l2 = 1066.8 mm, H = 1847.85 mm

• Translational mass (Mx=My):

Floor 1 = 1.42 kN*sec2/cm, Floor 2 = 0.799 kN*sec2/cm

Floor 3 = 0.355 kN*sec2/cm, Floor 4 = 0.0888 kN*sec2/cm

• Rotational mass (Im):

Floor 1 = 690155 kN*sec2/cm2, Floor 2 = 218369 kN*sec2/cm2

Floor 3 = 43134.7 kN*sec2/cm2, Floor 4 = 2695.92 kN*sec2/cm2

• Section properties (horizontal beams):


57

S T K O

A = 87.0966 cm2

Torsional stiffness (Ixx) = 29635.67750272 cm4

Moment of inertia (Iyy) = 936.5207076 cm4

Moment of inertia (Izz) = 50.7802339232cm4

• Material. Linear elastic, E = 2.034E+011 N/m2, weight density 7.823E+03 kg/m3.

• Section properties (diagonal beams):

A = 75.4837 cm2

Torsional stiffness (Ixx)= 6076.97881376 cm4

Moment of inertia (Iyy)= 2043.696299696 cm4

Moment of inertia (Izz) = 46.6179196672 cm4

• Material. Linear elastic, E = 2.034E+011 N/m2, weight density 7.823E+03 kg/m3.

• Boundary condition:

Nodes 6 ~ 41 (at an increment of 5): Constrain all rigidDiaphram

Nodes 42 ~ 45 (Master nodes): Constrain Dx, Dy, Rz of all nodes at each level to

these node

Test Results


FVB3 Load case 1:

Table 16. Displacement ux (mm)

MIDAS Gen STKO

Value Value Diff.

mm mm %

ElasticBeamColumn Element 0.181 0.181 0.00

FVB3 Load case 2:

Table 16. Displacement ux (mm)

MIDAS Gen STKO

Value Value Diff.

mm mm %

ElasticBeamColumn Element 0.128 0.128 0.00

58

S T K O

Table 27. Natural periods, T (sec)

1st mode 2nd mode 3rd mode

value diff. value diff. value diff.

s % s % s %

Midas Gen 0.12098 0.12098 0.06309

ElasticBeamColumn Element 0.12107 0.07 0.12107 0.07 0.06318 0.14

4th mode 5th mode 6th mode


s % s % s %

Midas Gen 0.06309 0.06099 0.03976

ElasticBeamColumn Element 0.06318 0.14 0.06091 -0.13 0.03982 0.15


value diff. value value diff. value

s % s % s %

Midas Gen 0.03976 0.02963 0.02388

ElasticBeamColumn Element 0.03982 0.15 0.02957 -0.20 0.02398 0.42

References

• Eigen 09: Midas verification examples. MIDAS Information Technology Co..

Input files

FVB3_Load_case_1.scd

FVB3_Load_case_2.scd

FVB3_Load_Eigenvalue.scd

59

S T K O

4.5 FVB4, BEAM AND SHELL ELEMENT: CANTILEVER MODEL

Elements tested

ElasticBeamColumn

ASDShellQ4

ShellMITC4


Figure 79: model description, adapted from Midas Figure 80: STKO FE Models

• Problem description. Compare the natural frequencies of a cantilever modelled

with plate elements and beam elements separately. l= 0.1 m, h = 0.05 m, thickness

= 0.1 m.


• Boundary conditions. Fixed at node 1. uy = z = 0 at all nodes.

Test Results


Figure 81: 1° Mode, ElasticBeamColumn Figure 82: 1° Mode, ASDShellQ4



60

S T K O

Table 28. 1st Mode Angular Velocity, ω (rad/sec)

THEORETICAL STKO

Value Value Diff.

rad/sec rad/sec %

ElasticBeamColumn 81.80 81.84 -0.05

ASDShellQ4 81.80 81.53 0.33

ShellMITC4 81.80 81.79 0.01

References


Input files

FVB4_ElasticBeamColumn.scd

FVB4_ASDShellQ4.scd

FVB4_ShellMITC4.scd




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4.6 FVS1, CANTILEVER SHELL MODEL

Elements tested

ASDShellQ4

ShellMITC4

ShellMITC9

ShellDKGQ

ShellDKGT


Figure 83: model description, adapted from Midas Figure 84: STKO FE Models

• Problem description. Eigenvalue analysis of a square cantilever plate.

L = 0.608 m, thickness (t) = 0.0254 m


• Boundary conditions. Fixed along edge y=0. ux = uy = z = 0 at all nodes.

• Loading. Self-weight is converted into nodal masses.

Test Results

Mode shapes are shown in the following images.

Figure 85: Mode 1 - ASDShellQ4 Figure 86: Mode 2 - ASDShellQ4




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Figure 89: Mode 5 - ASDShellQ4

Period values obtained and percentage differences with respect to the reference

solution are shown in the following table.

Table 29. Mode Periods T (sec) – other software



s % s % s %

THEORETICAL SOLUTION 0.0179 0.00732 0.00292

SAP2000

0.0178 -0.50 0.0065 -11.48 0.0029 -2.40

MIDAS/GEN 0.0172 -3.69 0.0071 -3.01 0.0028 -2.74

4th mode 5th mode

value diff. value diff.

s % s %

THEORETICAL SOLUTION 0.00228 0.00201

SAP2000 0.0022 -2.19 0.0019 -6.97




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MIDAS/GEN 0.0023 0.88 0.0020 -1.99

Table 30. Mode Periods T (sec) – STKO



s % s % s %

THEORETICAL SOLUTION 0.0179 0.00732 0.00292

ASDShellQ4 1.7201E-02 -3.90 7.0962E-03 -3.06 2.8263E-03 -3.21

ShellMITC4 1.7182E-02 -4.01 7.0718E-03 -3.39 2.8037E-03 -3.98

ShellMITC9 1.7194E-02 -3.95 7.1025E-03 -2.97 2.8224E-03 -3.34

ShellDKGQ 1.7160E-02 -4.14 7.0011E-03 -4.36 2.7899E-03 -4.46

ShellDKGT

1.7158E-02 -4.15 6.9955E-03 -4.43 2.7916E-03 -4.40

4th mode 5th mode


s % s %

THEORETICAL SOLUTION 0.00228 0.00201

ASDShellQ4 2.2264E-03 -2.35 1.9637E-03 -2.30

ShellMITC4 2.2013E-03 -3.45 1.9449E-03 -3.24

ShellMITC9 2.2148E-03 -2.86 1.9619E-03 -2.39

ShellDKGQ 2.1836E-03 -4.23 1.9189E-03 -4.53

ShellDKGT

2.1824E-03 -4.28 1.9168E-03 -4.64

References


Input files

VS1_ASDShellQ4.scd

FVS1_ShellMITC4.scd

FVS1_ShellMITC9.scd

FVS1_ShellDKGQ.scd

FVS1_ShellDKGT.scd








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4.7 FVS2, SKEWED CANTILEVER PLATE

Elements tested

ASDShellQ4

ShellMITC4

ShellMITC9

ShellDKGQ



Figure 91: STKO FE Models

• Problem description. l = 1 m, thickness = 0.01 m. Four different geometries

are considered: =0°, =15°, =30°, and =45°.

• Material. Linear elastic, E = 2.1E+11 N/m2 , = 0.3, weight density 7.8E+03

kg/m3.

• Boundary conditions. Fixed along edge y=0.

Test Results

Figure 92: 1° Mode, ASDShellQ4, =45° Figure 93: 2° Mode, ASDShellQ4,, =45




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Table 31. Mode frequencies, f (Hz) – other software

THEORETICAL SAP2000 MIDAS GEN

value value diff. value diff.

Hz Hz % Hz %

Mode 1

= 0 ° 8.727 8.633 -1.08 8.632 -1.09

= 15° 8.999 8.906 -1.03 8.906 -1.03

= 30 ° 9.899 9.748 -1.53 9.748 -1.53

= 45° 11.150 11.159 0.08 11.159 0.08

Mode 2

= 0 ° 21.304 20.989 -1.48 20.989 -1.48

= 15° 22.171 21.449 -3.26 21.449 -3.26

= 30 ° 25.465 23.168 -9.02 23.168 -9.02

= 45° 27.000 27.579 2.14 27.579 2.14

Table 32. Mode frequencies, f (Hz) – STKO

THEOR. ASDShellQ4 ShellMITC4 ShellMITC9 ShellDKGQ

value value diff. value diff. value diff. value diff.

Hz Hz % Hz % Hz % Hz %

Mode 1

= 0 ° 8.727 8.740 0.15 8.774 0.54 8.780 0.61 8.751 0.28

= 15° 8.999 8.931 -0.76 8.967 -0.36 8.945 -0.60 9.031 0.36

= 30 ° 9.899 9.797 -1.03 9.844 -0.56 9.857 -0.42 9.795 -1.05

= 45° 11.150 11.296 1.31 11.368 1.96 11.391 2.16 11.230 0.72

Mode 2

= 0 ° 21.304 21.270 -0.16 21.535 1.08 21.553 1.17 21.459 0.73

= 15° 22.171 20.888 -5.79 21.153 -4.59 20.927 -5.61 21.935 -1.06

= 30 ° 25.465 23.373 -8.22 23.689 -6.97 23.676 -7.03 23.480 -7.80

= 45° 27.000 28.031 3.82 28.450 5.37 28.586 5.87 27.984 3.64

References


Input files

FVS2_ASDShellQ4_0.scd




FVS2_ShellMITC4_0.scd


















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FVS2_ShellDKGQ_0.scd





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5 FREE VIBRATION TESTS: NAFEMS BENCHMARKS

5.1 FV2: PIN-ENDED DOUBLE CROSS: IN-PLANE VIBRATION ELEMENTS

TESTED

Elements tested

ElasticBeamColumn



Elastic Forced Based



• Problem description. Beam element vibrations.

Square cross section = 0.125x0.125 m


• Boundary conditions. Beams are fixed at A, B, C, D, E, F, G, H.

Test Results


Figure 96: Mode 1 Figure 97: Mode 2 Figure 98: Mode 3 Figure 99: Mode 4





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Frequency values obtained and percentage differences with respect to the reference


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Table 33. Mode frequencies, f (Hz)

1st mode 2nd mode 3rd mode 4th mode

value diff. value diff. value diff. value diff.

Hz % Hz % Hz % Hz %

NAFEMS 11.336 17.709 17.709 17.709

ElasticBeamColumn

11.333 -0.03 17.662 -0.27 17.662 -0.27 17.690 -0.11


11.333 -0.03 17.662 -0.27 17.662 -0.27 17.690 -0.11


11.333 -0.03 17.662 -0.27 17.662 -0.27 17.690 -0.11

ElasticForceBeamColumn

11.333 -0.03 17.662 -0.27 17.662 -0.27 17.690 -0.11

5th mode 6th mode 7th mode 8th mode


Hz % Hz % Hz % Hz %

NAFEMS 17.709 17.709 17.709 17.709

ElasticBeamColumn

17.690 -0.11 17.690 -0.11 17.690 -0.11 17.690 -0.11


17.690 -0.11 17.690 -0.11 17.690 -0.11 17.690 -0.11


17.690 -0.11 17.690 -0.11 17.690 -0.11 17.690 -0.11


17.690 -0.11 17.690 -0.11 17.690 -0.11 17.690 -0.11



Hz % Hz % Hz % Hz %

NAFEMS 45.345 57.390 57.390 57.390

ElasticBeamColumn

45.016 -0.73 56.058 -2.32 56.058 -2.32 56.344 -1.82


45.016 -0.73 56.058 -2.32 56.058 -2.32 56.344 -1.82


45.016 -0.73 56.058 -2.32 56.058 -2.32 56.344 -1.82


45.016 -0.73 56.058 -2.32 56.058 -2.32 56.344 -1.82



Hz % Hz % Hz % Hz %

NAFEMS 57.390 57.390 57.390 57.390

ElasticBeamColumn

56.344 -1.82 56.344 -1.82 56.344 -1.82 56.344 -1.82


56.344 -1.82 56.344 -1.82 56.344 -1.82 56.344 -1.82


56.344 -1.82 56.344 -1.82 56.344 -1.82 56.344 -1.82


56.344 -1.82 56.344 -1.82 56.344 -1.82 56.344 -1.82

References




Input files

FV2_ElasticBeamColumn.scd

FV2_DispBeamColumnElement.scd

FV2_ForceBeamColumnElement.scd

FV2_ElasticForceBeamColumn.scd

















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5.2 FV4: CANTILEVER WITH OFF-CENTER POINT MASSES

Elements tested

ElasticBeamColumn





• Problem description. Beam element vibrations.

Circular cross section r(radius)= 0.25 m.


• Boundary conditions. Fixed at A.

Test Results


Figure 114: Mode 1 Figure 115: Mode 2 Figure 116: Mode 3

Figure 117: Mode 4 Figure 118: Mode 5 Figure 119: Mode 6



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1st mode 2nd mode 3rd mode 4th mode

value diff. value diff. value diff. value diff

.

Hz % Hz % Hz % Hz %

NAFEMS 1.723 1.727 7.413 9.972

ElasticBeamColumn

1.7147 -0.48 1.7189 -0.47 7.416 0.05 9.978 0.06


1.7147 -0.48 1.7189 -0.47 7.416 0.05 9.978 0.06


1.7147 -0.48 1.7189 -0.47 7.416 0.05 9.978 0.06

5th mode 6th mode


Hz % Hz %

NAFEMS 18.155 26.957

ElasticBeamColumn

17.774 -2.10 27.058 0.37


17.774 -2.10 27.058 0.37


17.774 -2.10 27.058 0.37

References




Input files

FV4_ElasticBeamColumn.scd

FV4_DispBeamColumnElement.scd

FV4_ForceBeamColumnElement.scd







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5.3 FV12: FREE THIN SQUARE PLATE

Elements tested

ASDShellQ4

ShellMITC4

ShellMITC9

ShellDKGQ

ShellDKGT



• Problem description. Thin square plate vibration.

Length l = 10 m



• Boundary conditions. ux=uy=z=0 at all nodes.

Test Results





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Figure 122: Mode 4 – ASDShellQ4 Figure 123: Mode 5– ASDShellQ4

Figure 124: Mode 6 – ASDShellQ4 Figure 125: Mode 7– ASDShellQ4

Figure 126: Mode 8– ASDShellQ4 Figure 127: Mode 9– ASDShellQ4




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1st 2nd 3rd

mode 4th mode 5th mode 6th mode

value diff. valu

e diff. value diff.

valu

e diff.

Hz % Hz % Hz % Hz %

NAFEMS RBM 1.622 2.36 2.922

ASDShellQ4 RBM 0.00 1.486 -8.38 2.011 -14.79 2.599 -11.04

ShellMITC4 RBM 0.00 1.705 5.15 2.579 9.27 3.320 13.61

ShellMITC9 RBM 0.00 1.674 3.22 2.479 5.02 3.114 6.56

ShellDKGQ RBM 0.00 1.651 1.78 2.458 4.15 3.115 6.62

ShellDKGT RBM 0.00 1.638 0.97 2.446 3.66 3.076 5.27


value diff. valu

e diff. value diff.

Hz % Hz % Hz %

NAFEMS 4.23 4.233 7.42

ASDShellQ4 3.424 -19.06 3.424 -19.11 5.628 -24.16

ShellMITC4 4.631 9.49 4.631 9.41 8.953 20.66

ShellMITC9 4.384 3.65 4.384 3.57 8.044 8.41

ShellDKGQ 4.427 4.65 4.427 4.58 8.493 14.46

ShellDKGT 4.366 3.22 4.366 3.15 7.536 1.56

A finer mesh (10x10) is tested for each element. Mesh type is shown in the following

figure.

Figure 128: STKO FE Model: 10x10 mesh


solution for finer mesh models are shown in the following table.

Table 36. Mode frequencies with finer mesh, f (Hz)







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1st 2nd 3rd

mode 4th mode 5th mode 6th mode

value diff. valu

e diff. value diff.

valu

e diff.

Hz % Hz % Hz % Hz %

NAFEMS RBM 1.622 2.36 2.922

ASDShellQ4 RBM 0.00 1.599 -1.40 2.290 -2.96 2.863 -2.03

ShellMITC4 RBM 0.00 1.624 0.10 2.369 0.37 2.940 0.63

ShellMITC9 RBM 0.00 1.629 0.43 2.388 1.17 2.977 1.88

ShellDKGQ RBM 0.00 1.627 0.29 2.378 0.76 2.958 1.23

ShellDKGT RBM 0.00 1.628 0.34 2.376 0.67 2.953 1.07


value diff. valu

e diff. value diff.

Hz % Hz % Hz %

NAFEMS 4.23 4.233 7.42

ASDShellQ4 4.051 -4.24 4.051 -4.31 7.134 -3.86

ShellMITC4 4.206 -0.56 4.206 -0.63 7.495 1.01

ShellMITC9 4.252 0.53 4.252 0.46 7.788 4.96

ShellDKGQ 4.235 0.11 4.235 0.04 7.632 2.86

ShellDKGT 4.250 0.47 4.250 0.40 7.599 2.42

Observations

Differences given by the ASDShellQ4 element are due to the use of a lumped mass

matrix neglecting rotational masses. Differences diminish with mesh refinement as

expected, and don’t have any practical impact.

References




Input files

FV12_ASDShellQ4.scd

FV12_ShellMITC4.scd

FV12_ShellMITC9.scd

FV12_ShellDKGQ.scd

FV12_ShellDKGT.scd

FV12_ASDShellQ4_fine_mesh.scd

FV12_ShellMITC4_fine_mesh.scd


FV12_ShellDKGQ_fine_mesh.scd
















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FV12_ShellDKGT_fine_mesh.scd


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5.4 FV15: FIXED THIN RHOMBIC PLATE

Elements tested

ASDShellQ4

ShellMITC4

ShellMITC9

ShellDKGQ

ShellDKGT



• Problem description. The behavior of distorted thin elements in normal

modes analysis is examined.

Length: l = 10 m, Thickness: t = 0.05 m, =45°.

• Material. Material. Linear elastic, E = 2 E+11 N/m2, = 0.3, = 8.0E+03

kg/m3.

• Boundary conditions. ux=uy=z=0 at all nodes.

Fixed along all edges, uz= y =x=0.

Test Results






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Table 37. Mode frequencies. f (Hz)



Hz % Hz % Hz %

NAFEMS 7.938 12.835 17.941

ASDShellQ4 8.001 0.79 13.305 3.66 18.684 4.14

ShellMITC4 8.143 2.58 13.893 8.24 20.041 11.70

ShellMITC9 8.171 2.93 13.693 6.69 19.843 10.60

ShellDKGQ 7.551 -4.87 12.307 -4.11 17.235 -3.93

ShellDKGT 7.844 -1.18 12.913 0.61 18.313 2.07



Hz % Hz % Hz %

NAFEMS 19.133 24.009 27.922

ASDShellQ4 19.436 1.58 25.207 4.99 29.729 6.47

ShellMITC4 20.170 5.42 27.959 16.45 32.055 14.80

ShellMITC9 19.870 3.85 27.543 14.72 30.341 8.66

ShellDKGQ 17.439 -8.86 23.170 -3.49 25.720 -7.88

ShellDKGT 18.628 -2.64 24.869 3.58 27.858 -0.23

A finer mesh (20x20) is tested for each element. Mesh type is shown in the following

figure.







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Figure 137: STKO FE Model: 20x20 mesh






Hz % Hz % Hz %

NAFEMS 7.938 12.835 17.941

ASDShellQ4 7.935 -0.03 12.980 1.13 18.120 1.00

ShellMITC4 7.987 0.62 13.190 2.76 18.599 3.67

ShellMITC9 7.983 0.56 13.078 1.89 18.439 2.78

ShellDKGQ 7.766 -2.16 12.615 -1.71 17.595 -1.93

ShellDKGT 7.884 -0.68 12.850 0.12 18.026 0.47



Hz % Hz % Hz %

NAFEMS 19.133 24.009 27.922

ASDShellQ4 19.093 -0.21 24.147 0.57 28.298 1.34

ShellMITC4 19.360 1.19 25.098 4.54 29.124 4.30

ShellMITC9 19.211 0.41 24.810 3.33 28.430 1.82

ShellDKGQ 18.323 -4.23 23.444 -2.35 26.817 -3.96

ShellDKGT 18.840 -1.53 24.133 0.52 27.702 -0.79

Observations




References










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Input files

FV15_ASDShellQ4.scd

FV15_ShellMITC4.scd

FV15_ShellMITC9.scd

FV15_ShellDKGQ.scd

FV15_ShellDKGT.scd

FV15_ASDShellQ4_fine_mesh.scd



FV15_ShellDKGQ_fine_mesh.scd

FV15_ShellDKGT_fine_mesh.scd











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5.5 FV16: CANTILEVERED THIN SQUARE PLATE

Elements tested

ASDShellQ4

ShellMITC4

ShellMITC9

ShellDKGQ

ShellDKGT


Test 1

Test 2

Test 3

Test 4


• Problem description. The behavior of a cantilevered thin square plate is

examined.

Length: l=10 m.



• Boundary conditions. ux=uy=uz=y=0 along edge x=0.

• Mesh. Mesh type are shown in the previous images. Node coordinates are

shown in the following table.

Table 39. Node Coordinates

NODE COORDINATES

1 2 3 4 5 6 7 8 9

x 4 2.25 4.75 7.25 7.5 7.75 5.25 2.25 2.5

y 4 2.25 2.5 2.75 4.75 7.25 7.25 7.25 4.75





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Test Results


Figure 140: TEST 2, Mode 1 - ASDShellQ4 Figure 141: TEST 2, Mode 2 - ASDShellQ4




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Hz % Hz % Hz %

NAFEMS 0.421 1.029 2.582

TEST 1

ASDShellQ4 0.412 -2.20 0.969 -5.80 2.445 -5.29

ShellMITC4 0.422 0.18 1.045 1.57 2.940 13.86

ShellMITC9 0.425 0.91 1.048 1.84 2.781 7.72

ShellDKGQ 0.415 -1.37 1.020 -0.90 2.695 4.39

ShellDKGT 0.414 -1.71 1.038 0.91 2.684 3.94

TEST 2

ASDShellQ4 0.411 -2.48 0.965 -6.21 2.428 -5.97

ShellMITC4 0.422 0.16 1.045 1.53 2.940 13.87

ShellMITC9 0.430 2.05 1.051 2.12 2.888 11.85

ShellDKGQ 0.415 -1.49 1.020 -0.90 2.704 4.72

ShellDKGT 0.414 -1.57 1.030 0.12 2.699 4.52

TEST 3

ASDShellQ4 0.391 -7.06 0.833 -19.04 2.036 -21.13

ShellMITC4 0.428 1.58 1.100 6.88 3.692 42.99

ShellMITC9 0.425 0.91 1.048 1.84 2.781 7.72

ShellDKGQ 0.406 -3.66 1.009 -1.98 2.759 6.85

ShellDKGT 0.399 -5.21 1.018 -1.07 2.789 8.03

TEST 4

ASDShellQ4 0.382 -9.24 0.822 -20.11 1.928 -25.32

ShellMITC4 0.423 0.42 1.094 6.34 3.436 33.06

ShellMITC9 0.459 8.98 1.161 12.86 3.878 50.18

ShellDKGQ 0.403 -4.25 0.997 -3.10 2.670 3.41

ShellDKGT 0.400 -4.88 0.998 -3.04 2.636 2.07



Hz % Hz % Hz %

NAFEMS 3.306 3.753 6.555

TEST 1

ASDShellQ4 2.906 -12.09 3.392 -9.63 5.211 -20.51

ShellMITC4 3.596 8.77 4.214 12.30 7.511 14.58

ShellMITC9 3.414 3.25 3.998 6.53 6.943 5.91

ShellDKGQ 3.439 4.03 3.896 3.82 6.994 6.70

ShellDKGT 3.439 4.04 3.947 5.16 6.833 4.23

TEST 2

ASDShellQ4 2.905 -12.12 3.366 -10.31 5.195 -20.74

ShellMITC4 3.629 9.75 4.210 12.18 7.537 14.98

ShellMITC9 3.449 4.34 4.053 7.99 7.096 8.25

ShellDKGQ 3.457 4.56 3.904 4.03 7.021 7.11

ShellDKGT 3.489 5.54 3.981 6.07 7.107 8.42

TEST 3

ASDShellQ4 3.096 -6.35 3.331 -11.25 3.363 -48.69

ShellMITC4 5.022 51.91 5.641 50.31 8.684 32.48

ShellMITC9 3.414 3.25 3.998 6.53 6.943 5.91

ShellDKGQ 3.456 4.54 3.932 4.76 7.077 7.96

ShellDKGT 3.312 0.19 4.017 7.03 6.030 -8.00

TEST 4

ASDShellQ4 2.886 -12.69 3.230 -13.93 3.673 -43.96

ShellMITC4 5.179 56.66 5.446 45.11 8.778 33.91

ShellMITC9 4.590 38.84 6.012 60.20 10.539 60.78

ShellDKGQ 3.526 6.65 3.940 4.99 7.124 8.68

ShellDKGT 3.483 5.37 3.645 -2.87 6.071 -7.38

























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A finer mesh is tested for each element. Mesh type is shown in the following figure.

Test 1

Test 2

Test 3

Test 4

Figure 146: STKO FE Model: iner mesh



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Hz % Hz % Hz %

NAFEMS 0.421 1.029 2.582

TEST 1

ASDShellQ4 0.418 -0.81 1.021 -0.82 2.557 -0.99

ShellMITC4 0.418 -0.66 1.026 -0.34 2.586 0.15

ShellMITC9 0.418 -0.62 1.026 -0.34 2.575 -0.28

ShellDKGQ 0.418 -0.77 1.024 -0.50 2.572 -0.37

ShellDKGT 0.418 -0.75 1.025 -0.42 2.570 -0.48

TEST 2

ASDShellQ4 0.416 -1.13 1.009 -1.95 2.535 -1.82

ShellMITC4 0.419 -0.46 1.030 0.08 2.659 2.96

ShellMITC9 0.420 -0.23 1.031 0.15 2.615 1.26

ShellDKGQ 0.417 -0.91 1.023 -0.58 2.603 0.83

ShellDKGT 0.417 -0.93 1.027 -0.22 2.597 0.59

TEST 3

ASDShellQ4 0.417 -1.07 1.010 -1.83 2.537 -1.73

ShellMITC4 0.419 -0.46 1.030 0.07 2.656 2.88

ShellMITC9 0.420 -0.28 1.031 0.15 2.615 1.27

ShellDKGQ 0.417 -0.88 1.023 -0.60 2.600 0.71

ShellDKGT 0.417 -0.91 1.027 -0.22 2.592 0.37

TEST 4

ASDShellQ4 0.416 -1.19 1.008 -2.03 2.519 -2.45

ShellMITC4 0.419 -0.49 1.030 0.06 2.651 2.68

ShellMITC9 0.420 -0.29 1.030 0.14 2.610 1.10

ShellDKGQ 0.417 -0.90 1.023 -0.61 2.598 0.64

ShellDKGT 0.417 -0.92 1.026 -0.27 2.591 0.36



Hz % Hz % Hz %

NAFEMS 3.306 3.753 6.555

TEST 1

ASDShellQ4 3.243 -1.90 3.706 -1.26 6.431 -1.89

ShellMITC4 3.295 -0.33 3.756 0.09 6.586 0.47

ShellMITC9 3.282 -0.73 3.741 -0.31 6.547 -0.12

ShellDKGQ 3.287 -0.58 3.739 -0.37 6.557 0.03

ShellDKGT 3.289 -0.51 3.744 -0.25 6.586 0.47

TEST 2

ASDShellQ4 3.146 -4.83 3.635 -3.13 6.162 -6.00

ShellMITC4 3.363 1.73 3.848 2.52 6.811 3.91

ShellMITC9 3.310 0.11 3.795 1.11 6.647 1.41

ShellDKGQ 3.326 0.61 3.776 0.61 6.677 1.87

ShellDKGT 3.339 0.98 3.798 1.21 6.801 3.75

TEST 3

ASDShellQ4 3.155 -4.56 3.643 -2.94 6.165 -5.94

ShellMITC4 3.357 1.54 3.847 2.50 6.778 3.40

ShellMITC9 3.308 0.07 3.792 1.04 6.632 1.17

ShellDKGQ 3.321 0.47 3.775 0.59 6.655 1.53

ShellDKGT 3.331 0.76 3.797 1.17 6.768 3.25

TEST 4

ASDShellQ4 3.148 -4.79 3.627 -3.35 6.160 -6.02

ShellMITC4 3.358 1.58 3.847 2.51 6.810 3.89

ShellMITC9 3.305 -0.02 3.791 1.02 6.634 1.20

ShellDKGQ 3.324 0.56 3.776 0.62 6.678 1.88

ShellDKGT 3.336 0.92 3.799 1.23 6.794 3.65

























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Observations




References




Input files

FV16_Test 1_ASDShellQ4.scd

FV16_Test 1_ShellMITC4.scd


FV16_Test 1_ShellDKGQ.scd

FV16_Test 1_ShellDKGT.scd
















FV16_Test 1_ASDShellQ4_fine_mesh.scd

FV16_Test 1_ShellMITC4_fine_mesh.scd























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FV16_Test 1_ShellDKGQ_fine_mesh.scd

FV16_Test 1_ShellDKGT_fine_mesh.scd


































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5.6 FV22: CLAMPED THICK RHOMBIC PLATE

Elements tested

ASDShellQ4

ShellMITC4

ShellMITC9

ShellDKGQ

ShellDKGT



• Problem description. The behavior of distorted thick elements in normal modes

analysis is examined. Thickness 1.0 m, l=10 m, =45°.


• Boundary conditions. ux=uy=z=0 at all nodes. uz= x = y=0 along all edges.

Test Results






89

S T K O


Figure 153: Mode 5 - ASDShellQ4






Hz % Hz % Hz %

NAFEMS 133.95 201.41 265.81

ASDShellQ4 135.93 1.48 209.16 3.85 277.15 4.27

ShellMITC4 138.16 3.14 217.62 8.05 295.36 11.12

ShellMITC9 134.84 0.66 205.44 2.00 273.03 2.72

ShellDKGQ 151.02 12.75 246.14 22.21 344.71 29.68

ShellDKGT 156.89 17.12 258.25 28.22 366.26 37.79

4th mode 5th mode


Hz % Hz %

NAFEMS 282.74 334.45

ASDShellQ4 289.23 2.29 349.80 4.59

ShellMITC4 299.22 5.83 384.09 14.84

ShellMITC9 286.90 1.47 345.56 3.32

ShellDKGQ 348.77 23.35 463.40 38.56

ShellDKGT 372.57 31.77 497.39 48.72







90

S T K O

Observations




References




Input files

FV22_ASDShellQ4.scd

FV22_ShellMITC4.scd

FV22_ShellMITC9.scd

FV22_ShellDKGQ.scd

FV22_ShellDKGT.scd






91

S T K O

5.7 FV32: CANTILEVERED TAPERED MEMBRANE

Elements tested

ASDShellQ4

Tri31 Element

Quad Element

SSPquad Element

ShellMITC4

ShellMITC9

ShellDKGQ

ShellDKGT



• Problem description. The behavior of tapered membrane problem with

irregular mesh. Thickness 0.05 m, h1=5 m, h2=1 m, and l=10 m.

• Material. Linear elastic, E = 2 E+11 N/m2, = 0.3, = 8.0E+03 kg/m3.

• Boundary conditions. ux=uy =0 along edge x=0. uz= at all nodes.

Test Results








92

S T K O








Hz % Hz % Hz %

NAFEMS 44.62 130.03 162.70

ASDShellQ4 44.23 -0.88 128.08 -1.50 162.12 -0.36

Tri31 Element 47.94 7.42 140.34 7.93 162.60 -0.06

Quad Element 45.23 1.36 132.18 1.65 162.24 -0.28

SSPquad Element 43.50 -2.52 126.38 -2.81 162.00 -0.43

ShellMITC4 45.72 2.45 138.08 6.19 163.21 0.31

ShellMITC9 44.63 0.02 130.11 0.06 162.71 0.00

ShellDKGQ 43.24 -3.11 130.66 0.48 160.69 -1.24

ShellDKGT 44.74 0.26 134.02 3.07 163.33 0.38



Hz % Hz % Hz %







93

S T K O

NAFEMS 246.05 379.90 391.44

ASDShellQ4 238.59 -3.03 359.93 -5.26 384.07 -1.88

Tri31 Element 264.54 7.51 385.68 1.52 398.29 1.75

Quad Element 248.18 0.87 375.50 -1.16 385.00 -1.65

SSPquad Element 235.34 -4.35 354.17 -6.77 382.94 -2.17

ShellMITC4 272.82 10.88 398.81 4.98 443.07 13.19

ShellMITC9 246.54 0.20 381.89 0.52 391.51 0.02

ShellDKGQ 259.05 5.28 392.95 3.43 421.24 7.61

ShellDKGT 265.40 7.87 398.75 4.96 433.16 10.66

References




Input files

FV32_ASDShellQ4.scd

FV32_Tri31 Element.scd

FV32_Quad Element.scd

FV32_SSPquad Element.scd

FV32_ShellMITC4.scd

FV32_ShellMITC9.scd

FV32_ShellDKGQ.scd

FV32_ShellDKGT.scd







94

S T K O

5.8 FV52: SIMPLY SUPPORTED “SOLID” SQUARE PLATE

Elements tested


Bbar Brick Element


SSPbrick Element



• Problem description. Thick square plate vibration. l = 10 m, thickness = 1m.

• Material. Linear elastic, E = 2 E+11 N/m2, = 0.3, = 8.0E+03 kg/m3.

• Boundary conditions. uz= at the midplane along four edges.

Test Results

Figure 164: Mode 4 – 20NodeBrick Figure 165: Mode 5 - 20NodeBrick





95

S T K O

Figure 166: Mode 6 - 20NodeBrick Figure 167: Mode 7 - 20NodeBrick

Figure 168: Mode 8 - 20NodeBrick Figure 169: Mode 9 - 20NodeBrick

Figure 170: Mode 10 - 20NodeBrick



96

S T K O


RBM 4th mode 5th mode 6th mode

value diff. value diff. value diff. valu

e diff.

Hz % Hz % Hz % Hz %

NAFEMS RBM 44.092 106.66 106.66

Standard Brick Element 0.000 0.00 69.42 57.45 206.06 93.19 206.06 93.19

Bbar Brick Element 0.000 0.00 51.37 16.50 132.72 24.43 132.72 24.43

Twenty Node Brick Element 0.000 0.00 44.22 0.30 107.40 0.69 107.40 0.69

SSPbrick Element 0.000 0.00 44.39 0.67 107.86 1.13 107.86 1.13


value diff. value diff. value diff. valu

e diff.

Hz % Hz % Hz % Hz %

NAFEMS 156.23 193.58 200.13 200.13

Standard Brick Element 206.26 32.02 222.12 14.74 222.12 10.99 228.55 14.20

Bbar Brick Element 195.63 25.22 196.70 1.61 209.61 4.74 209.61 4.74

Twenty Node Brick Element 163.12 4.41 193.60 0.01 203.95 1.91 204.22 2.04

SSPbrick Element 158.93 1.73 193.58 0.00 200.21 0.04 200.21 0.04

References




Input files

FV52_Standard Brick Element.scd

FV52_Bbar Brick Element.scd

FV52_Twenty Node Brick Element.scd

FV52_SSPbrick Element.scd









97

S T K O

6 NONLINEAR GEOMETRY TESTS

6.1 NLG1. COROTATIONAL TRUSS ELEMENT

Elements tested

Corotational Truss Element


Figure 171: model description, adapted

from Midas Figure 172: STKO FE Model

• Problem description. Cable structure subjected to vertical loads.

l1 = 12.802 m

l2 = 18.288 m

h1 = 0.402 m

h2 = 0.536 m

h3 = 0.603 m

h4 = 0.804 m

A = 0.00092903 m2

• Material. Linear elastic, E = 1.724E+11 N/m2.

• Boundary conditions. Fixed as indicated in the figure above.

• Loading. F = 60528 N as indicated in the figure above.


98

S T K O

Test Results

Displacement values obtained are shown in the following table.

Table 45. Displacement results, (m)

Midas Gen STKO Abaqus

Displacement Ux

Node 1 -0.0069 -0.0069 -0.0069

Node 2 0.0000 0.0000 0.0000

Node 3 -0.0072 -0.0072 -0.0072

Node 4 0.0000 0.0000 0.0000

Displacement Uy

Node 1 0.0022 0.0022 0.0022

Node 2 0.0030 0.0029 0.0029

Node 3 0.0083 0.0083 0.0083

Node 4 0.0112 0.0112 0.0112

Displacement Uz

Node 1 -0.0713 -0.0714 -0.0714

Node 2 -0.0732 -0.0732 -0.0732

Node 3 -0.1036 -0.1037 -0.1037

Node 4 -0.1073 -0.1072 -0.1072

References

• GNL-02: Midas verification examples. MIDAS Information Technology Co.

Input files

NLG1.scd

99

S T K O

6.2 NLG2: SNAP THROUGH

Elements tested

Force Beam Column



• Problem description. 2D geometrical nonlinear analysis of truss element

subjected to a vertical load at the node 2.

Dimension: L= 2.5 E+03 m, H= 2.5 E+01 m.

Section Property: A = 1 m2.

• Material. E = 5 E+07 N/m2, = 0.

• Boundary conditions. ux = uz = 0 on node 1, ux = 0 on node 2.

• Loading. Concentrated load F of 12 N is applied at node 2 in the –Z direction.

Test Results


Figure 175: Z deflection, forceBeamColumn

Table 46. Deflection z at node 2, δz (m)

100

S T K O

MIDAS Gen

STKO –

ForceBeamColumn

m m

Node 3 -5.45E+01 -5.46E+01

References

• GNL 05: Midas verification examples. MIDAS Information Technology Co.

• Crisfield, M.A., “Non-linear Finite Element Analysis of Solids and Structures”,

Volume 1: Advanced Topics, 1991.

Input files

NLG2_forceBeamColumn.scd

101

S T K O

6.3 NLG3: STATIC LARGE DISPLACEMENT ANALYSIS OF A TOWER

CABLE

Elements tested

Catenary Cable



• Problem description. A cable stretched between a ground anchor point and

tower attach point was analyzed for static displacements. The cable is modeled

using 12 truss elements of linear elastic material. Insulators are located at nodes

named 2, 4 and 6, and a cluster of 6 insulators are located at node named 8.

Nodes named 3, 5, 7 and 9 through 12 are intermediate nodes located along the

cable without insulators.

Dimension: L= 8.191 E+03 m, H= 7.321 E+03 m.

Section Property: A = 3.61 E-1 m2.

• Material. E = 1.9 E+08 N/m2, = 0.2, = 1.0667 E-01 kg/m.

• Boundary conditions. ux = uz = 0 on node 1 and 13.

• Loading. The initial tension in the cable is 7.52 E+04 N. Insulators weighing 5.1

E+03 N each are located at nodes named 2, 4 and 6. A cluster of 6 insulators

totaling 3.06 E+04 N is located at node named 8.

Test Results

Define the nonlinear response for node 8. Values obtained and percentage differences

with respect to the reference solution are shown in the following figure and table.

102

S T K O

Figure 178: Load Factor-Displacement Ux curve, Catenary Cable

Figure 179: Load Factor-Displacement Uz curve, Catenary Cable 4

Table 47. Deflection z at node 2, δz (m)

THEOR. MIDAS STKO

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 50 100 150 200 250

Load

Fac

tor

Displacement - Ux

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-270 -220 -170 -120 -70 -20

Load

Fac

tor

Displacement - Uz

103

S T K O

LOAD

PERC.

δx δz δx δz δx δz

m m m % m % m % m %

0.2 107.71 -121.59 106.17 -1.43 -118.3 -2.71 135.09 25.42 -150.31 23.62

0.4 159.13 -180.28 156.86 -1.43 -174.3 -3.32 171.04 7.49 -189.88 5.33

0.6 193.56 -219.87 190.89 -1.38 -211.8 -3.67 198.23 2.41 -219.72 -0.07

0.8 220.15 -250.61 217.22 -1.33 -240.76 -3.93 220.44 0.13 -244.06 -2.62

1 242.14 -276.16 239.02 -1.29 -264.72 -4.14 239.40 -1.13 -264.82 -4.11

References

• GNL 7: Midas verification examples. MIDAS Information Technology Co.

• Bathe, K-j., Ozdemir, H., Wilson, E. L. (1974) “Static and Dynamic Geometric

and Material Nonlinear Analysis”, UCSESM Report No. 74-4, University of

California at Berkeley, Berkeley, Ca.

Input files

NLG3_CatenaryCable.scd

104

S T K O

6.4 NLG4: CANTILIVER SUBJECTED TO BENDING MOMENT

Elements tested

ASDShellQ4



• Problem description. The cantilever length is L = 12.

• Mesh. Three different structured meshed, thati is a coarse one (12x1), an

intermediate one (16x1) and a fine one (24x1), were considered. Mesh type are

shown in the figures below.

• Boundary conditions. Fixed as indicated in the figure above.

• Loading. The deformed configuration is a circular arc with radius R = EI / M. The

analytical deflections are

𝑢𝑡𝑖𝑝 = (sin(𝑀 𝑀0⁄ )

𝑀 𝑀0⁄− 1) 𝐿, 𝑤𝑡𝑖𝑝 =

1 − cos(𝑀 𝑀0⁄ )

𝑀 𝑀0⁄𝐿

with M0=EI / L. The maximum end moment Mmax is taken to be 2πM0, at which

the beam will be bent into a circle.

Mesh type

Figure 182: coarse mesh

Figure 183: intermediate mesh

Figure 184: fine mesh


105

S T K O

Test Results

The relevant load–deflection curves are plotted in Figure 185, where the analytical

solution is re.ported for comparison.

Figure 185: Load–deflection curves

Table 48. Deformed shape

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.50 1.00 1.50

End

Mo

men

t (x

L/2

πEI

)

Tip Deflection (/L)

u_tip theoretical

w_tip theoretical

u_tip coarse mesh

w_tip coarse mesh

u_tip intermediatemesh

w_tip intermediate mesh

u_tip fine mesh

w_tip fine mesh

106

S T K O

References

• F. Caselli, P. Bisegna (2013) “Polar decomposition based corotational framework

for triangular shell elements with distributed loads”, International Journal For

Numerical Methods, Wiley Online Library.

Input files

NLG4_ASDShellQ4_coarse_mesh.scd

NLG4_ASDShellQ4_intermediate_mesh.scd

NLG4_ASDShellQ4_fine_mesh.scd

107

S T K O

7 NONLINEAR GEOMETRY TESTS: NAFEMS BENCHMARKS

7.1 3DNLG-1: ELASTIC LARGE DEFLECTION RESPONSE OF A

CANTILEVER UNDER AN END LOAD

Elements tested

Corotational ElasticBeamColumn

ASDShellQ4

ShellMITC4

ShellNLDKGQ



• Problem description. Elastic large deflection response of a Z-shaped cantilever

under an end load.

l = 60

h = 30

Element section 20x1.7.

• Material. Linear elastic, E = 2.05E+5, = 0.3.

• Boundary conditions. Fixed at node 1.

• Loading. F = 4000 as indicated in the figure above.




108

S T K O

Test Results

Figure 188: Displacement at applied load = 4000

Figure 189: Applied load-tip displacement curve Figure 190: Section Moment in A-Applied load

Table 49. Displacement beam results, Uz at node 2

STKO

72 El.

Abaqus

72 El. diff.

STKO

9 El.

Abaqus

9 El. diff.

B31 143.5

B32 143.4

Cor. ElasticBeamColumn

143.44 0.04% 144.33 -0.58%

Cor. DispBeamColumn 143.44 0.04% 144.33 -0.58%

Cor. ForceBeamColumn 143.44 0.04% 144.33 -0.58%

Table 50. Displacement shell results, Uz at node 2

-4000

-3000

-2000

-1000

0

0 50 100 150

RF

z

uz

-12000

-6000

0

6000

12000

-4000-3000-2000-10000

RM

y, A

RFz




109

S T K O

STKO

72 elements

Abaqus

72 elements Diff.

S4 143.5

MITC4- updateBasis Failed to converge

ASDShellQ4 143.2 -0.21%

ShellNLDKGQ

142.65 143.5 -0.59%

References

• National Agency for Finite Element Methods and Standards (U.K.): Test 3DNLG-

7 from NAFEMS Publication R0024. A Review of Benchmark Problems for

Geometric Non-linear Behaviour of 3D Beams and Shells (Summary).

• Abaqus Benchmark’s Guide, Dassault Systèmes.

Input files

3DNLG-01_ASDShellQ4.scd

3DNLG-01_shellDKGQNL.scd

3DNLG-2_ShellMITC4.scd

3DNLG-01_Corotational ElasticBeamColumn_9el.scd

3DNLG-01_Corotational ElasticBeamColumn_72el.scd

3DNLG-01_CorotationalDispBeamColumn_9el.scd

3DNLG-01_CorotationalDispBeamColumn_72el.scd

3DNLG-01_CorotationalForceBeamColumn_9el.scd

3DNLG-01_CorotationalForceBeamColumn_72el.scd




110

S T K O

7.2 3DNLG-7: ELASTIC LARGE DEFLECTION RESPONSE OF A HINGED

SPHERICAL SHELL UNDER PRESSURE LOADING

Elements tested

ASDShellQ4

ShellMITC4

ShellNLDKGQ



• Problem description. Large displacement elastic response of a spherical shell

under uniform pressure loading. l = 1570, h = 125.0012, element thickness=100.

• Material. Linear elastic, E = 69, = 0.3.

• Boundary conditions. Simply supported along all edges.

• Loading. Pressure = 0.1 as indicated in the figure above.

Test Results

Figure 193: Deform shape at displacement = 300

l lh



111

S T K O

Figure 194: Load factor-Displacement curve, shellNLDKGQ

Figure 195: Load factor-Displacement curve, shellMITC4

Figure 196: Load factor-Displacement curve, ASDShellQ4

0

0.025

0.05

0.075

0.1

0 100 200 300

Load

fac

tor

Displacement

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 100 200 300

Load

fac

tor

Displacement

0.00

0.02

0.04

0.06

0.08

0.10

0 50 100 150 200 250 300

Load

fac

tor

Displacement

112

S T K O

Table 51. Displacement results, limit points 1 and 2 correspond to the peak and local minimum

Software Element

LIMIT POINT 1 LIMIT POINT 2

Pressure Uz at center Pressure Uz at center

Abaqus S3/S3R 6.64E-02 78.15 3.30E-02 220.9

S4 6.58E-02 78.84 3.13E-02 223.5

S4R 6.58E-02 78.98 3.13E-02 223.5

S4R5 6.28E-02 79.62 2.96E-02 224.4

S8R 6.26E-02 79.14 2.85E-02 223.6

S8R5 6.26E-02 79.21 2.88E-02 223.7

S9R5 6.26E-02 79.08 2.88E-02 223.4

STRI3 6.40E-02 78.61 3.18E-02 221.9

STRI65 6.24E-02 79.26 2.89E-02 224

SC6R 6.75E-02 81 3.39E-02 217.1

SC8R 6.68E-02 81.9 3.21E-02 217.6

ShellNLDKGQ 6.39E-02 82.5 3.17E-02 224.6

ASDShellQ4 6.30E-02 81 2.86 E-02 228

ShellMITC4 Test not passed

References

• Abaqus Benchmark’s Guide, Dassault Systèmes.

• National Agency for Finite Element Methods and Standards (U.K.): Test 3DNLG-

7 from NAFEMS Publication R0024. A Review of Benchmark Problems for

Geometric Non-linear Behaviour of 3D Beams and Shells (Summary)

Input files

3DNLG-07_ASDShellQ4.scd

3DNLG-07_shellMITC4.scd

3DNLG-07_ShellNLDKGQ.scd



113

S T K O

8 NONLINEAR MATERIAL TESTS

8.1 NLM1: PLANE STRAIN PLASTICITY

Elements tested

SSP Quad




• Problem description. 2D material nonlinear analysis of a square plane strain

mesh under enforced bi-axial tension is analysed. A 4-node isoparametric

element is used to model a unit square plate. Two degrees of freedom are used,

one for the horizontal component (u) and the other for the vertical component

(w). Dimension: L= 1 mm.

• Material. Both perfect plasticity and isotropic hardening models are considered.

E = 2.5 E+05 N/mm2, = 0.25, Yield criteria: von Mises, σy = 5.0 N/mm2, after

yield for isotropic hardening: Et = 5 E+04 N/mm2

• Boundary conditions. ux = uz = 0 on node 1 and 4, uz = 0 on node 2.

• Loading. Eight load increments are considered shown in the following table

R = 2.5 E-05 mm

Table 52. Load increment

Increment Displacement

change u w Stress state

1 Δu = R R 0 First yield

2 Δu = R 2R 0 Plastic flow

3 Δw = R 2R R Elastic unloading

4 Δw = R 2R 2R Plastic reloading

5 Δu = -R R 2R Plastic flow

6 Δu = -R 0 2R Plastic flow

114

S T K O

7 Δw = -R 0 R Elastic unloading

8 Δw = R 0 0 Plastic flow

Test Results


Table 53. Stress for isotropic hardening material model, σxx and σyy (MPa)

Increment

THEOR. MIDAS STKO – SSPQUAD ELEMENT

σxx σzz σxx σzz σxx σzz

MPa MPa MPa % MPa % MPa % MPa %

1 7.5E+00 2.5E+00 7.5E+00 0.00 2.5E+00 0.00 7.5E+00 0.00 2.5E+00 0.00

2 1.2E+01 6.7E+00 1.2E+01 0.00 6.7E+00 0.00 1.2E+01 0.00 6.7E+00 0.00

3 1.4E+01 1.4E+01 1.4E+01 0.00 1.4E+01 0.00 1.4E+01 0.00 1.4E+01 0.00

4 1.7E+01 2.0E+01 1.7E+01 0.00 2.0E+01 0.00 1.7E+01 0.00 2.0E+01 0.00

5 1.0E+01 1.6E+01 1.0E+01 0.00 1.6E+01 0.00 1.0E+01 0.00 1.6E+01 0.00

6 5.3E+00 1.1E+01 5.3E+00 0.00 1.1E+01 0.00 5.3E+00 0.00 1.1E+01 0.00

7 2.8E+00 3.5E+00 2.8E+00 0.00 3.5E+00 0.00 2.8E+00 0.00 3.5E+00 0.00

8 1.9E-01 -3.0E+00 1.9E-01 0.05 -3.0E+00 0.00 1.9E-01 0.00 -3.0E+00 0.00

Table 54. Stress for perfect plasticity material model, σxx and σyy (MPa)

Increment

THEOR. MIDAS STKO – SSPQUAD ELEMENT

σxx σzz σxx σzz σxx σzz

MPa MPa MPa % MPa % MPa % MPa %

1 7.5E+00 2.5E+00 7.5E+00 0.00 2.5E+00 0.00 7.5E+00 0.00 2.5E+00 0.00

2 1.2E+01 6.4E+00 1.2E+01 0.00 6.4E+00 0.00 1.2E+01 -0.80 6.4E+00 0.77

3 1.5E+01 1.4E+01 1.5E+01 0.00 1.4E+01 0.00 1.5E+01 -0.67 1.4E+01 0.36

4 1.7E+01 2.0E+01 1.7E+01 0.00 2.0E+01 0.00 1.7E+01 -0.49 2.0E+01 -0.37

5 1.0E+01 1.7E+01 1.0E+01 0.00 1.7E+01 0.00 1.0E+01 -0.10 1.7E+01 -0.99

6 4.2E+00 1.3E+01 4.2E+00 0.00 1.3E+01 0.00 4.4E+00 2.98 1.2E+01 -2.00

7 1.7E+00 5.1E+00 1.7E+00 0.00 5.1E+00 0.00 1.9E+00 7.27 4.8E+00 -4.96

8 -7.6E-01 -2.4E+00 -7.6E-01 -0.01 -2.4E+00 0.00 -6.3E-01 16.67 -2.7E+00 10.38

References

• MNL2: Midas verification examples. MIDAS Information Technology Co.

• Becker, A.A. “Background to Material Non-Linear Benchamrks (Report R0049)”,

NAFEMS, Glasgow, UK.

Input files

NLM1_SSPquad_hardening.scd

NLM1_SSPquad_perfectplasticity.scd

115

S T K O

8.2 NLM2: 3D PLASTICITY

Elements tested

SSPbrick


Bbar Brick Element



Figure 199: model description, adapted from

Midas Figure 200: STKO FE Model

• Problem description. 3D cube model undergoing elastic-plastic deformation.

3D continuum elements are utilized to obtain nonlinear responses.

Dimension: L= 1 mm.

• Material. Both perfect plasticity and isotropic hardening models are considered.

E = 2.5 E+05 N/mm2, = 0.25, σy = 5 N/mm2, after yield for isotropic hardening:

Et = 5 E+04 N/mm2

• Boundary conditions. Boundary conditions are shown in Figure 199.

• Loading. Eight load increments are considered shown in the following table

R = 2.5 E-05 mm




116

S T K O

Table 55. Load increment

Increment Displacement

change δx δy δz

1 Δux = R R 0 0

2 Δux = R 2R 0 0

3 Δuy = R 2R R 0

4 Δuy = R 2R 2R 0

5 Δuz = R 2R 2R R

6 Δuz = R 2R 2R 2R

7 Δux = -R R 2R 2R

8 Δux = -R 0 2R 2R

9 Δuy = -R 0 R 2R

10 Δuy = -R 0 0 2R

11 Δuz = -R 0 0 R

12 Δuz = -R 0 0 0

Test Results


Table 56. Stress obtained at step 6 for perfect plasticity material model, σxx, σyy, σzz and σeff (MPa) – OTHER SOFTWARE

THEOR. MIDAS -Gen

Value Value Diff.

MPa MPa %

STRESS σxx 2.23E+01 2.23E+01 -0.03

STRESS σyy 2.47E+01 2.47E+01 0.05

STRESS σzz 2.80E+01 2.80E+01 -0.02

STRESS σeff 5.00E+00 5.00E+00 0.00

Table 57. Stress obtained at step 6 for perfect plasticity material model, σxx, σyy, σzz and σeff (MPa) – STKO

THEOR. Standard

Brick BbarBrick SSPBrick 20NodeBrick

Value Value Diff. Value Diff. Value Value Diff. Value

MPa MPa % MPa % MPa % MPa %

STRESS σxx 2.23E+01 2.23E+01 0.16 2.23E+01 0.16 2.23E+01 0.16 2.23E+01 0.16

STRESS σyy 2.47E+01 2.46E+01 -0.24 2.46E+01 -0.24 2.46E+01 -0.24 2.46E+01 -0.24

STRESS σzz 2.80E+01 2.80E+01 0.09 2.80E+01 0.09 2.80E+01 0.09 2.80E+01 0.09

STRESS σeff 5.00E+00 5.00E+00 0.00 5.00E+00 0.00 5.00E+00 0.00 5.00E+00 0.00

Table 58. Stress obtained at step 6 for hardening material model, σxx σyy (MPa) – OTHER SOFTWARE

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THEOR. MIDAS -Gen

Value Value Diff.

MPa MPa %

STRESS σxx 2.18E+01 2.19E+01 0.10

STRESS σyy 2.52E+01 2.52E+01 -0.05

STRESS σzz 2.79E+01 2.79E+01 -0.03

STRESS σeff 5.27E+00 5.24E+00 -0.54

Table 59. Stress obtained at step 6 for perfect plasticity material model, σxx σyy (MPa) – STKO

THEOR. Standard

Brick BbarBrick SSPBrick 20NodeBrick

Value Value Diff. Value Diff. Value Value Diff. Value

MPa MPa % MPa % MPa % MPa %

STRESS σxx 2.18E+01 2.20E+01 0.74 2.20E+01 0.74 2.20E+01 0.74 2.20E+01 0.74

STRESS σyy 2.52E+01 2.50E+01 -0.86 2.50E+01 -0.86 2.50E+01 -0.86 2.50E+01 -0.86

STRESS σzz 2.79E+01 2.80E+01 0.20 2.80E+01 0.20 2.80E+01 0.20 2.80E+01 0.20

STRESS σeff 5.27E+00 5.16E+00 -1.96 5.16E+00 -1.96 5.16E+00 -1.96 5.16E+00 -1.96

References

• 3D Plasticity: Midas verification examples. MIDAS Information Technology Co.

Input files

NLM2_3D_standardBrick_hardening.scd

NLM2_3D_standardBrick_perfectplasticity.scd

NLM2_3D_BBarBrick_hardening.scd.scd

NLM2_3D_BBarBrick_perfectplasticity.scd

NLM2_3D_SSPBrick_hardening.scd

NLM2_3D_SSPBrick_perfectplasticity.scd

NLM2_3D_20NodeBrickElement_hardening.scd

NLM2_3D_20NodeBrickElement_perfectplasticity.scd

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9 NONLINEAR ANALYSIS TESTS

9.1 TH1: DYNAMIC MODAL RESPONSE FOR 2-D RIGID FRAME

Elements tested

Elastic Beam Column




• Problem description. 2D Time history analysis of a structure under lateral

dynamic loads.

Analysis time t is equal to 0.2 s and time step is equal to 0.001 s.

Dimension: L= 9.144 m; H1= 3.048 m; H2= 4.572 m.

Section Property: C1st_F = 1.03 E-04 m; C2nd_F = 4.42 E-05 m; B = 4.16 E+08 m.

Rigid diaphram at each floor, master nodes: 7 and 8.

• Material. E = 2.07 E+11 N/m2.

• Boundary conditions. Constrain all DOFs on node 1 and 2 and uz = Rx = 0 on

node from 3 to 6. Constrain uy of all nodes at each floor to node 7 and 8.

• Masses. M1 = 2.38 E+04 kg; M2 = 1.16 E+04 kg.

• Loading. Impulse loads are applied in the Y direction. Impulse loads are shown

in the following figures.

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Figure 203: Impulse at the 1st floor Figure 204: Impulse at the 2nd floor

Test Results


Figure 205: lateral displacement ad node 3 (m)

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Figure 206: lateral displacement ad node 5 (m

Table 60. Modal natural frequency (Hz)



Hz Hz % Hz % Hz %

1st mode 11.80 11.80 0.0 11.80 0.0 11.83 0.2

2nd mode 32.90 32.90 0.0 32.90 0.0 32.90 0.0

Table 61. Maximum displacement in y direction (m)

Time at which the

maximum displacement

occurs (t)

Maximum displacement

δy_max (m)

SAP

2000

MIDAS

Gen STKO

SAP

2000

MIDAS

Gen STKO

NODE 3 0.1710 0.1710 0.1710 0.0171 0.0171 0.0169

NODE 5 0.1230 0.1230 0.1230 0.0171 0.0171 0.0170

References

• TH 4: Midas verification examples. MIDAS Information Technology Co.

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Input files

TH1_ElasticBeamColumn.scd

A REVOLUTIONARY TOOLKIT FOR OPENSEES ...- Midas Gen 2020, MIDAS Information Technology Co., Ltd. 1.2...

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