a revolutionary toolkit for opensees ...- midas gen 2020, midas information technology co., ltd. 1.2...
TRANSCRIPT
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S T K O
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
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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
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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
Benchmark description
Figure 4: model description, adapted from Midas Figure 5: STKO FE Model
• 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.
• Material. Linear elastic, E = 8.27 E+10 N/m2.
• 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
Values obtained and percentage differences with respect to the reference solution.
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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
• Static 05: Midas verification examples. MIDAS Information Technology Co.
Input files
ST2_ElasticBeamColumn.scd
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2.3 ST3: BEAM WITH ELASTIC SUPPORTS AND AN INTERNAL HINGE
Elements tested
Elastic Beam Column
Benchmark description
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
• Material. Linear elastic, E = 2.07 E+11 N/m2.
• 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.
Values obtained and percentage differences with respect to the reference solution.
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)
THEOR. SAP2000 MIDAS -Gen STKO
Value Value Diff. Value Diff. Value Diff.
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)
THEOR. SAP200 MIDAS -Gen STKO
Value Value Diff. Value Diff. Value Diff.
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
• Static 07: Midas verification examples. MIDAS Information Technology Co.
Input files
ST3_ElasticBeamColumn.scd
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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
Benchmark description
Figure 11: model description, adapted from Midas Figure 12: STKO FE Model
• 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
• Material. Linear elastic, E = 7.38 E+10 N/m2.
• 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
Values obtained and percentage differences with respect to the reference solution.
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
• Static 12: Midas verification examples. MIDAS Information Technology Co.
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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
Benchmark description
Figure 14: model description, adapted from Midas Figure 15: STKO FE Model
• 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
Values obtained and percentage differences with respect to the reference solution.
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
• Static 18: Midas verification examples. MIDAS Information Technology Co.
• 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
Benchmark description
Figure 17: model description, adapted from Midas Figure 18: STKO FE Model
• 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.
• Material. Linear elastic, E = 2.1 E+11 N/m2, = 0.3.
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• Mesh. Element mesh is shown in Figure 17.
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.
Values obtained and percentage differences with respect to the reference solution.
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
• Static 19: Midas verification examples. MIDAS Information Technology Co.
• 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
Benchmark description
Figure 20: model description, adapted from Midas Figure 21: STKO FE Model
• 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.
• Mesh. Element mesh is shown in Figure 20.
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.
Values obtained and percentage differences with respect to the reference solution.
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
• Static 23: Midas verification examples. MIDAS Information Technology Co.
• 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
Benchmark description
Figure 23: model description, adapted from Midas
Figure 24: STKO FE Model
• 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.
• Material. Linear elastic, E = 2.07 E+05 N/mm2, = 0.3.
• Mesh. Element mesh is shown in Figure 23.
• 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.
Values obtained and percentage differences with respect to the reference solution.
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Figure 25: Y displacement, ShellANDeS
Table 12. z Displacement, δz (m)
THEOR. MIDAS -Gen
STKO
shellANdes
STKO
shellDKGT
Value Value Diff. Value Diff. Value Diff.
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
• Static 32: Midas verification examples. MIDAS Information Technology Co.
Input files
ST8_ShellANdeS.scd
ST8_ShellDKGT.scd
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2.9 ST9: TWISTED SOLID CANTILEVER BEAM
Elements tested
SSPbrick
Standard Brick Element
Bbar Brick Element
Twenty Node Brick Element
FourNodeTetrahedron
Benchmark description
Figure 26: model description, adapted from Midas Figure 27: STKO FE Model
• 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.
• Material. Linear elastic, E = 2.9 E+08 N/m2, = 0.22.
• Mesh. Element mesh is shown in Figure 26.
• 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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Values obtained and percentage differences with respect to the reference solution.
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.
28
S T K O
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
• Static 37: Midas verification examples. MIDAS Information Technology Co.
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
Benchmark description
Figure 30: model description, adapted from Midas Figure 31: STKO FE Model
• 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.
• Material. Linear elastic, E = 2.9 E+08 N/m2, = 0.22.
• 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.
Values obtained and percentage differences with respect to the reference solution.
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
Coarse Mesh Fine Mesh Very Fine Mesh
Value Diff. Value Diff. Value Diff.
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
• Static 37: Midas verification examples. MIDAS Information Technology Co.
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
Mesh global seed (-)
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
Benchmark description
Figure 39: model description, adapted from
Midas
Figure 40: STKO FE Model
• Problem description. 3D Static analysis of a circular slab subjected to a
pressure load.
Dimension: R= 1000 mm.
Thickness: t= 5 mm.
• Material. Linear elastic, E = 2.1 E+5 N/mm2, = 0.3.
• Mesh. Element mesh is shown in Figure 39.
• 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.
Values obtained and percentage differences with respect to the reference solution.
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
Benchmark description
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.
• 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. 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).
Values obtained and percentage differences with respect to the reference solution.
Table 19. Direct stress, yy at D (Pa)
Element
Coarse Mesh Fine Mesh Very Fine Mesh
Value Diff. Value Diff. Value Diff.
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
Le1_ShellMITC9_coarse_mesh.scd
Le1_ShellMITC9_fine_mesh.scd
Le1_ShellMITC9_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)
Mesh global seed (-)
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
Benchmark description
Figure 49: model description Figure 50: STKO FE Model
• 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.
• 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= 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
Figure 51: coarse mesh Figure 52: fine mesh Figure 53: very fine mesh
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
Coarse Mesh Fine Mesh Very Fine Mesh
Value Diff. Value Diff. Value Diff.
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
• 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
Le3_ASDShellQ4_coarse_mesh.scd
Le3_ASDShellQ4_fine_mesh.scd
Le3_ASDShellQ4_very_fine_mesh.scd
Le3_ShellMITC4_coarse_mesh.scd
Le3_ShellMITC4_fine_mesh.scd
Le3_ShellMITC4_very_fine_mesh.scd
Le3_ShellMITC9_coarse_mesh.scd
Le3_ShellMITC9_fine_mesh.scd
Le3_ShellMITC9_very_fine_mesh.scd
Le3_ShellDKGQ_coarse_mesh.scd
Le3_ShellDKGQ_fine_mesh.scd
Le3_ShellDKGQ_very_fine_mesh.scd
Le3_ShellDKGT_coarse_mesh.scd
Le3_ShellDKGT_fine_mesh.scd
Le3_ShellDKGT_very_fine_mesh.scd
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)
Mesh global seed (-)
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
Benchmark description
Figure 55: model description Figure 56: STKO FE Model
• Problem description. Z-section cantilever under torsional loading. All units in
the figures are in meters.
• Material. Linear elastic, E = 2.1E+11 N/m2, = 0.3, = 7.8E+03 kg/m3.
• 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. 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
Figure 57: coarse mesh Figure 58: fine mesh Figure 59: very fine mesh
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
Coarse Mesh Fine Mesh Very Fine Mesh
Value Diff. Value Diff. Value Diff.
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)
Mesh global seed (-)
THEOR. RESULT
ASDShellQ4
sheLLMITC4
shellMITC9
shellDKGQ
shellDKGT
46
S T K O
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
Le5_ASDShellQ4_coarse.scd
Le5_ASDShellQ4_fine_mesh.scd
Le5_ASDShellQ4_very_fine_mesh.scd
Le5_ShellMITC4_coarse.scd
Le5_ShellMITC4_fine_mesh.scd
Le5_ShellMITC4_very_fine_mesh.scd
Le5_ShellMITC9_coarse.scd
Le5_ShellMITC9_fine_mesh.scd
Le5_ShellMITC9_very_fine_mesh.scd
Le5_ShellDKGQ_coarse.scd
Le5_ShellDKGQ_fine_mesh.scd
Le5_ShellDKGQ_very_fine_mesh.scd
Le5_ShellDKGT_coarse.scd
Le5_ShellDKGT_fine_mesh.scd
Le5_ShellDKGT_very_fine_mesh.scd
47
S T K O
3.4 LE6: SKEW PLATE UNDER NORMAL PRESSURE
Elements tested
ASDShellQ4
ShellMITC4
ShellMITC9
ShellDKGQ
ShellDKGT
Benchmark description
Figure 61: model description Figure 62: STKO FE Model
• 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
• 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
Le6_ASDShellQ4_coarse_mesh.scd
Le6_ASDShellQ4_fine_mesh.scd
Le6_ShellMITC4_coarse_mesh.scd
Le6_ShellMITC4_fine_mesh.scd
Le6_ShellMITC9_coarse_mesh.scd
Le6_ShellMITC9_fine_mesh.scd
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
)
Mesh global seed (-)
THEOR. RESULT
shellMITC4
shellMITC9
shellDKGQ
shellDKGT
ASDShellQ4
49
S T K O
Le6_ShellDKGQ_coarse_mesh.scd
Le6_ShellDKGQ_fine_mesh.scd
Le6_ShellDKGT_coarse_mesh.scd
Le6_ShellDKGT_fine_mesh.scd
50
S T K O
3.5 LE10: THICK PLATE UNDER PRESSURE
Elements tested
Standard Brick Element
SSPbrick Element
Bbar Brick Element
Twenty Node Brick Element
FourNodeTetrahedron
Benchmark description
Figure 66: model description Figure 67: STKO FE Model
• 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
• Material. Linear elastic, E = 2.1E+11 N/m2, = 0.3, = 7.8E+03 kg/m3.
• 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).
Values obtained and percentage differences with respect to the reference solution.
Table 23. Direct stress, yy at D (MPa)
Coarse Mesh Fine Mesh
Value Diff. Value Diff.
Element N/m2 % N/m2 %
Standard Brick Element
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
Twenty Node Brick Element
5.42E+06 0.74 5.35E+06 -0.56
FourNodeTetrahedron
3.09E+06 -42.57 4.80E+06 -10.78
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
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
Benchmark description
Figure 70: model description, adapted from Midas Figure 71: STKO FE Model
• 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
Values obtained and percentage differences with respect to the reference solution.
Table 24. Angular velocity, ω (rad/sec)
1st mode 2nd mode
Value Diff. Value Diff.
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
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4.2 FVB1, BEAM ELEMENT: BEAMS ON SPRINGS
Elements tested
ElasticBeamColumn
2 nodes link
Benchmark description
Figure 72: model description, adapted from Midas Figure 73: STKO FE Model
• 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
Values obtained and percentage differences with respect to the reference solution.
Table 25. Angular velocity, ω (rad/sec)
MIDAS GEN STKO
rad/sec rad/sec
Mode 1, ω 11.7833 11.7833
References
• Eigen 02: Midas verification examples. MIDAS Information Technology Co.
Input files
FVB1.scd
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4.3 FVB2, BEAM ELEMENT: ANALYSIS OF A SHAFT WITH THREE DISKS
Elements tested
ElasticBeamColumn
Benchmark description
Figure 74: model description, adapted from Midas Figure 75: STKO FE Model
• 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
Values obtained and percentage differences with respect to the reference solution.
Table 26. Angular velocity, ω (rad/sec)
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
• Eigen 03: Midas verification examples. MIDAS Information Technology Co.
Input files
FVB2.scd
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4.4 FVB3, BEAM ELEMENT: PYRAMID
Elements tested
Elastic Beam Column Element
Benchmark description
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):
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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
Values obtained and percentage differences with respect to the reference solution.
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
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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
value diff. value diff. value diff.
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
7th mode 8th mode 9th mode
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
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4.5 FVB4, BEAM AND SHELL ELEMENT: CANTILEVER MODEL
Elements tested
ElasticBeamColumn
ASDShellQ4
ShellMITC4
Benchmark description
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.
• Material. Linear elastic, E = 2.0E+11 N/m2, = 0.3, = 7.8E+03 kg/m3.
• Boundary conditions. Fixed at node 1. uy = z = 0 at all nodes.
Test Results
Values obtained and percentage differences with respect to the reference solution.
Figure 81: 1° Mode, ElasticBeamColumn Figure 82: 1° Mode, ASDShellQ4
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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
• Eigen 05: Midas verification examples. MIDAS Information Technology Co.
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
Benchmark description
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
• Material. Linear elastic, E = 2.03E+11 N/m2, = 0.3, = 7.7E+03 kg/m3.
• 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 87: Mode 3 - ASDShellQ4 Figure 88: Mode 4 - ASDShellQ4
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
1st mode 2nd mode 3rd mode
value diff. value diff. value diff.
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
1st mode 2nd mode 3rd mode
value diff. value diff. value diff.
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
value diff. value diff.
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
• Eigen 06: Midas verification examples. MIDAS Information Technology Co.
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
Benchmark description
Figure 90: model description, adapted from Midas
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
• Eigen 10: Midas verification examples. MIDAS Information Technology Co.
Input files
FVS2_ASDShellQ4_0.scd
FVS2_ASDShellQ4_15.scd
FVS2_ASDShellQ4_30.scd
FVS2_ASDShellQ4_45.scd
FVS2_ShellMITC4_0.scd
FVS2_ShellMITC4_15.scd
FVS2_ShellMITC4_30.scd
FVS2_ShellMITC4_45.scd
FVS2_ShellMITC9_0.scd
FVS2_ShellMITC9_15.scd
FVS2_ShellMITC9_30.scd
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FVS2_ShellMITC9_45.scd
FVS2_ShellDKGQ_0.scd
FVS2_ShellDKGQ_15.scd
FVS2_ShellDKGQ_30.scd
FVS2_ShellDKGQ_45.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
DispBeamColumnElement
ForceBeamColumnElement
Elastic Forced Based
Benchmark description
Figure 94: model description Figure 95: STKO FE Model
• Problem description. Beam element vibrations.
Square cross section = 0.125x0.125 m
• Material. Linear elastic, E = 2.1E+11 N/m2, = 0.3, = 8.0E+03 kg/m3.
• Boundary conditions. Beams are fixed at A, B, C, D, E, F, G, H.
Test Results
Mode shapes are shown in the following images.
Figure 96: Mode 1 Figure 97: Mode 2 Figure 98: Mode 3 Figure 99: Mode 4
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Figure 100: Mode 5 Figure 101: Mode 6 Figure 102: Mode 7 Figure 103: Mode 8
Figure 104: Mode 9 Figure 105: Mode 10 Figure 106: Mode 11 Figure 107: Mode 12
Figure 108: Mode 13 Figure 109: Mode 14 Figure 110: Mode 15 Figure 111: Mode 16
Frequency values obtained and percentage differences with respect to the reference
solution are shown in the following table.
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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
DispBeamColumnElement
11.333 -0.03 17.662 -0.27 17.662 -0.27 17.690 -0.11
ForceBeamColumnElement
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
value diff. value diff. value diff. value diff.
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
DispBeamColumnElement
17.690 -0.11 17.690 -0.11 17.690 -0.11 17.690 -0.11
ForceBeamColumnElement
17.690 -0.11 17.690 -0.11 17.690 -0.11 17.690 -0.11
ElasticForceBeamColumn
17.690 -0.11 17.690 -0.11 17.690 -0.11 17.690 -0.11
9th mode 10th mode 11th mode 12th mode
value diff. value diff. value diff. value diff.
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
DispBeamColumnElement
45.016 -0.73 56.058 -2.32 56.058 -2.32 56.344 -1.82
ForceBeamColumnElement
45.016 -0.73 56.058 -2.32 56.058 -2.32 56.344 -1.82
ElasticForceBeamColumn
45.016 -0.73 56.058 -2.32 56.058 -2.32 56.344 -1.82
13th mode 14th mode 15th mode 16th mode
value diff. value diff. value diff. value diff.
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
DispBeamColumnElement
56.344 -1.82 56.344 -1.82 56.344 -1.82 56.344 -1.82
ForceBeamColumnElement
56.344 -1.82 56.344 -1.82 56.344 -1.82 56.344 -1.82
ElasticForceBeamColumn
56.344 -1.82 56.344 -1.82 56.344 -1.82 56.344 -1.82
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
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
DispBeamColumnElement
ForceBeamColumnElement
Benchmark description
Figure 112: model description, adapted from Midas Figure 113: STKO FE Model
• Problem description. Beam element vibrations.
Circular cross section r(radius)= 0.25 m.
• Material. Linear elastic, E = 2.1E+11 N/m2, = 0.3, = 8.0E+03 kg/m3.
• Boundary conditions. Fixed at A.
Test Results
Mode shapes are shown in the following images.
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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Frequency values obtained and percentage differences with respect to the reference
solution are shown in the following table.
Table 34. 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 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
DispBeamColumnElement
1.7147 -0.48 1.7189 -0.47 7.416 0.05 9.978 0.06
ForceBeamColumnElement
1.7147 -0.48 1.7189 -0.47 7.416 0.05 9.978 0.06
5th mode 6th mode
value diff. value diff.
Hz % Hz %
NAFEMS 18.155 26.957
ElasticBeamColumn
17.774 -2.10 27.058 0.37
DispBeamColumnElement
17.774 -2.10 27.058 0.37
ForceBeamColumnElement
17.774 -2.10 27.058 0.37
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
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
Benchmark description
Figure 120: model description Figure 121: STKO FE Model
• Problem description. Thin square plate vibration.
Length l = 10 m
Thickness: t = 0.05 m.
• Material. Linear elastic, E = 2.1E+11 N/m2, = 0.3, = 8.0E+03 kg/m3.
• Boundary conditions. ux=uy=z=0 at all nodes.
Test Results
Mode shapes are shown in the following images.
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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
Frequency values obtained and percentage differences with respect to the reference
solution are shown in the following table.
Table 35. Mode frequencies, f (Hz)
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S T K O
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
7th mode 8th mode 9th mode
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
Frequency values obtained and percentage differences with respect to the reference
solution for finer mesh models are shown in the following table.
Table 36. Mode frequencies with finer mesh, f (Hz)
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S T K O
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
7th mode 8th mode 9th mode
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
• 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
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_ShellMITC9_fine_mesh.scd
FV12_ShellDKGQ_fine_mesh.scd
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5.4 FV15: FIXED THIN RHOMBIC PLATE
Elements tested
ASDShellQ4
ShellMITC4
ShellMITC9
ShellDKGQ
ShellDKGT
Benchmark description
Figure 129: model description Figure 130: STKO FE Model
• 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
Mode shapes are shown in the following images.
Figure 131: Mode 1 - ASDShellQ4 Figure 132: Mode 2 - ASDShellQ4
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Figure 133: Mode 3 - ASDShellQ4 Figure 134: Mode 4 - ASDShellQ4
Figure 135: Mode 5 - ASDShellQ4 Figure 136: Mode 6 - ASDShellQ4
Frequency values obtained and percentage differences with respect to the reference
solution are shown in the following table.
Table 37. Mode frequencies. f (Hz)
1st mode 2nd mode 3rd mode
value diff. value diff. value diff.
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
4th mode 5th mode 6th mode
value diff. value diff. value diff.
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
Frequency values obtained and percentage differences with respect to the reference
solution for finer mesh models are shown in the following table.
Table 38. Mode frequencies with finer mesh, f (Hz)
1st mode 2nd mode 3rd mode
value diff. value diff. value diff.
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
4th mode 5th mode 6th mode
value diff. value diff. value diff.
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
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
• NAFEMS Ltd, The Standard NAFEMS BENCHMARKS TNSB Rev. 3, NAFEMS Ltd,
Scottish Enterprise Technology Park, Whitworth Building, East Kilbride, Glasgow,
United Kingdom,1990.
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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_ShellMITC4_fine_mesh.scd
FV15_ShellMITC9_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
Benchmark description
Test 1
Test 2
Test 3
Test 4
Figure 138: model description Figure 139: STKO FE Model
• Problem description. The behavior of a cantilevered thin square plate is
examined.
Length: l=10 m.
Thickness: t = 0.05 m.
• Material. Linear elastic, E = 2.1E+11 N/m2, = 0.3, = 8.0E+03 kg/m3.
• 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
Mode shapes are shown in the following images.
Figure 140: TEST 2, Mode 1 - ASDShellQ4 Figure 141: TEST 2, Mode 2 - ASDShellQ4
Figure 142: TEST 2, Mode 3 - ASDShellQ4 Figure 143: TEST 2, Mode 4 - ASDShellQ4
Figure 144: TEST 2, Mode 5 - ASDShellQ4 Figure 145: TEST 2, Mode 6 - ASDShellQ4
Table 40. Mode frequencies, f (Hz)
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1st mode 2nd mode 3rd mode
value diff. value diff. value diff.
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
4th mode 5th mode 6th mode
value diff. value diff. value diff.
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
Frequency values obtained and percentage differences with respect to the reference
solution for finer mesh models are shown in the following table.
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Table 41. Mode frequencies with finer mesh, f (Hz)
1st mode 2nd mode 3rd mode
value diff. value diff. value diff.
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
4th mode 5th mode 6th mode
value diff. value diff. value diff.
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
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
• 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
FV16_Test 1_ASDShellQ4.scd
FV16_Test 1_ShellMITC4.scd
FV16_Test 1_ShellMITC9.scd
FV16_Test 1_ShellDKGQ.scd
FV16_Test 1_ShellDKGT.scd
FV16_Test 2_ASDShellQ4.scd
FV16_Test 2_ShellMITC4.scd
FV16_Test 2_ShellMITC9.scd
FV16_Test 2_ShellDKGQ.scd
FV16_Test 2_ShellDKGT.scd
FV16_Test 3_ASDShellQ4.scd
FV16_Test 3_ShellMITC4.scd
FV16_Test 3_ShellMITC9.scd
FV16_Test 3_ShellDKGQ.scd
FV16_Test 3_ShellDKGT.scd
FV16_Test 4_ASDShellQ4.scd
FV16_Test 4_ShellMITC4.scd
FV16_Test 4_ShellMITC9.scd
FV16_Test 4_ShellDKGQ.scd
FV16_Test 4_ShellDKGT.scd
FV16_Test 1_ASDShellQ4_fine_mesh.scd
FV16_Test 1_ShellMITC4_fine_mesh.scd
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FV16_Test 1_ShellMITC9_fine_mesh.scd
FV16_Test 1_ShellDKGQ_fine_mesh.scd
FV16_Test 1_ShellDKGT_fine_mesh.scd
FV16_Test 2_ASDShellQ4_fine_mesh.scd
FV16_Test 2_ShellMITC4_fine_mesh.scd
FV16_Test 2_ShellMITC9_fine_mesh.scd
FV16_Test 2_ShellDKGQ_fine_mesh.scd
FV16_Test 2_ShellDKGT_fine_mesh.scd
FV16_Test 3_ASDShellQ4_fine_mesh.scd
FV16_Test 3_ShellMITC4_fine_mesh.scd
FV16_Test 3_ShellMITC9_fine_mesh.scd
FV16_Test 3_ShellDKGQ_fine_mesh.scd
FV16_Test 3_ShellDKGT_fine_mesh.scd
FV16_Test 4_ASDShellQ4_fine_mesh.scd
FV16_Test 4_ShellMITC4_fine_mesh.scd
FV16_Test 4_ShellMITC9_fine_mesh.scd
FV16_Test 4_ShellDKGQ_fine_mesh.scd
FV16_Test 4_ShellDKGT_fine_mesh.scd
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5.6 FV22: CLAMPED THICK RHOMBIC PLATE
Elements tested
ASDShellQ4
ShellMITC4
ShellMITC9
ShellDKGQ
ShellDKGT
Benchmark description
Figure 147: model description Figure 148: STKO FE Model
• Problem description. The behavior of distorted thick elements in normal modes
analysis is examined. Thickness 1.0 m, l=10 m, =45°.
• Material. Linear elastic, E = 2.1E+11 N/m2, = 0.3, = 8.0E+03 kg/m3.
• Boundary conditions. ux=uy=z=0 at all nodes. uz= x = y=0 along all edges.
Test Results
Mode shapes are shown in the following images.
Figure 149: Mode 1 - ASDShellQ4 Figure 150: Mode 2 - ASDShellQ4
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Figure 151: Mode 3 - ASDShellQ4 Figure 152: Mode 4 - ASDShellQ4
Figure 153: Mode 5 - ASDShellQ4
Frequency values obtained and percentage differences with respect to the reference
solution are shown in the following table.
Table 42. Mode frequencies, f (Hz)
1st mode 2nd mode 3rd mode
value diff. value diff. value diff.
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
value diff. value diff.
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
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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
• 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
FV22_ASDShellQ4.scd
FV22_ShellMITC4.scd
FV22_ShellMITC9.scd
FV22_ShellDKGQ.scd
FV22_ShellDKGT.scd
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5.7 FV32: CANTILEVERED TAPERED MEMBRANE
Elements tested
ASDShellQ4
Tri31 Element
Quad Element
SSPquad Element
ShellMITC4
ShellMITC9
ShellDKGQ
ShellDKGT
Benchmark description
Figure 154: model description Figure 155: STKO FE Model
• 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
Figure 156: Mode 1 - ASDShellQ4 Figure 157: Mode 2 - ASDShellQ4
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Figure 158: Mode 3 - ASDShellQ4 Figure 159: Mode 4 - ASDShellQ4
Figure 160: Mode 5 - ASDShellQ4 Figure 161: Mode 6 - ASDShellQ4
Frequency values obtained and percentage differences with respect to the reference
solution are shown in the following table.
Table 43. Mode frequencies, f (Hz)
1st mode 2nd mode 3rd mode
value diff. value diff. value diff.
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
4th mode 5th mode 6th mode
value diff. value diff. value diff.
Hz % Hz % Hz %
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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
• 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
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
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5.8 FV52: SIMPLY SUPPORTED “SOLID” SQUARE PLATE
Elements tested
Standard Brick Element
Bbar Brick Element
Twenty Node Brick Element
SSPbrick Element
Benchmark description
Figure 162: model description Figure 163: STKO FE Model
• 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
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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
Frequency values obtained and percentage differences with respect to the reference
solution are shown in the following table.
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Table 44. Mode frequencies, f (Hz)
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
7th mode 8th mode 9th mode 10th mode
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
• 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
FV52_Standard Brick Element.scd
FV52_Bbar Brick Element.scd
FV52_Twenty Node Brick Element.scd
FV52_SSPbrick Element.scd
97
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6 NONLINEAR GEOMETRY TESTS
6.1 NLG1. COROTATIONAL TRUSS ELEMENT
Elements tested
Corotational Truss Element
Benchmark description
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.
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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
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6.2 NLG2: SNAP THROUGH
Elements tested
Force Beam Column
Benchmark description
Figure 173: model description, adapted from Midas Figure 174: STKO FE Model
• 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
Values obtained and percentage differences with respect to the reference solution.
Figure 175: Z deflection, forceBeamColumn
Table 46. Deflection z at node 2, δz (m)
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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
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6.3 NLG3: STATIC LARGE DISPLACEMENT ANALYSIS OF A TOWER
CABLE
Elements tested
Catenary Cable
Benchmark description
Figure 176: model description, adapted from Midas Figure 177: STKO FE Model
• 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
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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
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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
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6.4 NLG4: CANTILIVER SUBJECTED TO BENDING MOMENT
Elements tested
ASDShellQ4
Benchmark description
Figure 180: model description Figure 181: STKO FE Model
• 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
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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
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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
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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
Benchmark description
Figure 186: model description Figure 187: STKO FE Model
• 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
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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
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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
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7.2 3DNLG-7: ELASTIC LARGE DEFLECTION RESPONSE OF A HINGED
SPHERICAL SHELL UNDER PRESSURE LOADING
Elements tested
ASDShellQ4
ShellMITC4
ShellNLDKGQ
Benchmark description
Figure 191: model description Figure 192: STKO FE Model
• 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
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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
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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
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8 NONLINEAR MATERIAL TESTS
8.1 NLM1: PLANE STRAIN PLASTICITY
Elements tested
SSP Quad
Benchmark description
Figure 197: model description, adapted from Midas
Figure 198: STKO FE Model
• 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
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7 Δw = -R 0 R Elastic unloading
8 Δw = R 0 0 Plastic flow
Test Results
Values obtained and percentage differences with respect to the reference solution.
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
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8.2 NLM2: 3D PLASTICITY
Elements tested
SSPbrick
Standard Brick Element
Bbar Brick Element
Twenty Node Brick Element
Benchmark description
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
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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
Values obtained and percentage differences with respect to the reference solution.
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
117
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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
118
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9 NONLINEAR ANALYSIS TESTS
9.1 TH1: DYNAMIC MODAL RESPONSE FOR 2-D RIGID FRAME
Elements tested
Elastic Beam Column
Benchmark description
Figure 201: model description, adapted from Midas
Figure 202: STKO FE Model
• 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.
119
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Figure 203: Impulse at the 1st floor Figure 204: Impulse at the 2nd floor
Test Results
Values obtained and percentage differences with respect to the reference solution.
Figure 205: lateral displacement ad node 3 (m)
120
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Figure 206: lateral displacement ad node 5 (m
Table 60. Modal natural frequency (Hz)
THEOR. SAP2000 MIDAS -Gen STKO
Value Value Diff. Value Diff. Value Diff.
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.
121
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Input files
TH1_ElasticBeamColumn.scd