Analysis Type:

Static

Model Type:

2D Axisymmetric

Comparison:

NASTRAN No. V2411

Reference:

• P.E. Grafton and D.R. Strome, "Analysis of Axisymmetrical Shells by the Direct Stiffness Method," AIAA Journal, 1(10): 23422347.
• J.W. Jones and H.H. Fong, "Evaluation of NASTRAN," Structural Mechanics Software Series, Vol. IV (N. Perrone and W. Pilkey, eds.), 1982.

Description:

Find the radial deflection at the loaded end of a cantilever cylinder that is modeled axisymmetrically.

Element B is optional, but has been included here to increase the accuracy of results in the area local to the loaded end and to reduce computation time. 
Element Type:  2D shell (2)  
Units:  IPS  
Dimensions:  length: 6 radius: 5 thickness: 0.01  
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 1e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraint:  placed on point A: fixed in all DOF  
Load:  placed on point C: FX = 1 Distribution: N/A Spatial Variation: N/A 
Theory  MSC/ NASTRAN  Structure  % Difference  
Radial Deflection @ Load (a=disp_x_radial)  2.8769e3  2.8715e3  2.8725e3  0.15% 
Convergence %: 0.5% on Local Disp and SE  Max P: 7  No. Equations: 33 
Analysis Type:  Static 
Model Type:  2D Axisymmetric 
Comparison:  ANSYS No. 15 
Reference:  Timoshenko, S. Strength of Materials, Part II, Advanced Theory and Problems. 3rd ed. NY: D. Van Nostrand Co., Inc. 1956, pp. 96, 97, and 103. 
Description:  A flat circular plate, modeled axisymmetrically, is subjected to various edge constraints and surface loadings. Determine the maximum stress for each case. 
Element Type:  2D shell (1)  
Units:  IPS  
Dimensions:  radius: 40 thickness: 1  
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 3e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints:  Location  Degrees of Freedom 
clamped  placed on point B:  fixed in all DOF 
simple  placed on point B:  fixed in TransX and TransY 
Loads:  Location/Magnitude:  Distribution:  Spatial Variation: 
clamped  placed on edge AB: FY = 6  per unit area  uniform 
simple  placed on edge AB: FY = 1.5  per unit area  uniform 
Theory  ANSYS  Structure  % Difference  
Maximum Stress (m=max_prin_mag, a=clamped)  7200  7152  7200  0.0% 
Convergence %: 0.0% on Local Disp and SE  Max P: 5  No. Equations: 15  
Maximum Stress (m=max_prin_mag, a=simple)  2970  2989  29701  0.0% 
Convergence %: 0.0% on Local Disp and SE  Max P: 5  No. Equations: 16 
Analysis Type:  Static 
Model Type:  2D Plane Stress 
Comparison:  NASTRAN No. V2408A 
Reference:  Singer, Ferdinand L. Strength of Materials. Harper & Row, 1962, Art. 52, p. 133. 
Description:  Find the bending stress at the fixed end for a cantilever plate subjected to an inplane shear load. 
Element Type:  2D plate (1)  
Units:  IPS  
Dimensions:  length: 3 height: 0.6 thickness: 0.1  
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 1.07e7  Poisson's Ratio: 0 Thermal Expansion: 0 Conductivity: 0 
Constraints:  placed on edge AB: fixed in TransX, TransY  
Loads:  placed on edge CD: FY= –200 Distribution: per unit length Spatial Variation: uniform  
The theoretical results are based on elementary beam theory. Structure models the actual physical structure, capturing the singular stresses present at the constrained corners. Setting Poisson's ratio equal to zero reduces the model to its elementary form. 
Theory  MSC/ NASTRAN  Structure  % Difference  
Bending Stress @ Node A (m=max_stress_xx)  6.0e4  5.5190e4  6.0121e4  0.20% 
Convergence %: 0.0% on Local Disp and SE  Max P: 4  No. Equations: 22 
Analysis Type:  Static 
Model Type:  2D Plane Strain 
Comparison:  The MacNealHarder Accuracy Tests 
Reference:  MacNeal, R.H., and Harder, R.L. "A Proposed Standard Set of Problems to Test Finite Element Accuracy." Finite Elements in Analysis and Design I. Elsevier Science Publishers, 1985. 
Description:  A thickwalled cylinder, modeled symmetrically, is loaded with unit internal pressure. Find the radial displacement at the inner radius for two nearly incompressible materials. 
Element Type:  2D solid (1)  
Units:  IPS  
Dimensions:  outer radius: 9.0 inner radius: 3.0  
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 1000  Poisson's Ratio: • 0.49 (case 1) • 0.499 (case 2) 
Constraints (UCS):  placed on edges AB & CD: fixed in all DOF except TransR  
Loads:  placed on edge AD: pressure load = 1 Distribution: N/A Spatial Variation: uniform 
Theory  Structure  % Difference  
Radial Displacement @ Inner Radius (case 1) (m=rad_disp)  5.0399e3  5.0394e3  <0.01% 
Convergence %:1% on Local Disp and SE  Max P: 6  No. Equations: 38  
Radial Displacement @ Inner Radius (case 2) (m=rad_disp)  5.0602e3  5.0553e3  0.09% 
Convergence %: 1.0% on Local Disp and SE  Max P: 6  No. Equations: 38 
Analysis Type:  Static 
Model Type:  2D Axisymmetric 
Comparison:  NASTRAN No. V2410 
Reference:  Crandall S.H., Dahl N.C. , and Larnder T.J. An Introduction to the Mechanics of Solids. 2nd ed. NY: McGrawHill Book Co., 1972, pp. 293297. 
Description:  Find the stress at radii r = 6.5" and r = 11.5". A thickwalled cylinder is modeled axisymmetrically and subjected to internal pressure. 
Element Type:  2D solid (3)  
Units:  IPS  
Dimensions:  inner radius: 6 height: 8 thickness: 6  
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 3e7  Poisson's Ratio: 0 Thermal Expansion: 0 Conductivity: 0 
Constraints (UCS):  placed on edges AD & BC: fixed in TransY and RotZ  
Loads:  placed on edge AB: pressure load = 10 Distribution: per unit area Spatial Variation: uniform 
Theory  MSC/ NASTRAN  Structure  % Difference  
@ r = 6.5  Stress Radial (m=r6_5_radial)  8.03  8.05  7.9720  0.72% 
Stress Hoop (m=r6_5_hoop)  14.69  14.73  14.69  0.0%  
@ r = 11.5  Stress Radial (m=r11_5_radial)  0.30  0.30  2.6636e1  0.0% 
Stress Hoop (m=r11_5_hoop)  6.96  6.96  6.96  0.0%  
Convergence %: 0.25% on Local Disp and SE  Max P: 4  No. Equations: 54 
Analysis Type:  Static 
Model Type:  3D 
Comparison:  NASTRAN No. V2405 
Reference:  Roark, R.J., and Young, W.C. Formulas for Stress and Strain. NY: McGrawHill Book Co., 1982, p. 96. 
Description:  A cantilever beam is subjected to a load at the free end. Find the deflection at the free end and the bending stress at the fixed end. 
Element Type:  beam (1)  
Units:  IPS  
Dimensions:  length: 30  
Beam Properties:  Area: 0.310 IYY: 0.0241 Shear FY: 1000 1 CY: 0.5  J: 0.0631 IZZ: 0.0390 Shear FZ: 1000 1 CZ: 0.375 
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 1.0e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints:  placed on point A: fixed in all DOF  
Loads:  placed on point B: FY = 100 Distribution: N/A Spatial Variation: N/A 
Theory  MSC/ NASTRAN  Structure  % Difference  
Deflection @ Tip (m=max_disp_y)  2.3077  2.3077  2.3094  0.073% 
Bending Stress @ Fixed End (m=max_beam_bending)  38461  38461  38461  0.0% 
Convergence %: 0.0% on Local Disp and SE  Max P: 4  No. Equations: 24 
Analysis Type:  Static 
Model Type:  3D 
Comparison:  ANSYS No. 2 
Reference:  Timoshenko, S. Strength of Materials, Part I, Elementary Theory and Problems. 3rd ed. NY: D. Van Nostrand Co., Inc., 1955, p. 98, Problem 4. 
Description:  A standard 30" WF beam, supported as shown below, is loaded on the overhangs uniformly. Find the maximum bending stress and deflection at the middle of the beam. 
Element Type:  beam (4)  
Units:  IPS  
Dimensions:  length: 480  
Beam Properties:  Area: 50.65 IYY: 1 Shear FY: 0.8333 CY: 15  J: 7893 IZZ: 7892 Shear FZ: 0.8333 CZ: 15 
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 3e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints  Location  Degrees of Freedom 
placed on point B: placed on point D:  fixed in all DOF except RotY and RotZ fixed in TransY and TransZ 
Loads  Location/Magnitude  Distribution  Spatial Variation 
placed on edge AB: FY = 833.33 placed on edge DE: FY = 833.33  per unit length per unit length  uniform uniform 
Theory  ANSYS  Structure  % Difference  
Max Bending Stress @ Middle (m=max_beam_bending)  11400  11404  11403.91  0.03% 
Max Deflection @ Middle (m=disp_center)  0.182  0.182  0.182  0.0% 
Convergence %: 0.0% on Local Disp and SE  Max P: 4  No. Equations: 96 
Analysis Type:  Static 
Model Type:  3D 
Comparison:  The MacNealHarder Accuracy Tests 
Reference:  MacNeal, R.H., and Harder, R.L. "A Proposed Standard Set of Problems to Test Finite Element Accuracy." Finite Elements in Analysis and Design I. Elsevier Science Publishers, 1985. 
Description:  A straight cantilever beam, constructed of parallelogramshaped elements, is subjected to four different unit loads at the free end, including • extension • inplane shear • outofplane shear • twisting loads Find the tip displacement in the direction of the load for each case. 
Element Type:  shell (3)  
Units:  IPS  
Dimensions:  length: 6 width: 0.2 thickness: 0.1  
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 1e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints:  placed on edge AD: fixed in all DOF 
Loads:  Location/Magnitude:  Distribution:  Spatial Variation: 
extension  placed on edge BC: FX = 1  total load  uniform 
in_plane  placed on edge BC: FY = 1  total load  uniform 
out_plane  placed on edge BC: FZ = 1  total load  uniform 
twist  placed on point E: MX = 1  total load  N/A 
Theory  Structure  % Difference  
Tip Disp. in Direction of Load (l=extension, m=max_disp_x)  3e5  2.998e5  0.06% 
Tip Disp. in Direction of Load (l=in_plane, m=max_disp_y)  0.1081  0.1078  0.27% 
Tip Disp. in Direction of Load (l=out_plane, m=max_disp_z)  0.4321  0.4309  0.27% 
Tip Disp. in Direction of Load (l=twist, m=max_rot_x)  0.03408 1  0.03424  0.46% 
Convergence %: 0.9% on Local Disp and SE  Max P: 6  No. Equations: 396 
Analysis Type:  Static 
Model Type:  3D 
Comparison:  The MacNealHarder Accuracy Tests 
Reference:  MacNeal, R.H., and Harder, R.L. "A Proposed Standard Set of Problems to Test Finite Element Accuracy." Finite Elements in Analysis and Design I. Elsevier Science Publishers, 1985. 
Description:  A straight cantilever beam, constructed of trapezoidalshaped elements, is subjected to four different unit loads at the free end, including • extension • inplane shear • outofplane shear • twisting Find the tip displacement in the direction of the load for each case. 
Element Type:  shell (3)  
Units:  IPS  
Dimensions:  length: 6 width: 0.2 thickness: 0.1  
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 1e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints:  placed on edge AD: fixed in all DOF 
Loads:  Location/Magnitude:  Distribution:  Spatial Variation: 
extension  placed on edge BC: FX = 1  total load  uniform 
in_plane  placed on edge BC: FY = 1  total load  uniform 
out_plane  placed on edge BC: FZ = 1  total load  uniform 
twist  placed on point E: MX = 1  total load  N/A 
Theory  Structure  % Difference  
Tip Disp. in Direction of Load (l=extension, m=max_disp_x)  3e5  2.998e5  0.08% 
Tip Disp. in Direction of Load (l=in_plane, m=max_disp_y)  0.1081  0.1079  0.32% 
Tip Disp. in Direction of Load (l=out_plane, m=max_disp_z)  0.4321  .4311  0.23% 
Tip Disp. in Direction of Load (l=twist, m=max_rot_x)  0.03408 1  0.03381  0.79% 
Convergence %: 0.7% on Local Disp and SE  Max P: 6  No. Equations: 906 
Analysis Type:  Static 
Model Type:  3D 
Comparison:  The MacNealHarder Accuracy Tests 
Reference:  MacNeal, R.H., and Harder, R.L. "A Proposed Standard Set of Problems to Test Finite Element Accuracy." Finite Elements in Analysis and Design I. Elsevier Science Publishers, 1985. 
Description:  A curved beam, spanning a 90arc, is fixed at one end and free at the other. If the beam is subjected to inplane and outofplane loads at the free end, find the tip displacement in the direction of the load for both cases. 
Element Type:  shell (2)  
Units:  IPS  
Dimensions:  outer radius: 4.32 inner radius: 4.12 thickness: 0.1  
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 1e7  Poisson's Ratio: 0.25 Thermal Expansion: 0 Conductivity: 0 
Constraints:  placed on edge AD: fixed in all DOF 
Loads:  Location/Magnitude:  Distribution:  Spatial Variation: 
in_plane  placed on edge BC: FY = 1  total load  uniform 
out_plane  placed on edge BC: FZ = 1  total load  uniform 
Theory  Structure  % Difference  
Tip Displacement in Direction of Load (l=in_plane, m=tip_disp_y)  0.08734  0.08833  1.13% 
Tip Displacement in Direction of Load (l=out_plane, m=tip_disp_z)  0.5022  0.50057  0.32% 
Convergence %: 0.3% on Local Disp and SE  Max P: 6  No. Equations: 234 
Analysis Type:  Static 
Model Type:  3D 
Comparison:  The MacNealHarder Accuracy Tests 
Reference:  MacNeal, R.H., and Harder, R.L. "A Proposed Standard Set of Problems to Test Finite Element Accuracy." Finite Elements in Analysis and Design I. Elsevier Science Publishers, 1985. 
Description:  A flat plate is simply supported on all four edges. One quarter of the plate is modeled using symmetry. The plate is loaded with two different loads, including uniform pressure and a point load at the center. Find the displacement at the center of the plate. 
Element Type:  shell (2)  
Units:  IPS  
Dimensions:  length: 5 width: 1 thickness: 0.0001  
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 1.7472e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints  Location  Degrees of Freedom 
placed on edges AD, CD: placed on edge AB: placed on edge BC:  fixed in TransX, TransY, and TransZ fixed in TransY, RotX, and RotZ fixed in TransX, RotY, and RotZ 
Loads:  Location/Magnitude:  Distribution:  Spatial Variation: 
pressure  placed on all shells: pressure = 1e4  total load per unit area  uniform 
point  placed on B: FZ = 1e4  N/A  N/A 
Theory  Structure  % Difference  
Displacement @ Center (l=pressure, m=disp_z_cen)  –12.97  –12.97  0.0% 
Displacement @ Center (l=point, m=disp_z_cen)  16.96  16.81  0.88% 
Convergence %: 0.8% on Local Disp and SE  Max P: 9  No. Equations: 438 
Analysis Type:  Static 
Model Type:  3D 
Comparison:  The MacNealHarder Accuracy Tests 
Reference:  MacNeal, R.H., and Harder, R.L. "A Proposed Standard Set of Problems to Test Finite Element Accuracy." Finite Elements in Analysis and Design I. Elsevier Science Publishers, 1985. 
Description:  One quarter of a rectangular plate, clamped on four edges, is modeled using symmetry. The plate is loaded with two different loads, including uniform pressure and a point load at center. Find the displacement at the center of the plate. 
Element Type:  shell (2)  
Units:  IPS  
Dimensions:  length: 5 width: 1 thickness: 0.0001  
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 1.7472e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints  Location  Degrees of Freedom 
placed on edges AD, DC: placed on edge AB: placed on edge BC:  fixed in all DOF fixed in TransY, RotX, and RotZ fixed in TransX, RotY, and RotZ 
Loads:  Location/Magnitude:  Distribution:  Spatial Variation: 
pressure  placed on all shells: pressure = 1e4  per unit area  uniform 
point  placed on B: FZ = 1e4  N/A  N/A 
Theory  Structure  % Difference  
Displacement @ Center (l=pressure, m=measure1)  –2.56  –2.604  1.71% 
Displacement @ Center (l=point, m=measure1)  7.23  7.168  0.85% 
Convergence %: 1.3% on Local Disp and SE  Max P: 9  No. Equations: 625 
Analysis Type:  Static 
Model Type:  3D 
Comparison:  The MacNealHarder Accuracy Tests 
Reference:  MacNeal, R.H., and Harder, R.L. "A Proposed Standard Set of Problems to Test Finite Element Accuracy." Finite Elements in Analysis and Design I. Elsevier Science Publishers, 1985. 
Description:  One quarter of an open hemisphere is modeled with symmetry and loaded with alternating point loads at 90 intervals on the equator. Find the radial displacement at any load point. 
Element Type:  shell (4)  
Units:  IPS  
Dimensions: (using a onequarter model)  radius: 10 arc span: 90o thickness: 0.04  
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 6.825e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints  Location  Degrees of Freedom 
placed on curve AC: placed on curve GE: placed on point D  fixed in TransP, RotR, and RotT fixed in TransP, RotR, and RotT fixed in TransT 
Loads:  Location/Magnitude:  Distribution:  Spatial Variation 
placed on point C: FR = 1 placed on E: FR = 1  N/A N/A  N/A N/A 
Theory  Structure  % Difference  
Radial Displacement @ Load (m=disp_rad)  –0.0924  –0.0933  0.97% 
Convergence %: 0.6% on Local Disp and SE  Max P: 9  No. Equations: 1965 
Analysis Type:  Static 
Model Type:  3D 
Comparison:  The MacNealHarder Accuracy Tests 
Reference:  MacNeal, R.H., and Harder, R.L. "A Proposed Standard Set of Problems to Test Finite Element Accuracy." Finite Elements in Analysis and Design I. Elsevier Science Publishers, 1985. 
Description:  A cantilever beam, twisted by 90, is subjected to inplane and outofplane loads at the free end. Find the tip displacement in the direction of the load for each case. 
Element Type:  solid (2)  
Units:  IPS  
Dimensions:  length: 12 width: 1.1 thickness: 0.32 angle of twist 90o (from fixed to free end)  
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 29e6  Poisson's Ratio: 0.22 Thermal Expansion: 0 Conductivity: 0 
Constraints:  placed on root surface: fixed in all DOF 
Loads:  Location/Magnitude:  Distribution:  Spatial Variation: 
in_plane  placed on free end surface: FY = 1  total load  uniform 
out_plane  placed on free end surface: FZ = 1  total load  uniform 
Theory  Structure  % Difference  
Tip Displacement in Direction of Load (l=in_plane, m=disp_tip_y1)  0.005424  0.005428  0.73% 
Tip Displacement in Direction of Load (l=out_of_plane, m=disp_tip_z1)  0.001754  0.001760  0.342% 
Convergence %: 0.8% on Local Disp and SE  Max P: 5  No. Equations: 590 
Analysis Type:  Static 
Model Type:  3D 
Comparison:  The MacNealHarder Accuracy Tests 
Reference:  MacNeal, R.H., and Harder, R.L. "A Proposed Standard Set of Problems to Test Finite Element Accuracy." Finite Elements in Analysis and Design I. Elsevier Science Publishers, 1985. 
Description:  A ScordelisLo roof is onequarter of an arched roof modeled using symmetry and loaded uniformly. Find the vertical displacement at the midpoint of the straight side (of the whole roof). 
Element Type:  shell (1)  
Units:  IPS  
Dimensions: (using a onequarter model)  length: 25 radius: 25 arc span: 40o thickness: 0.25  
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 4.32e8  Poisson's Ratio: 0 Thermal Expansion: 0 Conductivity: 0 
Constraints  Location  Degrees of Freedom 
(UCS) (UCS) (UCS)  placed on curve AB: placed on curve AD: placed on curve CD  fixed in TransZ, RotR, and RotT fixed in TransT, RotZ, and RotR fixed in TransR and TransT 
Loads:  Location/Magnitude:  Distribution:  Spatial Variation: 
placed on face ABCD: FZ = 90  per unit area  uniform 
Theory  Structure  % Difference  
Vertical Displacement @ Point B (m=disp_z_mid)  –0.3024  –0.3008  0.53% 
Convergence %: 0.2% on Local Disp and SE  Max P: 7  No. Equations: 148 
Analysis Type:  Static 
Model Type:  2D Axisymmetric 
Reference:  NAFEMS, LSB1, No. IC 39 
Description:  An axisymmetric cylinder and halfsphere vessel is loaded with uniform internal pressure. Find the hoop stress on the outer surface at point D. 
Element Type:  2D shell (4)  
Units:  MKS  
Dimensions:  radius: 1 thickness: 0.025  
Material Properties:  Mass Density: 0.007 Cost Per Unit Mass: 0 Young's Modulus: 210000  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints  Location  Degrees of Freedom 
constraint1  placed on point A: placed on point E:  fixed in TransX and RotZ fixed in TransY 
Loads:  Location/Magnitude:  
load1  placed on all 2D shell elements: internal pressure = 1 
Theory  Structure1  % Difference  
Szz on outer surface  38.5  38.62  0.3% 
Convergence %: 0.8% on Local Disp and SE  Max P: 7  No. Equations: 72 
Analysis Type:  Static 
Model Type:  Plane Stress 
Reference:  NAFEMS, LSB1, No. IC 2 
Description:  A tapered membrane has uniform acceleration in the global X direction. Find the direct stress Sxx at point B. 
Element Type:  2D plate (2)  
Units:  MKS  
Dimensions:  thickness: 0.1  
Material Properties:  Mass Density: 0.007 Cost Per Unit Mass: 0 Young's Modulus: 210000  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints:  Location  Degrees of Freedom 
constraint1  placed on curves AB, BC: placed on point B:  fixed in TransX fixed in TransX, TransY 
Loads:  Location/Magnitude: 
load1  Global acceleration: GX=9.81 
Theory  Structure  % Difference  
Stress XX at point B (m=measure1)  0.247  0.247  0% 
Convergence %: 0.7% on Local Disp and SE  Max P: 7  No. Equations: 248 
Analysis Type:  Static 
Model Type:  3D 
Reference:  NAFEMS, LSB1, No. IC 29 
Description:  A Zsection cantilevered plate is subjected to a torque at the free end by two uniformly distributed edge shears. Find the direct stress Sxx at the midplane of the plate. 
Element Type:  shell (6)  
Units:  MKS  
Dimensions:  length: 10 thickness: 0.1  
Material Properties:  Mass Density: 0.007 Cost Per Unit Mass: 0 Young's Modulus: 210000  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints  Location  Degrees of Freedom 
constraint1  placed on curves AB, BC, and CD:  fixed in TransX, TransY, and TransZ 
Loads:  Location/Magnitude:  Distribution  Spatial Variation 
load1  placed on curve EF: FZ=0.6 placed on curve GH: FZ=0.6  total load total load  uniform uniform 
Theory  Structure1  % Difference  
Sxx at midsurface at point M  –108.8  –110.02  1.1% 
Convergence %: 0.4% on Local Disp and SE  Max P: 7  No. Equations: 870 
Analysis Type:  Static 
Model Type:  3D 
Reference:  NAFEMS, LSB1, No. IC 19 
Description:  A cylindrical shell in 3D space is loaded with a uniform normal edge moment on one edge. Find the outer surface tangential stress at point E. 
Element Type:  shell (1)  
Units:  MKS  
Dimensions:  radius: 1 thickness: 0.01  
Material Properties:  Mass Density: 0.007 Cost Per Unit Mass: 0 Young's Modulus: 210000  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints  Location  Degrees of Freedom 
constraint1  placed on curve AB: placed on curves AD and BC:  fixed in all DOF fixed in TransZ, RotX, and RotY 
Loads:  Location/Magnitude:  Distribution  Spatial Variation 
load1  placed on curve CD: MZ=0.001  force per unit length  uniform 
Theory  Structure1  % Difference  
Sxx on outer surface at point E  60.0  59.6  .67% 
Convergence %: 0.9% on Local Disp and SE  Max P: 5  No. Equations: 66 
Analysis Type:  Static 
Model Type:  3D 
Comparison:  Theory 
Reference:  Roark, R.J., and Young, W.C. Formulas for Stress and Strain. 5th Edition. NY: McGrawHill Book Co. 1982, p. 64. 
Description:  A cantilever beam is subjected to transverse loads in Y and Z and axial load in X . Find the deflection at the free end, the bending stress at the fixed end, and the axial stress along the beam. 
In all cases, the displacement results are dependent upon the direction of the load. Thus, in this problem, all the results listed as Deflection at Tip may be interpreted as positive or negative. 
Element Type:  Square Beam  
Units:  IPS  
Dimensions:  a: 0.25  
Beam Properties:  Area: 0.0625 IYY: 0.000325521 Shear FY: 10001 CY: 0.125  J: 0.000549316 IZZ: 0.000325521 Shear FZ: 10001 CZ: 0.125 
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 3e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints:  placed on point A: fixed in all DOF  
Load:  Location:  Magnitude: 
axial  placed on point B  FX=100 
transverse y  placed on point B  FY=100 
transverse z  placed on point B  FZ=100 
Load  Measure Name  Theory  Structure  % Difference 
Deflection at Tip:  
axial  sq_d_x  1.6e3  1.6e3  0% 
transverse y  sq_d_y  9.216e1  9.216e1  0% 
transverse z  sq_d_z  9.216e1  9.216e1  0% 
Stress:  
axial  sq_s_ten  1.6e3  1.6e3  0% 
transverse y  sq_s_bnd  1.152003e6  1.15200e6  0% 
transverse z  sq_s_bnd  1.152003e6  1.15200e6  0% 
Load  Lcl Disp & SE  Max P  No. Equations  
Convergence:  
axial  0%  2  264  
transverse y  0%  2  264  
transverse z  0%  2  264 
Element Type:  Rectangle Beam  
Units:  IPS  
Dimensions:  b: 1 d: 0.25  
Beam Properties:  Area: 0.25 IYY: 0.0208333 Shear FY: 10001 CY: 0.125  J: 0.00438829 IZZ: 0.00130208 Shear FZ: 10001 CZ: 0.5 
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 3e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints:  placed on point A: fixed in all DOF  
Load:  Location:  Magnitude: 
axial  placed on point B  FX=100 
transverse y  placed on point B  FY=100 
transverse z  placed on point B  FZ=100 
Load  Measure Name  Theory  Structure  % Difference 
Deflection at Tip:  
axial  rct_d_x  4.0e4  4.0e4  0% 
transverse y  rct_d_y  2.304e1  2.304e1  0% 
transverse z  rct_d_z  1.44  1.44  0% 
Stress:  
Load  Measure Name  Theory  Structure  % Difference 
axial  rct_s_ten  4.0e2  4.0e2  0% 
transverse y  rct_s_bnd  2.880e5  2.880e5  0% 
transverse z  rct_s_bnd  7.200e4  7.200e4  0% 
Load  Lcl Disp & SE  Max P  No. Equations  
Convergence:  
axial  0%  2  264  
transverse y  0%  2  264  
transverse z  0%  2  264 
Element Type:  Hollow Rectangle Beam  
Units:  IPS  
Dimensions:  b: 1 bi: 0.875 d: 0.25 di: 0.125  
Beam Properties:  Area: 0.140625 IYY: 0.013855 Shear FY: 10001 CY: 0.125  J: 0.00343323 IZZ: 0.00115967 Shear FZ: 10001 CZ: 0.5 
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 3e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints:  placed on point A: fixed in all DOF  
Load:  Location:  Magnitude: 
axial  placed on point B  FX=100 
transverse y  placed on point B  FY=100 
transverse z  placed on point B  FZ=100 
Load  Measure Name  Theory  Structure  % Difference 
Deflection at Tip:  
axial  hrct_d_x  7.112e4  7.111e4  0.02% 
transverse y  hrct_d_y  2.5869e1  2.5876e1  0.027% 
transverse z  hrct_d_z  2.1653  2.1677  0.10% 
Stress:  
Load  Measure Name  Theory  Structure  % Difference 
axial  hrct_s_ten  7.112e2  7.111e2  0.01% 
transverse y  hrct_s_bnd  3.2337e5  3.2336e5  0.003% 
transverse z  hrct_s_bnd  1.0826e5  1.0826e5  0% 
Load  Lcl Disp & SE  Max P  No. Equations  
Convergence:  
axial  0%  2  264  
transverse y  0%  2  264  
transverse z  0%  2  264 
Element Type:  Channel Beam  
Units:  IPS  
Dimensions:  b: 1 di: 1 t: 0.125 tw: 0.125  
Beam Properties:  Area: 0.375 IYY: 0.0369466 Shear FY: 10001 CY: 0.625  J: 0.00179932 IZZ: 0.0898438 Shear FZ: 10001 CZ: 0.645833 
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 3e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints:  placed on point A: fixed in all DOF  
Load:  Location:  Magnitude: 
axial  placed on point B  FX=100 
transverse y  placed on point B  FY=100 
transverse z  placed on point B  FZ=100 
Load  Measure Name  Theory  Structure  % Difference 
Deflection at Tip:  
axial  chnl_d_x  2.6667e4  6.061674e04  0% 
transverse y  chnl_d_y  3.339e1  3.339e1  0% 
transverse z  chnl_d_z  8.1198e1  8.1198e1  0% 
Stress:  
Load  Measure Name  Theory  Structure  % Difference 
axial  chnl_s_ten  2.6667e2  2.6667e2  0% 
transverse y  chnl_s_bnd  2.087e4  2.087e4  0% 
transverse z  chnl_s_bnd  5.244e4  5.244e4  0% 
Convergence:  
Load  Lcl Disp & SE  Max P  No. Equations  
axial  0%  4  264  
transverse y  0%  4  264  
transverse z  0%  4  264 
Element Type:  ISection Beam  
Units:  IPS  
Dimensions:  b: 1 di: 1 t: 0.125 tw: 0.125  
Beam Properties:  Area: 0.375 IYY: 0.0209961 Shear FY: 10001 CY: 0.625  J: 0.00179932 IZZ: 0.0898438 Shear FZ: 10001 CZ: 0.5 
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 3e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints:  placed on point A: fixed in all DOF  
Load:  Location:  Magnitude: 
axial  placed on point B  FX=100 
transverse y  placed on point B  FY=100 
transverse z  placed on point B  FZ=100 
Load  Measure Name  Theory  Structure  % Difference 
Deflection at Tip:  
axial  I_d_x  2.6667e4  2.6667e4  0% 
transverse y  I_d_y  3.3391e1  3.3573e1  0.54% 
transverse z  I_d_z  1.4288  1.4296  0.05% 
Stress:  
Load  Measure Name  Theory  Structure  % Difference 
axial  I_s_ten  2.6667e2  2.6667e2  0% 
transverse y  I_s_bnd  2.0870e4  2.0869e4  0.004% 
transverse z  I_s_bnd  7.1442e4  7.14418e4  0.001% 
Load  Lcl Disp & SE  Max P  No. Equations  
Convergence:  
axial  0%  2  264  
transverse y  0%  2  264  
transverse z  0%  2  264 
Element Type:  LSection Beam  
Units:  IPS  
Dimensions:  b: 1 d: 1 t: 0.125 tw: 0.125  
Beam Properties:  Area: 0.25 IYY: 0.0105794 Shear FY: 10001 CY: 0.789352  J: 0.00119955 IZZ: 0.0423177 Shear FZ: 10001 CZ: 0.433047 
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 3e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints:  placed on point A: fixed in all DOF  
Load:  Location:  Magnitude: 
axial  placed on point B  FX=100 
transverse y  placed on point B  FY=100 
transverse z  placed on point B  FZ=100 
Load  Measure Name  Theory  Structure  % Difference 
Deflection at Tip:  
axial  L_d_x  4.0e4  4.0e4  0% 
transverse y  L_d_y  7.0892e1  7.1017e1  0.17% 
transverse z  L_d_z  2.8357  2.8369  0.04% 
Stress:  
Load  Measure Name  Theory  Structure  % Difference 
axial  L_s_ten  4e2  4e2  0% 
transverse y  L_s_ben  5.5611e4  0  — 
transverse z  L_s_ben  1.228e5  0  — 
Load  Lcl Disp & SE  Max P  No. Equations  
Convergence:  
axial  0%  2  264  
transverse y  0%  2  264  
transverse z  0%  2  264 
Element Type:  Diamond Beam  
Units:  IPS  
Dimensions:  b: 0.25 d: 0.25  
Beam Properties:  Area: 0.03125 IYY: 8.13802e5 Shear FY: 10001 CY: 0.125  J: 0.000146484 IZZ: 8.13802e5 Shear FZ: 10001 CZ: 0.125 
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 3e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints:  placed on point A: fixed in all DOF  
Load:  Location:  Magnitude: 
axial  placed on point B  FX=100 
transverse y  placed on point B  FY=100 
transverse z  placed on point B  FZ=100 
Load  Measure Name  Theory  Structure  % Difference 
Deflection at Tip:  
axial  dmnd_d_x  3.2e3  3.2e3  0% 
transverse y  dmnd_d_y  3.6864e2  3.6864e2  0% 
transverse z  dmnd_d_z  3.6864e2  3.6864e2  0% 
Stress:  
Load  Measure Name  Theory  Structure  % Difference 
axial  dmnd_s_ten  3.2e3  3.2e3  0% 
transverse y  dmnd_s_bnd  4.608e6  4.608e6  0% 
transverse z  dmnd_s_bnd  4.608e6  4.608e6  0% 
Load  Lcl Disp & SE  Max P  No. Equations  
Convergence:  
axial  0%  2  264  
transverse y  0%  2  264  
transverse z  0%  2  264 
Element Type:  Solid Circle Beam  
Units:  IPS  
Dimensions:  r: 0.25  
Beam Properties:  Area: 0.19635 IYY: 0.00306796 Shear FY: 10001 CY: 0.25  J: 0.00613592 IZZ: 0.00306796 Shear FZ: 10001 CZ: 0.25 
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 3e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints:  placed on point A: fixed in all DOF  
Load:  Location:  Magnitude: 
axial  placed on point B  FX=100 
transverse y  placed on point B  FY=100 
transverse z  placed on point B  FZ=100 
Load  Measure Name  Theory  Structure  % Difference 
Deflection at Tip:  
axial  crcl_d_x  5.093e4  5.092e4  0.019% 
transverse y  crcl_d_y  9.77848  9.77995  0.015% 
transverse z  crcl_d_z  9.77848  9.77995  0.015% 
Stress:  
axial  crcl_s_ten  5.093e2  5.092e2  0.019% 
transverse y  crcl_s_bnd  2.44462e5  2.44462e5  0% 
transverse z  crcl_s_bnd  2.44462e5  2.44462e5  0% 
Load  Lcl Disp & SE  Max P  No. Equations  
Convergence:  
axial  0%  2  264  
transverse y  0%  2  264  
transverse z  0%  2  264 
Element Type:  Hollow Circle Beam  
Units:  IPS  
Dimensions:  ri: 0.25  
Beam Properties:  Area: 0.147262 IYY: 0.00287621 Shear FY: 100001 CY: 0.25  J: 0.00575243 IZZ: 0.00287621 Shear FZ: 10001 CZ: 0.25 
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 3e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints:  placed on point A: fixed in all DOF  
Load:  Location:  Magnitude: 
axial  placed on point B  FX=100 
transverse y  placed on point B  FY=100 
transverse z  placed on point B  FZ=100 
Load  Measure Name  Theory  Structure  % Difference 
Deflection at Tip:  
axial  hcr_d_x  6.7906e4  6.7906e4  0% 
transverse y  hcr_d_y  1.04304e1  1.04331e1  0.025% 
transverse z  hcr_d_z  1.04304e1  1.04332e1  0.026% 
Stress:  
axial  hcr_s_ten  6.7906e2  6.7906e2  0% 
transverse y  hcr_s_bnd  2.6076e5  2.6075e5  0.003% 
transverse z  hcr_s_bnd  2.6076e5  2.6076e5  — 
Load  Lcl Disp & SE  Max P  No. Equations  
Convergence:  
axial  0%  2  264  
transverse y  0%  2  264  
transverse z  0%  2  264 
Element Type:  Ellipse Beam  
Units:  IPS  
Dimensions:  a: 1 b: 0.25  
Beam Properties:  Area: 0.785398 IYY: 0.19635 Shear FY: 10001 CY: 0.25  J: 0.0461999 IZZ: 0.0122718 Shear FZ: 10001 CZ: 1 
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 3e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints:  placed on point A: fixed in all DOF  
Load:  Location:  Magnitude: 
axial  placed on point B  FX=100 
transverse y  placed on point B  FY=100 
transverse z  placed on point B  FZ=100 
Load  Measure Name  Theory  Structure  % Difference 
Deflection at Tip:  
axial  elps_d_x  1.2732e4  1.2732e4  0% 
transverse y  elps_d_y  2.4446  2.4445  0.004% 
transverse z  elps_d_z  1.527887e1  1.531516e1  0.23% 
Stress:  
axial  elps_s_ten  1.273239e2  1.27324e2  0% 
transverse y  elps_s_bnd  6.11155e4  6.111550e4  0% 
transverse z  elps_s_bnd  1.527887e4  1.527887e+04  0% 
Load  Lcl Disp & SE  Max P  No. Equations  
Convergence:  
axial  0%  2  264  
transverse y  0%  2  264  
transverse z  0%  2  264 
Element Type:  Hollow Ellipse Beam  
Units:  IPS  
Dimensions:  a: 1 b: 0.25 ai: 0.875  
Beam Properties:  Area: 0.184078 IYY: 0.081253 Shear FY: 10001 CY: 0.25  J: 0.0191184 IZZ: 0.00507832 Shear FZ: 10001 CZ: 1 
Material Properties:  Mass Density: 0 Cost Per Unit Mass: 0 Young's Modulus: 3e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints:  placed on point A: fixed in all DOF  
Load:  Location:  Magnitude: 
axial  placed on point B  FX=100 
transverse y  placed on point B  FY=100 
transverse z  placed on point B  FZ=100 
Load  Measure Name  Theory  Structure  % Difference 
Deflection at Tip:  
axial  hel_d_x  5.4325e4  5.4324e4  0.0018% 
transverse y  hel_d_y  5.9075  5.9091  0.45% 
transverse z  hel_d_z  3.6922e1  3.7091e1  0.027% 
Stress:  
axial  hel_s_ten  5.4325e2  5.4324e2  0.0018% 
transverse y  hel_s_bnd  1.4769e5  1.4768e5  0.0027% 
transverse z  hel_s_bnd  3.6922e4  3.6921e4  0.0067% 
Load  Lcl Disp & SE  Max P  No. Equations  
Convergence:  
axial  0%  2  264  
transverse y  0%  2  264  
transverse z  0%  2  264 
Analysis Type:  Static 
Model Type:  3D 
Reference:  Roark, R.J., and Young, W.C. Formulas for Stress and Strain. NY: McGrawHill Book Co., 5th edition, Table 32, Case 1. 
Description:  A thickwalled cylinder subjected to an internal pressure is free to expand in all directions. Obtain maximum radial and circumferential stresses. 
Element Type:  tets (133)  
Units:  IPS  
Dimensions:  length: 20 Ro: 6 Ri: 4  
Material Properties:  Mass Density: 0.0002614 Cost Per Unit Mass: 0 Young's Modulus: 1.06e7  Poisson's Ratio: 0.33 Thermal Expansion: 1.25e05 Conductivity: 9.254 
Constraints  Location  Degrees of Freedom 
constraint1  placed on point A: placed on point B: placed on point D:  fixed in TransX, TransY, and TransZ fixed in TransY fixed in TransY and TransZ 
Loads:  Location/Magnitude:  Distribution  Spatial Variation 
pressure  placed on all internal surfaces: pressure = 1000  total load/unit area  uniform 
Theory  Structure  % Difference  
yy along edges CE & FG  2600  2603.7325  0.14% 
xx along edges CE & FG  1000  999.1724  0.08% 
MultiPass Convergence %: The analysis converged to within 1% on measures.  Max P: 6  No. Equations: 1875 
Analysis Type:  Static 
Model Type:  3D Cyclic Symmetric 
Reference:  Roark, R.J., and Young, W.C. Formulas for Stress and Strain. NY: McGrawHill Book Co., 5th edition, Table 29, Case 3c. 
Description:  A thinwalled halfspherical vessel is subjected to its own weight (gravity load). Obtain the hoop stress at points A and B. 
Element Type:  shells (3)  
Units:  IPS  
Dimensions:  R: 10  
Material Properties:  Mass Density: 0.0002588 Cost Per Unit Mass: 0 Young's Modulus: 1.0e7  Poisson's Ratio: 0.3 Thermal Expansion: 0 Conductivity: 0 
Constraints  Location  Degrees of Freedom 
constraint1  Edges @ = 0 & = 90: Edge @ z = 0: Placed on point C @ r = 10, = 0, z = 0:  cyclical symmetry fixed in TransZ fixed in TransR, TransT, and TransZ 
Load:  Direction:  Magnitude: 
gravity  x y z  0.0 386.4 0.0 
Theory  Structure  % Difference  
zz at point A:  1  0.987  1.3% 
tt at point B:  1  0.982  1.8% 
MultiPass Adaptive Convergence %: The analysis converged to within 4.9% on Local Displacement and Element Strain Energy. It converged to 1.7% on Global RMS Stress.  Max P: 9  No. Equations: 773 
Analysis Type:  Static with Orthotropic Material Properties 
Model Type:  3D 
Comparison:  Theory 
Reference:  Noor, A.K. and Mathers, M.D., "ShearFlexible FiniteElement Models of Laminated Composite Plates and Shells." NASA TN D8044; Langley Research Center, Hampton, Va. Dec. 1975. 
Description:  Determine maximum resultant bending moment and transverse deformation in a clamped, ninelayered, orthotropic square plate. 
Element Type:  shell (4)  
Units:  IPS  
Dimensions:  length: 2.5 width: 2.5 thickness: 0.5  
Shell Properties:  
Extensional Stiffness  A11=10.266  A12=0.1252  A16=0 
A22=10.266  A26=0  
A66=0.3  
ExtensionalBending Coupling Stiffness  B11=0  B12=0  B16=0 
B22=0  B26=0  
B66=0  
Bending Stiffness  D11=0.25965  D12=0.0026082  D16=0 
D22=0.1681  D26=0  
D66=0.00625  
Transverse Shear Stiffnesses  A55=0.275004  A45=0  A44=0.275004 
Mass per Unit Area  7.2915e5  
Rotary Inertia per Unit Area  1.5191e5  
Thermal Resultant Coefficients:  
Force  N11=0  N22=0  N12=0 
Moment  M11=0  M22=0  M12=0 
Stress Recovery Locations  CZ  Ply Orientation (Degrees)  Material 
Location Reported for "Top" in Results  0.25  0  trniso1 
Location Reported for "Bottom" in Results  0.25  0  trniso1 
Material Properties:  
Mass Density: 0.00014583  Cost Per Unit Mass: 0  
Young's Moduli  E1=4e1  E2=1  E3=1 
Poisson's Ratio  Nu21=0.25  Nu31=0.25  Nu32=0 
Shear Moduli  G21=0.6  G31=0.6  G32= E2/[2*(1+Nu32)] 
Coefficients of Thermal Expansion  a1=0  a2=0  a3=0 
Constraints:  symmetry constraints on edges BC and CD clamped on edges AB and AD  
Loads:  uniform pressure load over the entire surface = 1 
Theory  Structure  % Difference  
Displacement  11.596  11.84151  2.11% 
Bending Moment 1  1.4094  1.41307  0.26% 
Convergence %: 1.1% on local displacement and element strain energy and 2.2 % on global RMS stress.  Max P: 3  No. Equations: 76 