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Static Analysis Problems
This chapter contains static analysis problems and Structure's results. In a Static analysis, Structure calculates deformations, stresses, and strains on your model in response to specified loads and subject to specified constraints. Structure also automatically calculates all predefined measures. This list of measures differs based on the analysis type.
This chapter contains the following models:
mvss001: 2D Axisymmetric Cylindrical Shell
mvss002: 2D Axisymmetric Flat Circular Plate
mvss003: 2D Plane Stress Cantilever Plate
mvss004: 2D Plane Strain Thick-Walled Cylinder
mvss005: 2D Axisymmetric Thick-Walled Cylinder
mvss006: 3D Cantilever Beam
mvss007: 3D Beam with Multiple Constraints
mvss008: 3D Beam with Parallelogram-Shaped Shell Elements
mvss009: 3D Beam with Trapezoidal-Shaped Shell Elements
mvss010: 3D Curved Beam Modeled with Shells
mvss011: 3D Simply Supported Rectangular Plate
mvss012: 3D Clamped Rectangular Plate
mvss013: 3D Hemispherical Shell
mvss014: 3D Cantilever Beam Twisted by 900
mvss015: 3D Scordelis-Lo Roof
mvss016: 2D Axisymmetric Cylinder/Sphere
mvss017: 2D Tapered Membrane with Gravity Load
mvss018: 3D Z-Section Cantilevered Plate
mvss019: 3D Cylindrical Shell with Edge Moment
mvss020: Beam Sections
mvss021:Thick-Walled Cylinder Under Internal Pressure
mvss022: Thin-Walled Spherical Vessel Under Its Own Weight
mvsl001: Static Analysis of Composite Lay-up
mvss001: 2D Axisymmetric Cylindrical Shell
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): 2342-2347.
• 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.
Specifications
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
Comparison of Results Data
Theory
MSC/ NASTRAN
Structure
% Difference
Radial Deflection @ Load (a=disp_x_radial)
2.8769e-3
2.8715e-3
2.8725e-3
0.15%
Convergence %: 0.5% on Local Disp and SE
Max P: 7
No. Equations: 33
mvss002: 2D Axisymmetric Flat Circular Plate
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.
Specifications
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 A-B: FY = 6
per unit area
uniform
simple
placed on edge A-B: FY = 1.5
per unit area
uniform
Comparison of Results Data
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
1 Sign of result is dependent upon direction of load.
mvss003: 2D Plane Stress Cantilever Plate
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 in-plane shear load.
Specifications
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 A-B: fixed in TransX, TransY
Loads:
placed on edge C-D: 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.
Comparison of Results Data
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
mvss004: 2D Plane Strain Thick-Walled Cylinder
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 thick-walled cylinder, modeled symmetrically, is loaded with unit internal pressure. Find the radial displacement at the inner radius for two nearly incompressible materials.
Specifications
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 A-B & C-D:
fixed in all DOF except TransR
Loads:
placed on edge A-D: pressure load = 1
Distribution: N/A
Spatial Variation: uniform
Comparison of Results Data
Theory
Structure
% Difference
Radial Displacement @ Inner Radius (case 1) (m=rad_disp)
5.0399e-3
5.0394e-3
<0.01%
Convergence %:1% on Local Disp and SE
Max P: 6
No. Equations: 38
Radial Displacement @ Inner Radius (case 2) (m=rad_disp)
5.0602e-3
5.0553e-3
0.09%
Convergence %: 1.0% on Local Disp and SE
Max P: 6
No. Equations: 38
mvss005: 2D Axisymmetric Thick-Walled Cylinder
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: McGraw-Hill Book Co., 1972, pp. 293-297.
Description:
Find the stress at radii r = 6.5" and r = 11.5". A thick-walled cylinder is modeled axisymmetrically and subjected to internal pressure.
Specifications
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 A-D & B-C: fixed in TransY and RotZ
Loads:
placed on edge A-B: pressure load = 10
Distribution: per unit area
Spatial Variation: uniform
Comparison of Results Data
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
-2.6636e-1
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
mvss006: 3D Cantilever Beam
Analysis Type:
Static
Model Type:
3D
Comparison:
NASTRAN No. V2405
Reference:
Roark, R.J., and Young, W.C. Formulas for Stress and Strain. NY: McGraw-Hill 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.
Specifications
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
1 Structure beams consider shear; however, the represented theoretical problem does not. The values for shear factor compensate for this.
Comparison of Results Data
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
mvss007: 3D Beam with Multiple Constraints
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.
Specifications
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 A-B: FY = 833.33
placed on edge D-E: FY = 833.33
per unit length
per unit length
uniform
uniform
Comparison of Results Data
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
mvss008: 3D Beam with Parallelogram-Shaped Shell Elements
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 parallelogram-shaped elements, is subjected to four different unit loads at the free end, including
• extension
• in-plane shear
• out-of-plane shear
• twisting loads
Find the tip displacement in the direction of the load for each case.
Specifications
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 A-D: fixed in all DOF
Loads:
Location/Magnitude:
Distribution:
Spatial Variation:
extension
placed on edge B-C: FX = 1
total load
uniform
in_plane
placed on edge B-C: FY = 1
total load
uniform
out_plane
placed on edge B-C: FZ = 1
total load
uniform
twist
placed on point E: MX = 1
total load
N/A
Comparison of Results Data
Theory
Structure
% Difference
Tip Disp. in Direction of Load (l=extension, m=max_disp_x)
3e-5
2.998e-5
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
1 There is a typographical error in Table 3 (p. 10) of MacNeal-Harder for the twist load on a straight beam. It should read 0.03408.
mvss009: 3D Beam with Trapezoidal-Shaped Shell Elements
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 trapezoidal-shaped elements, is subjected to four different unit loads at the free end, including
• extension
• in-plane shear
• out-of-plane shear
• twisting
Find the tip displacement in the direction of the load for each case.
Specifications
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 A-D: fixed in all DOF
Loads:
Location/Magnitude:
Distribution:
Spatial Variation:
extension
placed on edge B-C: FX = 1
total load
uniform
in_plane
placed on edge B-C: FY = 1
total load
uniform
out_plane
placed on edge B-C: FZ = 1
total load
uniform
twist
placed on point E: MX = 1
total load
N/A
Comparison of Results Data
Theory
Structure
% Difference
Tip Disp. in Direction of Load
(l=extension, m=max_disp_x)
3e-5
2.998e-5
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
1 There is a typographical error in Table 3 (p. 10) of the McNeal-Harder reference for the twist load on a straight beam. It should read 0.03408.
mvss010: 3D Curved Beam Modeled with Shells
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 in-plane and out-of-plane loads at the free end, find the tip displacement in the direction of the load for both cases.
Specifications
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 A-D: fixed in all DOF
Loads:
Location/Magnitude:
Distribution:
Spatial Variation:
in_plane
placed on edge B-C: FY = 1
total load
uniform
out_plane
placed on edge B-C: FZ = 1
total load
uniform
Comparison of Results Data
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
mvss011: 3D Simply Supported Rectangular Plate
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.
Specifications
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 A-D, C-D:
placed on edge A-B:
placed on edge B-C:
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
Comparison of Results Data
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
mvss012: 3D Clamped Rectangular Plate
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.
Specifications
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 A-D, D-C:
placed on edge A-B:
placed on edge B-C:
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
Comparison of Results Data
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
mvss013: 3D Hemispherical Shell
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.
Specifications
Element Type:
shell (4)
Units:
IPS
Dimensions:
(using a one-quarter 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 A-C:
placed on curve G-E:
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
Comparison of Results Data
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
mvss014: 3D Cantilever Beam Twisted by 90
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 in-plane and out-of-plane loads at the free end. Find the tip displacement in the direction of the load for each case.
Specifications
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
Comparison of Results Data
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
mvss015: 3D Scordelis-Lo Roof
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 Scordelis-Lo roof is one-quarter 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).
Specifications
Element Type:
shell (1)
Units:
IPS
Dimensions:
(using a one-quarter 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 A-B:
placed on curve A-D:
placed on curve C-D
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 A-B-C-D: FZ = 90
per unit area
uniform
Comparison of Results Data
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
mvss016: 2D Axisymmetric Cylinder/Sphere
Analysis Type:
Static
Model Type:
2D Axisymmetric
Reference:
NAFEMS, LSB1, No. IC 39
Description:
An axisymmetric cylinder and half-sphere vessel is loaded with uniform internal pressure. Find the hoop stress on the outer surface at point D.
Specifications
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
Comparison of Results Data
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
1 You cannot view the results information in the summary file. To view the results, you must define a result window for the Stress ZZ (Bottom), and query the value at point D.
mvss017: 2D Tapered Membrane with Gravity Load
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.
Specifications
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 A-B, B-C:
placed on point B:
fixed in TransX
fixed in TransX, TransY
Loads:
Location/Magnitude:
load1
Global acceleration: GX=9.81
Comparison of Results Data
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
mvss018: 3D Z-Section Cantilevered Plate
Analysis Type:
Static
Model Type:
3D
Reference:
NAFEMS, LSB1, No. IC 29
Description:
A Z-section cantilevered plate is subjected to a torque at the free end by two uniformly distributed edge shears. Find the direct stress Sxx at the mid-plane of the plate.
Specifications
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 A-B, B-C, and C-D:
fixed in TransX, TransY, and TransZ
Loads:
Location/Magnitude:
Distribution
Spatial Variation
load1
placed on curve E-F: FZ=0.6
placed on curve G-H: FZ=0.6
total load
total load
uniform
uniform
Comparison of Results Data
Theory
Structure1
% Difference
Sxx at mid-surface at point M
–108.8
–110.02
1.1%
Convergence %: 0.4% on Local Disp and SE
Max P: 7
No. Equations: 870
1 You cannot view the results information in the summary file. To view the results, you must define a result window for the measure Stress XX (Top and Bottom), and query the value at point M. Then average the top (118) and bottom (105.56) values.
mvss019: 3D Cylindrical Shell with Edge Moment
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.
Specifications
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 A-B:
placed on curves A-D and B-C:
fixed in all DOF
fixed in TransZ, RotX, and RotY
Loads:
Location/Magnitude:
Distribution
Spatial Variation
load1
placed on curve C-D: MZ=0.001
force per unit length
uniform
Comparison of Results Data
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
1 You cannot view the results information in the summary file. To view the results, you must define a result window for measure Stress XX (Top) with Face Grid on, and query the value at point E.
mvss020: Beam Sections
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.
This Beam Sections model contains the following element types and corresponding results:
Square Beam
Rectangle Beam
Hollow Rectangle Beam
Channel Beam
I-Section Beam
L-Section Beam
Diamond Beam
Solid Circle Beam
Hollow Circle Beam
Ellipse Beam
Hollow Ellipse 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.
Square Beam
Specifications
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
1 Structure beams consider shear; however, the represented theoretical problem does not. The values for shear factor compensate for this.
Comparison of Results Data (Square Beam)
Load
Measure Name
Theory
Structure
% Difference
Deflection at Tip:
axial
sq_d_x
1.6e-3
1.6e-3
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
Rectangle Beam
Specifications
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
1Structure beams consider shear; however, the represented theoretical problem does not. The values for shear factor compensate for this.
Comparison of Results Data (Rectangle Beam)
Load
Measure Name
Theory
Structure
% Difference
Deflection at Tip:
axial
rct_d_x
4.0e-4
4.0e-4
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
Hollow Rectangle Beam
Specifications
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
1 Structure beams consider shear; however, the represented theoretical problem does not. The values for shear factor compensate for this.
Comparison of Results Data (Hollow Rectangle Beam)
Load
Measure Name
Theory
Structure
% Difference
Deflection at Tip:
axial
hrct_d_x
7.112e-4
7.111e-4
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
Channel Beam
Specifications
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
1 Structure beams consider shear; however, the represented theoretical problem does not. The values for shear factor compensate for this.
Comparison of Results Data (Channel Beam)
Load
Measure Name
Theory
Structure
% Difference
Deflection at Tip:
axial
chnl_d_x
2.6667e-4
6.061674e-04
0%
transverse y
chnl_d_y
3.339e-1
-3.339e-1
0%
transverse z
chnl_d_z
8.1198e-1
-8.1198e-1
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
I-Section Beam
Specifications
Element Type:
I-Section 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
1 Structure beams consider shear; however, the represented theoretical problem does not. The values for shear factor compensate for this.
Comparison of Results Data (I-Section Beam)
Load
Measure Name
Theory
Structure
% Difference
Deflection at Tip:
axial
I_d_x
2.6667e-4
2.6667e-4
0%
transverse y
I_d_y
3.3391e-1
3.3573e-1
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
L-Section Beam
Specifications
Element Type:
L-Section 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
1 Structure beams consider shear; however, the represented theoretical problem does not. The values for shear factor compensate for this.
Comparison of Results Data (L-Section Beam)
Load
Measure Name
Theory
Structure
% Difference
Deflection at Tip:
axial
L_d_x
4.0e-4
4.0e-4
0%
transverse y
L_d_y
7.0892e-1
7.1017e-1
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
Diamond Beam
Specifications
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
1 Structure beams consider shear; however, the represented theoretical problem does not. The values for shear factor compensate for this.
Comparison of Results Data (Diamond Beam)
Load
Measure Name
Theory
Structure
% Difference
Deflection at Tip:
axial
dmnd_d_x
3.2e-3
3.2e-3
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
Solid Circle Beam
Specifications
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
1 Structure beams consider shear; however, the represented theoretical problem does not. The values for shear factor compensate for this.
Comparison of Results Data (Solid Circle Beam)
Load
Measure Name
Theory
Structure
% Difference
Deflection at Tip:
axial
crcl_d_x
5.093e-4
5.092e-4
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
Hollow Circle Beam
Specifications
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
1 Structure beams consider shear; however, the represented theoretical problem does not. The values for shear factor compensate for this.
Comparison of Results Data (Hollow Circle Beam)
Load
Measure Name
Theory
Structure
% Difference
Deflection at Tip:
axial
hcr_d_x
6.7906e-4
6.7906e-4
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
Ellipse Beam
Specifications
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
1 Structure beams consider shear; however, the represented theoretical problem does not. The values for shear factor compensate for this.
Comparison of Results Data (Ellipse Beam)
Load
Measure Name
Theory
Structure
% Difference
Deflection at Tip:
axial
elps_d_x
1.2732e-4
1.2732e-4
0%
transverse y
elps_d_y
2.4446
2.4445
0.004%
transverse z
elps_d_z
1.527887e-1
1.531516e-1
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
Hollow Ellipse Beam
Specifications
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
1 Structure beams consider shear; however, the represented theoretical problem does not. The values for shear factor compensate for this.
Comparison of Results Data (Hollow Ellipse Beam)
Load
Measure Name
Theory
Structure
% Difference
Deflection at Tip:
axial
hel_d_x
5.4325e-4
5.4324e-4
0.0018%
transverse y
hel_d_y
5.9075
5.9091
0.45%
transverse z
hel_d_z
3.6922e-1
3.7091e-1
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
mvss021:Thick-Walled Cylinder Under Internal Pressure
Analysis Type:
Static
Model Type:
3D
Reference:
Roark, R.J., and Young, W.C. Formulas for Stress and Strain. NY: McGraw-Hill Book Co., 5th edition, Table 32, Case 1.
Description:
A thick-walled cylinder subjected to an internal pressure is free to expand in all directions. Obtain maximum radial and circumferential stresses.
Specifications
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
Comparison of Results Data
Theory
Structure
% Difference
yy along edges C-E & F-G
2600
2603.7325
0.14%
xx along edges C-E & F-G
1000
999.1724
0.08%
Multi-Pass Convergence %: The analysis converged to within 1% on measures.
Max P: 6
No. Equations: 1875
mvss022: Thin-Walled Spherical Vessel Under Its Own Weight
Analysis Type:
Static
Model Type:
3D Cyclic Symmetric
Reference:
Roark, R.J., and Young, W.C. Formulas for Stress and Strain. NY: McGraw-Hill Book Co., 5th edition, Table 29, Case 3c.
Description:
A thin-walled half-spherical vessel is subjected to its own weight (gravity load). Obtain the hoop stress at points A and B.
Specifications
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
Comparison of Results Data
Theory
Structure
% Difference
zz at point A:
1
0.987
1.3%
tt at point B:
-1
-0.982
1.8%
Multi-Pass 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
mvsl001: Static Analysis of Composite Lay-up
Analysis Type:
Static with Orthotropic Material Properties
Model Type:
3D
Comparison:
Theory
Reference:
Noor, A.K. and Mathers, M.D., "Shear-Flexible Finite-Element Models of Laminated Composite Plates and Shells." NASA TN D-8044; Langley Research Center, Hampton, Va. Dec. 1975.
Description:
Determine maximum resultant bending moment and transverse deformation in a clamped, nine-layered, orthotropic square plate.
Specifications
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 B-C and C-D
clamped on edges A-B and A-D
Loads:
uniform pressure load over the entire surface = 1
Comparison of Results Data
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
1 To verify this Creo Simulate result, create a query result window for the quantity Moment:Shell Resultant:XX. Show the result window and query for the value in the upper left corner of the model. This is obtained using View:Model Min. The absolute value of this negative number is greater than the value reported using View:Model Max and is reported here.