A circular plate is fixed along its perimeter. Using
plate/shell elements, find the deflection at the center, maximum bending stress
due to a uniformly distributed load, and a concentrated load at the center.
Reference
Timoshenko, S.,
Strength of Materials, Part II, 3rd Edition, Van
Nostrand Co., 1956, pp.96-97,
103.
Problem
The circular plate shown below is subject to two load cases.
Load 1 is a uniform pressure, w, and load 2 is a concentrated force, P, at the
center. Determine:
- deflection at the
center for both load cases
- bending stress at
the support for both load cases
- moment at the center
for load case 1
Finite element model of a circular
plate
Comparison
Table 1. Comparison of results
Result
Type
|
Theory
|
STAAD.Pro
|
Difference
|
Comments
|
Load Case 1
|
σbend (psi)
|
7,200
|
6,887a
|
4.3%
|
The theoretical results
from classical Plate theory was compared with the results from Finite Element
model - hence the difference in results.
|
δmax (in) (Y translation at
Node 127)
|
-0.0874
|
-0.08618
|
1.4%
|
Moment at center (in·lb/in)
|
780
|
806.1b
|
3.4%
|
Load Case 2
|
σbend (psi)
|
3,600
|
3,700c
|
2.8%
|
δmax (in) (Y translation at
Node 127)
|
-0.0874
|
-0.08646
|
1.0%
|
- In element 9 at
node 10, My = 1,147.91 in·lb/in. This yields σbend = 1,147.91(6) =
6,887 psi.
- In element 109 at
node 127, Mx = 811.88 in·lb/in and My = 800.33 in·lb/in, the average of which
is 806.1 in·lb/in.
- In element 9 at node
10, My = 616.69 in·lb/in. This yields σbend = 616.19(6) = 3,700 psi.
STAAD Output
JOINT DISPLACEMENT (INCH RADIANS) STRUCTURE TYPE = SPACE
------------------
JOINT LOAD X-TRANS Y-TRANS Z-TRANS X-ROTAN Y-ROTAN Z-ROTAN
127 1 0.00000 -0.08618 0.00000 -0.00000 0.00000 -0.00000
2 0.00000 -0.08646 0.00000 0.00000 0.00000 -0.00001
************** END OF LATEST ANALYSIS RESULT **************
93. LOAD LIST 1
94. PRINT ELEMENT JOINT STRESSES LIST 9 109
ELEMENT JOINT STRESSES LIST
:A CIRCULAR PLATE-FIXED ALONG ITS PERIMETER -- PAGE NO. 5
ELEMENT STRESSES FORCE,LENGTH UNITS= POUN INCH
----------------
STRESS = FORCE/UNIT WIDTH/THICK, MOMENT = FORCE-LENGTH/UNIT WIDTH
ELEMENT LOAD SQX SQY MX MY MXY
VONT VONB SX SY SXY
TRESCAT TRESCAB
9 1 0.00 -103.62 115.36 719.90 0.01
4018.27 4018.27 0.00 0.00 0.00
4319.39 4319.39
TOP : SMAX= 4319.39 SMIN= 692.18 TMAX= 1813.61 ANGLE= 90.0
BOTT: SMAX= -692.18 SMIN= -4319.39 TMAX= 1813.61 ANGLE= -0.0
JOINT 14.39 -113.81 338.29 1147.77 123.92
9 6263.05 6263.05 0.00 0.00 0.00
TOP : SMAX= 6997.90 SMIN= 1918.50 TMAX= 2539.70 ANGLE= 81.5
BOTT: SMAX= -1918.50 SMIN= -6997.90 TMAX= 2539.70 ANGLE= -8.5
JOINT -14.38 -113.83 338.35 1147.91 -123.91
10 6263.75 6263.75 0.00 0.00 0.00
TOP : SMAX= 6998.72 SMIN= 1918.87 TMAX= 2539.93 ANGLE=-81.5
BOTT: SMAX= -1918.87 SMIN= -6998.72 TMAX= 2539.93 ANGLE= 8.5
JOINT -10.79 -93.43 -124.84 381.27 -175.70
28 3292.73 3292.73 0.00 0.00 0.00
TOP : SMAX= 2617.74 SMIN= -1079.14 TMAX= 1848.44 ANGLE=-72.6
BOTT: SMAX= 1079.14 SMIN= -2617.74 TMAX= 1848.44 ANGLE= 17.4
JOINT 10.79 -93.42 -124.84 381.32 175.75
27 3293.22 3293.22 0.00 0.00 0.00
TOP : SMAX= 2618.18 SMIN= -1079.25 TMAX= 1848.71 ANGLE= 72.6
BOTT: SMAX= 1079.25 SMIN= -2618.18 TMAX= 1848.71 ANGLE=-17.4
109 1 6.37 1.93 -799.52 -799.67 -0.21
4797.59 4797.59 0.00 0.00 0.00
4798.94 4798.94
TOP : SMAX= -4796.24 SMIN= -4798.94 TMAX= 1.35 ANGLE=-35.3
BOTT: SMAX= 4798.94 SMIN= 4796.24 TMAX= 1.35 ANGLE= 54.7
JOINT 6.37 1.93 -811.88 -800.33 -0.21
127 4837.01 4837.01 0.00 0.00 0.00
TOP : SMAX= -4801.97 SMIN= -4871.30 TMAX= 34.67 ANGLE=-88.9
BOTT: SMAX= 4871.30 SMIN= 4801.97 TMAX= 34.67 ANGLE= 1.1
JOINT 6.37 1.93 -792.77 -800.33 -0.21
109 4779.46 4779.46 0.00 0.00 0.00
TOP : SMAX= -4756.58 SMIN= -4802.02 TMAX= 22.72 ANGLE= -1.6
BOTT: SMAX= 4802.02 SMIN= 4756.58 TMAX= 22.72 ANGLE= 88.4
JOINT 6.37 1.93 -793.92 -798.36 -0.21
110 4776.89 4776.89 0.00 0.00 0.00
TOP : SMAX= -4763.47 SMIN= -4790.19 TMAX= 13.36 ANGLE= -2.7
BOTT: SMAX= 4790.19 SMIN= 4763.47 TMAX= 13.36 ANGLE= 87.3
:A CIRCULAR PLATE-FIXED ALONG ITS PERIMETER -- PAGE NO. 6
**** MAXIMUM STRESSES AMONG SELECTED PLATES AND CASES ****
MAXIMUM MINIMUM MAXIMUM MAXIMUM MAXIMUM
PRINCIPAL PRINCIPAL SHEAR VONMISES TRESCA
STRESS STRESS STRESS STRESS STRESS
6.998721E+03 -6.998721E+03 2.539927E+03 4.797589E+03 4.798940E+03
PLATE NO. 9 9 9 109 109
CASE NO. 1 1 1 1 1
********************END OF ELEMENT FORCES********************
95. LOAD LIST 2
96. PRINT ELEMENT JOINT STRESSES LIST 9
ELEMENT JOINT STRESSES LIST
:A CIRCULAR PLATE-FIXED ALONG ITS PERIMETER -- PAGE NO. 7
ELEMENT STRESSES FORCE,LENGTH UNITS= POUN INCH
----------------
STRESS = FORCE/UNIT WIDTH/THICK, MOMENT = FORCE-LENGTH/UNIT WIDTH
ELEMENT LOAD SQX SQY MX MY MXY
VONT VONB SX SY SXY
TRESCAT TRESCAB
9 2 0.00 -34.32 75.98 484.41 0.01
2707.43 2707.43 0.00 0.00 0.00
2906.43 2906.43
TOP : SMAX= 2906.43 SMIN= 455.87 TMAX= 1225.28 ANGLE= 90.0
BOTT: SMAX= -455.87 SMIN= -2906.43 TMAX= 1225.28 ANGLE= -0.0
JOINT 4.34 -37.38 188.37 616.60 52.05
9 3328.03 3328.03 0.00 0.00 0.00
TOP : SMAX= 3737.03 SMIN= 1092.82 TMAX= 1322.10 ANGLE= 83.2
BOTT: SMAX= -1092.82 SMIN= -3737.03 TMAX= 1322.10 ANGLE= -6.8
JOINT -4.33 -37.39 188.41 616.69 -52.04
10 3328.46 3328.46 0.00 0.00 0.00
TOP : SMAX= 3737.54 SMIN= 1093.08 TMAX= 1322.23 ANGLE=-83.2
BOTT: SMAX= -1093.08 SMIN= -3737.54 TMAX= 1322.23 ANGLE= 6.8
JOINT -3.25 -31.25 -36.94 382.08 -70.33
28 2519.30 2519.30 0.00 0.00 0.00
TOP : SMAX= 2361.41 SMIN= -290.59 TMAX= 1326.00 ANGLE=-80.7
BOTT: SMAX= 290.59 SMIN= -2361.41 TMAX= 1326.00 ANGLE= 9.3
JOINT 3.25 -31.24 -36.95 382.09 70.36
27 2519.50 2519.50 0.00 0.00 0.00
TOP : SMAX= 2361.54 SMIN= -290.69 TMAX= 1326.12 ANGLE= 80.7
BOTT: SMAX= 290.69 SMIN= -2361.54 TMAX= 1326.12 ANGLE= -9.3
**** MAXIMUM STRESSES AMONG SELECTED PLATES AND CASES ****
MAXIMUM MINIMUM MAXIMUM MAXIMUM MAXIMUM
PRINCIPAL PRINCIPAL SHEAR VONMISES TRESCA
STRESS STRESS STRESS STRESS STRESS
3.737542E+03 -3.737542E+03 1.326119E+03 2.707434E+03 2.906431E+03
PLATE NO. 9 9 9 9 9
CASE NO. 2 2 2 2 2