V. 2D Circular Surface Displacements and Stresses

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

E = 30,000.0 ksi

Poisson’s ratio = 0.3

r = 40 in.

t = 1 in.

w = 6 psi.

P = 7,539.82 lbs

Finite element model of a circular plate

Comparison

Table 1. Comparison of results
Result Type		Theory	STAAD.Pro	Difference	Comments
Load Case 1	σ_bend (psi)	7,200	6,887^a	4.3%	The theoretical results from classical Plate theory was compared with the results from Finite Element model - hence the difference in results.
	δ_max (in) (Y translation at Node 127)	-0.0874	-0.08618	1.4%
	Moment at center (in·lb/in)	780	806.1^b	3.4%
Load Case 2	σ_bend (psi)	3,600	3,700^c	2.8%
Load Case 2	δ_max (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 σ_bend = 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 σ_bend = 616.19(6) = 3,700 psi.

STAAD Input

Tip: You can copy and paste this content directly into a .std file to run in STAAD.Pro.

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\04 Plates Shells\2D Circular Surface Displacements and Stresses.STD is typically installed with the program.

STAAD SPACE :A CIRCULAR PLATE-FIXED ALONG ITS PERIMETER
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
*
* REFERENCE: TIMOSHENKO, S., STRENGTH OF MATERIALS
*            PART II, ADVANCED THEORY AND PROBLEMS
*            PAGES 96, 97, AND 103.
*
INPUT WIDTH 79
UNIT INCHES POUND
JOINT COORDINATES
1 40 0 0; 2 37.588 0 -13.681; 3 30.642 0 -25.712; 4 20 0 -34.641;
5 6.946 0 -39.392; 6 -6.946 0 -39.392; 7 -20 0 -34.641; 8 -30.642 0 -25.711;
9 -37.588 0 -13.681; 10 -40 0 0; 11 -37.588 0 13.681; 12 -30.642 0 25.712;
13 -20 0 34.641; 14 -6.946 0 39.392; 15 6.946 0 39.392; 16 20 0 34.641;
17 30.642 0 25.711; 18 37.588 0 13.681; 19 30 0 0; 20 28.191 0 -10.261;
21 22.981 0 -19.284; 22 15 0 -25.981; 23 5.209 0 -29.544; 24 -5.209 0 -29.544;
25 -15 0 -25.981; 26 -22.981 0 -19.284; 27 -28.191 0 -10.261; 28 -30 0 0;
29 -28.191 0 10.261; 30 -22.981 0 19.284; 31 -15 0 25.981; 32 -5.209 0 29.544;
33 5.21 0 29.544; 34 15 0 25.981; 35 22.981 0 19.284; 36 28.191 0 10.261;
37 20 0 0; 38 18.794 0 -6.84; 39 15.321 0 -12.856; 40 10 0 -17.321;
41 3.473 0 -19.696; 42 -3.473 0 -19.696; 43 -10 0 -17.32; 44 -15.321 0 -12.856;
45 -18.794 0 -6.84; 46 -20 0 0; 47 -18.794 0 6.84; 48 -15.321 0 12.856;
49 -10 0 17.321; 50 -3.473 0 19.696; 51 3.473 0 19.696; 52 10 0 17.32;
53 15.321 0 12.856; 54 18.794 0 6.84; 55 15 0 0; 56 14.095 0 -5.13;
57 11.491 0 -9.642; 58 7.5 0 -12.99; 59 2.605 0 -14.772; 60 -2.605 0 -14.772;
61 -7.5 0 -12.99; 62 -11.491 0 -9.642; 63 -14.095 0 -5.13; 64 -15 0 0;
65 -14.095 0 5.13; 66 -11.491 0 9.642; 67 -7.5 0 12.99; 68 -2.605 0 14.772;
69 2.605 0 14.772; 70 7.5 0 12.99; 71 11.491 0 9.642; 72 14.095 0 5.13;
73 10 0 0; 74 9.397 0 -3.42; 75 7.66 0 -6.428; 76 5 0 -8.66; 77 1.736 0 -9.848;
78 -1.736 0 -9.848; 79 -5 0 -8.66; 80 -7.66 0 -6.428; 81 -9.397 0 -3.42;
82 -10 0 0; 83 -9.397 0 3.42; 84 -7.66 0 6.428; 85 -5 0 8.66;
86 -1.736 0 9.848; 87 1.737 0 9.848; 88 5 0 8.66; 89 7.66 0 6.428;
90 9.397 0 3.42; 91 7 0 0; 92 6.578 0 -2.394; 93 5.362 0 -4.5; 94 3.5 0 -6.062;
95 1.216 0 -6.894; 96 -1.216 0 -6.894; 97 -3.5 0 -6.062; 98 -5.362 0 -4.5;
99 -6.578 0 -2.394; 100 -7 0 0; 101 -6.578 0 2.394; 102 -5.362 0 4.5;
103 -3.5 0 6.062; 104 -1.216 0 6.894; 105 1.216 0 6.894; 106 3.5 0 6.062;
107 5.362 0 4.5; 108 6.578 0 2.394; 109 3 0 0; 110 2.819 0 -1.026;
111 2.298 0 -1.928; 112 1.5 0 -2.598; 113 0.521 0 -2.954; 114 -0.521 0 -2.954;
115 -1.5 0 -2.598; 116 -2.298 0 -1.928; 117 -2.819 0 -1.026; 118 -3 0 0;
119 -2.819 0 1.026; 120 -2.298 0 1.928; 121 -1.5 0 2.598; 122 -0.521 0 2.954;
123 0.521 0 2.954; 124 1.5 0 2.598; 125 2.298 0 1.928; 126 2.819 0 1.026;
127 0 0 0;
ELEMENT INCIDENCES SHELL
1 1 2 20 19; 2 2 3 21 20; 3 3 4 22 21; 4 4 5 23 22; 5 5 6 24 23; 6 6 7 25 24;
7 7 8 26 25; 8 8 9 27 26; 9 9 10 28 27; 10 10 11 29 28; 11 11 12 30 29;
12 12 13 31 30; 13 13 14 32 31; 14 14 15 33 32; 15 15 16 34 33; 16 16 17 35 34;
17 17 18 36 35; 18 18 1 19 36; 19 19 20 38 37; 20 20 21 39 38; 21 21 22 40 39;
22 22 23 41 40; 23 23 24 42 41; 24 24 25 43 42; 25 25 26 44 43; 26 26 27 45 44;
27 27 28 46 45; 28 28 29 47 46; 29 29 30 48 47; 30 30 31 49 48; 31 31 32 50 49;
32 32 33 51 50; 33 33 34 52 51; 34 34 35 53 52; 35 35 36 54 53; 36 36 19 37 54;
37 37 38 56 55; 38 38 39 57 56; 39 39 40 58 57; 40 40 41 59 58; 41 41 42 60 59;
42 42 43 61 60; 43 43 44 62 61; 44 44 45 63 62; 45 45 46 64 63; 46 46 47 65 64;
47 47 48 66 65; 48 48 49 67 66; 49 49 50 68 67; 50 50 51 69 68; 51 51 52 70 69;
52 52 53 71 70; 53 53 54 72 71; 54 54 37 55 72; 55 55 56 74 73; 56 56 57 75 74;
57 57 58 76 75; 58 58 59 77 76; 59 59 60 78 77; 60 60 61 79 78; 61 61 62 80 79;
62 62 63 81 80; 63 63 64 82 81; 64 64 65 83 82; 65 65 66 84 83; 66 66 67 85 84;
67 67 68 86 85; 68 68 69 87 86; 69 69 70 88 87; 70 70 71 89 88; 71 71 72 90 89;
72 72 55 73 90; 73 73 74 92 91; 74 74 75 93 92; 75 75 76 94 93; 76 76 77 95 94;
77 77 78 96 95; 78 78 79 97 96; 79 79 80 98 97; 80 80 81 99 98;
81 81 82 100 99; 82 82 83 101 100; 83 83 84 102 101; 84 84 85 103 102;
85 85 86 104 103; 86 86 87 105 104; 87 87 88 106 105; 88 88 89 107 106;
89 89 90 108 107; 90 90 73 91 108; 91 91 92 110 109; 92 92 93 111 110;
93 93 94 112 111; 94 94 95 113 112; 95 95 96 114 113; 96 96 97 115 114;
97 97 98 116 115; 98 98 99 117 116; 99 99 100 118 117; 100 100 101 119 118;
101 101 102 120 119; 102 102 103 121 120; 103 103 104 122 121;
104 104 105 123 122; 105 105 106 124 123; 106 106 107 125 124;
107 107 108 126 125; 108 108 91 109 126; 109 127 109 110; 110 110 111 127;
111 127 111 112; 112 112 113 127; 113 127 113 114; 114 114 115 127;
115 127 115 116; 116 116 117 127; 117 127 117 118; 118 127 109 126;
119 126 125 127; 120 127 125 124; 121 124 123 127; 122 127 123 122;
123 122 121 127; 124 127 121 120; 125 120 119 127; 126 127 119 118;
ELEMENT PROPERTY
1 TO 126 THICKNESS 1
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 3e+07
POISSON 0.3
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 TO 18 FIXED
LOAD 1 UNIFORM LOAD
ELEMENT LOAD
1 TO 126 PR GY -6
LOAD 2 POINT LOAD
JOINT LOAD
127 FY -7539.82
PERFORM ANALYSIS
PRINT JOINT DISPLACEMENTS LIST 127
LOAD LIST 1
PRINT ELEMENT JOINT STRESSES LIST 9 109
LOAD LIST 2
PRINT ELEMENT JOINT STRESSES LIST 9
FINISH

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