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V. 2D Circular Plate In-Plane Stresses

A thick cylindrical plate supported along 2 radial edges. Find the radial displacement, radial stress, tangential stress and longitudinal stress at inner surface due to a unit pressure applied at the inner surface.

Reference

MacNeal, R.H. and Harder, R.C., A Proposed Standard Set of Problems to Test Finite Element Accuracy, Finite Element in Analysis and Design 1, 1985.

Problem

Loading is unit pressure at inner radius

E = 1×(10)6 psi

Poisson’s ratio = 0.3

Inner radius = 3.0 in

Outer radius = 9.0 in

Semi-circular plate finite element model

Comparison

Table 1. Comparison of results
Result Type Theory STAAD.Pro Difference
Radial deflection (10-3 in) 4.582 4.650a 1.5%
Radial stress (psi) -1.00 -0.95b 5.0%
Tangential stress (psi) 1.25 1.26b 0.8%
Longitudinal stress (psi) 0.075 0.080 7.0%
  1. Radial displacements are measured along FY at node 102 and FX at node 101.
  2. On element 82, at node 102, SX is tangential stress, SY is radial stress.

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 Plate In-Plane Stresses.STD is typically installed with the program.

STAAD SPACE :A THICK WALLED CYLINDER PLATE SUPPORTED ALONG 2 RADIAL EDGES
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
*
* REFERENCE: MACNEAL AND HARDER, A PROPOSED STANDARD SET OF PROBLEMS
*            TO TEST FINITE ELEMENT ACCURACY,
*            FINITE ELEMENT IN ANALYSIS AND DESIGN 1, NORTH HOLLAND
*            1985
*
INPUT WIDTH 72
UNIT INCHES POUND
JOINT COORDINATES
1 9 0 0; 2 0 9 0; 3 1.563 8.863 0; 4 3.078 8.457 0; 5 4.5 7.794 0;
6 5.785 6.894 0; 7 6.894 5.785 0; 8 7.794 4.5 0; 9 8.457 3.078 0;
10 8.863 1.563 0; 11 7.875 0 0; 12 0 7.875 0; 13 1.367 7.755 0;
14 2.693 7.4 0; 15 3.938 6.82 0; 16 5.062 6.033 0; 17 6.033 5.062 0;
18 6.82 3.937 0; 19 7.4 2.693 0; 20 7.755 1.367 0; 21 6.75 0 0;
22 0 6.75 0; 23 1.172 6.647 0; 24 2.309 6.343 0; 25 3.375 5.846 0;
26 4.339 5.171 0; 27 5.171 4.339 0; 28 5.846 3.375 0; 29 6.343 2.309 0;
30 6.647 1.172 0; 31 5.975 0 0; 32 0 5.975 0; 33 1.038 5.884 0;
34 2.044 5.615 0; 35 2.988 5.174 0; 36 3.841 4.577 0; 37 4.577 3.841 0;
38 5.175 2.987 0; 39 5.615 2.044 0; 40 5.884 1.038 0; 41 5.2 0 0;
42 0 5.2 0; 43 0.903 5.121 0; 44 1.779 4.886 0; 45 2.6 4.503 0;
46 3.343 3.983 0; 47 3.983 3.342 0; 48 4.503 2.6 0; 49 4.886 1.778 0;
50 5.121 0.903 0; 51 4.7 0 0; 52 0 4.7 0; 53 0.816 4.629 0;
54 1.608 4.417 0; 55 2.35 4.07 0; 56 3.021 3.6 0; 57 3.6 3.021 0;
58 4.07 2.35 0; 59 4.417 1.607 0; 60 4.629 0.816 0; 61 4.2 0 0;
62 0 4.2 0; 63 0.729 4.136 0; 64 1.436 3.947 0; 65 2.1 3.637 0;
66 2.7 3.217 0; 67 3.217 2.7 0; 68 3.637 2.1 0; 69 3.947 1.436 0;
70 4.136 0.729 0; 71 3.85 0 0; 72 0 3.85 0; 73 0.669 3.792 0;
74 1.317 3.618 0; 75 1.925 3.334 0; 76 2.475 2.949 0; 77 2.949 2.475 0;
78 3.334 1.925 0; 79 3.618 1.317 0; 80 3.792 0.669 0; 81 3.5 0 0;
82 0 3.5 0; 83 0.608 3.447 0; 84 1.197 3.289 0; 85 1.75 3.031 0;
86 2.25 2.681 0; 87 2.681 2.25 0; 88 3.031 1.75 0; 89 3.289 1.197 0;
90 3.447 0.608 0; 91 3.25 0 0; 92 0 3.25 0; 93 0.564 3.201 0;
94 1.112 3.054 0; 95 1.625 2.815 0; 96 2.089 2.49 0; 97 2.49 2.089 0;
98 2.815 1.625 0; 99 3.054 1.112 0; 100 3.201 0.564 0; 101 3 0 0;
102 0 3 0; 103 0.521 2.954 0; 104 1.026 2.819 0; 105 1.5 2.598 0;
106 1.928 2.298 0; 107 2.298 1.928 0; 108 2.598 1.5 0;
109 2.819 1.026 0; 110 2.954 0.521 0;
ELEMENT INCIDENCES SHELL
1 2 3 13 12; 2 3 4 14 13; 3 4 5 15 14; 4 5 6 16 15; 5 6 7 17 16;
6 7 8 18 17; 7 8 9 19 18; 8 9 10 20 19; 9 10 1 11 20; 10 12 13 23 22;
11 13 14 24 23; 12 14 15 25 24; 13 15 16 26 25; 14 16 17 27 26;
15 17 18 28 27; 16 18 19 29 28; 17 19 20 30 29; 18 20 11 21 30;
19 22 23 33 32; 20 23 24 34 33; 21 24 25 35 34; 22 25 26 36 35;
23 26 27 37 36; 24 27 28 38 37; 25 28 29 39 38; 26 29 30 40 39;
27 30 21 31 40; 28 32 33 43 42; 29 33 34 44 43; 30 34 35 45 44;
31 35 36 46 45; 32 36 37 47 46; 33 37 38 48 47; 34 38 39 49 48;
35 39 40 50 49; 36 40 31 41 50; 37 42 43 53 52; 38 43 44 54 53;
39 44 45 55 54; 40 45 46 56 55; 41 46 47 57 56; 42 47 48 58 57;
43 48 49 59 58; 44 49 50 60 59; 45 50 41 51 60; 46 52 53 63 62;
47 53 54 64 63; 48 54 55 65 64; 49 55 56 66 65; 50 56 57 67 66;
51 57 58 68 67; 52 58 59 69 68; 53 59 60 70 69; 54 60 51 61 70;
55 62 63 73 72; 56 63 64 74 73; 57 64 65 75 74; 58 65 66 76 75;
59 66 67 77 76; 60 67 68 78 77; 61 68 69 79 78; 62 69 70 80 79;
63 70 61 71 80; 64 72 73 83 82; 65 73 74 84 83; 66 74 75 85 84;
67 75 76 86 85; 68 76 77 87 86; 69 77 78 88 87; 70 78 79 89 88;
71 79 80 90 89; 72 80 71 81 90; 73 82 83 93 92; 74 83 84 94 93;
75 84 85 95 94; 76 85 86 96 95; 77 86 87 97 96; 78 87 88 98 97;
79 88 89 99 98; 80 89 90 100 99; 81 90 81 91 100; 82 92 93 103 102;
83 93 94 104 103; 84 94 95 105 104; 85 95 96 106 105; 86 96 97 107 106;
87 97 98 108 107; 88 98 99 109 108; 89 99 100 110 109;
90 100 91 101 110;
ELEMENT PROPERTY
1 TO 90 THICKNESS 1
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 1e+06
POISSON 0.3
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
2 12 22 32 42 52 62 72 82 92 102 FIXED BUT FY
1 11 21 31 41 51 61 71 81 91 101 FIXED BUT FX
LOAD 1 UNIT PRESSURE AT INNER RADIUS
JOINT LOAD
101 FX 0.2618
110 FX 0.5156 FY 0.0909
109 FX 0.492 FY 0.1791
108 FX 0.4534 FY 0.2618
107 FX 0.4011 FY 0.3366
106 FX 0.3366 FY 0.4011
105 FX 0.2618 FY 0.4534
104 FX 0.1791 FY 0.492
103 FX 0.0909 FY 0.5156
102 FY 0.2618
* CREATED LOAD 2 IN ORDER TO PRINT DISPLACEMENT VALUES
LOAD 2 MULTIPLY LOAD BY 1000
REPEAT LOAD
1 1000.0 
PERFORM ANALYSIS
LOAD LIST 2
PRINT JOINT DISPLACEMENTS LIST 101 102 107
LOAD LIST 1
PRINT ELEMENT JOINT STRESSES LIST 82 83 89 90
FINISH

STAAD Output

   JOINT DISPLACEMENT (INCH RADIANS)    STRUCTURE TYPE = SPACE
   ------------------
 JOINT  LOAD   X-TRANS   Y-TRANS   Z-TRANS   X-ROTAN   Y-ROTAN   Z-ROTAN
    101    2    0.00465   0.00000   0.00000   0.00000   0.00000   0.00000
    102    2    0.00000   0.00465   0.00000   0.00000   0.00000   0.00000
    107    2    0.00357   0.00299   0.00000   0.00000   0.00000   0.00000
   ************** END OF LATEST ANALYSIS RESULT **************
    97. LOAD LIST 1
    98. PRINT ELEMENT JOINT STRESSES LIST 82 83 89 90
  ELEMENT  JOINT    STRESSES LIST     
      :A THICK WALLED CYLINDER PLATE SUPPORTED ALONG 2 RADIAL  -- 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
     82      1         0.00        0.00        0.00        0.00        0.00
                       1.81        1.81        1.17       -0.91        0.00
                       2.08        2.08
       TOP : SMAX=       1.17 SMIN=      -0.91 TMAX=       1.04 ANGLE=  0.0
       BOTT: SMAX=       1.17 SMIN=      -0.91 TMAX=       1.04 ANGLE=  0.0
         JOINT         0.00        0.00        0.00        0.00        0.00
           92          1.69        1.69        1.07       -0.88       -0.08
       TOP : SMAX=       1.07 SMIN=      -0.88 TMAX=       0.98 ANGLE= -2.3
       BOTT: SMAX=       1.07 SMIN=      -0.88 TMAX=       0.98 ANGLE= -2.3
         JOINT         0.00        0.00        0.00        0.00        0.00
           93          1.69        1.69        1.07       -0.88        0.08
       TOP : SMAX=       1.07 SMIN=      -0.88 TMAX=       0.98 ANGLE=  2.3
       BOTT: SMAX=       1.07 SMIN=      -0.88 TMAX=       0.98 ANGLE=  2.3
         JOINT         0.00        0.00        0.00        0.00        0.00
          103          1.93        1.93        1.26       -0.95        0.08
       TOP : SMAX=       1.26 SMIN=      -0.95 TMAX=       1.11 ANGLE=  2.1
       BOTT: SMAX=       1.26 SMIN=      -0.95 TMAX=       1.11 ANGLE=  2.1
         JOINT         0.00        0.00        0.00        0.00        0.00
          102          1.93        1.93        1.26       -0.95       -0.08
       TOP : SMAX=       1.26 SMIN=      -0.95 TMAX=       1.11 ANGLE= -2.0
       BOTT: SMAX=       1.26 SMIN=      -0.95 TMAX=       1.11 ANGLE= -2.0
     83      1         0.00        0.00        0.00        0.00        0.00
                       1.81        1.81        1.17       -0.92       -0.00
                       2.08        2.08
       TOP : SMAX=       1.17 SMIN=      -0.92 TMAX=       1.04 ANGLE= -0.0
       BOTT: SMAX=       1.17 SMIN=      -0.92 TMAX=       1.04 ANGLE= -0.0
         JOINT         0.00        0.00        0.00        0.00        0.00
           93          1.69        1.69        1.07       -0.88       -0.08
       TOP : SMAX=       1.07 SMIN=      -0.88 TMAX=       0.98 ANGLE= -2.3
       BOTT: SMAX=       1.07 SMIN=      -0.88 TMAX=       0.98 ANGLE= -2.3
         JOINT         0.00        0.00        0.00        0.00        0.00
           94          1.69        1.69        1.06       -0.88        0.08
       TOP : SMAX=       1.07 SMIN=      -0.88 TMAX=       0.98 ANGLE=  2.3
       BOTT: SMAX=       1.07 SMIN=      -0.88 TMAX=       0.98 ANGLE=  2.3
         JOINT         0.00        0.00        0.00        0.00        0.00
          104          1.93        1.93        1.26       -0.95        0.08
       TOP : SMAX=       1.26 SMIN=      -0.96 TMAX=       1.11 ANGLE=  2.1
       BOTT: SMAX=       1.26 SMIN=      -0.96 TMAX=       1.11 ANGLE=  2.1
         JOINT         0.00        0.00        0.00        0.00        0.00
          103          1.93        1.93        1.26       -0.95       -0.08
       TOP : SMAX=       1.26 SMIN=      -0.96 TMAX=       1.11 ANGLE= -2.1
       BOTT: SMAX=       1.26 SMIN=      -0.96 TMAX=       1.11 ANGLE= -2.1
     89      1         0.00        0.00        0.00        0.00        0.00
                       1.81        1.81        1.17       -0.92        0.00
                       2.08        2.08
       TOP : SMAX=       1.17 SMIN=      -0.92 TMAX=       1.04 ANGLE=  0.0
       BOTT: SMAX=       1.17 SMIN=      -0.92 TMAX=       1.04 ANGLE=  0.0
      :A THICK WALLED CYLINDER PLATE SUPPORTED ALONG 2 RADIAL  -- PAGE NO.    6
   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
         JOINT         0.00        0.00        0.00        0.00        0.00
           99          1.69        1.69        1.06       -0.88       -0.08
       TOP : SMAX=       1.07 SMIN=      -0.88 TMAX=       0.98 ANGLE= -2.3
       BOTT: SMAX=       1.07 SMIN=      -0.88 TMAX=       0.98 ANGLE= -2.3
         JOINT         0.00        0.00        0.00        0.00        0.00
          100          1.69        1.69        1.07       -0.88        0.08
       TOP : SMAX=       1.07 SMIN=      -0.88 TMAX=       0.98 ANGLE=  2.3
       BOTT: SMAX=       1.07 SMIN=      -0.88 TMAX=       0.98 ANGLE=  2.3
         JOINT         0.00        0.00        0.00        0.00        0.00
          110          1.93        1.93        1.26       -0.95        0.08
       TOP : SMAX=       1.26 SMIN=      -0.96 TMAX=       1.11 ANGLE=  2.1
       BOTT: SMAX=       1.26 SMIN=      -0.96 TMAX=       1.11 ANGLE=  2.1
         JOINT         0.00        0.00        0.00        0.00        0.00
          109          1.93        1.93        1.26       -0.95       -0.08
       TOP : SMAX=       1.26 SMIN=      -0.96 TMAX=       1.11 ANGLE= -2.1
       BOTT: SMAX=       1.26 SMIN=      -0.96 TMAX=       1.11 ANGLE= -2.1
     90      1         0.00        0.00        0.00        0.00        0.00
                       1.81        1.81        1.17       -0.91       -0.00
                       2.08        2.08
       TOP : SMAX=       1.17 SMIN=      -0.91 TMAX=       1.04 ANGLE= -0.0
       BOTT: SMAX=       1.17 SMIN=      -0.91 TMAX=       1.04 ANGLE= -0.0
         JOINT         0.00        0.00        0.00        0.00        0.00
          100          1.69        1.69        1.07       -0.88       -0.08
       TOP : SMAX=       1.07 SMIN=      -0.88 TMAX=       0.98 ANGLE= -2.3
       BOTT: SMAX=       1.07 SMIN=      -0.88 TMAX=       0.98 ANGLE= -2.3
         JOINT         0.00        0.00        0.00        0.00        0.00
           91          1.69        1.69        1.07       -0.88        0.08
       TOP : SMAX=       1.07 SMIN=      -0.88 TMAX=       0.98 ANGLE=  2.3
       BOTT: SMAX=       1.07 SMIN=      -0.88 TMAX=       0.98 ANGLE=  2.3
         JOINT         0.00        0.00        0.00        0.00        0.00
          101          1.93        1.93        1.26       -0.95        0.08
       TOP : SMAX=       1.26 SMIN=      -0.95 TMAX=       1.11 ANGLE=  2.0
       BOTT: SMAX=       1.26 SMIN=      -0.95 TMAX=       1.11 ANGLE=  2.0
         JOINT         0.00        0.00        0.00        0.00        0.00
          110          1.93        1.93        1.26       -0.95       -0.08
       TOP : SMAX=       1.26 SMIN=      -0.95 TMAX=       1.11 ANGLE= -2.1
       BOTT: SMAX=       1.26 SMIN=      -0.95 TMAX=       1.11 ANGLE= -2.1
      :A THICK WALLED CYLINDER PLATE SUPPORTED ALONG 2 RADIAL  -- PAGE NO.    7
          **** MAXIMUM STRESSES AMONG SELECTED PLATES AND CASES ****
             MAXIMUM       MINIMUM       MAXIMUM       MAXIMUM       MAXIMUM
            PRINCIPAL     PRINCIPAL       SHEAR       VONMISES       TRESCA
             STRESS        STRESS        STRESS        STRESS        STRESS
          1.263807E+00 -9.557953E-01  1.109365E+00  1.807750E+00  2.082330E+00
 PLATE NO.      82            83            82            89            89
 CASE  NO.       1             1             1             1             1
Parent topic: V.04 Plate and Shell Elements