V. 2D Rectangular slab with simply supported edges

To find the vertical deflection and bending moments at several points due to a unit pressure on a thin rectangular plate simply supported along 4 edges.

Reference

Timoshenko, S. and Woinowsky-Kreiger, S., Theory of Plates and Shells, McGraw-Hill, 2nd Edition, 1959, Pages 113-117.

Details

Loading is unit pressure (1 psi) over entire surface.

Surface geometry:

Length = 16 in.
Width = 10 in.
Thickness = 0.2 in.

Material properties:

E = 1×(10)⁶ psi
Poisson’s ratio = 0.3

Surface model

Comparison

Table 1. Comparison of results
Result Type		Theory	STAAD.Pro	Difference
Vertical deflection, δ (in) at joint	162	0.036	0.03549	1.2%
	166	0.113	0.11230	1.0%
	306	0.025	0.02478	1.4%
Bending moment, Mx (in·lb) at joint	9	1.763	1.64	7.0%
	89	8.513	8.41	1.2%
	96	1.098	1.05	4.4%
Bending moment, My (in·lb) at joint	9	0.897	0.86	4.1%
	89	4.873	4.81	1.3%
	96	1.108	0.99	10.7%

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 Rectangular slab with simply supported edges.std is typically installed with the program.

STAAD SPACE :UNIFORM PRESSURE ON RECTANGULAR PLATE ELEMENTS
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
* FILE:  2D Rectangular slab with simply supported edges.STD
*
* REFERENCE:  "THEORY OF PLATES AND SHELLS", TIMOSHENKO, S. AND
*             WOINOWSKY-KREIGER, S., SECOND EDITION, MCGRAW-HILL,
*             PAGES 113-117.
*
UNIT INCHES POUND
JOINT COORDINATES
1 0 0 0; 2 1 0 0; 3 2 0 0; 4 3 0 0; 5 4 0 0; 6 5 0 0; 7 6 0 0; 8 7 0 0;
9 8 0 0; 10 9 0 0; 11 10 0 0; 21 0 0 1; 22 1 0 1; 23 2 0 1; 24 3 0 1;
25 4 0 1; 26 5 0 1; 27 6 0 1; 28 7 0 1; 29 8 0 1; 30 9 0 1; 31 10 0 1;
41 0 0 2; 42 1 0 2; 43 2 0 2; 44 3 0 2; 45 4 0 2; 46 5 0 2; 47 6 0 2;
48 7 0 2; 49 8 0 2; 50 9 0 2; 51 10 0 2; 61 0 0 3; 62 1 0 3; 63 2 0 3;
64 3 0 3; 65 4 0 3; 66 5 0 3; 67 6 0 3; 68 7 0 3; 69 8 0 3; 70 9 0 3;
71 10 0 3; 81 0 0 4; 82 1 0 4; 83 2 0 4; 84 3 0 4; 85 4 0 4; 86 5 0 4;
87 6 0 4; 88 7 0 4; 89 8 0 4; 90 9 0 4; 91 10 0 4; 101 0 0 5; 102 1 0 5;
103 2 0 5; 104 3 0 5; 105 4 0 5; 106 5 0 5; 107 6 0 5; 108 7 0 5;
109 8 0 5; 110 9 0 5; 111 10 0 5; 121 0 0 6; 122 1 0 6; 123 2 0 6;
124 3 0 6; 125 4 0 6; 126 5 0 6; 127 6 0 6; 128 7 0 6; 129 8 0 6;
130 9 0 6; 131 10 0 6; 141 0 0 7; 142 1 0 7; 143 2 0 7; 144 3 0 7;
145 4 0 7; 146 5 0 7; 147 6 0 7; 148 7 0 7; 149 8 0 7; 150 9 0 7;
151 10 0 7; 161 0 0 8; 162 1 0 8; 163 2 0 8; 164 3 0 8; 165 4 0 8;
166 5 0 8; 167 6 0 8; 168 7 0 8; 169 8 0 8; 170 9 0 8; 171 10 0 8;
181 0 0 9; 182 1 0 9; 183 2 0 9; 184 3 0 9; 185 4 0 9; 186 5 0 9;
187 6 0 9; 188 7 0 9; 189 8 0 9; 190 9 0 9; 191 10 0 9; 201 0 0 10;
202 1 0 10; 203 2 0 10; 204 3 0 10; 205 4 0 10; 206 5 0 10; 207 6 0 10;
208 7 0 10; 209 8 0 10; 210 9 0 10; 211 10 0 10; 221 0 0 11; 222 1 0 11;
223 2 0 11; 224 3 0 11; 225 4 0 11; 226 5 0 11; 227 6 0 11; 228 7 0 11;
229 8 0 11; 230 9 0 11; 231 10 0 11; 241 0 0 12; 242 1 0 12; 243 2 0 12;
244 3 0 12; 245 4 0 12; 246 5 0 12; 247 6 0 12; 248 7 0 12; 249 8 0 12;
250 9 0 12; 251 10 0 12; 261 0 0 13; 262 1 0 13; 263 2 0 13; 264 3 0 13;
265 4 0 13; 266 5 0 13; 267 6 0 13; 268 7 0 13; 269 8 0 13; 270 9 0 13;
271 10 0 13; 281 0 0 14; 282 1 0 14; 283 2 0 14; 284 3 0 14; 285 4 0 14;
286 5 0 14; 287 6 0 14; 288 7 0 14; 289 8 0 14; 290 9 0 14; 291 10 0 14;
301 0 0 15; 302 1 0 15; 303 2 0 15; 304 3 0 15; 305 4 0 15; 306 5 0 15;
307 6 0 15; 308 7 0 15; 309 8 0 15; 310 9 0 15; 311 10 0 15; 321 0 0 16;
322 1 0 16; 323 2 0 16; 324 3 0 16; 325 4 0 16; 326 5 0 16; 327 6 0 16;
328 7 0 16; 329 8 0 16; 330 9 0 16; 331 10 0 16;
ELEMENT INCIDENCES SHELL
1 1 2 22 21; 2 21 22 42 41; 3 41 42 62 61; 4 61 62 82 81;
5 81 82 102 101; 6 101 102 122 121; 7 121 122 142 141;
8 141 142 162 161; 9 161 162 182 181; 10 181 182 202 201;
11 201 202 222 221; 12 221 222 242 241; 13 241 242 262 261;
14 261 262 282 281; 15 281 282 302 301; 16 301 302 322 321;
21 2 3 23 22; 22 22 23 43 42; 23 42 43 63 62; 24 62 63 83 82;
25 82 83 103 102; 26 102 103 123 122; 27 122 123 143 142;
28 142 143 163 162; 29 162 163 183 182; 30 182 183 203 202;
31 202 203 223 222; 32 222 223 243 242; 33 242 243 263 262;
34 262 263 283 282; 35 282 283 303 302; 36 302 303 323 322;
41 3 4 24 23; 42 23 24 44 43; 43 43 44 64 63; 44 63 64 84 83;
45 83 84 104 103; 46 103 104 124 123; 47 123 124 144 143;
48 143 144 164 163; 49 163 164 184 183; 50 183 184 204 203;
51 203 204 224 223; 52 223 224 244 243; 53 243 244 264 263;
54 263 264 284 283; 55 283 284 304 303; 56 303 304 324 323;
61 4 5 25 24; 62 24 25 45 44; 63 44 45 65 64; 64 64 65 85 84;
65 84 85 105 104; 66 104 105 125 124; 67 124 125 145 144;
68 144 145 165 164; 69 164 165 185 184; 70 184 185 205 204;
71 204 205 225 224; 72 224 225 245 244; 73 244 245 265 264;
74 264 265 285 284; 75 284 285 305 304; 76 304 305 325 324;
81 5 6 26 25; 82 25 26 46 45; 83 45 46 66 65; 84 65 66 86 85;
85 85 86 106 105; 86 105 106 126 125; 87 125 126 146 145;
88 145 146 166 165; 89 165 166 186 185; 90 185 186 206 205;
91 205 206 226 225; 92 225 226 246 245; 93 245 246 266 265;
94 265 266 286 285; 95 285 286 306 305; 96 305 306 326 325;
101 6 7 27 26; 102 26 27 47 46; 103 46 47 67 66; 104 66 67 87 86;
105 86 87 107 106; 106 106 107 127 126; 107 126 127 147 146;
108 146 147 167 166; 109 166 167 187 186; 110 186 187 207 206;
111 206 207 227 226; 112 226 227 247 246; 113 246 247 267 266;
114 266 267 287 286; 115 286 287 307 306; 116 306 307 327 326;
121 7 8 28 27; 122 27 28 48 47; 123 47 48 68 67; 124 67 68 88 87;
125 87 88 108 107; 126 107 108 128 127; 127 127 128 148 147;
128 147 148 168 167; 129 167 168 188 187; 130 187 188 208 207;
131 207 208 228 227; 132 227 228 248 247; 133 247 248 268 267;
134 267 268 288 287; 135 287 288 308 307; 136 307 308 328 327;
141 8 9 29 28; 142 28 29 49 48; 143 48 49 69 68; 144 68 69 89 88;
145 88 89 109 108; 146 108 109 129 128; 147 128 129 149 148;
148 148 149 169 168; 149 168 169 189 188; 150 188 189 209 208;
151 208 209 229 228; 152 228 229 249 248; 153 248 249 269 268;
154 268 269 289 288; 155 288 289 309 308; 156 308 309 329 328;
161 9 10 30 29; 162 29 30 50 49; 163 49 50 70 69; 164 69 70 90 89;
165 89 90 110 109; 166 109 110 130 129; 167 129 130 150 149;
168 149 150 170 169; 169 169 170 190 189; 170 189 190 210 209;
171 209 210 230 229; 172 229 230 250 249; 173 249 250 270 269;
174 269 270 290 289; 175 289 290 310 309; 176 309 310 330 329;
181 10 11 31 30; 182 30 31 51 50; 183 50 51 71 70; 184 70 71 91 90;
185 90 91 111 110; 186 110 111 131 130; 187 130 131 151 150;
188 150 151 171 170; 189 170 171 191 190; 190 190 191 211 210;
191 210 211 231 230; 192 230 231 251 250; 193 250 251 271 270;
194 270 271 291 290; 195 290 291 311 310; 196 310 311 331 330;
ELEMENT PROPERTY
1 TO 16 21 TO 36 41 TO 56 61 TO 76 81 TO 96 THICKNESS 0.2
101 TO 116 121 TO 136 141 TO 156 161 TO 176 181 TO 196 THICKNESS 0.2
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 1e+06
POISSON 0.3
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 11 321 331 FIXED
2 TO 10 322 TO 330 FIXED BUT MX
21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201 -
211 221 231 241 251 261 271 281 291 301 311 FIXED BUT MZ
LOAD 1 UNIFORM PRESSURE
ELEMENT LOAD
1 TO 16 21 TO 36 41 TO 56 61 TO 76 81 TO 96 PR 1
101 TO 116 121 TO 136 141 TO 156 161 TO 176 181 TO 196 PR 1
PERFORM ANALYSIS
PRINT JOINT DISPLACEMENTS LIST 162 166 306
PRINT ELEMENT STRESSES LIST 9 89 96
FINISH

STAAD Output

   JOINT DISPLACEMENT (INCH RADIANS)    STRUCTURE TYPE = SPACE
   ------------------
 JOINT  LOAD   X-TRANS   Y-TRANS   Z-TRANS   X-ROTAN   Y-ROTAN   Z-ROTAN
    162    1    0.00000  -0.03549   0.00000   0.00000   0.00000  -0.03420
    166    1    0.00000  -0.11230   0.00000   0.00000   0.00000   0.00000
    306    1    0.00000  -0.02478   0.00000  -0.02424   0.00000   0.00000
   ************** END OF LATEST ANALYSIS RESULT **************
   115. PRINT ELEMENT STRESSES LIST 9 89 96
  ELEMENT  STRESSES LIST     9        
      :UNIFORM PRESSURE ON RECTANGULAR PLATE ELEMENTS          -- 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        19.24       -0.08        1.64        0.86        0.28
                     225.14      225.14        0.00        0.00        0.00
                     259.42      259.42
       TOP : SMAX=     259.42 SMIN=     115.15 TMAX=      72.14 ANGLE= 17.6
       BOTT: SMAX=    -115.15 SMIN=    -259.42 TMAX=      72.14 ANGLE=-72.4
     89      1         1.99       -0.51        8.41        4.81        0.04
                    1096.84     1096.84        0.00        0.00        0.00
                    1262.19     1262.19
       TOP : SMAX=    1262.19 SMIN=     721.80 TMAX=     270.20 ANGLE=  0.7
       BOTT: SMAX=    -721.80 SMIN=   -1262.19 TMAX=     270.20 ANGLE=-89.3
     96      1         0.25      -15.87        1.05        0.99        0.58
                     214.93      214.93        0.00        0.00        0.00
                     240.24      240.24
       TOP : SMAX=     240.24 SMIN=      66.20 TMAX=      87.02 ANGLE= 43.5
       BOTT: SMAX=     -66.20 SMIN=    -240.24 TMAX=      87.02 ANGLE=-46.5
          **** MAXIMUM STRESSES AMONG SELECTED PLATES AND CASES ****
             MAXIMUM       MINIMUM       MAXIMUM       MAXIMUM       MAXIMUM
            PRINCIPAL     PRINCIPAL       SHEAR       VONMISES       TRESCA
             STRESS        STRESS        STRESS        STRESS        STRESS
          1.262189E+03 -1.262189E+03  2.701960E+02  1.096844E+03  1.262189E+03
 PLATE NO.      89            89            89            89            89
 CASE  NO.       1             1             1             1             1
   ********************END OF ELEMENT FORCES********************