V. Curved Roof Displacements and Stresses
A Cylindrical roof is supported along two circular edges. Using plate/shell elements, find the vertical deflection at the center of the free edge, principal stresses at the center of the support and center of the free edge (top and bottom of the roof plate) due to uniformly distributed gravity load.
Reference
Scordelis, A.C. and Lo, K.S., "Computer Analysis of Cylindrical Shells", Journal of the American Concrete Institute, Vol. 61, May 1964.
Problem
For the cylindrical roof shell calculate the following deflection and stresses due to the gravity load.
The vertical deflection, δy, at the center of the free edge.
Principal stresses, σmax and σmin, at the center line section at the vertical angle (top and bottom of the roof plate element). Principal stresses, σmax and σmin, at the center section of the free edge (top and bottom of the roof plate element).
| E = 4.32×(10)8 psf |
| t = 3.0 in. |
| Poisson’s ratio = 0.0 in theory (0.01001 in STAAD) |
| w = 90 psf (uniform on surface). |
| L = 50 feet. |
| r = 25 feet, 40º sector either side of vertical |
Boundary conditions: simply supported on circular edges
Comparison
| Result Type | Theory | STAAD.Pro | Difference | Comments | |
|---|---|---|---|---|---|
| δy, at the center of the free edge (in) (y translation at node 63) | -3.703 | 3.650 | 1.4% | A finer mesh may reduce the difference between the theoretical and software output. | |
| σ at center (ft·kip) | Bottom | 191.23 | 181.55a | 5.1% | |
| Top | 218.74 | 209.16a | 4.4% | ||
| σ at free edge (ft·kip) | Bottom | 215.57 | 208.33b | 3.4% | |
| Top | 340.7 | 334.95b | 1.7% | ||
STAAD Input
The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\04 Plates Shells\Curved Roof Displacements and Stresses.STD is typically installed with the program.
STAAD SPACE SCORDELIS-LO ROOF
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
* A CYLINDRICAL ROOF SUPPORTED ALONG TWO CIRCULAR EDGES
*
* REFERENCE: SCORDELIS, A.C., AND LO K. S., COMPUTER ANALYSIS
* OF CYLINDRICAL SHELLS, JOURNAL OF THE AMERICAN CONCRETE
* INSTITUTE, VOL. 61, MAY 1964.
*
* Theory displacement = .3086 feet or 3.7032 inches
* STAAD displacement with POIS = 0.30
* Joint 63&173 -3.79217 inches
* Joint 63&173 2.01523 -3.79217 0.00009 -0.00010 0.00019 0.03358
* STAAD displacement with POIS = 0.01001
* Joint 63&173 -3.65022 inches
* Joint 63&173 1.92962 -3.65022 0.00009 -0.00008 0.00015 0.03148
INPUT WIDTH 72
UNIT FEET POUND
JOINT COORDINATES
1 0 25 0; 2 2.179 24.905 0; 3 4.341 24.62 0; 4 6.471 24.148 0;
5 8.551 23.492 0; 6 10.566 22.658 0; 7 12.5 21.651 0; 8 14.339 20.479 0;
9 16.07 19.151 0; 10 0 25 4.167; 11 2.179 24.905 4.167;
12 4.341 24.62 4.167; 13 6.471 24.148 4.167; 14 8.551 23.492 4.167;
15 10.566 22.658 4.167; 16 12.5 21.651 4.167; 17 14.339 20.479 4.167;
18 16.07 19.151 4.167; 19 0 25 8.334; 20 2.179 24.905 8.334;
21 4.341 24.62 8.334; 22 6.471 24.148 8.334; 23 8.551 23.492 8.334;
24 10.566 22.658 8.334; 25 12.5 21.651 8.334; 26 14.339 20.479 8.334;
27 16.07 19.151 8.334; 28 0 25 12.501; 29 2.179 24.905 12.501;
30 4.341 24.62 12.501; 31 6.471 24.148 12.501; 32 8.551 23.492 12.501;
33 10.566 22.658 12.501; 34 12.5 21.651 12.501; 35 14.339 20.479 12.501;
36 16.07 19.151 12.501; 37 0 25 16.668; 38 2.179 24.905 16.668;
39 4.341 24.62 16.668; 40 6.471 24.148 16.668; 41 8.551 23.492 16.668;
42 10.566 22.658 16.668; 43 12.5 21.651 16.668; 44 14.339 20.479 16.668;
45 16.07 19.151 16.668; 46 0 25 20.835; 47 2.179 24.905 20.835;
48 4.341 24.62 20.835; 49 6.471 24.148 20.835; 50 8.551 23.492 20.835;
51 10.566 22.658 20.835; 52 12.5 21.651 20.835; 53 14.339 20.479 20.835;
54 16.07 19.151 20.835; 55 0 25 25.002; 56 2.179 24.905 25.002;
57 4.341 24.62 25.002; 58 6.471 24.148 25.002; 59 8.551 23.492 25.002;
60 10.566 22.658 25.002; 61 12.5 21.651 25.002; 62 14.339 20.479 25.002;
63 16.07 19.151 25.002; 64 0 25 29.169; 65 2.179 24.905 29.169;
66 4.341 24.62 29.169; 67 6.471 24.148 29.169; 68 8.551 23.492 29.169;
69 10.566 22.658 29.169; 70 12.5 21.651 29.169; 71 14.339 20.479 29.169;
72 16.07 19.151 29.169; 73 0 25 33.336; 74 2.179 24.905 33.336;
75 4.341 24.62 33.336; 76 6.471 24.148 33.336; 77 8.551 23.492 33.336;
78 10.566 22.658 33.336; 79 12.5 21.651 33.336; 80 14.339 20.479 33.336;
81 16.07 19.151 33.336; 82 0 25 37.503; 83 2.179 24.905 37.503;
84 4.341 24.62 37.503; 85 6.471 24.148 37.503; 86 8.551 23.492 37.503;
87 10.566 22.658 37.503; 88 12.5 21.651 37.503; 89 14.339 20.479 37.503;
90 16.07 19.151 37.503; 91 0 25 41.67; 92 2.179 24.905 41.67;
93 4.341 24.62 41.67; 94 6.471 24.148 41.67; 95 8.551 23.492 41.67;
96 10.566 22.658 41.67; 97 12.5 21.651 41.67; 98 14.339 20.479 41.67;
99 16.07 19.151 41.67; 100 0 25 45.837; 101 2.179 24.905 45.837;
102 4.341 24.62 45.837; 103 6.471 24.148 45.837;
104 8.551 23.492 45.837; 105 10.566 22.658 45.837;
106 12.5 21.651 45.837; 107 14.339 20.479 45.837;
108 16.07 19.151 45.837; 109 0 25 50.004; 110 2.179 24.905 50.004;
111 4.341 24.62 50.004; 112 6.471 24.148 50.004;
113 8.551 23.492 50.004; 114 10.566 22.658 50.004;
115 12.5 21.651 50.004; 116 14.339 20.479 50.004;
117 16.07 19.151 50.004; 118 -2.179 24.905 0; 119 -4.341 24.62 0;
120 -6.471 24.148 0; 121 -8.551 23.492 0; 122 -10.566 22.658 0;
123 -12.5 21.651 0; 124 -14.339 20.479 0; 125 -16.07 19.151 0;
126 -2.179 24.905 4.167; 127 -4.341 24.62 4.167;
128 -6.471 24.148 4.167; 129 -8.551 23.492 4.167;
130 -10.566 22.658 4.167; 131 -12.5 21.651 4.167;
132 -14.339 20.479 4.167; 133 -16.07 19.151 4.167;
134 -2.179 24.905 8.334; 135 -4.341 24.62 8.334;
136 -6.471 24.148 8.334; 137 -8.551 23.492 8.334;
138 -10.566 22.658 8.334; 139 -12.5 21.651 8.334;
140 -14.339 20.479 8.334; 141 -16.07 19.151 8.334;
142 -2.179 24.905 12.501; 143 -4.341 24.62 12.501;
144 -6.471 24.148 12.501; 145 -8.551 23.492 12.501;
146 -10.566 22.658 12.501; 147 -12.5 21.651 12.501;
148 -14.339 20.479 12.501; 149 -16.07 19.151 12.501;
150 -2.179 24.905 16.668; 151 -4.341 24.62 16.668;
152 -6.471 24.148 16.668; 153 -8.551 23.492 16.668;
154 -10.566 22.658 16.668; 155 -12.5 21.651 16.668;
156 -14.339 20.479 16.668; 157 -16.07 19.151 16.668;
158 -2.179 24.905 20.835; 159 -4.341 24.62 20.835;
160 -6.471 24.148 20.835; 161 -8.551 23.492 20.835;
162 -10.566 22.658 20.835; 163 -12.5 21.651 20.835;
164 -14.339 20.479 20.835; 165 -16.07 19.151 20.835;
166 -2.179 24.905 25.002; 167 -4.341 24.62 25.002;
168 -6.471 24.148 25.002; 169 -8.551 23.492 25.002;
170 -10.566 22.658 25.002; 171 -12.5 21.651 25.002;
172 -14.339 20.479 25.002; 173 -16.07 19.151 25.002;
174 -2.179 24.905 29.169; 175 -4.341 24.62 29.169;
176 -6.471 24.148 29.169; 177 -8.551 23.492 29.169;
178 -10.566 22.658 29.169; 179 -12.5 21.651 29.169;
180 -14.339 20.479 29.169; 181 -16.07 19.151 29.169;
182 -2.179 24.905 33.336; 183 -4.341 24.62 33.336;
184 -6.471 24.148 33.336; 185 -8.551 23.492 33.336;
186 -10.566 22.658 33.336; 187 -12.5 21.651 33.336;
188 -14.339 20.479 33.336; 189 -16.07 19.151 33.336;
190 -2.179 24.905 37.503; 191 -4.341 24.62 37.503;
192 -6.471 24.148 37.503; 193 -8.551 23.492 37.503;
194 -10.566 22.658 37.503; 195 -12.5 21.651 37.503;
196 -14.339 20.479 37.503; 197 -16.07 19.151 37.503;
198 -2.179 24.905 41.67; 199 -4.341 24.62 41.67;
200 -6.471 24.148 41.67; 201 -8.551 23.492 41.67;
202 -10.566 22.658 41.67; 203 -12.5 21.651 41.67;
204 -14.339 20.479 41.67; 205 -16.07 19.151 41.67;
206 -2.179 24.905 45.837; 207 -4.341 24.62 45.837;
208 -6.471 24.148 45.837; 209 -8.551 23.492 45.837;
210 -10.566 22.658 45.837; 211 -12.5 21.651 45.837;
212 -14.339 20.479 45.837; 213 -16.07 19.151 45.837;
214 -2.179 24.905 50.004; 215 -4.341 24.62 50.004;
216 -6.471 24.148 50.004; 217 -8.551 23.492 50.004;
218 -10.566 22.658 50.004; 219 -12.5 21.651 50.004;
220 -14.339 20.479 50.004; 221 -16.07 19.151 50.004;
ELEMENT INCIDENCES SHELL
1 1 2 10; 2 10 2 11; 3 11 2 3; 4 11 3 12; 5 12 3 4; 6 4 13 12; 7 4 5 14;
8 14 13 4; 9 5 15 14; 10 5 6 15; 11 15 7 16; 12 16 7 17; 13 17 7 8;
14 8 18 17; 15 8 9 18; 16 6 7 15; 17 10 11 19; 18 19 11 20; 19 20 11 12;
20 20 12 21; 21 21 12 13; 22 13 22 21; 23 13 14 23; 24 23 22 13;
25 14 24 23; 26 14 15 24; 27 24 16 25; 28 25 16 26; 29 26 16 17;
30 17 27 26; 31 17 18 27; 32 15 16 24; 33 19 20 28; 34 28 20 29;
35 29 20 21; 36 29 21 30; 37 30 21 22; 38 22 31 30; 39 22 23 32;
40 32 31 22; 41 23 33 32; 42 23 24 33; 43 33 25 34; 44 34 25 35;
45 35 25 26; 46 26 36 35; 47 26 27 36; 48 24 25 33; 49 28 29 37;
50 37 29 38; 51 38 29 30; 52 38 30 39; 53 39 30 31; 54 31 40 39;
55 31 32 41; 56 41 40 31; 57 32 42 41; 58 32 33 42; 59 42 34 43;
60 43 34 44; 61 44 34 35; 62 35 45 44; 63 35 36 45; 64 33 34 42;
65 37 38 46; 66 46 38 47; 67 47 38 39; 68 47 39 48; 69 48 39 40;
70 40 49 48; 71 40 41 50; 72 50 49 40; 73 41 51 50; 74 41 42 51;
75 51 43 52; 76 52 43 53; 77 53 43 44; 78 44 54 53; 79 44 45 54;
80 42 43 51; 81 46 47 55; 82 55 47 56; 83 56 47 48; 84 56 48 57;
85 57 48 49; 86 49 58 57; 87 49 50 59; 88 59 58 49; 89 50 60 59;
90 50 51 60; 91 60 52 61; 92 61 52 62; 93 62 52 53; 94 53 63 62;
95 53 54 63; 96 51 52 60; 97 55 56 64; 98 64 56 65; 99 65 56 57;
100 65 57 66; 101 66 57 58; 102 58 67 66; 103 58 59 68; 104 68 67 58;
105 59 69 68; 106 59 60 69; 107 69 61 70; 108 70 61 71; 109 71 61 62;
110 62 72 71; 111 62 63 72; 112 60 61 69; 113 64 65 73; 114 73 65 74;
115 74 65 66; 116 74 66 75; 117 75 66 67; 118 67 76 75; 119 67 68 77;
120 77 76 67; 121 68 78 77; 122 68 69 78; 123 78 70 79; 124 79 70 80;
125 80 70 71; 126 71 81 80; 127 71 72 81; 128 69 70 78; 129 73 74 82;
130 82 74 83; 131 83 74 75; 132 83 75 84; 133 84 75 76; 134 76 85 84;
135 76 77 86; 136 86 85 76; 137 77 87 86; 138 77 78 87; 139 87 79 88;
140 88 79 89; 141 89 79 80; 142 80 90 89; 143 80 81 90; 144 78 79 87;
145 82 83 91; 146 91 83 92; 147 92 83 84; 148 92 84 93; 149 93 84 85;
150 85 94 93; 151 85 86 95; 152 95 94 85; 153 86 96 95; 154 86 87 96;
155 96 88 97; 156 97 88 98; 157 98 88 89; 158 89 99 98; 159 89 90 99;
160 87 88 96; 161 91 92 100; 162 100 92 101; 163 101 92 93;
164 101 93 102; 165 102 93 94; 166 94 103 102; 167 94 95 104;
168 104 103 94; 169 95 105 104; 170 95 96 105; 171 105 97 106;
172 106 97 107; 173 107 97 98; 174 98 108 107; 175 98 99 108;
176 96 97 105; 177 100 101 109; 178 109 101 110; 179 110 101 102;
180 110 102 111; 181 111 102 103; 182 103 112 111; 183 103 104 113;
184 113 112 103; 185 104 114 113; 186 104 105 114; 187 114 106 115;
188 115 106 116; 189 116 106 107; 190 107 117 116; 191 107 108 117;
192 105 106 114; 193 1 118 10; 194 10 118 126; 195 126 118 119;
196 126 119 127; 197 127 119 120; 198 120 128 127; 199 120 121 129;
200 129 128 120; 201 121 130 129; 202 121 122 130; 203 130 123 131;
204 131 123 132; 205 132 123 124; 206 124 133 132; 207 124 125 133;
208 122 123 130; 209 10 126 19; 210 19 126 134; 211 134 126 127;
212 134 127 135; 213 135 127 128; 214 128 136 135; 215 128 129 137;
216 137 136 128; 217 129 138 137; 218 129 130 138; 219 138 131 139;
220 139 131 140; 221 140 131 132; 222 132 141 140; 223 132 133 141;
224 130 131 138; 225 19 134 28; 226 28 134 142; 227 142 134 135;
228 142 135 143; 229 143 135 136; 230 136 144 143; 231 136 137 145;
232 145 144 136; 233 137 146 145; 234 137 138 146; 235 146 139 147;
236 147 139 148; 237 148 139 140; 238 140 149 148; 239 140 141 149;
240 138 139 146; 241 28 142 37; 242 37 142 150; 243 150 142 143;
244 150 143 151; 245 151 143 144; 246 144 152 151; 247 144 145 153;
248 153 152 144; 249 145 154 153; 250 145 146 154; 251 154 147 155;
252 155 147 156; 253 156 147 148; 254 148 157 156; 255 148 149 157;
256 146 147 154; 257 37 150 46; 258 46 150 158; 259 158 150 151;
260 158 151 159; 261 159 151 152; 262 152 160 159; 263 152 153 161;
264 161 160 152; 265 153 162 161; 266 153 154 162; 267 162 155 163;
268 163 155 164; 269 164 155 156; 270 156 165 164; 271 156 157 165;
272 154 155 162; 273 46 158 55; 274 55 158 166; 275 166 158 159;
276 166 159 167; 277 167 159 160; 278 160 168 167; 279 160 161 169;
280 169 168 160; 281 161 170 169; 282 161 162 170; 283 170 163 171;
284 171 163 172; 285 172 163 164; 286 164 173 172; 287 164 165 173;
288 162 163 170; 289 55 166 64; 290 64 166 174; 291 174 166 167;
292 174 167 175; 293 175 167 168; 294 168 176 175; 295 168 169 177;
296 177 176 168; 297 169 178 177; 298 169 170 178; 299 178 171 179;
300 179 171 180; 301 180 171 172; 302 172 181 180; 303 172 173 181;
304 170 171 178; 305 64 174 73; 306 73 174 182; 307 182 174 175;
308 182 175 183; 309 183 175 176; 310 176 184 183; 311 176 177 185;
312 185 184 176; 313 177 186 185; 314 177 178 186; 315 186 179 187;
316 187 179 188; 317 188 179 180; 318 180 189 188; 319 180 181 189;
320 178 179 186; 321 73 182 82; 322 82 182 190; 323 190 182 183;
324 190 183 191; 325 191 183 184; 326 184 192 191; 327 184 185 193;
328 193 192 184; 329 185 194 193; 330 185 186 194; 331 194 187 195;
332 195 187 196; 333 196 187 188; 334 188 197 196; 335 188 189 197;
336 186 187 194; 337 82 190 91; 338 91 190 198; 339 198 190 191;
340 198 191 199; 341 199 191 192; 342 192 200 199; 343 192 193 201;
344 201 200 192; 345 193 202 201; 346 193 194 202; 347 202 195 203;
348 203 195 204; 349 204 195 196; 350 196 205 204; 351 196 197 205;
352 194 195 202; 353 91 198 100; 354 100 198 206; 355 206 198 199;
356 206 199 207; 357 207 199 200; 358 200 208 207; 359 200 201 209;
360 209 208 200; 361 201 210 209; 362 201 202 210; 363 210 203 211;
364 211 203 212; 365 212 203 204; 366 204 213 212; 367 204 205 213;
368 202 203 210; 369 100 206 109; 370 109 206 214; 371 214 206 207;
372 214 207 215; 373 215 207 208; 374 208 216 215; 375 208 209 217;
376 217 216 208; 377 209 218 217; 378 209 210 218; 379 218 211 219;
380 219 211 220; 381 220 211 212; 382 212 221 220; 383 212 213 221;
384 210 211 218;
ELEMENT PROPERTY
1 TO 384 THICKNESS 0.25
UNIT FEET KIP
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 432000
POISSON 0.01001
DENSITY 0.36
END DEFINE MATERIAL
UNIT FEET POUND
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 TO 9 109 TO 125 214 TO 221 FIXED BUT FZ MX MY
55 FIXED BUT FX FY MX MY MZ KFZ 12
LOAD 1 DEAD LOAD (SELFWEIGHT OF 90 PSF)
SELFWEIGHT Y -1
PERFORM ANALYSIS
UNIT FEET KIP
PRINT JOINT DISPLACEMENTS LIST 55 63 173
PRINT ELEMENT JOINT STRESSES LIST 97 111
FINISH
STAAD Output
JOINT DISPLACEMENT (INCH RADIANS) STRUCTURE TYPE = SPACE ------------------ JOINT LOAD X-TRANS Y-TRANS Z-TRANS X-ROTAN Y-ROTAN Z-ROTAN 55 1 0.00000 0.54458 0.00000 0.00001 0.00000 0.00000 63 1 -1.92962 -3.65022 0.00009 -0.00008 -0.00015 -0.03148 173 1 1.92962 -3.65022 0.00009 -0.00008 0.00015 0.03148 ************** END OF LATEST ANALYSIS RESULT ************** 221. PRINT ELEMENT JOINT STRESSES LIST 97 111 ELEMENT JOINT STRESSES LIST SCORDELIS-LO ROOF -- PAGE NO. 7 ELEMENT STRESSES FORCE,LENGTH UNITS= KIP FEET ---------------- STRESS = FORCE/UNIT WIDTH/THICK, MOMENT = FORCE-LENGTH/UNIT WIDTH ELEMENT LOAD SQX SQY MX MY MXY VONT VONB SX SY SXY TRESCAT TRESCAB 97 1 0.14 0.06 -2.01 -0.10 -0.00 200.58 176.28 -14.22 -4.43 1.22 207.05 178.61 TOP : SMAX= -13.63 SMIN= -207.05 TMAX= 96.71 ANGLE= 89.7 BOTT: SMAX= 178.61 SMIN= 4.77 TMAX= 86.92 ANGLE= 0.4 JOINT 0.14 0.06 -2.03 -0.12 -0.00 55 202.01 178.11 -13.81 -4.02 1.22 TOP : SMAX= -15.15 SMIN= -209.16 TMAX= 97.00 ANGLE= 89.7 BOTT: SMAX= 181.55 SMIN= 7.10 TMAX= 87.22 ANGLE= 0.4 JOINT 0.14 0.06 -1.96 -0.12 -0.00 56 193.92 171.12 -13.81 -5.25 1.22 TOP : SMAX= -16.38 SMIN= -201.59 TMAX= 92.61 ANGLE= 89.7 BOTT: SMAX= 173.98 SMIN= 5.88 TMAX= 84.05 ANGLE= 0.5 JOINT 0.14 0.06 -2.03 -0.06 -0.00 64 205.87 179.66 -15.04 -4.02 1.22 TOP : SMAX= -9.36 SMIN= -210.40 TMAX= 100.52 ANGLE= 89.7 BOTT: SMAX= 180.31 SMIN= 1.31 TMAX= 89.50 ANGLE= 0.4 111 1 0.27 -0.24 0.01 0.58 0.06 318.91 210.31 -15.62 254.40 -16.47 326.36 218.90 TOP : SMAX= 310.89 SMIN= -15.47 TMAX= 163.18 ANGLE=-88.1 BOTT: SMAX= 200.52 SMIN= -18.38 TMAX= 109.45 ANGLE=-84.2 JOINT 0.27 -0.24 -0.09 0.67 0.06 62 296.45 162.76 -8.74 222.47 -16.47 TOP : SMAX= 287.08 SMIN= -17.93 TMAX= 152.51 ANGLE=-88.0 BOTT: SMAX= 161.26 SMIN= -2.96 TMAX= 82.11 ANGLE=-82.2 JOINT 0.27 -0.24 0.05 0.67 0.06 63 336.92 216.84 -8.74 270.37 -16.47 TOP : SMAX= 334.95 SMIN= -3.91 TMAX= 169.43 ANGLE=-88.2 BOTT: SMAX= 208.33 SMIN= -16.10 TMAX= 112.22 ANGLE=-84.3 JOINT 0.27 -0.24 0.05 0.42 0.06 72 323.64 252.42 -29.38 270.37 -16.47 TOP : SMAX= 310.66 SMIN= -24.56 TMAX= 167.61 ANGLE=-88.1 BOTT: SMAX= 232.26 SMIN= -36.38 TMAX= 134.32 ANGLE=-85.3 SCORDELIS-LO ROOF -- PAGE NO. 8 **** MAXIMUM STRESSES AMONG SELECTED PLATES AND CASES **** MAXIMUM MINIMUM MAXIMUM MAXIMUM MAXIMUM PRINCIPAL PRINCIPAL SHEAR VONMISES TRESCA STRESS STRESS STRESS STRESS STRESS 3.349464E+02 -2.103963E+02 1.694301E+02 3.189089E+02 3.263608E+02 PLATE NO. 111 97 111 111 111 CASE NO. 1 1 1 1 1 ********************END OF ELEMENT FORCES********************