V. 2D Surface with Hole Edge Stress
To find the normal stress on the edge of a circular hole in the center of a rectangular plate.
Reference
Young, W. C., Roark’s Formulas for Stress and Strain, McGraw-Hill Inc., 6th Edition, 1989 (Page 732, Type 7).
Problem
Find the normal stress on the edge of the circular hole for the plate shown, when an in-plane load causes tension. Use a one-quarter, doubly symmetric model.
One quarter of rectangular plate with hole
| E = 30,000.0 ksi |
| Size = 12.10 in × 7.0 in |
| Thickness = 0.1 in |
| Fillet radius = 1 in |
| Poisson’s ratio = 0.3 |
| P = 2,000 lbs |
Model with nodes and elements labeled
Comparison
| Result Type | Theory | STAAD.Pro | Difference |
|---|---|---|---|
| Stress on fillet (node 1, plate 1) (psi) | 9.475 | 9.489 | <1% |
STAAD Input
The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\04 Plates Shells\2D Surface with Hole Edge Stress.STD is typically installed with the program.
STAAD PLANE :STRESS CONCENTRATION IN A PLATE
START JOB INFORMATION
ENGINEER DATE 14-Sep-18
END JOB INFORMATION
*
* REFERENCE: "ROARK'S FORMULAS FOR STRESS AND STRAIN", YOUNG W. C.,
* SIXTH EDITION, MCGRAW-HILL, PAGE 732.
* 7. GENERAL CIRCULAR HOLE IN CENTER OF A MEMBER OF RECTANGULAR CROSS SECTION
* 1/4 MODEL DOUBLY SYMMETRIC
* D = 7.0 INCHES WIDTH
* R = 1.0 INCH HOLE RADIUS
* T = 0.1 INCH THICKNESS
* L = 12.1 INCH LENGTH
* LOAD = 2000 POUNDS
* THEORETICAL RESULTS
* AVERAGE STRESS AT ENDS = 2857 PSI (7 INCH WIDTH)
* AVERAGE NOMINAL STRESS AT CENTER = 4000 PSI (7-5 INCH WIDTH)
* K STRESS CONCENTRATION FACTOR = 2.3688
* MAXIMUM STRESS AT EDGE OF HOLE = 2.3688 * 4000 = 9475 PSI
* STAAD RESULTS
* MAXIMUM STRESS AT EDGE OF HOLE = 9489 PSI
* K STRESS CONCENTRATION FACTOR = 2.37225
* PERCENT ERROR = 0.15%
*
UNIT INCHES POUND
JOINT COORDINATES
1 0 1 0; 2 -0.174 0.985 0; 3 -0.342 0.94 0; 4 -0.5 0.866 0;
5 -0.643 0.766 0; 6 -0.766 0.643 0; 7 -0.866 0.5 0; 8 -0.94 0.342 0;
9 -0.985 0.174 0; 10 -1 0 0; 11 0 1.2 0; 12 -0.208 1.182 0;
13 -0.41 1.128 0; 14 -0.6 1.039 0; 15 -0.771 0.919 0; 16 -0.919 0.771 0;
17 -1.039 0.6 0; 18 -1.128 0.41 0; 19 -1.182 0.208 0; 20 -1.2 0 0;
21 0 1.4 0; 22 -0.243 1.379 0; 23 -0.479 1.316 0; 24 -0.7 1.212 0;
25 -0.9 1.072 0; 26 -1.072 0.9 0; 27 -1.212 0.7 0; 28 -1.316 0.479 0;
29 -1.379 0.243 0; 30 -1.4 0 0; 31 0 1.6 0; 32 -0.278 1.576 0;
33 -0.547 1.504 0; 34 -0.8 1.386 0; 35 -1.028 1.226 0;
36 -1.226 1.028 0; 37 -1.386 0.8 0; 38 -1.504 0.547 0;
39 -1.576 0.278 0; 40 -1.6 0 0; 41 -1.101 1.5 0; 42 -1.5 1.101 0;
43 -1.5 1.5 0; 44 0 1.8 0; 45 -0.3 1.8 0; 46 -0.6 1.8 0; 47 -0.9 1.8 0;
48 -1.2 1.8 0; 49 -1.5 1.8 0; 50 0 2.225 0; 51 -0.3 2.225 0;
52 -0.6 2.225 0; 53 -0.9 2.225 0; 54 -1.2 2.225 0; 55 -1.5 2.225 0;
56 0 2.65 0; 57 -0.3 2.65 0; 58 -0.6 2.65 0; 59 -0.9 2.65 0;
60 -1.2 2.65 0; 61 -1.5 2.65 0; 62 0 3.075 0; 63 -0.3 3.075 0;
64 -0.6 3.075 0; 65 -0.9 3.075 0; 66 -1.2 3.075 0; 67 -1.5 3.075 0;
68 0 3.5 0; 69 -0.3 3.5 0; 70 -0.6 3.5 0; 71 -0.9 3.5 0; 72 -1.2 3.5 0;
73 -1.5 3.5 0; 74 -1.8 0 0; 75 -2.225 0 0; 76 -2.65 0 0; 77 -3.075 0 0;
78 -3.5 0 0; 79 -3.925 0 0; 80 -4.35 0 0; 81 -4.775 0 0; 82 -5.2 0 0;
83 -5.625 0 0; 84 -6.05 0 0; 85 -1.8 0.3 0; 86 -2.225 0.3 0;
87 -2.65 0.3 0; 88 -3.075 0.3 0; 89 -3.5 0.3 0; 90 -3.925 0.3 0;
91 -4.35 0.3 0; 92 -4.775 0.3 0; 93 -5.2 0.3 0; 94 -5.625 0.3 0;
95 -6.05 0.3 0; 96 -1.8 0.6 0; 97 -2.225 0.6 0; 98 -2.65 0.6 0;
99 -3.075 0.6 0; 100 -3.5 0.6 0; 101 -3.925 0.6 0; 102 -4.35 0.6 0;
103 -4.775 0.6 0; 104 -5.2 0.6 0; 105 -5.625 0.6 0; 106 -6.05 0.6 0;
107 -1.8 0.9 0; 108 -2.225 0.9 0; 109 -2.65 0.9 0; 110 -3.075 0.9 0;
111 -3.5 0.9 0; 112 -3.925 0.9 0; 113 -4.35 0.9 0; 114 -4.775 0.9 0;
115 -5.2 0.9 0; 116 -5.625 0.9 0; 117 -6.05 0.9 0; 118 -1.8 1.2 0;
119 -2.225 1.2 0; 120 -2.65 1.2 0; 121 -3.075 1.2 0; 122 -3.5 1.2 0;
123 -3.925 1.2 0; 124 -4.35 1.2 0; 125 -4.775 1.2 0; 126 -5.2 1.2 0;
127 -5.625 1.2 0; 128 -6.05 1.2 0; 129 -1.8 1.5 0; 130 -2.225 1.5 0;
131 -2.65 1.5 0; 132 -3.075 1.5 0; 133 -3.5 1.5 0; 134 -3.925 1.5 0;
135 -4.35 1.5 0; 136 -4.775 1.5 0; 137 -5.2 1.5 0; 138 -5.625 1.5 0;
139 -6.05 1.5 0; 140 -1.8 1.8 0; 141 -2.225 1.8 0; 142 -2.65 1.8 0;
143 -3.075 1.8 0; 144 -3.5 1.8 0; 145 -3.925 1.8 0; 146 -4.35 1.8 0;
147 -4.775 1.8 0; 148 -5.2 1.8 0; 149 -5.625 1.8 0; 150 -6.05 1.8 0;
151 -1.8 2.225 0; 152 -2.225 2.225 0; 153 -2.65 2.225 0;
154 -3.075 2.225 0; 155 -3.5 2.225 0; 156 -3.925 2.225 0;
157 -4.35 2.225 0; 158 -4.775 2.225 0; 159 -5.2 2.225 0;
160 -5.625 2.225 0; 161 -6.05 2.225 0; 162 -1.8 2.65 0;
163 -2.225 2.65 0; 164 -2.65 2.65 0; 165 -3.075 2.65 0; 166 -3.5 2.65 0;
167 -3.925 2.65 0; 168 -4.35 2.65 0; 169 -4.775 2.65 0; 170 -5.2 2.65 0;
171 -5.625 2.65 0; 172 -6.05 2.65 0; 173 -1.8 3.075 0;
174 -2.225 3.075 0; 175 -2.65 3.075 0; 176 -3.075 3.075 0;
177 -3.5 3.075 0; 178 -3.925 3.075 0; 179 -4.35 3.075 0;
180 -4.775 3.075 0; 181 -5.2 3.075 0; 182 -5.625 3.075 0;
183 -6.05 3.075 0; 184 -1.8 3.5 0; 185 -2.225 3.5 0; 186 -2.65 3.5 0;
187 -3.075 3.5 0; 188 -3.5 3.5 0; 189 -3.925 3.5 0; 190 -4.35 3.5 0;
191 -4.775 3.5 0; 192 -5.2 3.5 0; 193 -5.625 3.5 0; 194 -6.05 3.5 0;
201 -1.301 1.301 0;
ELEMENT INCIDENCES SHELL
1 1 11 12 2; 2 2 12 13 3; 3 3 13 14 4; 4 4 14 15 5; 5 5 15 16 6;
6 6 16 17 7; 7 7 17 18 8; 8 8 18 19 9; 9 9 19 20 10; 11 11 21 22 12;
12 12 22 23 13; 13 13 23 24 14; 14 14 24 25 15; 15 15 25 26 16;
16 16 26 27 17; 17 17 27 28 18; 18 18 28 29 19; 19 19 29 30 20;
21 21 31 32 22; 22 22 32 33 23; 23 23 33 34 24; 24 24 34 35 25;
25 25 35 36 26; 26 26 36 37 27; 27 27 37 38 28; 28 28 38 39 29;
29 29 39 40 30; 31 31 44 45 32; 32 32 45 33; 33 33 47 34; 34 34 47 41;
35 34 41 35; 36 35 41 201; 37 35 201 36; 38 36 201 42; 39 36 42 37;
40 37 42 107; 41 37 107 38; 42 38 85 39; 43 39 85 74 40; 44 45 46 33;
45 46 47 33; 46 47 48 41; 47 41 48 43; 48 41 43 201; 49 201 43 42;
50 42 43 118; 51 42 118 107; 52 38 107 96; 53 38 96 85; 54 48 49 43;
55 43 129 118; 56 43 49 140 129; 57 74 85 86 75; 58 75 86 87 76;
59 76 87 88 77; 60 77 88 89 78; 61 78 89 90 79; 62 79 90 91 80;
63 80 91 92 81; 64 81 92 93 82; 65 82 93 94 83; 66 83 94 95 84;
77 85 96 97 86; 78 86 97 98 87; 79 87 98 99 88; 80 88 99 100 89;
81 89 100 101 90; 82 90 101 102 91; 83 91 102 103 92; 84 92 103 104 93;
85 93 104 105 94; 86 94 105 106 95; 97 96 107 108 97; 98 97 108 109 98;
99 98 109 110 99; 100 99 110 111 100; 101 100 111 112 101;
102 101 112 113 102; 103 102 113 114 103; 104 103 114 115 104;
105 104 115 116 105; 106 105 116 117 106; 117 107 118 119 108;
118 108 119 120 109; 119 109 120 121 110; 120 110 121 122 111;
121 111 122 123 112; 122 112 123 124 113; 123 113 124 125 114;
124 114 125 126 115; 125 115 126 127 116; 126 116 127 128 117;
137 118 129 130 119; 138 119 130 131 120; 139 120 131 132 121;
140 121 132 133 122; 141 122 133 134 123; 142 123 134 135 124;
143 124 135 136 125; 144 125 136 137 126; 145 126 137 138 127;
146 127 138 139 128; 157 129 140 141 130; 158 130 141 142 131;
159 131 142 143 132; 160 132 143 144 133; 161 133 144 145 134;
162 134 145 146 135; 163 135 146 147 136; 164 136 147 148 137;
165 137 148 149 138; 166 138 149 150 139; 171 44 50 51 45;
172 45 51 52 46; 173 46 52 53 47; 174 47 53 54 48; 175 48 54 55 49;
176 49 55 151 140; 177 140 151 152 141; 178 141 152 153 142;
179 142 153 154 143; 180 143 154 155 144; 181 144 155 156 145;
182 145 156 157 146; 183 146 157 158 147; 184 147 158 159 148;
185 148 159 160 149; 186 149 160 161 150; 191 50 56 57 51;
192 51 57 58 52; 193 52 58 59 53; 194 53 59 60 54; 195 54 60 61 55;
196 55 61 162 151; 197 151 162 163 152; 198 152 163 164 153;
199 153 164 165 154; 200 154 165 166 155; 201 155 166 167 156;
202 156 167 168 157; 203 157 168 169 158; 204 158 169 170 159;
205 159 170 171 160; 206 160 171 172 161; 211 56 62 63 57;
212 57 63 64 58; 213 58 64 65 59; 214 59 65 66 60; 215 60 66 67 61;
216 61 67 173 162; 217 162 173 174 163; 218 163 174 175 164;
219 164 175 176 165; 220 165 176 177 166; 221 166 177 178 167;
222 167 178 179 168; 223 168 179 180 169; 224 169 180 181 170;
225 170 181 182 171; 226 171 182 183 172; 231 62 68 69 63;
232 63 69 70 64; 233 64 70 71 65; 234 65 71 72 66; 235 66 72 73 67;
236 67 73 184 173; 237 173 184 185 174; 238 174 185 186 175;
239 175 186 187 176; 240 176 187 188 177; 241 177 188 189 178;
242 178 189 190 179; 243 179 190 191 180; 244 180 191 192 181;
245 181 192 193 182; 246 182 193 194 183;
ELEMENT PROPERTY
1 TO 9 11 TO 19 21 TO 29 31 TO 66 77 TO 86 THICKNESS 0.1
97 TO 106 117 TO 126 137 TO 146 157 TO 166 171 TO 186 191 TO 205 -
206 THICKNESS 0.1
211 TO 226 231 TO 246 THICKNESS 0.1
ELEMENT PLANE STRESS
1 TO 9 11 TO 19 21 TO 29 31 TO 66 77 TO 86 97 TO 106 117 TO 126 137 -
138 TO 146 157 TO 166 171 TO 186 191 TO 206 211 TO 226 231 TO 246
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 3e+07
POISSON 0.3
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
10 20 30 40 74 TO 84 FIXED BUT FX
1 11 21 31 44 50 56 62 68 FIXED BUT FY
LOAD 1 TENSILE LOAD
JOINT LOAD
84 FX -42.857
95 106 117 128 139 FX -85.714
150 FX -103.571
161 172 183 FX -121.428
194 FX -60.714
PERFORM ANALYSIS PRINT STATICS CHECK
PRINT JOINT DISPLACEMENTS LIST 6 84
PRINT ELEMENT JOINT STRESSES LIST 1
FINISH
STAAD Output
JOINT DISPLACEMENT (INCH RADIANS) STRUCTURE TYPE = PLANE ------------------ JOINT LOAD X-TRANS Y-TRANS Z-TRANS X-ROTAN Y-ROTAN Z-ROTAN 6 1 -0.00024 -0.00008 0.00000 0.00000 0.00000 -0.00029 84 1 -0.00066 0.00000 0.00000 0.00000 0.00000 0.00000 ************** END OF LATEST ANALYSIS RESULT ************** 156. PRINT ELEMENT JOINT STRESSES LIST 1 ELEMENT JOINT STRESSES LIST :STRESS CONCENTRATION IN A PLATE -- 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 1 1 0.00 0.00 0.00 0.00 0.00 7461.40 7461.40 650.49 7726.25 -439.78 7753.48 7753.48 TOP : SMAX= 7753.48 SMIN= 623.26 TMAX= 3565.11 ANGLE=-86.5 BOTT: SMAX= 7753.48 SMIN= 623.26 TMAX= 3565.11 ANGLE=-86.5 JOINT 0.00 0.00 0.00 0.00 0.00 1 9229.88 9229.88 751.63 9488.87 -757.04 TOP : SMAX= 9553.97 SMIN= 686.52 TMAX= 4433.73 ANGLE=-85.1 BOTT: SMAX= 9553.97 SMIN= 686.52 TMAX= 4433.73 ANGLE=-85.1 JOINT 0.00 0.00 0.00 0.00 0.00 11 5912.86 5912.86 608.93 6174.80 -272.64 TOP : SMAX= 6188.12 SMIN= 595.61 TMAX= 2796.26 ANGLE=-87.2 BOTT: SMAX= 6188.12 SMIN= 595.61 TMAX= 2796.26 ANGLE=-87.2 JOINT 0.00 0.00 0.00 0.00 0.00 12 5802.41 5802.41 548.98 6051.20 -154.55 TOP : SMAX= 6055.53 SMIN= 544.64 TMAX= 2755.44 ANGLE=-88.4 BOTT: SMAX= 6055.53 SMIN= 544.64 TMAX= 2755.44 ANGLE=-88.4 JOINT 0.00 0.00 0.00 0.00 0.00 2 8940.02 8940.02 666.87 9167.41 -719.10 TOP : SMAX= 9227.81 SMIN= 606.47 TMAX= 4310.67 ANGLE=-85.2 BOTT: SMAX= 9227.81 SMIN= 606.47 TMAX= 4310.67 ANGLE=-85.2 **** MAXIMUM STRESSES AMONG SELECTED PLATES AND CASES **** MAXIMUM MINIMUM MAXIMUM MAXIMUM MAXIMUM PRINCIPAL PRINCIPAL SHEAR VONMISES TRESCA STRESS STRESS STRESS STRESS STRESS 9.553974E+03 5.446445E+02 4.433728E+03 7.461400E+03 7.753480E+03 PLATE NO. 1 1 1 1 1 CASE NO. 1 1 1 1 1 ********************END OF ELEMENT FORCES********************