Formulas for flat plates with straight boundaries and constant thickness
Case no., shape, and supports 1. Rectangular plate; all edges simply supported
Case no., loading
Formulas and tabulated specific values (At center) smax
1a. Uniform over entire plate
bqb2 ¼ sb ¼ 2 t
and ymax
aqb4 ¼ Et3
(At center of long sides) Rmax ¼ gqb a=b b a g
1.0
1.2
1.4
1.6
1.8
2.0
3.0
4.0
5.0
1
0.2874 0.0444 0.420
0.3762 0.0616 0.455
0.4530 0.0770 0.478
0.5172 0.0906 0.491
0.5688 0.1017 0.499
0.6102 0.1110 0.503
0.7134 0.1335 0.505
0.7410 0.1400 0.502
0.7476 0.1417 0.501
0.7500 0.1421 0.500
(At center) smax
1b. Uniform over small concentric circle of radius ro (note definition of r0o )
3W 2b ¼ ð1 þ nÞ ln 0 þ b 2pt2 pro
ymax ¼
aWb2 Et3
a=b
1.0
1.2
1.4
1.6
1.8
2.0
1
0.435 0.1267
0.650 0.1478
0.789 0.1621
0.875 0.1715
0.927 0.1770
0.958 0.1805
1.000 0.1851
b a
(Ref. 21 for n ¼ 0:3)
Formulas for Stress and Strain
NOTATION: The notation for Table 11.2 applies with the following modifications: a and b refer to plate dimensions, and when used as subscripts for stress, they refer to the stresses in directions parallel to the sides a and b, respectively. s is a bending stress which is positive when tensile on the bottom and compressive on the top if loadings are considered vertically downward. R is the reaction force per unit length normal to the plate surface exerted by the boundary support on the edge of the plate. r0o is the equivalent radius of contact for a load concentrated on a very small area pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and is given by r0o ¼ 1:6r2o þ t2 0:675t if ro < 0:5t and r0o ¼ ro if ro 5 0:5t
502
TABLE 11.4
(Ref. 21 for n ¼ 0:3)
[CHAP. 11
TABLE 11.4
Formulas for flat plates with straight boundaries and constant thickness (Continued ) smax ¼ sb ¼
a1 =b b1 =b
0
0.2
a¼b 0.4 0.6
0.8
1.0
1.82 1.39 1.12 0.92 0.76
1.82 1.28 1.07 0.90 0.76 0.63
1.38 1.08 0.84 0.72 0.62 0.52
0.93 0.76 0.62 0.52 0.42 0.35
0.76 0.63 0.52 0.43 0.36 0.30
0 0.2 0.4 0.6 0.8 1.0
where W ¼ qa1 b1
1.12 0.90 0.72 0.60 0.51 0.42
0
0.2
a ¼ 1:4b 0.4 0.8
1.2
1.4
1.78 1.39 1.10 0.90 0.75
2.0 1.43 1.13 0.91 0.76 0.62
1.55 1.23 1.00 0.82 0.68 0.57
0.84 0.74 0.62 0.53 0.45 0.38
0.75 0.64 0.55 0.47 0.40 0.33
1.12 0.95 0.80 0.68 0.57 0.47
0
0.4
a ¼ 2b 0.8 1.2
1.6
2.0
1.73 1.32 1.04 0.87 0.71
1.64 1.31 1.08 0.90 0.76 0.61
1.20 1.03 0.88 0.76 0.63 0.53
0.78 0.68 0.60 0.54 0.44 0.38
0.64 0.57 0.50 0.44 0.38 0.30
0.97 0.84 0.74 0.64 0.54 0.45
11.14]
bW t2
(At center)
SEC.
1c. Uniform over central rectangular area
(Values from charts of Ref. 8; n ¼ 0:3.) 1d. Uniformly increasing along length
smax ¼
bqb2 t2
a=b
1
1.5
2.0
2.5
3.0
3.5
4.0
0.16 0.022
0.26 0.043
0.34 0.060
0.38 0.070
0.43 0.078
0.47 0.086
0.49 0.091
3.5
4.0
b a
and
ymax ¼
aqb4 Et3
(Values from charts of Ref. 8; n ¼ 0:3.) 1e. Uniformly increasing along width
smax ¼
bqb2 t2
a=b
1
1.5
0.16 0.022
0.26 0.042
b a
and
ymax ¼ 2.0
aqb4 Et3 2.5
0.32 0.35 0.056 0.063
3.0 0.37 0.067
0.38 0.38 0.069 0.070
(Values from charts of Ref. 8; n ¼ 0:3.)
Flat Plates 503
TABLE 11.4
Formulas for flat plates with straight boundaries and constant thickness (Continued ) Case no., loading
Formulas and tabulated specific values qb4 qb2 ymax ¼ a 3 ðsa Þmax ¼ bx 2 Et t the following values: Coef.
P=PE
ðsb Þmax
qb2 a P p2 Et3 ¼ by 2 : Here a; bx ; and by depend on ratios ; where PE ¼ ; and have and b PE t 3ð1 n2 Þb2
0
0.15
1
2
3
4
5
a
1 112 2 3 4
0.044 0.084 0.110 0.133 0.140
0.039 0.075 0.100 0.125 0.136
0.023 0.045 0.067 0.100 0.118
0.015 0.0305 0.0475 0.081 0.102
0.011 0.024 0.0375 0.066 0.089
0.008 0.019 0.030 0.057 0.080
0.0075 0.0170 0.0260 0.0490 0.072
bx
1 112 2 3 4
0.287 0.300 0.278 0.246 0.222
0.135 0.150 0.162 0.180 0.192
0.096 0.105 0.117 0.150 0.168
0.072 0.078 0.093 0.126 0.156
0.054 0.066 0.075 0.105 0.138
0.045 0.048 0.069 0.093 0.124
by
1 112 2 3 4
0.287 0.487 0.610 0.713 0.741
0.132 0.240 0.360 0.510 0.624
0.084 0.156 0.258 0.414 0.540
0.054 0.114 0.198 0.348 0.480
0.036 0.090 0.162 0.294 0.420
0.030 0.072 0.138 0.258 0.372
a
1 112 2 3 4
0.044 0.084 0.110 0.131 0.140
0.060 0.109 0.139 0.145 0.142
0.094 0.155 0.161 0.150 0.142
0.180 0.237 0.181 0.150 0.138
bx
1 112 2 3 4
0.287 0.300 0.278 0.246 0.222
0.372 0.372 0.330 0.228 0.225
0.606 0.522 0.390 0.228 0.225
1.236 0.846 0.450 0.210 0.225
by
1 112 2 3 4
0.287 0.487 0.610 0.713 0.741
0.420 0.624 0.720 0.750 0.750
0.600 0.786 0.900 0.792 0.750
1.260 1.380 1.020 0.750 0.750
a=b
0.25
0.50
0.75
P, Tension 0.030 0.060 0.084 0.1135 0.1280
Formulas for Stress and Strain
1f. Uniform over entire plate plus uniform tension or compression P lb=linear in applied to short edges
504
Case no., shape, and supports
P, Compression
[CHAP. 11
In the above formulas sa and sb are stresses due to bending only; add direct stress P=t to sa
TABLE 11.4
(Ref. 41)
Formulas for flat plates with straight boundaries and constant thickness (Continued ) qb4 qb2 ðsa Þmax ¼ bx 2 Et3 t the following values: ymax ¼ a
ðsb Þmax ¼ by
qb2 a P p2 Et3 and : Here a; bx ; and by depend on ratios ; where PE ¼ ; and have b PE t2 3ð1 n2 Þb2
0
0.15
0.5
1
2
3
4
5
a
1 112 2 3 4
0.044 0.084 0.110 0.133 0.140
0.035 0.060 0.075 0.085 0.088
0.022 0.035 0.042 0.045 0.046
0.015 0.022 0.025 0.026 0.026
0.008 0.012 0.014 0.016 0.016
0.006 0.008 0.010 0.011 0.011
0.004 0.006 0.007 0.008 0.008
0.003 0.005 0.006 0.007 0.007
bx
1 112 2 3 4
0.287 0.300 0.278 0.246 0.222
0.216 0.204 0.189 0.183 0.183
0.132 0.117 0.111 0.108 0.108
0.084 0.075 0.072 0.070 0.074
0.048 0.045 0.044 0.043 0.047
0.033 0.031 0.031 0.031 0.032
0.026 0.024 0.024 0.025 0.027
0.021 0.020 0.020 0.020 0.024
by
1 112 2 3 4
0.287 0.487 0.610 0.713 0.741
0.222 0.342 0.302 0.444 0.456
0.138 0.186 0.216 0.234 0.240
0.090 0.108 0.132 0.141 0.144
0.051 0.066 0.072 0.078 0.078
0.036 0.042 0.051 0.054 0.054
0.030 0.036 0.042 0.042 0.042
0.024 0.030 0.036 0.036 0.036
In the above formulas sa and sb are stresses due to bending only; add direct stress P=t to sa and sb : 2. Rectangular plate; three edges simply supported, one edge (b) free
2a. Uniform over entire plate
smax
bqb2 ¼ 2 t
and ymax
(Ref. 42)
aqb4 ¼ Et3
0.50
0.667
1.0
1.5
2.0
4.0
b a
0.36 0.080
0.45 0.106
0.67 0.140
0.77 0.160
0.79 0.165
0.80 0.167 (Ref. 8 for v ¼ 0:3)
2d. Uniformly increasing along the a side
smax
and ymax
aqb4 ¼ Et3
0.50
0.667
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.11 0.026
0.16 0.033
0.20 0.040
0.28 0.050
0.32 0.058
0.35 0.064
0.36 0.067
0.37 0.069
0.37 0.070
505
a=b b a
Flat Plates
a=b
bqb2 ¼ 2 t
11.14]
P=PE a=b
Coef.
SEC.
1g. Uniform over entire plate plus uniform tension P lb=linear in applied to all edges
(Ref. 8 for v ¼ 0:3)
TABLE 11.4
Formulas for flat plates with straight boundaries and constant thickness (Continued ) Case no., loading
Formulas and tabulated specific values
3a. Uniform over entire plate
smax ¼ a=b b a
bqb t2
2
ymax ¼
and
aqb Et3
4
1
1.5
2.0
2.5
3.0
3.5
4.0
0.50 0.030
0.67 0.071
0.73 0.101
0.74 0.122
0.75 0.132
0.75 0.137
0.75 0.139
Formulas for Stress and Strain
3. Rectangular plate; three edges simply supported, one short edge (b) fixed
506
Case no., shape, and supports
(Values from charts of Ref. 8; n ¼ 0:3) 4. Rectangular plate; three edges simply supported, one long edge (a) fixed
4a. Uniform over entire plate
smax ¼ a=b b a
bqb2 t2
ymax ¼
and
aqb4 Et3
1
1.5
2.0
2.5
3.0
3.5
4.0
0.50 0.030
0.66 0.046
0.73 0.054
0.74 0.056
0.74 0.057
0.75 0.058
0.75 0.058
(Values from charts of Ref. 8; n ¼ 0:3) 5. Rectangular plate; two long edges simply supported, two short edges fixed
5a. Uniform over entire plate
(At center of short edges) smax ¼ ymax ¼
(At center) a=b b a
aqb4 Et3
bqb2 t2
1
1.2
1.4
1.6
1.8
2
1
0.4182 0.0210
0.5208 0.0349
0.5988 0.0502
0.6540 0.0658
0.6912 0.0800
0.7146 0.0922
0.750 (Ref. 21)
6. Rectangular plate; two long edges fixed, two short edges simply supported
6a. Uniform over entire plate
smax
(At center of long edges) (At center)
b a
aqb4 Et3
1
1.2
1.4
1.6
1.8
2
1
0.4182 0.0210
0.4626 0.0243
0.4860 0.0262
0.4968 0.0273
0.4971 0.0280
0.4973 0.0283
0.500 0.0285 (Ref. 21)
TABLE 11.4
[CHAP. 11
a=b
ymax ¼
bqb2 ¼ t2
Formulas for flat plates with straight boundaries and constant thickness (Continued )
(At center of free edge) (At end of free edge)
2 3
7aa. Uniform over of plate from fixed edge
b1 qb2 t2 b qb2 s¼ 22 t
(At center of fixed edge) s ¼
and
R ¼ g1 qb
11.14]
7a. Uniform over entire plate
SEC.
7. Rectangular plate; one edge fixed, opposite edge free, remaining edges simply supported
R ¼ g2 qb
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b1 b2 g1 g2
0.044 0.048 0.183 0.131
0.176 0.190 0.368 0.295
0.380 0.386 0.541 0.526
0.665 0.565 0.701 0.832
1.282 0.730 0.919 1.491
1.804 0.688 1.018 1.979
2.450 0.434 1.055 2.401
(At center of fixed edge) s ¼
(Ref. 49 for n ¼ 0:2)
bqb and R ¼ gqb t2 2
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b g
0.044 0.183
0.161 0.348
0.298 0.466
0.454 0.551
0.730 0.645
0.932 0.681
1.158 0.689 (Ref. 49 for n ¼ 0:2)
7aaa. Uniform over 13 of plate from fixed edge
(At center of fixed edge) s ¼
bqb2 t2
and
R ¼ gqb
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b g
0.040 0.172
0.106 0.266
0.150 0.302
0.190 0.320
0.244 0.334
0.277 0.338
0.310 0.338 (Ref. 49 for n ¼ 0:2Þ
(At center of fixed edge) s ¼
bqb2 t2
and
R ¼ gqb
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b g
0.037 0.159
0.120 0.275
0.212 0.354
0.321 0.413
0.523 0.482
0.677 0.509
0.866 0.517
Flat Plates
7d. Uniformly decreasing from fixed edge to free edge
(Ref. 49 for n ¼ 0:2Þ
507
TABLE 11.4
Formulas for flat plates with straight boundaries and constant thickness (Continued ) Case no., loading
Formulas and tabulated specific values (At center of fixed edge) s ¼
bqb t2
2
R ¼ gqb
and
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b g
0.033 0.148
0.094 0.233
0.146 0.277
0.200 0.304
0.272 0.330
0.339 0.339
0.400 0.340 (Ref. 49 for n ¼ 0:2)
7ddd. Uniformly decreasing from fixed edge to zero at 13 b
(At center of fixed edge) s ¼
bqb2 t2
R ¼ gqb
and
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b g
0.023 0.115
0.048 0.149
0.061 0.159
0.073 0.164
0.088 0.167
0.097 0.168
0.105 0.168 (Ref. 49 for n ¼ 0:2)
7f. Distributed line load w lb=in along free edge
b1 wb t2 b wb (At center of free edge) sa ¼ 2 2 t
(At center of fixed edge) sb ¼
Formulas for Stress and Strain
7dd. Uniformly decreasing from fixed edge to zero at 23 b
508
Case no., shape, and supports
and R ¼ g1 w
(At ends of free edge) R ¼ g2 w
8. Rectangular plate, all edges fixed
8a. Uniform over entire plate
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b1 b2 g1 g2
0.000 0.321 0.000 1.236
0.024 0.780 0.028 2.381
0.188 1.204 0.160 3.458
0.570 1.554 0.371 4.510
1.726 1.868 0.774 6.416
2.899 1.747 1.004 7.772
4.508 1.120 1.119 9.031
b1 b2 a
1.0
1.2
1.4
1.6
1.8
2.0
1
0.3078 0.1386 0.0138
0.3834 0.1794 0.0188
0.4356 0.2094 0.0226
0.4680 0.2286 0.0251
0.4872 0.2406 0.0267
0.4974 0.2472 0.0277
0.5000 0.2500 0.0284
[CHAP. 11
a=b
TABLE 11.4
(Ref. 49 for n ¼ 0:2)
b1 qb2 (At center of long edge) smax ¼ t2 2 b qb aqb4 (At center) s ¼ 2 2 and ymax ¼ t Et3
(Refs. 7 and 25 and Ref. 21 for n ¼ 0:3)
Formulas for flat plates with straight boundaries and constant thickness (Continued ) 3W 2b ð1 þ nÞ ln 0 þ b1 and 2pt2 pro
(At center of long edge) sb ¼ a=b b1 b2 a
ymax ¼
aWb2 Et3
11.14]
(At center) sb ¼
SEC.
8b. Uniform over small concentric circle of radius ro (note definition of r0o )
b2 W t2
1.0
1.2
1.4
1.6
1.8
2.0
1
0.238 0.7542 0.0611
0.078 0.8940 0.0706
0.011 0.9624 0.0754
0.053 0.9906 0.0777
0.068 1.0000 0.0786
0.067 1.004 0.0788
0.067 1.008 0.0791 (Ref. 26 and Ref. 21 for n ¼ 0:3Þ
8d. Uniformly decreasing parallel to side b
ðAt x ¼ 0; z ¼ 0
ðsb Þmax ¼
b1 qb t2
2
ðAt x ¼ 0; z ¼ 0:4bÞ
sb ¼
b2 qb2 b qb2 and sa ¼ 3 2 t2 t
ðAt x ¼ 0; z ¼ bÞ
sb ¼
b4 qb2 t2
ðsa Þmax ¼
b5 qb2 t2
a At x ¼ ; z ¼ 0:45b 2 ymax ¼ a=b
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.1132 0.0410 0.0637 0.0206 0.1304 0.0016
0.1778 0.0633 0.0688 0.0497 0.1436 0.0047
0.2365 0.0869 0.0762 0.0898 0.1686 0.0074
0.2777 0.1038 0.0715 0.1249 0.1800 0.0097
0.3004 0.1128 0.0610 0.1482 0.1845 0.0113
0.3092 0.1255 0.0509 0.1615 0.1874 0.0126
0.3100 0.1157 0.0415 0.1680 0.1902 0.0133
0.3068 0.1148 0.0356 0.1709 0.1908 0.0136
(Ref. 28 for n ¼ 0:3)
Flat Plates
b1 b2 b3 b4 b5 a
aqb4 Et3
509
TABLE 11.4
Formulas for flat plates with straight boundaries and constant thickness (Continued ) Case no., loading 9a. Uniform over entire plate
Formulas and tabulated specific values ðsb Þmax
ðAt x ¼ 0; z ¼ 0Þ
b1 qb2 ¼ t2
and R ¼ g1 qb
b2 qb t2 R ¼ g2 qb
ðAt x ¼ 0; z ¼ 0:6bÞ
sb ¼
ðAt x ¼ 0; z ¼ bÞ a At x ¼ ; z ¼ 0:6b 2
sa ¼
Formulas for Stress and Strain
9. Rectangular plate, three edges fixed, one edge (a) simply supported
510
Case no., shape, and supports
2
and sa ¼
b4 qb2 t2
b3 qb2 t2
and R ¼ g3 qb
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b1 b2 b3 b4 g1 g2 g3
0.020 0.004 0.016 0.031 0.115 0.123 0.125
0.081 0.018 0.061 0.121 0.230 0.181 0.256
0.173 0.062 0.118 0.242 0.343 0.253 0.382
0.307 0.134 0.158 0.343 0.453 0.319 0.471
0.539 0.284 0.164 0.417 0.584 0.387 0.547
0.657 0.370 0.135 0.398 0.622 0.397 0.549
0.718 0.422 0.097 0.318 0.625 0.386 0.530 (Ref. 49 for n ¼ 0:2Þ
9aa. Uniform over 23 of plate from fixed edge
ðsb Þmax
b2 qb2 t2 R ¼ g2 qb
ðAt x ¼ 0; z ¼ 0:6bÞ
sb ¼
ðAt x ¼ 0; z ¼ bÞ a At x ¼ ; z ¼ 0:4b 2
sa ¼
R ¼ g1 qb
and
sa ¼
and
b4 qb2 t2
b3 qb2 t2
and R ¼ g3 qb
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b1 b2 b3 b4 g1 g2 g3
0.020 0.003 0.012 0.031 0.115 0.002 0.125
0.080 0.016 0.043 0.111 0.230 0.015 0.250
0.164 0.044 0.081 0.197 0.334 0.048 0.345
0.274 0.093 0.108 0.255 0.423 0.088 0.396
0.445 0.193 0.112 0.284 0.517 0.132 0.422
0.525 0.252 0.091 0.263 0.542 0.139 0.417
0.566 0.286 0.066 0.204 0.543 0.131 0.405
(Ref. 49 for n ¼ 0:2)
[CHAP. 11
TABLE 11.4
ðAt x ¼ 0; z ¼ 0Þ
b1 qb2 ¼ t2
Formulas for flat plates with straight boundaries and constant thickness (Continued ) ðsb Þmax ¼
b1 qb2 t2
b qb2 sb ¼ 2 2 t R ¼ g2 qb
ðAt x ¼ 0; z ¼ 0:2bÞ ðAt x ¼ 0; z ¼ bÞ a At x ¼ ; z ¼ 0:2b 2
sa ¼
and
sa ¼
and
b4 qb2 t2
R ¼ g1 qb
and
11.14]
ðAt x ¼ 0; z ¼ 0Þ
SEC.
9aaa. Uniform over 13 of plate from fixed edge
b3 qb2 t2
R ¼ g3 qb
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b1 b2 b3 b4 g1 g2 g3
0.020 0.005 0.013 0.026 0.114 0.000 0.111
0.068 0.026 0.028 0.063 0.210 0.000 0.170
0.108 0.044 0.031 0.079 0.261 0.004 0.190
0.148 0.050 0.026 0.079 0.290 0.011 0.185
0.194 0.047 0.016 0.068 0.312 0.020 0.176
0.213 0.041 0.011 0.056 0.316 0.021 0.175
0.222 0.037 0.008 0.037 0.316 0.020 0.190 (Ref. 49 for n ¼ 0:2)
9d. Uniformly decreasing from fixed edge to simply supported edge
ðAt x ¼ 0; z ¼ 0
ðsb Þmax ¼
b1 qb t2
sa ¼
b2 qb t2
a At x ¼ ; z ¼ 0:4bÞ 2
2
2
and
R ¼ g1 qb
and R ¼ g2 qb
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b1 b2 g1 g2
0.018 0.019 0.106 0.075
0.064 0.068 0.195 0.152
0.120 0.124 0.265 0.212
0.192 0.161 0.323 0.245
0.303 0.181 0.383 0.262
0.356 0.168 0.399 0.258
0.382 0.132 0.400 0.250
Flat Plates
(Ref. 49 for n ¼ 0:2)
511
512
TABLE 11.4
Formulas for flat plates with straight boundaries and constant thickness (Continued ) Case no., loading 9dd. Uniformly decreasing from fixed edge to zero at 23 b
Formulas and tabulated specific values ðAt x ¼ 0; z ¼ 0Þ
ðsb Þmax
b1 qb2 ¼ t2
R ¼ g1 qb
and
a b2 qb2 At x ¼ ; z ¼ 0:4b if a 5 b or z ¼ 0:2b if a < b sa ¼ 2 t2
R ¼ g2 qb
and
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b1 b2 g1 g2
0.017 0.019 0.101 0.082
0.056 0.050 0.177 0.129
0.095 0.068 0.227 0.146
0.140 0.098 0.262 0.157
0.201 0.106 0.294 0.165
0.228 0.097 0.301 0.162
0.241 0.074 0.301 0.158 (Ref. 49 for n ¼ 0:2Þ
9ddd. Uniformly decreasing from fixed edge to zero at 13 b
ðAt x ¼ 0; z ¼ 0Þ
ðsb Þmax
a At x ¼ ; z ¼ 0:2b 2
b1 qb2 ¼ t2
sa ¼
R ¼ g1 qb
and
b2 qb2 t2
Formulas for Stress and Strain
Case no., shape, and supports
and R ¼ g2 qb
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b1 b2 g1 g2
0.014 0.010 0.088 0.046
0.035 0.024 0.130 0.069
0.047 0.031 0.146 0.079
0.061 0.030 0.155 0.077
0.075 0.025 0.161 0.074
0.080 0.020 0.162 0.074
0.082 0.013 0.162 0.082
(Ref. 49 for n ¼ 0:2
[CHAP. 11
TABLE 11.4
Formulas for flat plates with straight boundaries and constant thickness (Continued ) ðAt x ¼ 0; z ¼ 0Þ
ðsb Þmax ¼
b1 qb2 t2
b qb ðAt x ¼ 0; z ¼ bÞ sa ¼ 2 2 t a b3 qb2 At x ¼ ; z ¼ b sa ¼ 2 t2
R ¼ g1 qb
and
11.14]
10a. Uniform over entire plate
SEC.
10. Rectangular plate; three edges fixed, one edge (a) free
2
10aa. Uniform over 23 of plate from fixed edge
R ¼ g2 qb
and
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b1 b2 b3 g1 g2
0.020 0.016 0.031 0.114 0.125
0.081 0.066 0.126 0.230 0.248
0.173 0.148 0.286 0.341 0.371
0.321 0.259 0.511 0.457 0.510
0.727 0.484 1.073 0.673 0.859
1.226 0.605 1.568 0.845 1.212
2.105 0.519 1.982 1.012 1.627
b1 qb ðAt x ¼ 0; z ¼ 0Þ ðsb Þmax ¼ and R ¼ g1 qb t2 a b2 qb2 sa ¼ At x ¼ ; z ¼ 0:6b for a > b or z ¼ 0:4b for a 4 b 2 t2
(Ref. 49 for n ¼ 0:2Þ
2
and
r ¼ g2 qb
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b1 b2 g1 g2
0.020 0.031 0.115 0.125
0.080 0.110 0.230 0.250
0.164 0.198 0.334 0.344
0.277 0.260 0.424 0.394
0.501 0.370 0.544 0.399
0.710 0.433 0.615 0.409
1.031 0.455 0.674 0.393
ðsb Þmax
b1 qb2 ¼ t2
(Ref. 49 for n ¼ 0:2Þ 10aaa. Uniform over 1 3 of plate from fixed edge
ðAt x ¼ 0; z ¼ 0Þ
b2 qb2 sa ¼ t2
R ¼ g1 qb
and and
R ¼ g2 qb
0.25
0.50
0.75
1.0
1.5
2.0
3.0
0.020 0.026 0.115 0.111
0.068 0.063 0.210 0.170
0.110 0.084 0.257 0.194
0.148 0.079 0.291 0.185
0.202 0.068 0.316 0.174
0.240 0.057 0.327 0.170
0.290 0.040 0.335 0.180
(Ref. 49 for n ¼ 0:2Þ
513
a=b b1 b2 g1 g2
Flat Plates
a At x ¼ ; z ¼ 0:2b 2
514
TABLE 11.4
Formulas for flat plates with straight boundaries and constant thickness (Continued ) Case no., loading 10d. Uniformly decreasing from fixed edge to zero at free edge
10dd. Uniformly decreasing from fixed edge to zero at 23 b
Formulas and tabulated specific values ðAt x ¼ 0; z ¼ 0Þ
b1 qb2 ¼ t2
ðsb Þmax
R ¼ g1 qb
and
a At x ¼ ; z ¼ b if a > b or z ¼ 0:4b if a < b 2
sa ¼
b2 qb2 t2
and
R ¼ g2 qb
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b1 b2 g1 g2
0.018 0.019 0.106 0.075
0.064 0.068 0.195 0.151
0.120 0.125 0.265 0.211
0.195 0.166 0.324 0.242
0.351 0.244 0.406 0.106
0.507 0.387 0.458 0.199
0.758 0.514 0.505 0.313
ðAt x ¼ 0; z ¼ 0Þ
b1 qb2 ¼ t2
ðsb Þmax
(Ref. 49 for n ¼ 0:2Þ
Formulas for Stress and Strain
Case no., shape and supports
R ¼ g1 qb
and
a b2 qb2 At x ¼ ; z ¼ 0:4b if a 5 b or z ¼ 0:2b if a < b sb ¼ t2 2
and R ¼ g2 qb
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b1 b2 g1 g2
0.017 0.019 0.102 0.082
0.056 0.050 0.177 0.129
0.095 0.068 0.227 0.146
0.141 0.099 0.263 0.157
0.215 0.114 0.301 0.163
0.277 0.113 0.320 0.157
0.365 0.101 0.336 0.146 (Ref. 49 for n ¼ 0:2Þ
[CHAP. 11
TABLE 11.4
Formulas for flat plates with straight boundaries and constant thickness (Continued ) ðsb Þmax ¼
a At x ¼ ; z ¼ 0:2b 2
b1 qb2 t2
R ¼ g1 qb
and
b2 qb2 sa ¼ t2
11.14]
ðAt x ¼ 0; z ¼ 0Þ
SEC.
10ddd. Uniformly decreasing from fixed edge to zero at 13 b
R ¼ g2 qb
and
a=b
0.25
0.50
0.75
1.0
1.5
2.0
3.0
b1 b2 g1 g2
0.014 0.010 0.088 0.046
0.035 0.024 0.130 0.069
0.047 0.031 0.146 0.079
0.061 0.030 0.156 0.077
0.076 0.025 0.162 0.073
0.086 0.020 0.165 0.073
0.100 0.014 0.167 0.079
ðsb Þmax
b1 qb2 ¼ t2
(Ref. 49 for n ¼ 0:2Þ 11. Rectangular plate; two adjacent edges fixed, two remaining edges free
11a. Uniform over entire plate
ðAt x ¼ a; z ¼ 0Þ
and
R ¼ g1 qb
b b At x ¼ 0; z ¼ b if a > or a ¼ 0:8b if a 4 2 2
sa ¼
a=b
0.125
0.25
0.375
0.50
0.75
1.0
b1 b2 g1 g2
0.050 0.047 0.312 0.127
0.182 0.188 0.572 0.264
0.353 0.398 0.671 0.413
0.631 0.632 0.874 0.557
1.246 1.186 1.129 0.829
1.769 1.769 1.183 1.183
b2 qb2 t2
and R ¼ g2 qb
(Ref. 49 for n ¼ 0:2) b1 qb ðAt x ¼ a; z ¼ 0Þ ðsb Þmax ¼ and R ¼ g1 qb t2 b b b2 qb2 sa ¼ At x ¼ 0; z ¼ 0:6b if a > or z ¼ 0:4b if a 4 and R ¼ g2 qb 2 2 t2 2
11aa. Uniform over plate from z ¼ 0 to z ¼ 23 b
0.125
0.25
0.375
0.50
0.75
1.0
0.050 0.044 0.311 0.126
0.173 0.143 0.543 0.249
0.297 0.230 0.563 0.335
0.465 0.286 0.654 0.377
0.758 0.396 0.741 0.384
0.963 0.435 0.748 0.393
Flat Plates
a=b b1 b2 g1 g2
(Ref. 49 for n ¼ 0:2Þ
515
Formulas for flat plates with straight boundaries and constant thickness (Continued )
Case no., shape, and supports
Case no., loading
516
TABLE 11.4
Formulas and tabulated specific values
11aaa. Uniform over plate from z ¼ 0 to z ¼ 13 b
11d. Uniformly decreasing from z ¼ 0 to z ¼ b
a=b
0.125
0.25
0.375
0.50
0.75
1.0
b1 b2 g1 g2
0.034 0.034 0.222 0.109
0.099 0.068 0.311 0.162
0.143 0.081 0.335 0.180
0.186 0.079 0.343 0.117
0.241 0.085 0.349 0.109
0.274 0.081 0.347 0.105
and
R ¼ g2 qb
(Ref. 49 for n ¼ 0:2Þ
b1 qb2 ¼ t2
ðAt x ¼ a; z ¼ 0Þ ðsb Þmax and R ¼ g1 qb b b b2 qb2 sa ¼ At x ¼ 0; z ¼ b if a ¼ b; or z ¼ 0:6b if 4 a < b; or z ¼ 0:4b if a < 2 2 t2 a=b
0.125
0.25
0.375
0.50
0.75
1.0
b1 b2 g1 g2
0.043 0.028 0.271 0.076
0.133 0.090 0.423 0.151
0.212 0.148 0.419 0.205
0.328 0.200 0.483 0.195
0.537 0.276 0.551 0.230
0.695 0.397 0.559 0.192
and
Formulas for Stress and Strain
b1 qb ðAt x ¼ a; z ¼ 0Þ ðsb Þmax ¼ and R ¼ g1 qb t2 b b b2 qb2 sa ¼ At x ¼ 0; z ¼ 0:4b if a > or z ¼ 0:2b if a 4 2 2 t2 2
R ¼ g2 qb
(Ref. 49 for n ¼ 0:2Þ 11dd. Uniformly decreasing from z ¼ 0 to z ¼ 23 b
ðsb Þmax
and
R ¼ g1 qb
ðAt x ¼ 0; z ¼ 0:4b if a 5 0:375b; or z ¼ 0:2b if a < 0:375bÞ sa ¼ a=b
0.125
0.25
0.375
0.50
0.75
1.0
b1 b2 g1 g2
0.040 0.026 0.250 0.084
0.109 0.059 0.354 0.129
0.154 0.089 0.316 0.135
0.215 0.107 0.338 0.151
0.304 0.116 0.357 0.156
0.362 0.113 0.357 0.152
b2 qb2 t2
and
R ¼ g2 qb
(Ref. 49 for n ¼ 0:2Þ
[CHAP. 11
TABLE 11.4
ðAt x ¼ a; z ¼ 0Þ
b1 qb2 ¼ t2
Formulas for flat plates with straight boundaries and constant thickness (Continued )
13. Continuous plate; supported continuously on an elastic foundation of modulus k (lb=in2=in)
12a. Uniform over entire surface
13b. Uniform over a small circle of radius ro , remote from edges
ðAt x ¼ 0; z ¼ 0:2bÞ
sa ¼
b1 qb2 t2
and
R ¼ g1 qb
11.14]
12. Continuous plate; supported at equal intervals a on circular supports of radius ro
ðAt x ¼ a; z ¼ 0Þ ðsb Þmax ¼
SEC.
11ddd. Uniformly decreasing from z ¼ 0 to z ¼ 13 b
b2 qb and R ¼ g2 qb t2 2
a=b
0.125
0.25
0.375
0.50
0.75
1.0
b1 b2 g1 g2
0.025 0.014 0.193 0.048
0.052 0.028 0.217 0.072
0.071 0.031 0.170 0.076
0.084 0.029 0.171 0.075
0.100 0.025 0.171 0.072
0.109 0.020 0.171 0.072
(At edge of support) 2 0:15q 4 1 þ4 when 0:15 4 n < 0:30 sa ¼ 2 a ro t 3 n 3qa2 a r2 or sa ¼ ð1 þ nÞ ln 21ð1 nÞ o2 0:55 1:50n ro 2pt2 a 2r where n ¼ o a (Under the load) smax ¼
3W ð1 þ nÞ L ln e þ 0:6159 2pt2 ro
Max foundation pressure qo ¼ ymax ¼
where Le ¼
(Ref. 49 for n ¼ 0:2Þ
ðRef: 9Þ when n < 0:15
ðRef: 11Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 Et3 12ð1 n2 Þk
W 8L2e
W 8kL2e (Ref. 14)
Flat Plates 517
TABLE 11.4
Formulas for flat plates with straight boundaries and constant thickness (Continued ) 518
Case no., shape, and supports
Case no., loading
Formulas and tabulated specific values (Under the load) smax ¼
0:863W ð1 þ nÞ L ln e þ 0:207 t2 ro W kL2e
ymax ¼ 0:408ð1 þ 0:4nÞ
(Ref. 14) 13bbb. Uniform over a small circle of radius ro , adjacent to a corner
(At the corner)
ymax
r W ¼ 1:1 1:245 o Le kL2e
pffiffiffiffiffiffiffiffiffiffi (At a distance ¼ 2:38 ro Le from the corner along diagonal) " 0:6 # 3W r smax ¼ 2 1 1:083 o t Le
Formulas for Stress and Strain
13bb. Uniform over a small circle of radius ro , adjacent to edge but remote from corner
(Ref. 14) 14. Parallelogram plate (skew slab); all edges simply supported
15. Parallelogram plate (skew slab); shorter edges simply supported, longer edges free
14a. Uniform over entire plate
(At center of plate) smax
ymax
and
aqb4 ¼ Et3
For a=b ¼ 2:0
15a. Uniform over entire plate
y
0
30
45
60
75
b a
0.585 0.119
0.570 0.118
0.539 0.108
0.463 0.092
0.201 0.011
(Along free edge) smax ¼
b1 qb t2
(At center of plate) smax ¼ For a=b ¼ 2:0
2
ymax
and
b2 qb2 t2
and
a1 qb Et3
ymax ¼
y
0
30
45
60
b1 b2 a1 a2
3.05 2.97 2.58 2.47
2.20 2.19 1.50 1.36
1.78 1.75 1.00 0.82
0.91 1.00 0.46 0.21
(Ref. 24 for n ¼ 0:2Þ
4
a2 qb4 Et3
(Ref. 24 for n ¼ 0:2Þ
[CHAP. 11
TABLE 11.4
bqb2 ¼ 2 t
Formulas for flat plates with straight boundaries and constant thickness (Continued ) 16a. Uniform over entire plate
b1 qb2 t2 aqb4 ¼ Et3
Along longer edge toward obtuse angle) smax ¼
SEC.
(At center of plate) s ¼
b2 qb2 t2
y
a=b
1.00
1.25
1.50
1.75
2.00
0
b1 b2 a
0.308 0.138 0.0135
0.400 0.187 0.0195
0.454 0.220 0.0235
0.481 0.239 0.0258
0.497 0.247 0.0273
15
b1 b2 a
0.320 0.135 0.0127
0.412 0.200 0.0189
0.483 0.235 0.0232
0.531 0.253 0.0257
0.553 0.261 0.0273
30
b1 b2 a
0.400 0.198 0.0168
0.495 0.221 0.0218
0.547 0.235 0.0249
0.568 0.245 0.0268
0.580 0.252 0.0281
45
b1 b2 a
0.394 0.218 0.0165
0.470 0.244 0.0208
0.531 0.260 0.0242
0.575 0.265 0.0265
0.601 0.260 0.0284
60
b1 b2 a
0.310 0.188 0.0136
0.450 0.204 0.0171
0.538 0.214 0.0198
and
ymax
2.25
2.50
11.14]
16. Parallelogram plate (skew slab); all edges fixed
3.00
0.613 0.224 0.0245 (Ref. 53 for n ¼ 13Þ
17. Equilateral triangle; all edges simply supported
17a. Uniform over entire plate
ðAt x ¼ 0; z ¼ 0:062aÞ ðsz Þmax
0:1488qa2 ¼ t2
ðAt x ¼ 0; z ¼ 0:129aÞ ðsx Þmax ¼
qa4 ð1 n2 Þ 81Et3 (Refs. 21 and 23 for n ¼ 0:3)
ðAt x ¼ 0; z ¼ 0Þ smax
ymax ¼ 0:069W ð1 n2 Þa2 =Et3
519
17b. Uniform over small circle of radius ro at x ¼ 0; z ¼ 0
3W 1 n 0:377a þ ð1 þ nÞ ln ¼ 2pt2 2 r0o
Flat Plates
ðAt x ¼ 0; z ¼ 0Þ ymax ¼
0:1554qa2 t2
TABLE 11.4
Formulas for flat plates with straight boundaries and constant thickness (Continued )
18a. Uniform over entire plate
Formulas and tabulated specific values smax
0:262qa2 ¼ sz ¼ t2
ðsx Þmax ¼ ymax ¼
0:225qa2 t2
0:038qa4 Et3 (Ref. 21 for n ¼ 0:3Þ
19. Regular polygonal plate; all edges simply supported
19a. Uniform over entire plate
(At center)
bqa2 s¼ 2 t
ymax
and
aqa4 ¼ Et3
(At center of straight edge) Max slope ¼
xqa3 Et3
n
3
4
5
6
7
8
9
10
15
1
b a x
1.302 0.910 1.535
1.152 0.710 1.176
1.086 0.635 1.028
1.056 0.599 0.951
1.044 0.581 0.910
1.038 0.573 0.888
1.038 0.572 0.877
1.044 0.572 0.871
1.074 0.586 0.883
1.236 0.695 1.050
Number of sides ¼ n 20. Regular polygonal plate; all edges fixed
(Ref. 55 for n ¼ 0:3Þ 20a. Uniform over entire plate
(At center )
s¼
b1 qa t2
2
and
aqa Et3 b2 qa2 ¼ t2
ymax ¼
(At center of straight edge) smax
Number of sides ¼ n
Formulas for Stress and Strain
18. Right-angle isosceles triangle; all edges simply supported
Case no., loading
520
Case no., shape, and supports
4
n
3
4
5
6
7
8
9
10
1
b b2 a
0.589 1.423 0.264
0.550 1.232 0.221
0.530 1.132 0.203
0.518 1.068 0.194
0.511 1.023 0.188
0.506 0.990 0.184
0.503 0.964 0.182
0.500 0.944 0.180
0.4875 0.750 0.171 (Ref. 55 for n ¼ 0:3Þ
[CHAP. 11
