铁摩辛柯系列丛书（Timoshenko）Timoshenko - Theory of Plates and Shells, 2e

16 THEORY OF PLATES AND SHELLS The use of the curves in Figs.5 and 8 will now be illustrated by a numerical example.A long rectangular steel plate has the dimensions 1=50 in.,h=in.,and g 10 psi.In such a case we have VG=a295(份)--8090-0aa6 30·106 1og16104√=2.5181 From Fig.8 we now find u 1.894;and from Fig.5,=0.8212. Sub- stituting these values in Eqs.(16)and (17),we find 30·106.1.8942 1=3-0.310=3,940pi c2=·10·104.0.8212=41,060psi 0mnx=1+2=45,000psi Comparing these stress values with the maximum stresses obtained for a plate of the same size,but with twice the load,on the assumption of 60,000 4a0 /M=120 1/h=00 50,000 /h=80 40,000 牙 930,000 20,000 Stresses in steel plotes with built-in edges Rotio width thickness =L/h 10,000 67 89,10111213141516 Lood in Ib per sa in. FI.9 simply supported edges (see page 11),it can be concluded that,owing to clamping of the edges,the direct tensile stress decreases considerably, whereas the maximum bending stress increases several times,so that finally the maximum total stress in the case of clamped edges becomes larger than in the case of simply supported edges

The use of the curves in Figs. 5 and 8 will now be illustrated by a numerical example. A long rectangular steel plate has the dimensions I = 50 in., h = % in., and q = 10 psi. In such a case we have From Fig. 8 we now find u = 1.894; and from Fig. 5, fa = 0.8212. Sub￾stituting these values in Eqs. (16) and (17), we find Comparing these stress values with the maximum stresses obtained for a plate of the same size, but with twice the load, on the assumption of Stresses in steel plates with built-in edges Ratio width : thickness =l/ h Stress in Ib per sq in. Lood in Ib per sq in. FIG. 9 simply supported edges (see page 11), it can be concluded that, owing to clamping of the edges, the direct tensile stress decreases considerably, whereas the maximum bending stress increases several times, so that finally the maximum total stress in the case of clamped edges becomes larger than in the case of simply supported edges

BENDING TO A CYLINDRICAL SURFACE 17 Proceeding as in the previous article it can be shown that the maxi- mum stress in a plate depends only on the load g and the ratio l/h,and we can plot a set of curves giving maximum stress in terms of g,each curve in the set corresponding to a particular value of l/h.Such curves are given in Fig.9.It is seen that for small values of the intensity of the load g,when the effect of the axial force on the deflections of the strip is small,the maximum stress increases approximately in the same ratio as g increases.But for larger values of g the relation between the load and the maximum stress becomes nonlinear. In conclusion,we give in Table 1 the numerical values of all the func- tions plotted in Figs.4,5,and 8.This table can be used instead of the curves in calculating maximum stresses and maximum deflections of long, uniformly loaded rectangular plates. 4.Cylindrical Bending of Uniformly Loaded Rectangular Plates with Elastically Built-in Edges.Let us assume that when bending occurs, the longitudinal edges of the plate rotate through an angle proportional to the bending moment at the edges.In such a case the forces acting on an elemental strip will again be of the type shown in Fig.7,and we shall obtain expression(b)of the previous article for the deflections w.How- ever,the conditions at the edges,from which the constants of integration and the moment Mo are determined,are different;viz.,the slope of the deflection curve at the ends of the strip is no longer zero but is propor- tional to the magnitude of the moment Mo,and we have =-BMo (a) where B is a factor depending on the rigidity of restraint along the edges. If this restraint is very flexible,the quantity 8 is large,and the conditions at the edges approach those of simply supported edges.If the restraint is very rigid,the quantity B becomes small,and the edge conditions approach those of absolutely built-in edges.The remaining two end conditions are the same as in the previous article.Thus we have -6M0 dw =0 dx (6) ei/ (w)x0=0 Using these conditions,we find both the constants of integration and the magnitude of Mo in expression (b)of the previous article.Owing to flexibility of the boundary,the end moments Mo will be smaller than those given by Eq.(13)for absolutely built-in edges,and the final result can be put in the form M=-y5 (19)

Proceeding as in the previous article it can be shown that the maxi￾mum stress in a plate depends only on the load q and the ratio l/h, and we can plot a set of curves giving maximum stress in terms of q, each curve in the set corresponding to a particular value of l/h. Such curves are given in Fig. 9. It is seen that for small values of the intensity of the load q, when the effect of the axial force on the deflections of the strip is small, the maximum stress increases approximately in the same ratio as q increases. But for larger values of q the relation between the load and the maximum stress becomes nonlinear. In conclusion, we give in Table 1 the numerical values of all the func￾tions plotted in Figs. 4, 5, and 8. This table can be used instead of the curves in calculating maximum stresses and maximum deflections of long, uniformly loaded rectangular plates. 4. Cylindrical Bending of Uniformly Loaded Rectangular Plates with Elastically Built-in Edges. Let us assume that when bending occurs, the longitudinal edges of the plate rotate through an angle proportional to the bending moment at the edges. In such a case the forces acting on an elemental strip will again be of the type shown in Fig. 7, and we shall obtain expression (b) of the previous article for the deflections w. How￾ever, the conditions at the edges, from which the constants of integration and the moment M0 are determined, are different; viz^ the slope of the deflection curve at the ends of the strip is no longer zero but is propor￾tional to the magnitude of the moment M0, and we have (o) where /3 is a factor depending on the rigidity of restraint along the edges. If this restraint is very flexible, the quantity /3 is large, and the conditions at the edges approach those of simply supported edges. If the restraint is very rigid, the quantity /3 becomes small, and the edge conditions approach those of absolutely built-in edges. The remaining two end conditions are the same as in the previous article. Thus we have (b) Using these conditions, we find both the constants of integration and the magnitude of Mo in expression (b) of the previous article. Owing to flexibility of the boundary, the end moments M0 will be smaller than those given by Eq. (13) for absolutely built-in edges, and the final result can be put in the form (19)

18 THEORY OF PLATES AND SHELLS TABLE 1 1og1o10V可。 log1o10√可 1og110V可 fo(u) fi(u) Vo(u) 1(w) 0 6 .0001.000 1.000 1.000 0.5 3.889 3.217 3.801 0.9080.9760.905 0.984 0.5 406 331 425 1.0 3.483 2.886 3.376 0.7110.9090.704 0.939 1.0 310 223 336 1.5 3.173 2.663 3.040 0.5320.8170.511 0.876 1.5 262 182 292 2.0 2.911 2.481 2.748 0.3800.7150.367 0.806 2.0 227 161 257 2.5 2.684 2.320 2,491 0.2810.6170.268 0.736 2.5 198 146 228 3.0 2.486 2.174 2.263 0.2130.5290.200 0.672 3.0 175 134 202 3.5 2.311 2.040 2.061 0.1660.4530.153 0.614 3.5 156 124 180 4.0 2.155 1.916 1.881 0.1320.3880.120 0.563 4.0 141 115 163 4.5 2.014 1.801 1.718 0.1070.3350.097 0.519 4.5 128 107 148 5.0 1.886 1.694 1.570 0.0880.2910.079 0.480 5.0 118 100 135 5.5 1.768 .594 1.435 0.0740.2540.066 0.446 5.5 108 93 124 6.0 1.660 1.501 1.311 0.0630.2230.055 0.417 6.0 100 88 115 6.5 1.560 1.413 1.196 0.0540.1970.047 0.391 6.5 93 82 107 7.0 1,467 1.331 1.089 0.0470.1750.041 0.367 7.0 87 78 100 7.5 1.380 1.253 0.989 0.0410.1560.036 0.347 7.5 82 74 94 8.0 1.208 1.179 0.895 0.0360.1410.031 0.328 8.0 77 70 89 8.5 1.221 1.109 0.806 0.0320.1270.028 0.311 8.5 73 67 83 9.0 1.148 1.042 0.723 0.0290.1150.025 0.296 9.0 69 63 80 9.5 1.079 0.979 0.643 0.0260.1050.022 0.283 9.5 65 61 75 10.0 1.014 0.918 0,568 0.0240.0960.020 0.27010.0 63 58 72 10.5 0.951 0.860 0.496 0.0210.0880.018 0.25910.5 59 55 69 11.0 0.892 0.805 0.427 0.020.0.0810.017 0.24811.0 57 54 65 11.5 0.835 0.751 0.362 0.0180.0750.0150.23811.5 55 51 63 12.0 0.780 0.700 0.299 0.0160.0600.0140.22912.0

TABLE 1 u logio 104 V ^ logio 104 VU~i log10104 \Z~Ul Mu) Mu) Mu) *i(u) u _ _ _ _ 1.0001.000 1.000 1.000 ~0~ 0.5 3.889 3.217 3.801 0.9080.976 0.905 0.984 0.5 406 331 425 1.0 3.483 2.886 3.376 0.7110.909 0.704 0.939 1.0 310 223 336 1.5 3.173 2.663 3.040 0.5320.817 0.511 0.876 1.5 262 182 292 2.0 2.911 2.481 2.748 0.3800.715 0.367 0.806 2.0 227 161 257 2.5 2.684 2.320 2.491 0.2810.617 0.268 0.736 2.5 198 146 228 3.0 2.486 2.174 2.263 0.2130.529 0.200 0.672 3.0 175 134 202 3.5 2.311 2.040 2.061 0.1660.453 0.153 0.614 3.5 156 124 180 4.0 2.155 1.916 1.881 0.132 0.388 0.120 0.563 4.0 141 115 163 4.5 2.014 1.801 1.718 0.107 0.335 0.097 0.519 4.5 128 107 148 5.0 1.886 1.694 1.570 0.0880.291 0.079 0.480 5.0 118 100 135 5.5 1.768 1.594 1.435 0.074 0.254 0.066 0.446 5.5 108 93 124 6.0 1.660 1501 1.311 0.063 0.223 0.055 0.417 6.0 100 88 115 6.5 1.560 1.413 1.196 0.054 0.197 0.047 0.391 6.5 93 82 107 7.0 1.467 1.331 1.089 0.047 0.175 0.041 0.367 7.0 87 78 100 7.5 1.380 1253 0.989 0.0410.156 0.036 0.347 7.5 82 74 94 8.0 1.298 1179 0.895 0.036 0.1410.031 0.328 8.0 77 70 89 8.5 1.221 1.109 0.806 0.032 0.127 0.028 0.311 8.5 73 67 83 9.0 1.148 1.042 0.723 0.0290.115 0.025 0.296 9.0 69 63 80 9.5 1.079 0.979 0.643 0.026 0.105 0.022 0.283 9.5 65 61 75 10.0 1.014 0.918 0.568 0.024 0.096 0.020 0.270 10.0 63 58 72 10.5 0.951 0.860 0.496 0.0210.088 0.018 0.259 10.5 59 55 69 11.0 0.892 0.805 0.427 0.0200.0810.017 0.248 11.0 57 54 65 11.5 0.835 0.751 0.362 0.0180.075 0.015 0.238 11.5 55 51 63 12.0 0.780 0.700 0.299 0.0160.069 0014 0.229 12.0

BENDING TO A CYLINDRICAL SURFACE 19 where y is a numerical factor smaller than unity and given by the formula tanh u Y=2孕Du+tanh“ It is seen that the magnitude of the moments Mo at the edges depends upon the magnitude of the coefficient 8 defining the rigidity of the restraint.When B is very small,the coefficient y approaches unity, and the moment Mo approaches the value(13)calculated for absolutely built-in edges.When 8 is very large,the coefficient y and the moment Mo become small,and the edge conditions approach those of simply supported edges. The deflection curve in the case under consideration can be repre- sented in the following form: 0= gl tanh u-r(tanh u-u) 16u+D tanh u cosh u 8u2D(0-x) (20) For y =1 this expression reduces to expression (14)for deflections of a plate with absolutely built-in edges.For y=0 we obtain expression(6) for a plate with simply supported edges. In calculating the tensile parameter u we proceed as in the previous cases and determine the tensile force S from the condition that the exten- sion of the elemental strip is equal to the difference between the length of the arc along the deflection curve and the chord length l.Hence s-( hE Substituting expression(20)in this equation and performing the inte- gration,we obtain E2hs (-吗0=(1-YW。+yU1-y(1-WU, (21) where Uo and Ui denote the right-hand sides of Egs.(8)and (15),respec- tively,and h-器(tanh-u+anh0 The values of logio(10vUa)are given in Table 1.By using this table, Eq.(21)can be readily solved by the trial-and-error method.For any particular plate we first calculate the left-hand side of the equation and

where 7 is a numerical factor smaller than unity and given by the formula It is seen that the magnitude of the moments M0 at the edges depends upon the magnitude of the coefficient 0 defining the rigidity of the restraint. When /3 is very small, the coefficient 7 approaches unity, and the moment MQ approaches the value (13) calculated for absolutely built-in edges. When /3 is very large, the coefficient 7 and the moment Af0 become small, and the edge conditions approach those of simply supported edges. The deflection curve in the case under consideration can be repre￾sented in the following form: (20) For 7 = 1 this expression reduces to expression (14) for deflections of a plate with absolutely built-in edges. For 7 = 0 we obtain expression (6) for a plate with simply supported edges. In calculating the tensile parameter u we proceed as in the previous cases and determine the tensile force S from the condition that the exten￾sion of the elemental strip is equal to the difference between the length of the arc along the deflection curve and the chord length I. Hence Substituting expression (20) in this equation and performing the inte￾gration, we obtain (21) where Uo and Ui denote the right-hand sides of Eqs. (8) and (15), respec￾tively, and The values of logio (104 VT/2) are given in Table 1. By using this table, Eq. (21) can be readily solved by the trial-and-error method. For any particular plate we first calculate the left-hand side of the equation and

20 THEORY OF PLATES AND SHELLS by using the curves in Figs.4 and 8,determine the values of the parame- ter u(1)for simply supported edges and(2)for absolutely built-in edges. Naturally u for elastically built-in edges must have a value intermediate between these two.Assuming one such value for u,we calculate Uo,Ui, and Ua by using Table 1 and determine the value of the right-hand side of Eq.(21).Generally this value will be different from the value of the left-hand side calculated previously,and a new trial calculation with a new assumed value for u must be made.Two such trial calculations will usually be sufficient to determine by interpolation the value of u satisfying Eq.(21).As soon as the parameter u is determined,the bend- ing moments Mo at the ends may be calculated from Eq.(19).We can also calculate the moment at the middle of the strip and find the maxi- mum stress.This stress will occur at the ends or at the middle,depend- ing on the degree of rigidity of the constraints at the edges. 5.The Effect on Stresses and Deflections of Small Displacements of Longitudinal Edges in the Plane of the Plate.It was assumed in the previous discussion that,during bending,the longitudinal edges of the plate have no displacement in the plane of the plate.On the basis of this assumption the tensile force S was calculated in each particular case. Let us assume now that the edges of the plate undergo a displacement toward each other specified by A.Owing to this displacement the extension of the elemental strip will be diminished by the same amount, and the equation for calculating the tensile force S becomes 9-(僧-a hE (a) At the same time Eqs.(6),(14),and (20)for the deflection curve hold true regardless of the magnitude of the tensile force S.They may be differentiated and substituted under the integral sign in Eq.(a).After evaluating this integral and substituting S=4u2D/12,we obtain for simply supported edges 3l△ E2hs u2十 72(1-20 u2 =U0 (22) and for built-in edges 31△ E*hs w2+ h2 q2(1- 2u2 =0U1 (23) If A is made zero,Eqs.(22)and(23)reduce to Eys.(8)and (15),obtained previously for immovable edges. The simplest case is obtained by placing compression bars between the longitudinal sides of the boundary to prevent free motion of one edge of

by using the curves in Figs. 4 and 8, determine the values of the parame￾ter u (1) for simply supported edges and (2) for absolutely built-in edges. Naturally u for elastically built-in edges must have a value intermediate between these two. Assuming one such value for u, we calculate Uo, Ui1 and U2 by using Table 1 and determine the value of the right-hand side of Eq. (21). Generally this value will be different from th^ value of the left-hand side calculated previously, and a new trial calculation with a new assumed value for u must be made. Two such trial calculations will usually be sufficient to determine by interpolation tfye value of u satisfying Eq. (21). As soon as the parameter u is determined, the bend￾ing moments M0 at the ends may be calculated from Eq. (;19). We can also calculate the moment at the middle of the strip and |ind the maxi￾mum stress. This stress will occur at the ends or at the middle, depend￾ing on the degree of rigidity of the constraints at the edges. 5. The Effect on Stresses and Deflections of Small Displacements of Longitudinal Edges in the Plane of the Plate. It was assumed in the previous discussion that, during bending, the longitudinal edges of the plate have no displacement in the plane of the plate. On the basis of this assumption the tensile force S was calculated in each particular case. Let us assume now that the edges of the plate undergo a displacement toward each other specified by A. Owing to this displacement the extension of the elemental strip will be diminished by the same amount, and the equation for calculating the tensile force S becomes (o) At the same time Eqs. (6), (14), and (20) for the deflection curve hold true regardless of the magnitude of the tensile force S. They may be differentiated and substituted under the integral sign in Eq. (a). After evaluating this integral and substituting S — ^u2D/Z2 , we obtain for simply supported edges ..(22) and for built-in edges (23) If A is made zero, Eqs. (22) and (23) reduce to Eqs. (8) and (15), obtained previously for immovable edges. The simplest case is obtained by placing compression bars between the longitudinal sides of the boundary to prevent free motion of one edge of