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

6 THEORY OF PLATES AND SHELLS of beams,is called the flexural rigidity of the plate.It is seen that the calculation of deflections of the plate reduces to the integration of Eq.(4), which has the same form as the differential equation for deflection of beams.If there is only a lateral load acting on the plate and the edges are free to approach each other as deflection occurs,the expression for the bending moment M can be readily derived,and the deflection curve is then obtained by integrating Eq.(4).In practice the problem is more complicated,since the plate is usually attached to the boundary and its edges are not free to move.Such a method of support sets up tensile reactions along the edges as soon as deflection takes place.These reac. tions depend on the magnitude of the deflection and affect the magnitude of the bending moment M entering in Eq.(4).The problem reduces to the investigation of bending of an elemental strip submitted to the action of a lateral load and also an axial force which depends on the deflection of the strip.1 In the following we consider this problem for the particular case of uniform load acting on a plate and for various conditions along the edges. 2.Cylindrical Bending of Uniformly Loaded Rectangular Plates with Simply Supported Edges.Let us consider a uniformly loaded long rec- tangular plate with longitudinal edges which are free to rotate but can- not move toward each other during bending.An elemental strip cut out FI0.3 from this plate,as shown in Fig.1,is in the condition of a uniformly loaded bar submitted to the action of an axial force S(Fig.3).The magnitude of S is such as to prevent the ends of the bar from moving along the x axis.Denoting by g the intensity of the uniform load,the bending moment at any cross section of the strip is M-号-答-sm In such a form the problem was first discussed by I.G.Boobnov;see the English translation of his work in Trans.Inst.Naval Architects,vol.44,p.15,1902,and his "Theory of Structure of Ships,"vol.2,p.545,St.Petersburg,1914.See also the paper by Stewart Way presented at the National Meeting of Applied Mechanics, ASME,New Haven,Conn.,June,1932;from this paper are taken the curves used in Arts.2 and 3

of beams, is called the flexural rigidity of the plate. It is seen that the calculation of deflections of the plate reduces to the integration of Eq. (4), which has the same form as the differential equation for deflection of beams. If there is only a lateral load acting on the plate and the edges are free to approach each other as deflection occurs, the expression for the bending moment M can be readily derived, and the deflection curve is then obtained by integrating Eq. (4). In practice the problem is more complicated, since the plate is usually attached to the boundary and its edges are not free to move. Such a method of support sets up tensile reactions along the edges as soon as deflection takes place. These reac￾tions depend on the magnitude of the deflection and affect the magnitude of the bending moment M entering in Eq. (4). The problem reduces to the investigation of bending of an elemental strip submitted to the action of a lateral load and also an axial force which depends on the deflection of the strip.1 In the following we consider this problem for the particular case of uniform load acting on a plate and for various conditions along the edges. 2. Cylindrical Bending of Uniformly Loaded Rectangular Plates with Simply Supported Edges. Let us consider a uniformly loaded long rec￾tangular plate with longitudinal edges which are free to rotate but can￾not move toward each other during bending. An elemental strip cut out FIG. 3 from this plate, as shown in Fig. 1, is in the condition of a uniformly loaded bar submitted to the action of an axial force S (Fig. 3). The magnitude of S is such as to prevent the ends of the bar from moving along the x axis. Denoting by q the intensity of the uniform load, the bending moment at any cross section of the strip is 1 In such a form the problem was first discussed by I. G. Boobnov; see the English translation of his work in Trans. Inst. Naval Architects, vol. 44, p. 15, 1902, and his "Theory of Structure of Ships," vol. 2, p. 545, St. Petersburg, 1914. See also the paper by Stewart Way presented at the National Meeting of Applied Mechanics, ASME, New Haven, Conn., June, 1932; from this paper are taken the curves used in Arts. 2 and 3

BENDING TO A CYLINDRICAL SURFACE 7 Substituting in Eq.(4),we obtain d'w Sw D 一 dxi 器+器 (a) Introducing the notation S D4 =22 (5) the general solution of Eq.(a)can be written in the following form: o-Cmh华+coh咎+器 ql2x2 qla -8u2D-16u4D (b) The constants of integration Ci and C2 will be determined from the conditions at the ends.Since the deflections of the strip at the ends are zero,we have 0=0 for x =0 and x (c) Substituting for w its expression(b),we obtain from these two conditions gl4 1-cosh 2u qls C=16uD sinh 2u C:=16uD and the expression (b)for the deflection w becomes qlix qex2 16u4D Su"D SuD Substituting cosh 2u=cosh2 u+sinh2 u sinh 2u =2 sinh u cosh u coshu=1 sinh2u we can represent this expression in a simpler form: g 16u4D cosh u 8rD- or 0= gls 16u+D :小+ (6) Thus,deflections of the elemental strip depend upon the quantity u, which,as we see from Eq.(5),is a function of the axial force S.This force can be determined from the condition that the ends of the strip (Fig.3)do not move along the x axis.Hence the extension of the strip produced by the forces S is equal to the difference between the length of the arc along the deflection curve and the chord length l.This difference

Substituting in Eq. (4), we obtain (a) Introducing the notation (5) the general solution of Eq. (a) can be written in the following form: (b) The constants of integration C\ and C2 will be determined from the conditions at the ends. Since the deflections of the strip at the ends are zero, we have w = 0 for x = 0 and x = I (c) Substituting for w its expression (6), we obtain from these two conditions and the expression (b) for the deflection w becomes Substituting we can represent this expression in a simpler form: (6) Thus, deflections of the elemental strip depend upon the quantity u, which, as we see from Eq. (5), is a function of the axial force S. This force can be determined from the condition that the ends of the strip (Fig. 3) do not move along the x axis. Hence the extension of the strip produced by the forces S is equal to the difference between the length of the arc along the deflection curve and the chord length I. This difference

8 THEORY OF PLATES AND SHELLS for small deflections can be represented by the formulal (7) In calculating the extension of the strip produced by the forces S,we assume that the lateral strain of the strip in the y direction is prevented and use Eq.(2).Then 品”-（僧 hE dx (d) Substituting expression (6)for w and performing the integration, we obtain the following equation for calculating S: 81=g I tanh?u 5 hE 5 tamh 256 256m6+384n or substituting S=4u2D/12,from Eq.(5),and the expression for D from Eq.(3),we finally obtain the equation Ehs =a”+器n”- 0-9g0=16° -16m8十2 (8) For a given material,a given ratio h/l,and a given load g the left-hand side of this equation can be readily calculated,and the value of u satis- fying the equation can be found by a trial-and-error method.To simplify this solution,the curves shown in Fig.4 can be used.The abscissas of these curves represent the values of u and the ordinates represent the quantities log1(104vUo),where Uo denotes the numerical value of the right-hand side of Eq.(8).VUo is used because it is more easily calcu- lated from the plate constants and the load;and the factor 104 is intro- duced to make the logarithms positive.In each particular case we begin by calculating the square root of the left-hand side of Eq.(8),equal to Eh4/(1 -2)ql,which gives vUo.The quantity logio(104 vUo)then gives the ordinate which must be used in Fig.4,and the corresponding value of u can be readily obtained from the curve.Having u,we obtain the value of the axial force S from Eq.(5). In calculating stresses we observe that the total stress at any cross section of the strip consists of a bending stress proportional to the bend- ing moment and a tensile stress of magnitude S/h which is constant along the length of the strip.The maximum stress occurs at the middle of the strip,where the bending moment is a maximum.From the differential equation (4)the maximum bending moment is =-D d"w See Timoshenko,"Strength of Materials,"part I,3d ed.,p.178,1955

for small deflections can be represented by the formula1 (7) In calculating the extension of the strip produced by the forces S, we assume that the lateral strain of the strip in the y direction is prevented and use Eq. (2). Then (d) Substituting expression (6) for w and performing the integration, we obtain the following equation for calculating S: or substituting S = 4:2i2D/l2 , from Eq. (5), and the expression for D, from Eq. (3), we finally obtain the equation (8) For a given material, a given ratio h/l, and a given load q the left-hand side of this equation can be readily calculated, and the value of u satis￾fying the equation can be found by a trial-and-error method. To simplify this solution, the curves shown in Fig. 4 can be used. The abscissas of these curves represent the values of u and the ordinates represent the quantities logio (104 VU0), where Uo denotes the numerical value of the right-hand side of Eq. (8). V Uo is used because it is more easily calcu￾lated from the plate constants and the load; and the factor 104 is intro￾duced to make the logarithms positive. In each particular case we begin by calculating the square root of the left-hand side of Eq. (8), equal to Eh4 /(1 - v 2 )ql\ which gives 'VUo. The quantity logio (104 VfT0) then gives the ordinate which must be used in Fig. 4, and the corresponding value of u can be readily obtained from the curve. Having u, we obtain the value of the axial force S from Eq. (5). In calculating stresses we observe that the total stress at any cross section of the strip consists of a bending stress proportional to the bend￾ing moment and a tensile stress of magnitude S/h which is constant along the length of the strip. The maximum stress occurs at the middle of the strip, where the bending moment is a maximum. From the differential equation (4) the maximum bending moment is 1 See Timoshenko, "Strength of Materials," part I, 3d ed., p. 178, 1955

BENDING TO A CYLINDRICAL SURFACE 9 3 On Curve variation in u is trom 48 8to12 4.0 20 1.0 35 Curve B 车Curve A Curve C 18 091.73.0 L.6 l.5 0.8 2.5 14 13 L0g104U@ for 07112120 48 59 26 4812 Value of u Fra.4 Substituting expression (6)for w,we obtain qt2 M.=$vo(u) (9) where 0= 1-sech u (e) u2 2 The values of to are given by curves in Fig.5.It is seen that these values diminish rapidly with increase of u,and for large u the maximum

Value of u FIG. 4 Substituting expression (6) for W1 we obtain (9) where (e) The values of ^o are given by curves in Fig. 5. It is seen that these values diminish rapidly with increase of u, and for large u the maximum On Curve A variation in u is from 0 to 4- »» • i» B " » u " » 4 to 8 C " « u » » 8 to 12 Curve B Curve A Curve C Log l04 VUo(u.) for various values of a

0I0 1.0 Max.bending moment=Mmax Q09 Max.deflection =Wmax 09 Mmax with tensile reactions 0.08 mwithout tensile rections f Wmax with tensile reactions Q.07 Wmax without tensile reactions Q7 Subscript"o":Simply supported edges 006 Subscript"":Built-in edges Q6 0.05 a.5 6(@ 台 f,@以 004 olu) Q4 fo (u) 003 03 6)内 0.02 02 Q01 0.1 0 0 2 3 g 5 6 78 12-u-Simple Support +—u-Built-in Edges2" 109 876 4 2 0 FIG.5

Max. bending moment= Mmax Max. deflection = Wn10x Mmax with +ensile reactions ^ max without tensile reactions r_ Wmax with tensile reactions VVmax without tensile reactions Subscript "o": Simply supported edges Subscript V': Built-in edges FIG. 5 u-Built-in Edges