Copyrighted Materials Cgo2 Cr Ue奇Pre女m CHAPTER EIGHT Shells In this chapter we consider thin composite shells,which we analyze on the basis of the main assumptions employed in the theory of thin plates.However,there is a major difference in the behavior of plates and shells subjected to external loads.Plates resist transverse loads by bending and by transverse shear forces.On the other hand,thin shells resist the transverse loads mostly by membrane forces, which,at any given point,are in the plane tangential to the reference surface (Fig.8.1).These membrane forces are determined by the "membrane theory of shells,"which neglects bending moments.The resulting stresses,strains,and de- formations are reasonable except near supports and in the vicinities of abrupt changes in loads.For thick shells (whose thickness is comparable to the radii of curvature)or when regions near supports or concentrated loads are of interest, more complex analytical solutions or finite element methods must be employed. The decision as to which method to use rests with the individual and depends on his or her experience with analytical solutions and finite element calculations. Herein we treat thin shells whose thickness h is small compared with all other dimensions and with the radii of curvatures(Fig.8.2).The membrane forces N, Ny,Nv,and Nyx acting at the reference surface of an infinitesimal element arel -0动- (8.1) where R and Ry are the radii of curvature in the x-z and y-z planes,and x,y,z are local coordinates with x and y in the plane tangential and z perpendicular to the reference surface at the point of interest(Fig.8.2).The origin of the coordi- nate system is at the reference surface,which,conveniently,may be taken at the 1 W.Flugge,Stresses in Shells.2nd edition.Springer.Berlin,1973.pp.5-6. 365
CHAPTER EIGHT Shells In this chapter we consider thin composite shells, which we analyze on the basis of the main assumptions employed in the theory of thin plates. However, there is a major difference in the behavior of plates and shells subjected to external loads. Plates resist transverse loads by bending and by transverse shear forces. On the other hand, thin shells resist the transverse loads mostly by membrane forces, which, at any given point, are in the plane tangential to the reference surface (Fig. 8.1). These membrane forces are determined by the “membrane theory of shells,” which neglects bending moments. The resulting stresses, strains, and deformations are reasonable except near supports and in the vicinities of abrupt changes in loads. For thick shells (whose thickness is comparable to the radii of curvature) or when regions near supports or concentrated loads are of interest, more complex analytical solutions or finite element methods must be employed. The decision as to which method to use rests with the individual and depends on his or her experience with analytical solutions and finite element calculations. Herein we treat thin shells whose thickness h is small compared with all other dimensions and with the radii of curvatures (Fig. 8.2). The membrane forces Nx, Ny, Nxy, and Nyx acting at the reference surface of an infinitesimal element are1 Nx = ) ht −hb σx 1 + z Ry dz Ny = ) ht −hb σy 1 + z Rx dz Nxy = ) ht −hb τxy 1 + z Ry dz Nyx = ) ht −hb τyx 1 + z Rx dz, (8.1) where Rx and Ry are the radii of curvature in the x–z and y–z planes, and x, y, z are local coordinates with x and y in the plane tangential and z perpendicular to the reference surface at the point of interest (Fig. 8.2). The origin of the coordinate system is at the reference surface, which, conveniently, may be taken at the 1 W. Fl ¨ugge, Stresses in Shells. 2nd edition. Springer, Berlin, 1973, pp. 5–6. 365
366 SHELLS Figure 8.1:Membrane forces in a shell. midsurface.For thin shells the quantities z/Ry and z/Rr are small with respect to unity,and these expressions reduce to N= (8.2) h In the "membrane theory of shells"the membrane forces depend only on the geometry,on the boundary conditions,and on the applied loads and are inde- pendent of the properties of the material.Hence,the membrane forces can be determined by the equations of static equilibrium. The force-strain relationships are(Eq.3.21) N A A2 A16 B11 B12 B16 ∈ N A2 A2 A6 B12 Bn B26 A16 A66 B16 B26 B66 Y M Bi1 D16 (8.3) B12 B16 Di1 D12 B12 B22 B26 D12 D22 D26 Mey B16 B26 B66 D16 D26 D66 Kxy One of the major assumptions of the membrane theory is that changes in curvatures do not affect the stresses.With this assumption,from the preceding equation,the strains are (8.4) A6 A26 A66 where e,andy are the strains of the reference surface.This set of equations applies to symmetrical as well as to unsymmetrical layups even though the form of the equations is the same as for symmetrical laminates(Eq.3.26).In the membrane Figure 8.2:The membrane forces and the radii of curvatures of an element
366 SHELLS y z x Nx Ny Nxy Nyx Figure 8.1: Membrane forces in a shell. midsurface. For thin shells the quantities z/Ry and z/Rx are small with respect to unity, and these expressions reduce to Nx = ) ht −hb σxdz Ny = ) ht −hb σydz Nxy = Nyx = ) ht −hb τxydz. (8.2) In the “membrane theory of shells” the membrane forces depend only on the geometry, on the boundary conditions, and on the applied loads and are independent of the properties of the material. Hence, the membrane forces can be determined by the equations of static equilibrium. The force–strain relationships are (Eq. 3.21) Nx Ny Nxy Mx My Mxy = A11 A12 A16 B11 B12 B16 A12 A22 A26 B12 B22 B26 A16 A26 A66 B16 B26 B66 B11 B12 B16 D11 D12 D16 B12 B22 B26 D12 D22 D26 B16 B26 B66 D16 D26 D66 o x o y γ o xy κx κy κxy . (8.3) One of the major assumptions of the membrane theory is that changes in curvatures do not affect the stresses. With this assumption, from the preceding equation, the strains are o x o y γ o xy = A11 A12 A16 A12 A22 A26 A16 A26 A66 −1 Nx Ny Nxy , (8.4) where o x , o y , and γ o xy are the strains of the reference surface. This set of equations applies to symmetrical as well as to unsymmetrical layups even though the form of the equations is the same as for symmetrical laminates (Eq. 3.26). In the membrane y z x Nx Ny Nxy Nyx Ry Rx ht hb Figure 8.2: The membrane forces and the radii of curvatures of an element
8.1 SHELLS OF REVOLUTION WITH AXISYMMETRICAL LOADING 367 Figure 8.3:Stresses in an isotropic(left)and composite shell(right). theory the strains are independent of the moments because the effects of changes in curvatures are neglected.In symmetrical laminates the strains are independent of the moments because the B]matrix is zero. We neglect the variations of the strains across the thickness of the shell.Hence, the strains are (8.5) The stresses in each layer are then calculated by Eq.(2.126)as follows: Ox Q11 Q12 12 Q22 (8.6) 16 26 066 Note that the stress distributions differ in isotropic and composite shells.In an isotropic shell the stress distribution across the thickness is uniform,and the resul- tant of the stresses is in the midplane(Fig.8.3).In a composite shell the stresses vary from layer to layer,and the resultant of the stresses generally is not in the midplane. The stresses and strains resulting from the preceding analysis are used in the design of the membrane section. Membrane forces for isotropic shells can be found in texts2 and handbooks. These membrane forces also apply to composite shells.In the next section,we present results for thin composite shells of practical interest 8.1 Shells of Revolution with Axisymmetrical Loading A shell of revolution is obtained by rotating a curve,called the meridian,about an axis of revolution.We consider an element of the shell's reference surface formed by two adjacent meridians and two parallel circles(Fig.8.4). The load is axisymmetrical,and therefore there are no shear forces (My=0), and only Mr and Ny normal forces (per unit length)act.Force balance in the zdirection(perpendicular to the surface)gives3 N:Ny +R, =Pz (8.7) where Rr is the radius of curvature of the meridian(Fig.8.4)and Ry is along a line normal to the meridian with a length that is the distance between the reference 2 Ibid. 3 Ibid..p.23
8.1 SHELLS OF REVOLUTION WITH AXISYMMETRICAL LOADING 367 z x σx z x σx Figure 8.3: Stresses in an isotropic (left) and composite shell (right). theory the strains are independent of the moments because the effects of changes in curvatures are neglected. In symmetrical laminates the strains are independent of the moments because the [B] matrix is zero. We neglect the variations of the strains across the thickness of the shell. Hence, the strains are x y γxy = o x o y γ o xy . (8.5) The stresses in each layer are then calculated by Eq. (2.126) as follows: σx σy τxy = Q11 Q12 Q16 Q12 Q22 Q26 Q16 Q26 Q66 x y γxy . (8.6) Note that the stress distributions differ in isotropic and composite shells. In an isotropic shell the stress distribution across the thickness is uniform, and the resultant of the stresses is in the midplane (Fig. 8.3). In a composite shell the stresses vary from layer to layer, and the resultant of the stresses generally is not in the midplane. The stresses and strains resulting from the preceding analysis are used in the design of the membrane section. Membrane forces for isotropic shells can be found in texts2 and handbooks. These membrane forces also apply to composite shells. In the next section, we present results for thin composite shells of practical interest. 8.1 Shells of Revolution with Axisymmetrical Loading A shell of revolution is obtained by rotating a curve, called the meridian, about an axis of revolution. We consider an element of the shell’s reference surface formed by two adjacent meridians and two parallel circles (Fig. 8.4). The load is axisymmetrical, and therefore there are no shear forces (Nxy = 0), and only Nx and Ny normal forces (per unit length) act. Force balance in the z direction (perpendicular to the surface) gives3 Nx Rx + Ny Ry = pz, (8.7) where Rx is the radius of curvature of the meridian (Fig. 8.4) and Ry is along a line normal to the meridian with a length that is the distance between the reference 2 Ibid. 3 Ibid., p. 23
368 SHELLS meridian meridian -axis of revolution Figure 8.4:Shell of revolution. surface and the point where the line intersects the axis of rotation;p:is the com- ponent of the load normal to the surface at the point of interest (Fig.8.5,left). We now consider the portion of the shell above the parallel circle defined by (Fig.8.5,right).We denote by F the resultant of all the loads acting on the shell above the parallel circle.A force balance along the axis of rotation gives F+2rroN.sin中=0， (8.8) where ro is defined in Figure 8.5.From the preceding equation,N is F N=- (8.9) 2xro sin The forces Ny and Nx are calculated from Eqs.(8.7)and(8.9).Expressions for M and N,are given in Table 8.1 for selected problems. 8.2 Cylindrical Shells We consider thin-walled circular cylinders subjected to pressure pa(which does not vary circumferentially),axial load N,and torque T(Fig.8.6). 8.2.1 Membrane Theory By neglecting edge effects,one may calculate the membrane forces(Fig.8.7)by the membrane theory.Force balances in the x and z directions and moment balance N N Figure 8.5:Load on an element and the free-body diagram for a shell of revolution
368 SHELLS z x Nx Nx Ny Ny y Ry Rx meridian meridian axis of revolution Figure 8.4: Shell of revolution. surface and the point where the line intersects the axis of rotation; pz is the component of the load normal to the surface at the point of interest (Fig. 8.5, left). We now consider the portion of the shell above the parallel circle defined by φ (Fig. 8.5, right). We denote by F the resultant of all the loads acting on the shell above the parallel circle. A force balance along the axis of rotation gives F + 2πr0Nx sin φ = 0, (8.8) where r0 is defined in Figure 8.5. From the preceding equation, Nx is Nx = − F 2πr0 sin φ . (8.9) The forces Ny and Nx are calculated from Eqs. (8.7) and (8.9). Expressions for Nx and Ny are given in Table 8.1 for selected problems. 8.2 Cylindrical Shells We consider thin-walled circular cylinders subjected to pressure pz (which does not vary circumferentially), axial load N, and torque T (Fig. 8.6). 8.2.1 Membrane Theory By neglecting edge effects, one may calculate the membrane forces (Fig. 8.7) by the membrane theory. Force balances in the x and z directions and moment balance Nx Nx Ry φ r0 F p pz z Figure 8.5: Load on an element and the free-body diagram for a shell of revolution
8.2 CYLINDRICAL SHELLS 369 Table 8.1.Membrane forces in spherical domes subjected to internal pressure(a). self-weight(b),and cones subjected to internal pressure(c);p,pz,p are in N/m2; pa is in N/m3. (a) N=P:R Ny=p:R (b) R N=- Ny=pR(T -cosφ) (c) N=号s[P。+)+P1s(受+】 Ny =(p:o+spa)(s +so)cota Figure 8.6:Thin cylinder subjected to radial pressure p:(which does not vary circumferentially),axial load N,and torque T. 个N Figure 8.7:The membrane forces in a thin cylinder
8.2 CYLINDRICAL SHELLS 369 Table 8.1. Membrane forces in spherical domes subjected to internal pressure (a), self-weight (b), and cones subjected to internal pressure (c); p, pz , pz0 are in N/m2; pz1 is in N/m3. (a) p R z φ Nx = 1 2 pzR Ny = 1 2 pzR (b) p R φ Nx = − pR 1+cos φ Ny = pR 1 1+cos φ − cos φ (c) pzo so s pzo + spz1 α Nx = cot α s+so s " pzo so + s 2 + pz1s so 2 + s 3 # Ny = (pzo + spz1) (s + so) cot α R pz L N N T T Figure 8.6: Thin cylinder subjected to radial pressure pz (which does not vary circumferentially), axial load N, and torque T . Ny x z y x y z Nxy Nx Figure 8.7: The membrane forces in a thin cylinder
370 SHELLS 个五 Figure 8.8:The loads and the membrane forces on a cylinder. about the x-axis yield (Fig.8.8) N2nR=N 2Ny P:2R (8.10) (Nzy R)2nR=T. where R is the radius of the wall's reference surface.From these equations,the membrane forces are N Nx=2R Ny P:R Nsy =2R (8.11) The strains corresponding to these membrane forces are calculated by Eq.(8.4). The axial u°，radial w°，and circumferential v°displacements are °=ed=xe+ugw°=Regv°= yo,dx xy+vo. (8.12) where ue and ve represent rigid-body motion. 8.2.2 Built-In Ends As we noted previously,near boundary supports the membrane theory is inaccu- rate,and the forces,moments,and displacements of the shell must be calculated by other means.In the following,we consider thin-walled circular cylinders built-in at each end.The cylinder is subjected to pressure pa,axial load N.and torque Figure 8.9:Cylinder built-in at both ends subjected to pressure p:,axial load N,and torque T. k 2R
370 SHELLS Nxy Nx N pz Ny x x T Figure 8.8: The loads and the membrane forces on a cylinder. about the x-axis yield (Fig. 8.8) Nx2π R = N 2Ny = pz2R (8.10) (NxyR) 2π R = T , where R is the radius of the wall’s reference surface. From these equations, the membrane forces are Nx = N 2π R Ny = pzR Nxy = T 2π R2 . (8.11) The strains corresponding to these membrane forces are calculated by Eq. (8.4). The axial uo, radial wo, and circumferential vo displacements are uo = ) o xdx = xo x + uo o wo = Ro y vo = ) γ o xydx = xγ o xy + vo o, (8.12) where uo o and vo o represent rigid-body motion. 8.2.2 Built-In Ends As we noted previously, near boundary supports the membrane theory is inaccurate, and the forces, moments, and displacements of the shell must be calculated by other means. In the following, we consider thin-walled circular cylinders built-in at each end. The cylinder is subjected to pressure pz, axial load N, and torque T pz L N N T T 2R x Figure 8.9: Cylinder built-in at both ends subjected to pressure pz, axial load N, and torque T
8.2 CYLINDRICAL SHELLS 371 Figure 8.10:Forces and moments inside the wall of a thin cylinder. (Fig.8.9).The pressure may vary linearly along the cylinder's axis, Pz=pao+xpa, (8.13) where Pao and Pa are specified constants.(For the pressure distribution shown in Fig.8.9,pa is negative.)For the applied loads(Fig.8.9),neither the stresses nor the strains vary circumferentially.Accordingly,the equilibrium equations are4 dN dx =0 (8.14) d (RNy+My)=0 (8.15) dx Ny dM =Pa (8.16) R dx2 v.=dM (8.17) dx dMy Vy=dx (8.18) where the forces Nr,Ny,Ny,the moments Mr,Mry,and the transverse shear forces V and V are illustrated in Figure 8.10. The strains and the curvatures of the reference surface are5 du e- wo du° (8.19) dx e= R y89= dx dPw° wo Kx= Ky= 2dv° Kxy=一 (8.20) dx2 R dx where°，v°，andw°are the axial,.circumferential,.and radial displacements of the reference surface. The force-strain relationships are identical to those of laminated plates and are given by Eq.(8.3). The equilibrium equations(Egs.8.14-8.18),the strain-displacement relation- ships (Egs.8.19-8.20),and the force-strain relationships (Eq.8.3)provide the forces,moments,and displacements of the wall.In the following we reduce these equations to readily usable forms. 4Ibid,Pp.205-206. 5 Ibid.,p.211
8.2 CYLINDRICAL SHELLS 371 Ny Nxy Nx Mxy Mxy My Mx Vy Vx Figure 8.10: Forces and moments inside the wall of a thin cylinder. (Fig. 8.9). The pressure may vary linearly along the cylinder’s axis, pz = pz0 + xpz1, (8.13) where pz0 and pz1 are specified constants. (For the pressure distribution shown in Fig. 8.9, pz1 is negative.) For the applied loads (Fig. 8.9), neither the stresses nor the strains vary circumferentially. Accordingly, the equilibrium equations are4 dNx dx = 0 (8.14) d dx (RNxy + Mxy) = 0 (8.15) Ny R − d2Mx dx2 = pz (8.16) Vx = dMx dx (8.17) Vy = dMxy dx , (8.18) where the forces Nx, Ny, Nxy, the moments Mx, Mxy, and the transverse shear forces Vx and Vy are illustrated in Figure 8.10. The strains and the curvatures of the reference surface are5 o x = duo dx o y = wo R γ o xy = dvo dx (8.19) κx = −d2wo dx2 κy = −wo R2 κxy = − 2 R dvo dx , (8.20) where uo, vo, and wo are the axial, circumferential, and radial displacements of the reference surface. The force–strain relationships are identical to those of laminated plates and are given by Eq. (8.3). The equilibrium equations (Eqs. 8.14–8.18), the strain–displacement relationships (Eqs. 8.19–8.20), and the force–strain relationships (Eq. 8.3) provide the forces, moments, and displacements of the wall. In the following we reduce these equations to readily usable forms. 4 Ibid., pp. 205–206. 5 Ibid., p. 211
372 SHELLS Table 8.2.The parameters required in the equations 2- [a]= [as]= A6- B12- R A6+ A-管 A2- B11 6- [a2l= 6- B6+ [a= B16- 2D16 [A= 「H H21 H22 [a]-[aallas]-'[az] g- [aallas]- i= 2五=-(h1+H)5=0f无=P0-装后=p1 The starting point of the analysis is the integration of the first two equilibrium equations (Eqs.8.14 and 8.15).These integrations yield D=Nx ,M」 D.=Nxy+R (8.21) where D,D2 are as yet unknown constants.By substituting the force-strain rela- tionships (Eq.8.3)into Eq.(8.21)and by introducing Eqs.(8.19)and (8.20)into the resulting equations,we obtain a=}+} (8.22) The matrices [az]and [a3]are given in Table 8.2. From Eq.(8.22)the derivatives of u°andu°are (8.23) The internal forces Ny and Mr may be expressed as (Eq.8.3) 公 A26 B12 B22 B61 Bu B12 B16 (8.24) Di D12 Kx
372 SHELLS Table 8.2. The parameters required in the equations [a1] = / A22 − B22 R B12 B12 − D12 R D110 [a3] = / A11 A16 − 2B16 R A16 + B16 R A66 − B66 R − 2D66 R2 0 [a2]= / A12 − B12 R B11 A26 − D26 R2 B16 + D16 R 0 [a4] = / A12 A26 − 2B26 R B11 B16 − 2D16 R 0 [H] = H11 H12 H21 H22! = [a1] − [a4][a3] −1[a2] g = g1 g2 = [a4][a3] −1 D1 D2 f1 = H22 f2 = − 1 R (H21 + H12) f3 = H11 R2 f4 = pz0 − g1 R f5 = pz1 The starting point of the analysis is the integration of the first two equilibrium equations (Eqs. 8.14 and 8.15). These integrations yield D1 = Nx D2 = Nxy + Mxy R , (8.21) where D1, D2 are as yet unknown constants. By substituting the force–strain relationships (Eq. 8.3) into Eq. (8.21) and by introducing Eqs. (8.19) and (8.20) into the resulting equations, we obtain D1 D2 = [a2] 1 wo R −d2wo dx2 6 + [a3] 1 duo dx dvo dx 6 . (8.22) The matrices [a2] and [a3] are given in Table 8.2. From Eq. (8.22) the derivatives of uo and vo are 1duo dx dvo dx 6 = −[a3] −1 [a2] 1 wo R −d2wo dx2 6 + [a3] −1 D1 D2 . (8.23) The internal forces Ny and Mx may be expressed as (Eq. 8.3) Ny Mx = A12 A22 A26 B12 B22 B26 B11 B12 B16 D11 D12 D16! o x o y γ o xy κx κy κxy . (8.24)
8.2 CYLINDRICAL SHELLS 373 By substituting Eqs.(8.19)and(8.20)into Eq.(8.24),we obtain }-}+ (8.25) The matrices [a]and [a4]are given in Table 8.2.Substitution of Eq.(8.23)into Eq.(8.25)results in =}+ (8.26) where [H]and g are given in Table 8.2.This equation can be written as P2w =H1R-H2+81 (8.27) w dPw° M,=1R-d+82, (8.28) where H,H2,H1,H2.g1,g2are the elements of the matrix [and the vectorg. By introducing Eqs.(8.27)and (8.28)into the third equilibrium equation(Eq.8.16) we obtain d4w°.edPw° h4+hd+后w°=店+x5, (8.29) where fi.....,fs are given in Table 8.2.We note that f contains the two as yet unknown constants D and D2. Solution of this fourth-order differential equation yields the radial displace- ment of the reference surface5 wo=(e-ix[Ci cos(Bx)+C2 sin(Bx)]+e-(L-[C3 cos(B(L-x)) +cm(L-加+[房6+ (8.30) where L is the length of the cylinder and A and B are the real and imaginary parts of the roots of the characteristic polynomial, λ=Re(y) -五+√经-4iB where (8.31) B Im(y) 2f Equation (8.30)is the solution of interest.This equation contains six unknown constants D,D2,C1-C4.The constant Di is given by Eqs.(8.21)and(8.11) N D =Nx 27R (8.32) The second equality in this equation is written by virtue of the fact that M is per unit length,N is the total force,and 2m R is the circumference;D2 is given by 6 E.Kreyszig.Advanced Engineering Mathematics.7th edition.John Wiley Sons,New York,1993, Pp.136-144
8.2 CYLINDRICAL SHELLS 373 By substituting Eqs. (8.19) and (8.20) into Eq. (8.24), we obtain Ny Mx = [a1] 1 wo R −d2wo dx2 6 + [a4] 1duo dx dvo dx 6 . (8.25) The matrices [a1] and [a4] are given in Table 8.2. Substitution of Eq. (8.23) into Eq. (8.25) results in Ny Mx = [H] 1 wo R −d2wo dx2 6 + g1 g2 , (8.26) where [H] and g are given in Table 8.2. This equation can be written as Ny = H11 wo R − H12 d2wo dx2 + g1 (8.27) Mx = H21 wo R − H22 d2wo dx2 + g2, (8.28) where H11, H12, H21, H22, g1, g2 are the elements of the matrix [H] and the vector g. By introducing Eqs. (8.27) and (8.28) into the third equilibrium equation (Eq. 8.16) we obtain f1 d4wo dx4 + f2 d2wo dx2 + f3wo = f4 + x f5, (8.29) where f1,...., f5 are given in Table 8.2. We note that f4 contains the two as yet unknown constants D1 and D2. Solution of this fourth-order differential equation yields the radial displacement of the reference surface6 wo = {e−λx[C1 cos (βx) + C2 sin (βx)] + e−λ(L−x) [C3 cos (β(L− x)) + C4 sin (β(L− x))]} + 1 f3 ( f4 + x f5) ! , (8.30) where Lis the length of the cylinder and λ and β are the real and imaginary parts of the roots of the characteristic polynomial, λ = Re(γ ) β = Im(γ ) where γ = 7889− f2 + $ f 2 2 − 4 f1 f3 2 f1 . (8.31) Equation (8.30) is the solution of interest. This equation contains six unknown constants D1, D2, C1–C4. The constant D1 is given by Eqs. (8.21) and (8.11) D1 = Nx = N 2π R . (8.32) The second equality in this equation is written by virtue of the fact that Nx is per unit length, Nis the total force, and 2π R is the circumference; D2 is given by 6 E. Kreyszig, Advanced Engineering Mathematics. 7th edition. John Wiley & Sons, New York, 1993, pp. 136–144
374 SHELLS Figure 8.11:The shear force and the twist moment. Eq.(8.21).The total torque acting at the edge of the cylinder is(Fig.8.11) =(Nsy R)27R+Mxy2n R. (8.33) Thus,from Eqs.(8.21)and (8.33)D2 is D2=2R (8.34) The constants C-Ca are obtained from the boundary conditions,which state that at a built-in end the radial displacement and its slope are zero as follows: dw w°=0 =0 atx=0 dx dwo (8.35) w°=0 =0 atx=L. dx The derivative of wo is given in Table 8.4.With the displacement given in Eq.(8.30),the boundary conditions above give 4 1 0 Y13 -入 B Y3 4 f+Lfs (8.36) 31 Y 0 C3 Y41 Yi where Yii are listed in Table 8.3.Equations(8.36)provide C1-C4.The displacement w is calculated with the constants D,D,C-C thus determined. Table 8.3.The parameters in Eq.(8.36) Yis =e-AL cos BL Yia =e-AL sin BL Y =e-L cos BL+Bsin BL) Ya=e-L (-B cos BL+sin BL) Ys1=e-AL cos BL Ys2=e-AL sin BL Ya =-e-L cos BL+B sin BL) Y=eL (B cos BL-sin BL)
374 SHELLS Nxy Mxy Figure 8.11: The shear force and the twist moment. Eq. (8.21). The total torque acting at the edge of the cylinder is (Fig. 8.11) T = (NxyR) 2π R + Mxy2π R. (8.33) Thus, from Eqs. (8.21) and (8.33) D2 is D2 = T 2π R2 . (8.34) The constants C1–C4 are obtained from the boundary conditions, which state that at a built-in end the radial displacement and its slope are zero as follows: wo = 0 dwo dx = 0 at x = 0 wo = 0 dwo dx = 0 at x = L. (8.35) The derivative of wo is given in Table 8.4. With the displacement given in Eq. (8.30), the boundary conditions above give 1 0 Y13 Y14 −λ β Y23 Y24 Y31 Y32 1 0 Y41 Y42 λ β C1 C2 C3 C4 = − f4 f3 f5 f3 f4+Lf5 f3 f5 f3 , (8.36) where Yi j are listed in Table 8.3. Equations (8.36) provide C1–C4. The displacement wo is calculated with the constants D1, D2, C1–C4 thus determined. Table 8.3. The parameters in Eq. (8.36) Y13 = e−λL cos βL Y14 = e−λL sin βL Y23 = e−λL (λ cos βL+ β sin βL) Y24 = e−λL (−β cos βL+ λ sin βL) Y31 = e−λL cos βL Y32 = e−λL sin βL Y41 = −e−λL (λ cos βL+ β sin βL) Y42 = e−λL (β cos βL− λ sin βL)