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