《纺织复合材料》课程参考文献（Composite Materials Handbook，Volume 3）Chapter 5 Design and Analysis

MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis Ex=Qu 02 Ey=Q22 5.3.1(X) 1 Q2-21 022 Q22 Also, Gxy-Q66 From the terms in the matrix (Equation 5.3.1(q))and the stiffness relations (Equation 5.3.1(x)). the elastic constants in an arbitrary coordinate system are functions of all the elastic constants in the prin- cipal material directions as well as the angle of rotation. The variation of elastic modulus E with angle of rotation is depicted in Figure 5.3.1(c)for a typical graphite/epoxy material.For demonstration purposes,two different shear moduli have been used in gen- erating the figure.The differences between the two curves demonstrate the effect of the principal material shear modulus on the transformed extensional stiffness.The two curves are identical at 0 and 90,as expected since Ex is simply E or E2.Between these two endpoints,substantial differences are present. For the smaller shear modulus,the extensional stiffness is less than the E2 value from approximately 50 to just less than 90.For these angles,the material stiffness is more strongly governed by the principal material shear modulus than by the transverse extensional modulus.The curves of Figure 5.3.1(c)can also be used to determine the modulus E,by simply reversing the angle scale. With the transformed stress-strain relations,it is now possible to develop an analysis for an assem- blage of plies,i.e.,a laminate. 25 E,=25.0Msi(172.4GPa) 160 E2=1.7Msi(11.7GPa) 20 G.=1.3 Msi (9.0 GPa) 140 2=0.30 120 15 E,=25.0Msi(172.4GPa) 100 E2=1.7Msi(11.7GPa) 80 (c0) 10 G2=0.65Msi(4.5GPa) %2=0.30 60 40 5 20 0- 0 0 30 60 90 0(Degrees) FIGURE 5.3.1(c)Variation of E with angle and Gi for typical graphite/epoxy materials. 5-26

MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-26 x 11 12 2 22 y 22 12 2 11 xy 12 22 21 22 E = Q - Q Q E = Q - Q Q = Q Q = Q Q ν 5.3.1(x) Also, xy G 66 = Q From the terms in the Q matrix (Equation 5.3.1(q)) and the stiffness relations (Equation 5.3.1(x)), the elastic constants in an arbitrary coordinate system are functions of all the elastic constants in the prin￾cipal material directions as well as the angle of rotation. The variation of elastic modulus Ex with angle of rotation is depicted in Figure 5.3.1(c) for a typical graphite/epoxy material. For demonstration purposes, two different shear moduli have been used in gen￾erating the figure. The differences between the two curves demonstrate the effect of the principal material shear modulus on the transformed extensional stiffness. The two curves are identical at 0° and 90°, as expected since Ex is simply E1 or E2. Between these two endpoints, substantial differences are present. For the smaller shear modulus, the extensional stiffness is less than the E2 value from approximately 50° to just less than 90°. For these angles, the material stiffness is more strongly governed by the principal material shear modulus than by the transverse extensional modulus. The curves of Figure 5.3.1(c) can also be used to determine the modulus Ey by simply reversing the angle scale. With the transformed stress-strain relations, it is now possible to develop an analysis for an assem￾blage of plies, i.e., a laminate. FIGURE 5.3.1(c) Variation of Ex with angle and G12 for typical graphite/epoxy materials

MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis 5.3.2 Lamination theory The development of procedures to evaluate stresses and deformations of laminates is crucially de- pendent on the fact that the thickness of laminates is much smaller than the in-plane dimensions.Typical thickness values for individual plies range between 0.005 and 0.010 inch(0.13 and 0.25 mm).Conse- quently,laminates using from 8 to 50 plies are still generally thin plates and,therefore,can be analyzed on the basis of the usual simplifications of thin plate theory. In the analysis of isotropic thin plates it has become customary to analyze the cases of in-plane load- ing and bending separately.The former case is described by plane stress elastic theory and the latter by classical plate bending theory.This separation is possible since the two loadings are uncoupled for sym- metric laminates;when both occur,the results are superposed. The classical assumptions of thin plate theory are: 1.The thickness of the plate is much smaller than the in-plane dimensions; 2.The shapes of the deformed plate surface are small compared to unity; 3. Normals to the undeformed plate surface remain normal to the deformed plate surface; 4. Vertical deflection does not vary through the thickness;and 5. Stress normal to the plate surface is negligible. On the basis of assumptions(2)-(4),the displacement field can be expressed as: uz=uz(x,y) 4=u吠（飞y)-zu业 5.3.2(a) x uyuy (x,y)-z uz ay with the x-y-z coordinate system defined in Figure 5.3.2(a).These relations (Equation 5.3.2(a))indicate that the in-plane displacements consist of a mid-plane displacement,designated by the superscript(). plus a linear variation through the thickness.The two partial derivatives are bending rotations of the mid- surface.The use of assumption(4)prescribes that uz does not vary through the thickness. The linear strain displacement relations are Exx= dux duy ∂x 6=3u+u 5.3.2b) 2 ay ax and performing the required differentiations yields Exx=Exx ZKxx Eyy =Eyy +ZKyy 5.3.2(c) 2Exy=2 Exy+2ZKxy or {ex}={e)+z 5.3.2(d) where 5-27

MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-27 5.3.2 Lamination theory The development of procedures to evaluate stresses and deformations of laminates is crucially de￾pendent on the fact that the thickness of laminates is much smaller than the in-plane dimensions. Typical thickness values for individual plies range between 0.005 and 0.010 inch (0.13 and 0.25 mm). Conse￾quently, laminates using from 8 to 50 plies are still generally thin plates and, therefore, can be analyzed on the basis of the usual simplifications of thin plate theory. In the analysis of isotropic thin plates it has become customary to analyze the cases of in-plane load￾ing and bending separately. The former case is described by plane stress elastic theory and the latter by classical plate bending theory. This separation is possible since the two loadings are uncoupled for sym￾metric laminates; when both occur, the results are superposed. The classical assumptions of thin plate theory are: 1. The thickness of the plate is much smaller than the in-plane dimensions; 2. The shapes of the deformed plate surface are small compared to unity; 3. Normals to the undeformed plate surface remain normal to the deformed plate surface; 4. Vertical deflection does not vary through the thickness; and 5. Stress normal to the plate surface is negligible. On the basis of assumptions (2) - (4), the displacement field can be expressed as: z z x x z y y z u = u (x,y) u = u (x, y) - z u x u = u (x, y) - z u y ° ° ° ∂ ∂ ∂ ∂ 5.3.2(a) with the x-y-z coordinate system defined in Figure 5.3.2(a). These relations (Equation 5.3.2(a)) indicate that the in-plane displacements consist of a mid-plane displacement, designated by the superscript (°), plus a linear variation through the thickness. The two partial derivatives are bending rotations of the mid￾surface. The use of assumption (4) prescribes that uz does not vary through the thickness. The linear strain displacement relations are xx x yy y xy x y = u x = u y = 1 2 u y + u x ε ε ε ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ F H G I K J 5.3.2(b) and performing the required differentiations yields xx xx xx yy yy yy xy xy xy = + z = + z 2 =2 +2z ε ε κ εε κ ε ε κ ° ° ° 5.3.2(c) or {ε ε x} = { }+ z{ }κ ° 5.3.2(d) where

MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis MID-SURFACE 】 K PLY TOTAL LAMINATE THICKNESS 2h FIGURE 5.3.2(a)Laminate construction. u 。 {eo}= 心 5.3.2(e) duy dy ax and uz {K= dy2 5.3.2(0 2 2ux dxdy The strain at any point in the plate is defined as the sum of a mid-surface strain (e).and a curvature {K)multiplied by the distance from the mid-surface. For convenience,stress and moment resultants will be used in place of stresses for the remainder of the development of lamination theory(see Figure 5.3.2(b)).The stress resultants are defined as 5-28

MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-28 FIGURE 5.3.2(a) Laminate construction. k p ε D = ∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ L N M M O Q P P R S | | | | T | | | | U V | | | | W | | | | ° ° ° ° u x u y u y u x x y x y 5.3.2(e) and κ κ κ κ k p = F H G G G I K J J J = − ∂ ∂ − ∂ ∂ − ∂ ∂ ∂ F H G G G G G G G I K J J J J J J J xx yy xy z z x u x u y u x y 2 2 2 2 2 2 5.3.2(f) The strain at any point in the plate is defined as the sum of a mid-surface strain {ε D }, and a curvature {κ} multiplied by the distance from the mid-surface. For convenience, stress and moment resultants will be used in place of stresses for the remainder of the development of lamination theory (see Figure 5.3.2(b)). The stress resultants are defined as

MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis ())dz 网-()zdz FIGURE 5.3.2(b)Stress and moment resultants. Nxx h {N}= Nyy = ∫{ox}dz 5.3.2(g) -h Nxy and the moment resultants are defined as Mx h {M= Myy ox)zdz 5.3.2h) -h Mxy where the integrations are carried out over the plate thickness. Noting Equations 5.3.1(o)and 5.3.2(c),relations between the stress and moment resultants and the mid-plane strains and curvatures can be written as N)=jod=j[回e+zx)d血 5.3.20 h -h M-odz-1[回e+z)zd2 5.3.20) -h -h Since the transformed lamina stiffness matrices are constant within each lamina and the mid-plane strains and curvatures are constant with respect to the z-coordinate,the integrals in Equations 5.3.2(i) and(j)can be replaced by summations. Introducing three matrices equivalent to the necessary summations,the relations can be written as {N}=[A]e,+[B] 5.3.2(k) 5-29

MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-29 FIGURE 5.3.2(b) Stress and moment resultants. {N} = N N N = { }dz xx yy xy x F H G G G I K J J J z −h h σ 5.3.2(g) and the moment resultants are defined as {M} = M M M = { }zdz xx yy xy x F H G G G I K J J J z −h h σ 5.3.2(h) where the integrations are carried out over the plate thickness. Noting Equations 5.3.1(o) and 5.3.2(c), relations between the stress and moment resultants and the mid-plane strains and curvatures can be written as {N} = { }dz = Q ({ }+ z{ })dz x − − ° z z h h h h σ ε κ 5.3.2(i) {M} = { }zdz = Q ({ }+ z{ })zdz x − − ° z z h h h h σ ε κ 5.3.2(j) Since the transformed lamina stiffness matrices are constant within each lamina and the mid-plane strains and curvatures are constant with respect to the z-coordinate, the integrals in Equations 5.3.2(i) and (j) can be replaced by summations. Introducing three matrices equivalent to the necessary summations, the relations can be written as {N} = A { }+ B{ } ° ε κ 5.3.2(k)

MIL-HDBK-17-3F Volume 3,Chapter 5 Design and Analysis {M=[B]e9}+[D]x 5.3.20 where the stiffness matrix is composed of the following 3x3 matrices: [A]=Σ[Q](z-z-) i=l 1 B]-2含好) 5.3.2m) D2o动 where N is the total number of plies,zi is defined in Figure 5.3.2(a)and subscript i denotes a property of the ith ply.Note that zi-ziI equals the ply thickness.Here the reduced lamina stiffnesses for the ith ply are found from Equations 5.3.2(k)and(I)using the principal properties and orientation angle for each ply in turn.Thus,the constitutive relations for a laminate have been developed in terms of stress and mo- ment resultants. Classical lamination theory has been used to predict the internal stress state,stiffness and dimen- sional stability of laminated composites (e.g.,References 5.3.2(a)-(e)).The constitutive law for CLT cou- ples extensional,shear,bending and torsional loads with strains and curvatures.Residual strains or warpage due to differential shrinkage or swelling of plies in a laminate have also been incorporated in lamination theory using an environmental load analogy (See Sections 5.3.3.3 and 5.3.3.4.).The com- bined influence of various types of loads and moments on laminated plate response can be described using the ABD matrix from Equations 5.3.2(k)and(I).In combined form: Nx A11 A12 A16 B11 B12 B16 Ex Ny A12 A22 A26 B12 B22 B26 到 Nxy A16 A26 A66 B16 B26 B66 Exy 5.3.2(n) Mx B11 B12 B16 D11 D12 D16 Kx My B12 B22 B26 D12 D22 D26 Ky My」 B16 B26 B66 D16 D26 D66] LKXy」 where N are loads,M are moments,e are strains,K are curvatures and Aj=extensional and shear stiffnesses B extension-bending coupling stiffnesses D=bending and torsional stiffnesses Several observations regarding lay-up and laminate stacking sequence(LSS)can be made with the help of Equation 5.3.2(n).These include: (1)The stiffness matrix Aii in Equation 5.3.2(n)is independent of LSS.Inversion of the stiffness ma- trix [ABD]yields the compliance matrix [A'B'D'].This inversion is necessary in order to calculate strains and curvatures in terms of loads and moments.The inversion results in a relationship be- tween LSS and extension/shear compliances.However,this relationship is eliminated if the lami- nate is symmetric. (2)Nonzero values of A1s and A26 indicates that there is extension/shear coupling (e.g.,longitudinal loads will result in both extensional and shear strains).If a laminate is balanced A16 and A26 be- come zero,eliminating extension/shear coupling. 5-30

MIL-HDBK-17-3F Volume 3, Chapter 5 Design and Analysis 5-30 {M}= B{ }+ D { } ° ε κ 5.3.2(l) where the stiffness matrix is composed of the following 3x3 matrices: [A] = [Q] (z - z ) [B] = 1 2 [Q] (z - z ) [D] = 1 3 [Q] (z - z ) i=1 N i i i-1 i=1 N i i 2 i-1 2 i=1 N i i 3 i-1 3 ∑ ∑ ∑ 5.3.2(m) where N is the total number of plies, zi is defined in Figure 5.3.2(a) and subscript i denotes a property of the ith ply. Note that zi - zi-1 equals the ply thickness. Here the reduced lamina stiffnesses for the ith ply are found from Equations 5.3.2(k) and (l) using the principal properties and orientation angle for each ply in turn. Thus, the constitutive relations for a laminate have been developed in terms of stress and mo￾ment resultants. Classical lamination theory has been used to predict the internal stress state, stiffness and dimen￾sional stability of laminated composites (e.g., References 5.3.2(a) - (e)). The constitutive law for CLT cou￾ples extensional, shear, bending and torsional loads with strains and curvatures. Residual strains or warpage due to differential shrinkage or swelling of plies in a laminate have also been incorporated in lamination theory using an environmental load analogy (See Sections 5.3.3.3 and 5.3.3.4.). The com￾bined influence of various types of loads and moments on laminated plate response can be described using the ABD matrix from Equations 5.3.2(k) and (l). In combined form: x y xy x y xy 11 12 16 11 12 16 12 22 26 12 22 26 16 26 66 16 26 66 11 12 16 11 12 16 12 22 26 12 22 26 16 26 66 16 26 66 x y N N N M M M = A A A B B B A A A B B B A A A B B B B B B D D D B B B D D D B B B D D D L N M M M M M M M M M O Q P P P P P P P P P L N M M M M M M M M M O Q P P P P P P P P P ε ε xy x y xy ε κ κ κ L N M M M M M M M M M O Q P P P P P P P P P 5.3.2(n) where N are loads, M are moments, ε are strains, κ are curvatures and Aij = extensional and shear stiffnesses Bij = extension-bending coupling stiffnesses Dij = bending and torsional stiffnesses Several observations regarding lay-up and laminate stacking sequence (LSS) can be made with the help of Equation 5.3.2(n). These include: (1) The stiffness matrix Aij in Equation 5.3.2(n) is independent of LSS. Inversion of the stiffness ma￾trix [ABD] yields the compliance matrix [A'B'D']. This inversion is necessary in order to calculate strains and curvatures in terms of loads and moments. The inversion results in a relationship be￾tween LSS and extension/shear compliances. However, this relationship is eliminated if the lami￾nate is symmetric. (2) Nonzero values of A16 and A26 indicates that there is extension/shear coupling (e.g., longitudinal loads will result in both extensional and shear strains). If a laminate is balanced A16 and A 26 be￾come zero, eliminating extension/shear coupling