Design In the preceding chapters,we have discussed various aspects of fiber-reinforced polymers,including the constituent materials,mechanics,performance,and manufacturing methods.A number of unique characteristics of fiber-reinforced polymers that have emerged in these chapters are listed in Table 6.1.Many of these characteristics are due to the orthotropic nature of fiber-reinforced com- posites,which has also necessitated the development of new design approaches that are different from the design approaches traditionally used for isotropic materials,such as steel or aluminum alloys.This chapter describes some of the design methods and practices currently used for fiber-reinforced polymers including the failure prediction methods,the laminate design procedures,and the joint design considerations.A number of design examples are also included. 6.1 FAILURE PREDICTION Design analysis of a structure or a component is performed by comparing stresses (or strains)due to applied loads with the allowable strength (or strain capacity)of the material.In the case of biaxial or multiaxial stress fields,a suitable failure theory is used for this comparison.For an isotropic material that exhibits yielding,such as a mild steel or an aluminum alloy,either the maximum shear stress theory or the distortional energy theory(von Mises yield criterion)is commonly used for designing against yielding.Fiber-reinforced polymers are not isotropic,nor do they exhibit gross yielding.Thus,failure theories developed for metals or other isotropic materials are not applicable to composite materials.Instead,many new failure theories have been proposed for fiber-reinforced composites,some of which are discussed in this section. 6.1.1 FAILURE PREDICTION IN A UNIDIRECTIONAL LAMINA We consider the plane stress condition of a general orthotropic lamina contain- ing unidirectional fibers at a fiber orientation angle of a with respect to the x axis(Figure 6.1).In Chapter 3,we saw that four independent elastic constants, namely,E1,E22,Gi2,and vi2,are required to define its elastic characteristics Its strength properties are characterized by five independent strength values: SLt longitudinal tensile strength Srt transverse tensile strength 2007 by Taylor&Francis Group.LLC
6 Design In the preceding chapters, we have discussed various aspects of fiber-reinforced polymers, including the constituent materials, mechanics, performance, and manufacturing methods. A number of unique characteristics of fiber-reinforced polymers that have emerged in these chapters are listed in Table 6.1. Many of these characteristics are due to the orthotropic nature of fiber-reinforced composites, which has also necessitated the development of new design approaches that are different from the design approaches traditionally used for isotropic materials, such as steel or aluminum alloys. This chapter describes some of the design methods and practices currently used for fiber-reinforced polymers including the failure prediction methods, the laminate design procedures, and the joint design considerations. A number of design examples are also included. 6.1 FAILURE PREDICTION Design analysis of a structure or a component is performed by comparing stresses (or strains) due to applied loads with the allowable strength (or strain capacity) of the material. In the case of biaxial or multiaxial stress fields, a suitable failure theory is used for this comparison. For an isotropic material that exhibits yielding, such as a mild steel or an aluminum alloy, either the maximum shear stress theory or the distortional energy theory (von Mises yield criterion) is commonly used for designing against yielding. Fiber-reinforced polymers are not isotropic, nor do they exhibit gross yielding. Thus, failure theories developed for metals or other isotropic materials are not applicable to composite materials. Instead, many new failure theories have been proposed for fiber-reinforced composites, some of which are discussed in this section. 6.1.1 FAILURE PREDICTION IN A UNIDIRECTIONAL LAMINA We consider the plane stress condition of a general orthotropic lamina containing unidirectional fibers at a fiber orientation angle of u with respect to the x axis (Figure 6.1). In Chapter 3, we saw that four independent elastic constants, namely, E11, E22, G12, and n12, are required to define its elastic characteristics. Its strength properties are characterized by five independent strength values: SLt ¼ longitudinal tensile strength STt ¼ transverse tensile strength 2007 by Taylor & Francis Group, LLC
TABLE 6.1 Unique Characteristics of Fiber-Reinforced Polymer Composites Nonisotropic Orthotropic Directional properties Four independent elastic constants instead of two Principal stresses and principal strains not in the same direction Coupling between extensional and shear deformations Nonhomogeneous More than one macroscopic constituent Local variation in properties due to resin-rich areas,voids,fiber misorientation,etc. Laminated structure Laminated structure Extensional-bending coupling Planes of weakness between layers Interlaminar stresses Properties depend on the laminate type Properties may depend on stacking sequence Properties can be tailored according to requirements Poisson's ratio can be greater than 0.5 Nonductile behavior Lack of plastic yielding Nearly elastic or slightly nonelastic stress-strain behavior Stresses are not locally redistributed around bolted or riveted holes by yielding Low strains-to-failure in tension Noncatastrophic failure modes Delamination Localized damage (fiber breakage,matrix cracking.debonding,fiber pullout,etc.) Less notch sensitivity Progressive loss in stiffness during cyclic loading Interlaminar shear failure in bending Low coefficient of thermal expansion Dimensional stability Zero coefficient of thermal expansion possible Attachment problem with metals due to thermal mismatch High internal damping:High attenuation of vibration and noise Noncorroding SLe =longitudinal compressive strength Sre transverse compressive strength SLTs in-plane shear strength Experimental techniques for determining these strength properties have been presented in Chapter 4.Note that the in-plane shear strength SLrs in the principal material directions does not depend on the direction of the shear stress although both the longitudinal and transverse strengths may depend on the direction of the normal stress,namely,tensile or compressive. 2007 by Taylor Francis Group,LLC
SLc ¼ longitudinal compressive strength STc ¼ transverse compressive strength SLTs ¼ in-plane shear strength Experimental techniques for determining these strength properties have been presented in Chapter 4. Note that the in-plane shear strength SLTs in the principal material directions does not depend on the direction of the shear stress although both the longitudinal and transverse strengths may depend on the direction of the normal stress, namely, tensile or compressive. TABLE 6.1 Unique Characteristics of Fiber-Reinforced Polymer Composites Nonisotropic Orthotropic Directional properties Four independent elastic constants instead of two Principal stresses and principal strains not in the same direction Coupling between extensional and shear deformations Nonhomogeneous More than one macroscopic constituent Local variation in properties due to resin-rich areas, voids, fiber misorientation, etc. Laminated structure Laminated structure Extensional–bending coupling Planes of weakness between layers Interlaminar stresses Properties depend on the laminate type Properties may depend on stacking sequence Properties can be tailored according to requirements Poisson’s ratio can be greater than 0.5 Nonductile behavior Lack of plastic yielding Nearly elastic or slightly nonelastic stress–strain behavior Stresses are not locally redistributed around bolted or riveted holes by yielding Low strains-to-failure in tension Noncatastrophic failure modes Delamination Localized damage (fiber breakage, matrix cracking, debonding, fiber pullout, etc.) Less notch sensitivity Progressive loss in stiffness during cyclic loading Interlaminar shear failure in bending Low coefficient of thermal expansion Dimensional stability Zero coefficient of thermal expansion possible Attachment problem with metals due to thermal mismatch High internal damping: High attenuation of vibration and noise Noncorroding 2007 by Taylor & Francis Group, LLC
y y FIGURE 6.1 Two-dimensional stress state in a thin orthotropic lamina. Many phenomenological theories have been proposed to predict failure in a unidirectional lamina under plane stress conditions.Among these,the simplest theory is known as the maximum stress theory;however,the more commonly used failure theories are the maximum strain theory and the Azzi-Tsai-Hill failure theory.We discuss these three theories as well as a more generalized theory,known as the Tsai-Wu theory.To use them,applied stresses(or strains) are first transformed into principal material directions using Equation 3.30. The transformed stresses are denoted o11,022,and T12,and the applied stresses are denoted oxx,oy,and Tx. 6.1.1.1 Maximum Stress Theory According to the maximum stress theory,failure occurs when any stress in the principal material directions is equal to or greater than the corresponding ultimate strength.Thus to avoid failure, -SLe <011<SLt, -STe <22 STL -SLTs TI2<SLTs. (6.1) For the simple case of uniaxial tensile loading in the x direction,only oxx is present and oyy =Tx=0.Using Equation 3.30,the transformed stresses are 2007 by Taylor&Francis Group.LLC
Many phenomenological theories have been proposed to predict failure in a unidirectional lamina under plane stress conditions. Among these, the simplest theory is known as the maximum stress theory; however, the more commonly used failure theories are the maximum strain theory and the Azzi–Tsai–Hill failure theory. We discuss these three theories as well as a more generalized theory, known as the Tsai–Wu theory. To use them, applied stresses (or strains) are first transformed into principal material directions using Equation 3.30. The transformed stresses are denoted s11, s22, and t12, and the applied stresses are denoted sxx, syy, and txy. 6.1.1.1 Maximum Stress Theory According to the maximum stress theory, failure occurs when any stress in the principal material directions is equal to or greater than the corresponding ultimate strength. Thus to avoid failure, SLc < s11 < SLt, STc < s22 < STt, SLTs < t12 < SLTs: (6:1) For the simple case of uniaxial tensile loading in the x direction, only sxx is present and syy ¼ txy ¼ 0. Using Equation 3.30, the transformed stresses are y syy syy sxx q sxx txy txy x FIGURE 6.1 Two-dimensional stress state in a thin orthotropic lamina. 2007 by Taylor & Francis Group, LLC.
011=0xc0s20, 022=0 xx sin20, T12 =-Oxx sin0 cos 0. Thus,using the maximum stress theory,failure of the lamina is predicted if the applied stress oxx exceeds the smallest of (SLt/cos),(Srt/sin 0),and (SLTs/sine cose).Thus the safe value of ox depends on the fiber orientation angle 6,as illustrated in Figure 6.2.At small values of 6,longitudinal tensile failure is expected,and the lamina strength is calculated from (SL/cos )At high values of 0,transverse tensile failure is expected,and the lamina strength is calculated from(Sr/sin20).At intermediate values of 0,in-plane shear failure of the lamina is expected and the lamina strength is calculated from(SLTs/sine cose).The change from longitudinal tensile failure to in-plane shear failure occurs at =01=tan-1 SLTs/SL and the change from in-plane shear failure to 200 100 60 40 20 Maximum strain theory 10 Maximum stress theory 6 Azzi-Tsai- Hill theory 2 15 30 45 60 75 90 Fiber orientation angle,(deg) FIGURE 6.2 Comparison of maximum stress,maximum strain,and Azzi-Tsai-Hill theories with uniaxial strength data of a glass fiber-reinforced epoxy composite.(After Azzi,V.D.and Tsai,S.W.,Exp.Mech.,5,283,1965.) 2007 by Taylor Francis Group,LLC
s11 ¼ sxx cos2 u, s22 ¼ sxx sin2 u, t12 ¼ sxx sin u cos u: Thus, using the maximum stress theory, failure of the lamina is predicted if the applied stress sxx exceeds the smallest of (SLt=cos2 u), (STt=sin2 u), and (SLTs=sinu cosu). Thus the safe value of sxx depends on the fiber orientation angle u, as illustrated in Figure 6.2. At small values of u, longitudinal tensile failure is expected, and the lamina strength is calculated from (SLt=cos2 u). At high values of u, transverse tensile failure is expected, and the lamina strength is calculated from (STt=sin2 u). At intermediate values of u, in-plane shear failure of the lamina is expected and the lamina strength is calculated from (SLTs=sinu cosu). The change from longitudinal tensile failure to in-plane shear failure occurs at u ¼ u1 ¼ tan1 SLTs=SLt and the change from in-plane shear failure to Maximum stress theory Maximum strain theory Azzi–Tsai− Hill theory Uniaxial tensile strength (ksi) Fiber orientation angle, q (deg) 2 0 15 30 45 60 75 90 200 100 60 40 20 10 8 6 4 q FIGURE 6.2 Comparison of maximum stress, maximum strain, and Azzi–Tsai–Hill theories with uniaxial strength data of a glass fiber-reinforced epoxy composite. (After Azzi, V.D. and Tsai, S.W., Exp. Mech., 5, 283, 1965.) 2007 by Taylor & Francis Group, LLC.
transverse tensile failure occurs at =02=tan-Sn/SLTs.For example,for an E-glass fiber-epoxy composite with SLt =1100 MPa,Sn=96.5 MPa,and SLTs 83 MPa,01=4.3 and 02 =49.3.Thus,according to the maximum stress theory,longitudinal tensile failure of this composite lamina will occur for0°≤0<4.3°，in-plane shear failure will occur for4.3°≤0≤49.3oand transverse tensile failure will occur for 49.3<0<90. EXAMPLE 6.1 A unidirectional continuous T-300 carbon fiber-reinforced epoxy laminate is subjected to a uniaxial tensile load P in the x direction.The laminate width and thickness are 50 and 2 mm,respectively.The following strength properties are known: SL SLe =1447.5 MPa,ST =44.8 MPa,and SLTs =62 MPa Determine the maximum value of P for each of the following cases:(a)0=0, (b)0=30°，and(c0=60°. SOLUTION The laminate is subjected to a uniaxial tensile stress oxx due to the tensile load applied in thexdirection.In all three cases,where A is the cross-sectional area of the laminate. L.Since0=0°，11=gxx,o22=0,andT12=0. Therefore,in this case the laminate failure occurs whenou=oxx =Su 1447.5MPa. Since the tensile load p at which failure occurs is 144.75 kN.The mode of failure is the longitudinal tensile failure of the lamina. 2.Since0=30°，using Equation3.30, 11=0xc0s230°=0.750x 22=0asin230°=0.25rxx, T12=oxx sin30°cos30°=0.433ax. According to Equation 6.1,the maximum values of ou,o22,and Ti2 are (1)o =0.750xx SLt 1447.5 MPa,which gives oxx 1930 MPa (2)022 =0.250xx STt =44.8 MPa,which gives oxx 179.2 MPa (3)T12 =0.4330xx SLTs 62 MPa,which gives oxx =143.2 MPa Laminate failure occurs at the lowest value ofx In this case,the lowest value is 143.2 MPa.Using x==143.2 MPa,P=14.32 kN.The mode of failure is the in-plane shear failure of the lamina. 2007 by Taylor Francis Group.LLC
transverse tensile failure occurs at u ¼ u2 ¼ tan1 STt=SLTs. For example, for an E-glass fiber–epoxy composite with SLt ¼ 1100 MPa, STt ¼ 96:5 MPa, and SLTs ¼ 83 MPa, u1 ¼ 4:3 and u2 ¼ 49:3. Thus, according to the maximum stress theory, longitudinal tensile failure of this composite lamina will occur for 0 u < 4:3, in-plane shear failure will occur for 4:3 u 49:3 and transverse tensile failure will occur for 49:3 < u 90. EXAMPLE 6.1 A unidirectional continuous T-300 carbon fiber-reinforced epoxy laminate is subjected to a uniaxial tensile load P in the x direction. The laminate width and thickness are 50 and 2 mm, respectively. The following strength properties are known: SLt ¼ SLc ¼ 1447:5 MPa, STt ¼ 44:8 MPa, and SLTs ¼ 62 MPa: Determine the maximum value of P for each of the following cases: (a) u ¼ 08, (b) u ¼ 308, and (c) u ¼ 608. SOLUTION The laminate is subjected to a uniaxial tensile stress sxx due to the tensile load applied in the x direction. In all three cases, sxx ¼ P A, where A is the cross-sectional area of the laminate. 1. Since u ¼ 0, s11 ¼ sxx, s22 ¼ 0, and t12 ¼ 0. Therefore, in this case the laminate failure occurs when s11 ¼ sxx ¼ SLt ¼ 1447:5 MPa. Since sxx ¼ P A ¼ P (0:05 m)(0:002 m) , the tensile load P at which failure occurs is 144.75 kN. The mode of failure is the longitudinal tensile failure of the lamina. 2. Since u ¼ 308, using Equation 3.30, s11 ¼ sxx cos2 30 ¼ 0:75 sxx, s22 ¼ sxx sin2 30 ¼ 0:25 sxx, t12 ¼ sxx sin 30 cos 30 ¼ 0:433 sxx: According to Equation 6.1, the maximum values of s11, s22, and t12 are (1) s11 ¼ 0:75sxx ¼ SLt ¼ 1447:5 MPa, which gives sxx ¼ 1930 MPa (2) s22 ¼ 0:25sxx ¼ STt ¼ 44:8 MPa, which gives sxx ¼ 179:2 MPa (3) t12 ¼ 0:433sxx ¼ SLTs ¼ 62 MPa, which gives sxx ¼ 143:2 MPa Laminate failure occurs at the lowest value of sxx. In this case, the lowest value is 143.2 MPa. Using sxx ¼ P A ¼ 143:2 MPa, P ¼ 14.32 kN. The mode of failure is the in-plane shear failure of the lamina. 2007 by Taylor & Francis Group, LLC.