MIL-HDBK-17-3F Volume 3,Chapter 6 Structural Behavior of Joints 200T FM-300 ■DEBOND DNO DEBOND 150 6h=87J/m2 APPLIED CYCLIC PREDICTED STRESS 100 △S.MPa EC-3445 2 50 =38/m △ CLS SPECIMEN 30 60 90 TAPER ANGLE,a,deg FIGURE 6.2.2.7(a)Crack development in bonds of tapered composite doublers at 10 loading cycles. 6.2.3 Stresses and structural behavior of adhesive joints 6.2.3.1 General Stress analyses of adhesive joints have ranged from very simplistic "P over A"formulations in which only average shear stresses in the bond layer are considered,to extremely elegant elasticity approaches that consider fine details,e.g.,the calculation of stress singularities for application of fracture mechanics concepts.A compromise between these two extremes is desirable,since the adequacy of structural joints does not usually depend on a knowledge of details at the micromechanics level,but rather only at the scale of the bond thickness.Since practical considerations force bonded joints to incorporate adherends which are thin relative to their dimensions in the load direction,stress variations through the thickness of the adherend and the adhesive layer tend to be moderate.Such variations do tend to be more significant for polymer matrix composite adherends because of their relative softness with respect to transverse shear and thickness normal stresses.However,a considerable body of design procedure has been de- veloped based on ignoring thickness-wise adherend stress variations.Such approaches involve using one-dimensional models in which only variations in the axial direction are accounted for.Accordingly,the bulk of the material to be covered in this chapter is based on simplified one-dimensional approaches characterized by the work of Hart-Smith,and emphasizes the principles which have been obtained from that type of effort,since it represents most of what has been successfully applied to actual joint design, especially in aircraft components.The Hart-Smith approach makes extensive use of closed form and classical series solutions since these are ideally suited for making parametric studies of joint designs. The most prominent of these have involved modification of Volkersen(Reference 6.2.1(a))and Goland- Reissner(Reference 6.2.1(b))solutions to deal with ductile response of adhesives in joints with uniform adherend thicknesses along their lengths,together with classical series expressions to deal with variable adherend thicknesses encountered with tapered adherends,and scarf joints.Simple lap joint solutions described below calculate shear stresses in the adhesive for various adherend stiffnesses and applied loadings.For the more practical step lap joints,the described expressions can be adapted to treat the joint as a series of separate joints each having uniform adherend thickness. 6-11
MIL-HDBK-17-3F Volume 3, Chapter 6 Structural Behavior of Joints 6-11 FIGURE 6.2.2.7(a) Crack development in bonds of tapered composite doublers at 106 loading cycles. 6.2.3 Stresses and structural behavior of adhesive joints 6.2.3.1 General Stress analyses of adhesive joints have ranged from very simplistic "P over A" formulations in which only average shear stresses in the bond layer are considered, to extremely elegant elasticity approaches that consider fine details, e.g., the calculation of stress singularities for application of fracture mechanics concepts. A compromise between these two extremes is desirable, since the adequacy of structural joints does not usually depend on a knowledge of details at the micromechanics level, but rather only at the scale of the bond thickness. Since practical considerations force bonded joints to incorporate adherends which are thin relative to their dimensions in the load direction, stress variations through the thickness of the adherend and the adhesive layer tend to be moderate. Such variations do tend to be more significant for polymer matrix composite adherends because of their relative softness with respect to transverse shear and thickness normal stresses. However, a considerable body of design procedure has been developed based on ignoring thickness-wise adherend stress variations. Such approaches involve using one-dimensional models in which only variations in the axial direction are accounted for. Accordingly, the bulk of the material to be covered in this chapter is based on simplified one-dimensional approaches characterized by the work of Hart-Smith, and emphasizes the principles which have been obtained from that type of effort, since it represents most of what has been successfully applied to actual joint design, especially in aircraft components. The Hart-Smith approach makes extensive use of closed form and classical series solutions since these are ideally suited for making parametric studies of joint designs. The most prominent of these have involved modification of Volkersen (Reference 6.2.1(a)) and GolandReissner (Reference 6.2.1(b)) solutions to deal with ductile response of adhesives in joints with uniform adherend thicknesses along their lengths, together with classical series expressions to deal with variable adherend thicknesses encountered with tapered adherends, and scarf joints. Simple lap joint solutions described below calculate shear stresses in the adhesive for various adherend stiffnesses and applied loadings. For the more practical step lap joints, the described expressions can be adapted to treat the joint as a series of separate joints each having uniform adherend thickness
MIL-HDBK-17-3F Volume 3,Chapter 6 Structural Behavior of Joints 6.2.3.2 Adhesive shear stresses Figure 6.2.3.2(a)shows a joint with ideally rigid adherends,in which neighboring points on the upper and lower adherends align vertically before sliding horizontally with respect to each other when the joint is loaded.This causes a displacement difference 6=uu-uL related to the bond layer shear strain by Yb=6/tb.The corresponding shear stress,tb,is given by to=GbYb.The rigid adherend assumption implies that 6,yb and tb are uniform along the joint.Furthermore,the equilibrium relationship indicated in Figure 6.2.3.2(a)(C),which requires that the shear stress be related to the resultant distribution in the upper adherend by dTU/dx =Tb 6.2.3.2(a leads to a linear distribution of Tu and TL(upper and lower adherend resultants)as well as the adherend axial stresses oxu and oxL,as indicated in Figure 6.2.3.2(b).These distributions are described by the following expressions: i.e.Oxu=Oxu=Oxl-os 6.2.3.2(b) whereox=T/t.In actual joints,adherend deformations will cause shear strain variations in the bond layer which are illustrated in Figure 6.2.3.2(c).For the case of a deformable upper adherend in combina- tion with a rigid lower adherend shown in Figure 6.2.3.2(c)(A)(in practice,one for which ELtL>>Eutu). stretching elongations in the upper adherend lead to a shear strain increase at the right end of the bond layer.In the case shown in Figure 6.2.3.2(c)(B)in which the adherends are equally deformable,the bond shear strain increases at both ends of the joint.This is due to the increase in axial strain in whichever adherend is stressed(noting that only one adherend is under load)at a particular end of the joint.For both cases,the variation of shear strain in the bond results in an corresponding variation in shear stress which,when inserted into the equilibrium equation(Equation 6.2.3.2(a))leads to a nonlinear variation of the bond and adherend stresses.The Volkersen shear lag analysis (Reference 6.2.1(a))provides for cal- culations of these stresses for cases of deformable adherends. Introducing the notation(see Figure 6.2.3.2(d)) Eu,EL,tu,and tL=Young's moduli and thicknesses of upper and lower adherends Gp and tp=shear modulus and thickness of bond layer with BU =EutU,BL=ELtL while,denoting T as the applied axial resultant with OxU =T/tu and OxKL=T/tL 6-12
MIL-HDBK-17-3F Volume 3, Chapter 6 Structural Behavior of Joints 6-12 6.2.3.2 Adhesive shear stresses Figure 6.2.3.2(a) shows a joint with ideally rigid adherends, in which neighboring points on the upper and lower adherends align vertically before sliding horizontally with respect to each other when the joint is loaded. This causes a displacement difference δ = − u u U L related to the bond layer shear strain by γ b δ b = / t . The corresponding shear stress, τ b , is given by τ γ b = Gb b . The rigid adherend assumption implies that δ , γ b and τ b are uniform along the joint. Furthermore, the equilibrium relationship indicated in Figure 6.2.3.2(a) (C), which requires that the shear stress be related to the resultant distribution in the upper adherend by d TU / dx = τ b 6.2.3.2(a) leads to a linear distribution of TU and TL (upper and lower adherend resultants) as well as the adherend axial stresses σ xU and σ xL , as indicated in Figure 6.2.3.2(b). These distributions are described by the following expressions: T T x T T x i e x x U L xU = =− xU x xL x F H I K == = − F H I AA A A K ; . ., ; 1 1 σσσ σσ 6.2.3.2(b) where σ x = T t / . In actual joints, adherend deformations will cause shear strain variations in the bond layer which are illustrated in Figure 6.2.3.2(c). For the case of a deformable upper adherend in combination with a rigid lower adherend shown in Figure 6.2.3.2(c) (A) (in practice, one for which ELtL>>EUtU), stretching elongations in the upper adherend lead to a shear strain increase at the right end of the bond layer. In the case shown in Figure 6.2.3.2(c) (B) in which the adherends are equally deformable, the bond shear strain increases at both ends of the joint. This is due to the increase in axial strain in whichever adherend is stressed (noting that only one adherend is under load) at a particular end of the joint. For both cases, the variation of shear strain in the bond results in an corresponding variation in shear stress which, when inserted into the equilibrium equation (Equation 6.2.3.2(a)) leads to a nonlinear variation of the bond and adherend stresses. The Volkersen shear lag analysis (Reference 6.2.1(a)) provides for calculations of these stresses for cases of deformable adherends. Introducing the notation (see Figure 6.2.3.2(d)) EU, EL, tU, and tL = Young’s moduli and thicknesses of upper and lower adherends Gb and tb = shear modulus and thickness of bond layer with BU EUtU B E t = = L L L , while, denoting T as the applied axial resultant with σ σ xU = = T t and T t U xKL L / /
MIL-HDBK-17-3F Volume 3,Chapter 6 Structural Behavior of Joints (A)RIGID AOHERBO MODEL (B BOND SHEAR STRAIN, BOUILIBRIUM To +T =T DF=〔-)-T三0 FIGURE 6.2.3.2(a)Elementary joint analysis(Rigid adherend model). (A)AXIAL RESULTANT DISTRIBUTION (B)AXIAL STRESS DISTRIBUTION 1 1 09 UFER AHRED 09 UFFER AOHERED 08 07 07 IH 08 a5 5 04 客 0.4 aS 02 02 Q1 LoN织O8BQ LOWER AOHERDD 0 0 0 Q1 02 Q3 04 05 08 07 0S 09 1 0 0.1 a20804 050Ba70a 1 x/0 x/ FIGURE 6.2.3.2(b)Axial stresses in joint with rigid adherends. 6-13
MIL-HDBK-17-3F Volume 3, Chapter 6 Structural Behavior of Joints 6-13 FIGURE 6.2.3.2(a) Elementary joint analysis (Rigid adherend model). FIGURE 6.2.3.2(b) Axial stresses in joint with rigid adherends
MIL-HDBK-17-3F Volume 3,Chapter 6 Structural Behavior of Joints MINMUM MXIMM (A)RIGID LOVER A糕I度STRAIN AXIAL STRAIN ADHEREND MINIMN LAXIMy (B)BOTH ADHERENDS DEFORMABLE MXMIM WINIMM AXIA STRAIN A糕ILS米W FIGURE 6.2.3.2(c)Adherend deformations in idealized joints. T产P797只 Ev' L U FIGURE 6.2.3.2(d)Geometry for Volkersen solution. denoting the stresses in the two adherends at their loaded ends,together with ；i=uL;Pg=BL/BU 6.2.32(c) 2 6-14
MIL-HDBK-17-3F Volume 3, Chapter 6 Structural Behavior of Joints 6-14 FIGURE 6.2.3.2(c) Adherend deformations in idealized joints. FIGURE 6.2.3.2(d) Geometry for Volkersen solution. denoting the stresses in the two adherends at their loaded ends, together with β ρ = + F H G I K J L N M M O Q P P = + G = t tB B t t t b B B bU L U L B L LU 2 1 2 1 1 2 / ; ;/ 6.2.3.2(c)
MIL-HDBK-17-3F Volume 3,Chapter 6 Structural Behavior of Joints then the distribution for the axial stress in the upper adherend,osu(x),obtained from the Volkersen analy- sis is given by OxU =OxU Bu sinh B(x-()/ Bu sinh/ 6.2.3.2(d) BU +BL sinh Be/t Bu +BL sinh B/t A comparison of the distribution of axial stresses together with the bond shear stresses for the case of equal thicknesses in the adherends but a relatively rigid lower adherend(EL=10Eu)vs.that of two equally deformable adherends(EL=Eu)is given in Figure 6.2.3.2(e)below.The results in Figure 6.2.3.2(e)are for tu =tL(so that the loading stresses at the adherend ends are equal)and for a bond shear modulus and thickness chosen so that B=0.387 and (/t =20 for both cases(giving Bc/t =7.74)and a nominal adher- end stress oxu=oxL=10(in unspecified units).The maximum shear stresses are to a good approxima- tion given by x=0 folmas-T8 Bu t BU+BL X=(Folma、BL 6.2.3.2(e) t Bu+BL 10 HAXAL STRESS DISTRBUTION --E-10E SHEAR STRESS DESTR图UTIN-气-O气 LOWER ADHEREND 125 6」 WPPER ADHEREND ●2406811.21.41.81 2 0020.40第0第1121.4181g2 问AL STRESS DISTRIBUTION-气"气， 2 4 EAR STRESS DISTR0VnO-5"气 UPPER ADHEREND 14 LOWER ADHEREND 04 02●408●B11.2141.01.2 00204080B112141B1B2 FIGURE 6.2.3.2(e)Comparison of adherend stresses and bond shear stresses for E=E vs.E=10E B and adherend thicknesses equal for both cases. 6-15
MIL-HDBK-17-3F Volume 3, Chapter 6 Structural Behavior of Joints 6-15 then the distribution for the axial stress in the upper adherend, σxU(x), obtained from the Volkersen analysis is given by σ σ β β β β xU xU U U L U U L B B B x t t B B B x t t = + + L − N M O Q P + + R S | T | U V | W | 1 sinh ( ) / sinh / sinh / sinh / A A A 6.2.3.2(d) A comparison of the distribution of axial stresses together with the bond shear stresses for the case of equal thicknesses in the adherends but a relatively rigid lower adherend (EL = 10EU) vs. that of two equally deformable adherends (EL = EU) is given in Figure 6.2.3.2(e) below. The results in Figure 6.2.3.2(e) are for tU = tL (so that the loading stresses at the adherend ends are equal) and for a bond shear modulus and thickness chosen so that β = 0.387 and A / t = 20 for both cases (giving βA / t = 7.74) and a nominal adherend stress σxU = σxL = 10 (in unspecified units). The maximum shear stresses are to a good approximation given by x T t B B B x T t B B B b U U L b L U L = ≈ + = ≈ + 0 τ β τ β | | max max A 6.2.3.2(e) FIGURE 6.2.3.2(e) Comparison of adherend stresses and bond shear stresses for El=Eu vs. El=10Eu, β and adherend thicknesses equal for both cases