MIL-HDBK-17-3F Volume 3.Chapter 6 Structural Behavior of Joints Typical characteristics of the shear stress distributions are seen at the rights (parts (B)and (D))in Figure 6.2.3.2(e)in the form of peaks at both ends for equally deformable adherends(BL=Bu);for dis- similar adherends with the lower adherend more rigid(BL>Bu),the higher peak stress obtained from Equation 6.2.3.2(e)occurs at the right end of the joint where x=(.Because of the shear strain charac- teristics which are illustrated in Figure 6.2.3.2(c)part(A),the higher peak generally occurs at the loaded end of the more flexible adherend. As a practical consideration,we will be interested primarily in long joints for which Be/t>>1.For these chases Equation 6.2.3.2(e)reduces to B/t>>1 6.2.3.20 BL BU:Tolmas "Box:BL.=BU;TolmasBs i.e.,for long overlaps,the maximum shear stress for the rigid adherend case tends to be twice as great as that for the case of equally deformable adherends,again illustrating the adverse effect of adherend un- balance on shear stress peaks. An additional point of interest is a typical feature of bonded joints illustrated in Figure 6.2.3.2(e)Part (d)which gives the shear stress distribution for equal adherend stiffness;namely,the fact that high adhe- sive shear stresses are concentrated near the ends of the joint.Much of the joint length is subjected to relatively low levels of shear stress,which implies in a sense that region of the joint is structurally ineffi- cient since it doesn't provide much load transfer:however,the region of low stress helps to improve dam- age tolerance of the joint since defects such as voids,and weak bond strength may be tolerated in re- gions where the shear stresses are low,and in joints with long overlaps this may include most of the joint. As discussed in Section 6.2.2.7.Hart-Smith has suggested that when ductility and creep are taken into account,it is a good idea to have a minimum shear stress level no more than 10%of the yield strength of the adhesive,which requires the minimum value of overlap length given in Equation 6.2.2.7(a). One other point of interest here is illustrated in Figure 6.2.3.2(f),which compares the behavior of the maximum shear stress in the bond with the average shear stress as a function of the dimensionless joint length,(/t(for the particular case of equal adherend stiffnesses).The average shear stress in the bond line is always the same as the uniform shear stress in the hypothetical joint with rigid adherends dis- cussed earlier,and from equilibrium is given by nle=au=tLt=a1=） The point illustrated here is the fact that although the average shear stress continuously decreases as the joint length increases,for the maximum shear stress which controls the load that can be applied with- out failure of the adhesive,there is a diminishing effect of the increased joint length when Be/t gets much greater than about 2.Joint design has sometimes been considered only a matter of choosing the joint length long enough to reduce the average shear stress given in Equation 6.2.3.2(f)to a value less than the allowable shear stress in the bond layer.Obviously if the adhesive responds elastically to failure and if the joint is long enough,the peak stresses at the joint ends will be much larger than the average stress. and joint failure will occur much below the load for which the average is equal to the allowable.On the other hand,ductility tends to dominate the behavior of structural adhesives,and design based on setting the peak stress equal to the allowable is too conservative.The effect of ductility which has already been discussed in Section 6.2.2.4 will be considered in the subsequent discussion. 6-16
MIL-HDBK-17-3F Volume 3, Chapter 6 Structural Behavior of Joints 6-16 Typical characteristics of the shear stress distributions are seen at the rights (parts (B) and (D)) in Figure 6.2.3.2(e) in the form of peaks at both ends for equally deformable adherends (BL = B U); for dissimilar adherends with the lower adherend more rigid (BL > B U), the higher peak stress obtained from Equation 6.2.3.2(e) occurs at the right end of the joint where x = A. Because of the shear strain characteristics which are illustrated in Figure 6.2.3.2(c) part (A), the higher peak generally occurs at the loaded end of the more flexible adherend. As a practical consideration, we will be interested primarily in long joints for which βA / t >> 1. For these chases Equation 6.2.3.2(e) reduces to β τ βσ τ βσ A / ;| ; ;| max max t B B BB L Ub x LUb x >> >> ≈ = ≈ 1 1 2 6.2.3.2(f) i.e., for long overlaps, the maximum shear stress for the rigid adherend case tends to be twice as great as that for the case of equally deformable adherends, again illustrating the adverse effect of adherend unbalance on shear stress peaks. An additional point of interest is a typical feature of bonded joints illustrated in Figure 6.2.3.2(e) Part (d) which gives the shear stress distribution for equal adherend stiffness; namely, the fact that high adhesive shear stresses are concentrated near the ends of the joint. Much of the joint length is subjected to relatively low levels of shear stress, which implies in a sense that region of the joint is structurally inefficient since it doesn't provide much load transfer; however, the region of low stress helps to improve damage tolerance of the joint since defects such as voids, and weak bond strength may be tolerated in regions where the shear stresses are low, and in joints with long overlaps this may include most of the joint. As discussed in Section 6.2.2.7, Hart-Smith has suggested that when ductility and creep are taken into account, it is a good idea to have a minimum shear stress level no more than 10% of the yield strength of the adhesive, which requires the minimum value of overlap length given in Equation 6.2.2.7(a). One other point of interest here is illustrated in Figure 6.2.3.2(f), which compares the behavior of the maximum shear stress in the bond with the average shear stress as a function of the dimensionless joint length, A / t ( for the particular case of equal adherend stiffnesses). The average shear stress in the bond line is always the same as the uniform shear stress in the hypothetical joint with rigid adherends discussed earlier, and from equilibrium is given by b ave x U L xU xL x | = t τ σ σσσ (t = t t; = ) A ≡ ≡ The point illustrated here is the fact that although the average shear stress continuously decreases as the joint length increases, for the maximum shear stress which controls the load that can be applied without failure of the adhesive, there is a diminishing effect of the increased joint length when βA / t gets much greater than about 2. Joint design has sometimes been considered only a matter of choosing the joint length A long enough to reduce the average shear stress given in Equation 6.2.3.2(f) to a value less than the allowable shear stress in the bond layer. Obviously if the adhesive responds elastically to failure and if the joint is long enough, the peak stresses at the joint ends will be much larger than the average stress, and joint failure will occur much below the load for which the average is equal to the allowable. On the other hand, ductility tends to dominate the behavior of structural adhesives, and design based on setting the peak stress equal to the allowable is too conservative. The effect of ductility which has already been discussed in Section 6.2.2.4 will be considered in the subsequent discussion
MIL-HDBK-17-3F Volume 3,Chapter 6 Structural Behavior of Joints 0.7 bda0.387 0.6 0.5 0.4 0.3 Tb)max/Ox 0.2 0.1 Tb)ave/Ox 0 0.51.52.53.54.55.56.57.5859.510.11.12. 555 /t→ FIGURE 6.2.3.2(f)Comparison of average and maximum shear stress vs.C/t. 6.2.3.3 Peel stresses Peel stresses,i.e.,through the thickness extensional stresses in the bond,are present because the load path in most adhesive joint geometries is eccentric.It is useful to compare the effect of peel stresses in single and double lap joints with uniform adherend thickness,since peel stresses are most severe for joints with uniform adherend thickness.The load path eccentricity in the single lap joint (Figure 6.2.3.3(a))is relatively obvious due to the offset of the two adherends which leads to bending deflection as in Figure 6.2.3.3(a)(B).In the case of double lap joints,as exemplified by the configuration shown in Figure 6.2.3.3(b),the load path eccentricity is not as obvious,and there may be a tendency to assume that peel stresses are not present for this type of joint because,as a result of the lateral symmetry of such configurations,there is no overall bending deflection.However,a little reflection brings to mind the fact that while the load in the symmetric lap joint flows axially through the central adherend prior to reaching the overlap region,there it splits in two directions,flowing laterally through the action of bond shear stresses to the two outer adherends.Thus eccentricity of the load path is also present in this type of joint. As seen in Figure 6.2.3.3(b)(C),the shear force,designated as FsH,which represents the accumulated effect of p for one end of the joint,produces a component of the total moment about the neutral axis of the upper adherend equal to Fsut/2.(Note that FsH is equivalent to T/2,since the shear stresses react this amount of load at each end.)The peel stresses,which are equivalent to the forces in the restraining springs shown in Figure 6.2.3.3(b)(B)and(C)have to be present to react the moment produced by the offset of Fsi about the neutral axis of the outer adherend.Peel stresses are highly objectionable.Later discussion will indicate that effects of ductility significantly reduce the tendency for failure associated with shear stresses in the adhesive.On the other hand,the adherends tend to prevent lateral contraction in the in-plane direction when the bond is strained in the thickness direction,which minimizes the availability of ductility effects that could provide the same reduction of adverse effects for the peel stresses.This is illustrated by what happens in the butt-tensile test shown in Figure 6.2.3.3(c)in which the two adherend surfaces adjacent to the bond are pulled away from each other uniformly.Here the shear stresses asso- ciated with yielding are restricted to a small region whose width is about equal to the thickness of the bond layer,near the outer edges of the system;in most of the bond,relatively little yielding can take 6-17
MIL-HDBK-17-3F Volume 3, Chapter 6 Structural Behavior of Joints 6-17 FIGURE 6.2.3.2(f) Comparison of average and maximum shear stress vs. A / t. 6.2.3.3 Peel stresses Peel stresses, i.e., through the thickness extensional stresses in the bond, are present because the load path in most adhesive joint geometries is eccentric. It is useful to compare the effect of peel stresses in single and double lap joints with uniform adherend thickness, since peel stresses are most severe for joints with uniform adherend thickness. The load path eccentricity in the single lap joint (Figure 6.2.3.3(a)) is relatively obvious due to the offset of the two adherends which leads to bending deflection as in Figure 6.2.3.3(a) (B). In the case of double lap joints, as exemplified by the configuration shown in Figure 6.2.3.3(b), the load path eccentricity is not as obvious, and there may be a tendency to assume that peel stresses are not present for this type of joint because, as a result of the lateral symmetry of such configurations, there is no overall bending deflection. However, a little reflection brings to mind the fact that while the load in the symmetric lap joint flows axially through the central adherend prior to reaching the overlap region, there it splits in two directions, flowing laterally through the action of bond shear stresses to the two outer adherends. Thus eccentricity of the load path is also present in this type of joint. As seen in Figure 6.2.3.3(b) (C), the shear force, designated as FSH, which represents the accumulated effect of τ b for one end of the joint, produces a component of the total moment about the neutral axis of the upper adherend equal to FSHt/2. (Note that FSH is equivalent to T/2, since the shear stresses react this amount of load at each end.) The peel stresses, which are equivalent to the forces in the restraining springs shown in Figure 6.2.3.3(b) (B) and (C) have to be present to react the moment produced by the offset of FSH about the neutral axis of the outer adherend. Peel stresses are highly objectionable. Later discussion will indicate that effects of ductility significantly reduce the tendency for failure associated with shear stresses in the adhesive. On the other hand, the adherends tend to prevent lateral contraction in the in-plane direction when the bond is strained in the thickness direction, which minimizes the availability of ductility effects that could provide the same reduction of adverse effects for the peel stresses. This is illustrated by what happens in the butt-tensile test shown in Figure 6.2.3.3(c) in which the two adherend surfaces adjacent to the bond are pulled away from each other uniformly. Here the shear stresses associated with yielding are restricted to a small region whose width is about equal to the thickness of the bond layer, near the outer edges of the system; in most of the bond, relatively little yielding can take
MIL-HDBK-17-3F Volume 3,Chapter 6 Structural Behavior of Joints place.For polymer matrix composite adherends,the adherends may fail at a lower peel stress level than that at which the bond fails,which makes the peel stresses even more undesirable. (A)UNDEFOREO GEOMETRY (ZERO LOAD) (B)DEFORMED GEOMETRY (LOADEO JOINT) 「PEEL5 TRA INED80*0 FIGURE 6.2.3.3(a)Peel stress development in single lap joints. (A)DOUBLE LAP JOINT C8 MOMENT EFFECT OF SHEAR FORCES Restralning Effect of Bond CC)COMPONENTS OF MOMENT FIGURE 6.2.3.3(b)Peel stress development in double lap joints. 6-18
MIL-HDBK-17-3F Volume 3, Chapter 6 Structural Behavior of Joints 6-18 place. For polymer matrix composite adherends, the adherends may fail at a lower peel stress level than that at which the bond fails, which makes the peel stresses even more undesirable. FIGURE 6.2.3.3(a) Peel stress development in single lap joints. FIGURE 6.2.3.3(b) Peel stress development in double lap joints
MIL-HDBK-17-3F Volume 3,Chapter 6 Structural Behavior of Joints Bond Edge Region (Distortional Strains) FIGURE 6.2.3.3(c)Shear stresses near outer edges of butt tensile test. It is important to understand that peel stresses are unavoidable in most bonded joint configurations. However,it will be seen that they can often be reduced to acceptable levels by selecting the adherend geometry appropriately. 6.2.3.4 Single and double lap joints with uniform adherend thickness In this section,joints with uniform adherend thickness are considered,since most important features of structural behavior of adhesive joints are illustrated by this case.Section 6.2.3.4.1 below deals with joint behavior under elastic response of the bond layer for structural loading alone.The effect of thermal stresses is treated in Section 6.2.3.4.2,while effects of adhesive ductility in the bond layer and trans- verse shear deformations in composite adherends are discussed in Sections 6.2.3.4.3 and 6.2.3.4.4, respectively. 6.2.3.4.1 Joint behavior with elastic response of the bond layer Double lap joints will be considered first since they are somewhat simpler to discuss than single lap joints because of lateral deflection effects which occur in the latter.The following notation (see Figure 6.2.3.4.1(a))is introduced for reference in the discussion: 6-19
MIL-HDBK-17-3F Volume 3, Chapter 6 Structural Behavior of Joints 6-19 FIGURE 6.2.3.3(c) Shear stresses near outer edges of butt tensile test. It is important to understand that peel stresses are unavoidable in most bonded joint configurations. However, it will be seen that they can often be reduced to acceptable levels by selecting the adherend geometry appropriately. 6.2.3.4 Single and double lap joints with uniform adherend thickness In this section, joints with uniform adherend thickness are considered, since most important features of structural behavior of adhesive joints are illustrated by this case. Section 6.2.3.4.1 below deals with joint behavior under elastic response of the bond layer for structural loading alone. The effect of thermal stresses is treated in Section 6.2.3.4.2, while effects of adhesive ductility in the bond layer and transverse shear deformations in composite adherends are discussed in Sections 6.2.3.4.3 and 6.2.3.4.4, respectively. 6.2.3.4.1 Joint behavior with elastic response of the bond layer Double lap joints will be considered first since they are somewhat simpler to discuss than single lap joints because of lateral deflection effects which occur in the latter. The following notation (see Figure 6.2.3.4.1(a)) is introduced for reference in the discussion:
MIL-HDBK-17-3F Volume 3,Chapter 6 Structural Behavior of Joints (A)DOUBLE STRAP JOINT +27 (B)DOUBLE LAP JOINT FIGURE 6.2.3.4.1(a)Symmetric double strap/double lap joints. Ei,ti,Eo,to=axial moduli and thickness of inner and outer adherends Go,Eb,t=bond shear and peel modulus and thickness Oxo,xi=axial adherend stresses To=Oxo to,T=Cxiti--axial resultants Tbob--bond shear and peel stress 6.2.3.4.1(a) -w--6 1/2 :i=0+4/2 ;PB=B:/B0 B To=o(ao-ai)AT:Gx-T/i:Oo-T/i Bo+Bi (thermal expansion coefs ao,ai;temperature change AT) Shear and peel stresses in double lap joints with uniform adherend thickness,including thermal mis- match effects have been treated in a number of places,in particular by Hart-Smith in Reference 6.2.1(i) Using the notation of Equation 6.2.3.4.1(a),the structural response of the joint accounting for both shear and peel stresses in the bond layer can be modeled using a combination of the Volkersen shear lag analysis(Reference 6.2.1(a))which gives d2To= 6.2.3.4.1(b) together with a beam-on-elastic foundation equation modified for the effect of tangential shear loading on the beam: d4o+4日=ldn dx4 140b -2to dx 6.2.3.4.1(c) 4=32） 1/4 6.2.3.4.1(d) Eo tb 6-20
MIL-HDBK-17-3F Volume 3, Chapter 6 Structural Behavior of Joints 6-20 FIGURE 6.2.3.4.1(a) Symmetric double strap/double lap joints. Ei, ti, Eo, to≡ axial moduli and thickness of inner and outer adherends Gb, Eb, tb≡ bond shear and peel modulus and thickness σxo, σxi = axial adherend stresses ; To ≡σxo to, T≡ σxiti - - axial resultants τ bσ b - - bond shear and peel stress 6.2.3.4.1(a) B t E B tE G t tB B t t t B B T B B B B T Tt T t i ii b b i i B i th i i i x th th 0 00 2 0 1 2 0 0 0 0 0 1 1 2 2 = == + F H G I K J L N M M O Q P P = + = = + − == ;; ; / ; / ; /; / / β ρ b g αα σ σ ∆ (thermal expansion coefs α 0 ,α i ; temperature change ∆T) Shear and peel stresses in double lap joints with uniform adherend thickness, including thermal mismatch effects have been treated in a number of places, in particular by Hart-Smith in Reference 6.2.1(i). Using the notation of Equation 6.2.3.4.1(a), the structural response of the joint accounting for both shear and peel stresses in the bond layer can be modeled using a combination of the Volkersen shear lag analysis (Reference 6.2.1(a)) which gives 2 o 2 b b o i o i o i d T dx = G t 1 B + 1 B T - 1 B T+ T F H G I K J L N M O Q ∆ aα α fP 6.2.3.4.1(b) together with a beam-on-elastic foundation equation modified for the effect of tangential shear loading on the beam: 4 b 4 d 4 4 b o d b d x + 4 t = 1 2 t d dx σ γ σ τ 6.2.3.4.1(c) d 1/4 b o o b = 3 E E t t γ F H G I K J 6.2.3.4.1(d)