Edge moment factor

Due to load eccentricity in the SLJ of O Fig. 24.5a, large deflections must be considered, as shown in Fig. 24.5b. This means that the bending moment JVft and the shear force Vj at the overlap ends need to be updated with structural deformation. This geometric nonlinearity can be characterized by the bending moment factor or edge moment factor k and is defined as ... 

Hart-Smith (1973) showed that the uncoupled formulation of Goland and Reissner (1944) is inconsistent with the force conditions of an individual adherend at the overlap ends. To overcome this deficiency and consider the influence of the adhesive deformation, Hart-Smith (1973) modeled each adherend of the overlap as an individual beam and derived the coupled formulation for load update and the adhesive shear stress. By using continuity conditions at intersection I, Hart-Smith (1973) obtained the edge moment factor and then simplified it as ... 

In this formulation (Hart-Smith 1973), the force conditions at intersections I and II are consistent the influence of the adhesive shear strain is included and the large deflections of the outer adherends are also modeled. However, the equilibrium equations in O Eq. 24.7 and Wi (wi + m )/2 are used in his formulation. It is noted thatO Eq. 24.7 does not model the overlap geometric nonlinearity and hence the large deflection effect of the overlap is ignored. Because Wi i= (wi + W2)/2 is used, the effect of the adhesive peel strain on the edge moment factor is also neglected. 

Oplinger (1994) treated the adherends as individual beams and considered the influence of the overlap deflection. The governing equations with the coupling of the adhesive shear strain and the overlap large deflection were derived and then analytical solutions for SLJs were found. The edge moment factor was derived as ... 

By using the continuity conditions at intersection I, unknowns Mk, u i and can be determined. The adherend displacements, adhesive stresses, and the updating forces at the overlap ends can be obtained simultaneously. The simplified edge moment factor is given by ... 

The edge moment factors (k) of the isotropic SLJ predicted hy various analytical solutions are illustrated in O Fig. 24.7. It is noted that A < 0 when the Timoshenko heam is used to model isotropic adherends of this SLJ. It has heen shown that edge moment factor predicted hy O Eq. 24.33 correlates very well with that predicted hy the geometrically nonlinear FE analyses (Luo and Tong 2007, 2008). 

Figure 24.8 shows the edge moment factors of the composite SLJ calculated hy using the theoretical formulations. It is noted A > 0 for this composite SLJ. It has also heen illustrated that k calculated by Eq. 24.33 correlates well with that of the geometrically nonlinear FE analyses (Luo and Tong 2008) for the composite SLJs. 

Edge moment factor of the isotropic SU predicted by the beam-adhesive models... 

Oplinger s model, the peel effect can be observed by comparing k for O Eq. 24.33 and O Eq. 24.33a. It is seen that the influence of adhesive peel strain on the edge moment factor is significant and should not be ignored. 

In O Figs. 24.9 and O 24.10, the same edge moment factor at = 9 calculated using O Eq. 24.33 is used for both models. Figure 24.9 shows that there are slight differences in values of shear stress predicted by the linear adhesive-beam model based on the Euler beam theory and the nonlinear adhesive-beam model based on the Timoshenko beam theory. However, there exist significant differences in the adhesive peel stress for the two models as shown in Fig. 24.10, including peak values and distribution patterns. 

Figure 6.12 Stress distribution along the shear plane as a function of distance from the edge of the overlap. The bending moment factor, k, is equal to one [3].

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