Bending moment factor

Goland and Reissner took the bending behaviour into account by using a bending moment factor, k, which relates the bending moment on the adherend at the end of the overlap, Mo, to the in-plane loading by the relationship ... 

Figure 5.35 Goland and Reissner bending moment factor (reference 5.29).

K is defined at the bending moment factor. Its value is unity for undeformed systems yet as the substrates bend, K decreases toward zero. If we assume K = I then Eq. (15) becomes the following ... 

Sneddon [26] and Adams and Peppiat [3] found an error in the initial formulation of Goland and Reissner s theory and presented a correction which derived an alternative bending moment factor. However, Carpenter [30] has since shown that Goland and Reissner s original solution was, in fact, correct. This was somewhat fortuitously achieved through cumulative typographical errors in the original paper. 

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].

For Goland and Reissner s first case, the stress distributions are shown in Fig. 6. 12 and the variations of the maximum stresses with the bending moment factor, K, are shown in Fig. 6.13. As may be seen, the transverse (out-of-plane) tensile stresses, crn, which operate perpendicular to the plane of the joint are most significant and are highest at the ends of the overlap joint hence such induced stresses are often referred to as cleavage or peel stresses, or sometimes... 

Benson (1969) has indicated more fundamental limitations in the Goland and Reissner theory in that the derivation of the bending moment factor, fc, allows for the rotation of the overlap region, but the internal stresses in the joint are derived assuming no rotation. The solution for the adhesive shear and the tensile stresses are only strictly accurate for fc = 1-0, i.e. for very low loads. It should be pointed out that the derivations relating the bending moment factor to the load assume that the joint is monolithic, i.e. that the glue-line thickness is... 

A series of parametric studies of the Goland and Reissner solutions was carried out by Kutscha and Hofer (1969) and they noted that both the load on the joint and its width are not explicitly factorable from the functions for shear and normal stress (i.e. the applied adherend stress is not factorable). This is because of the change in value of the bending moment factor as the load is increased. 

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 ... 

This criterion has been applied with successful results to thin bondlines and a conservative value has always been obtained (Adams et al. 1997). The prediction is also conservative for thin bondlines (less than 1 mm) but overestimates the results of thicker bondlines (more than 1 mm). This could be explained by the fact that the model does not take into account strain hardening of the steel. For a more accurate solution, the bending moment factor k could be computed according to Hart-Smith s equation that takes into account the adhesive thickness (Hart-Smith 1973). Note, however, that Eq. (27.4) would now require a numerical solution because k also depends on the load applied P. Moreover, Hart-Smith s bending moment factor is inaccurate when the adherend deforms plastically. 

In many isotropic materials the shear modulus G is high compared to the elastic modulus E, and the shear distortion of a transversely loaded beam is so small that it can be neglected in calculating deflection. In a structural sandwich the core shear modulus G, is usually so much smaller than Ef of the facings that the shear distortion of the core may be large and therefore contribute significantly to the deflection of a transversely loaded beam. The total deflection of a beam is thus composed of two factors the deflection caused by the bending moment alone, and the deflection caused by shear, that is, S = m + Ss, where S = total deflection, Sm = moment deflection, and Ss = shear deflection. 

Under transverse loading, bending moment deflection is proportional to the load and the cube of the span and inversely proportional to the stiffness factor, El. Shear deflection is proportional to the load and span and inversely proportional to shear stiffness factor N, whose value for symmetrical sandwiches is ... 

Therefore, to prevent the crack from growing (with a safety factor of three) the maximum bending moment must not exceed- 35 in.-lb. 

Ob depends on Mbm,s which is statistically derived and masks the maxima that can occur in the spectra of fluctuating bending moments. In order to include these maxima in fatigue analysis a factor of safety, Fbs, is used and a value of 2.5 should cover most cases. Thus from equation (13.11) one gets... 

Taking the bending moment safety factor, Fb, = 2.5, 32 Afbrms... 

M = external bending moment, in.-lb m = gasket factor N = gasket width, in. n = number of bolts... 

---