# Shearing Strength of Loaded Joints

The literature shows that descriptions of tests of the state of stresses in the joints of connections of anisotropic bodies are nonlinear, and in its assumptions, they adopt many simplifications. Niskanen (1955, 1957) was the first to give the distribution of stresses in the joint of connected wooden elements. By using the model of a solid anisotropic two-sheared shield, the author ignored the impact of the elastic prop­erties of the glue-line, believing that a thin glue-line occurring at gluing wood does not have a significant impact on deformations in connectors. These tests were also undertaken by Wilczynski (1988), when analysing the impact of dimensional proportions of connected elements on the size and form of the distribution of tangential stresses. Therefore, the considered connection was treated as a solid wooden element, simplifying the solution to the problem greatly. In order to accurately determine the distribution of stresses in the glue-lines of joints, numerical methods were used (Apalak and Davies 1993, 1994; Biblis and Carino 1993; Godzimirski 1985; Groth and Nordlund 1991; Ieandrau 1991; Janowiak 1993; Kline 1984; Lindemann and Zimmerman 1996; Nakai and Takemura 1995, 1996a, b; Pellicane 1994; Pellicane et al. 1994; Smardzewski 1994, 1995). Wilczynski’s (1988) tests seem particularly interesting, as a result of which the state of tangential stresses in the joints of two-sheared wooden shields was computer-indicated using the finite element method, using a rectangular, four-node orthotropic element and by verifying these calculations with a laboratory mea­surement by using electrical strain gauges.

Assuming a constant distribution of stresses in the direction of the width of elements and adopting a glue-line as a single-layer system, the lap joint (Fig. 6.34) is reduced to the form of a flat task of the elasticity theory (Fig. 6.35).

The balance of forces in the connection elements for the bottom element can be written as follows:

(6.64)

hence, the stress in the lower lap is expressed as follows:

and for the upper element,

ГйЬЛ + sdxb – rX1 +-dX~dx *1^1 = 0,

hence,

where

S1, S2 thickness of the upper and lower laps,

ax1, ffx2 normal stress in the upper and lower laps, t tangential stress in the glue-line.

Fig. 6.36 Deformations of the elementary, extreme section of the glue-line

In analysing the deformation of a single extreme element of an adhesive joint from Fig. 6.36, we obtain the system:

U1 + dx + s1dx = U2 + dx + s2dx,

U1 = y + dx dx)Sk; (6.69)

U2 = kSk;

from which we have:

dy _ £2 – £1 dx Sk

where

e1, e2 normal strains of the upper and lower laps, у shear strain of the glue-line.

Bearing in mind the balance of forces acting between the single element of the lap and glue-line, the deformations of the upper lap can be expressed as follows:

de1 ds1

dx dx

and shear strain in the form:

ds1

dx

Therefore, the normal strains e1, e2 of the upper and lower laps are expressed as follows:

(6.73)

(6.74)

where

G shear modulus of the glue-line,

E1, E2 linear elasticity module of the upper and lower laps.

By differentiating on both sides the equation:

dy _ £2 – £1 dx Sk

and substituting the above equations, we obtain a differential equation of the second order with a constant coefficient, expressing the change in the shear strain у on length l of the glue-line:

The general integral of the differential equation of the second order takes the form:

in which constant integrations C1 and C2 should be determined from border con­ditions for deformations of the glue-line:

By solving the above system relative to C1, C2 and introducing the shear strain of a glue-line y, the value of tangential stresses in a glue-line can be written in the general form as follows:

where

If E1= E2 = E and b1 = b2 = b, then the value of tangential stresses in the glue-line is described by the equation:

where

Depending on the susceptibility of e1 and e2 of elements of the connection, the maximum stresses can concentrate for x = 0 or x = l (Fig. 6.37).

The average value of these stresses on the whole length of the glue-line is well described by the equation:

Q

bl

The correctness of the results calculated with the use of the above equations will depend on the correctness of the constant flexibilities of wood and glue-line, determined in the course of experimental studies. However, it is known that the elastic properties of wood depend, among others, on the anatomic direction, and hence, the method of cutting elements glued together is significant for the distri­bution of stresses in joints. The system of wooden fibres in relation to the axis of stretching projects on the value of extensions in laps and hence on the shear strain and distribution of tangential stresses in the glue-line. By analysing this state of stresses, the main methods of gluing woods in the following planes should be considered:

– tangential and radial LT-LR (Fig. 6.38a),

– tangential LT-LT (Fig. 6.38b),

In the discussed cases, in order to determine the distribution of tangential stresses in wooden lap joints, taking into account the natural system of wood fibres, it is necessary to determine the value of the linear elasticity module of any lap in the transformed system of coordinates, rotated by the angle ф relative to the axis of stretching. At the same time, the values of constant elasticities of wood in non-transformable systems should be determined. The results of such tests enable to determine the impact of the type of material, plane of gluing, direction of fibres in relation to the axis of stretching on the distribution of tangential stresses in the glue-line. Moreover results determine possibilities to use typical solutions for iso­tropic bodies in case of wooden adhesive joints.

 Fig. 6.38 Methods of joining wood by gluing planes together: a LR and LT, b LT and LT, c LR and LR