Angular joints are the dominant structural solutions of modern skeletal wooden furniture, whose shape and proportions have changed little since the days of ancient Egypt, Greece and Rome (Dzi^gielewski and Smardzewski 1995). For several decades, attempts have been made to provide their strength in the form of mathematical relationships. The first calculations, aiming to determine analytically the strength of rectangular glue-lines of angular wooden joints, were conducted by Ronai (1969), however mistakenly assuming that the state of tangential stresses in the glue-line corresponds to the state of stresses in any cross section of a prismatic rod under torsion. It was not until the work of Haberzak (1975) and later works (Matsui 1990a, b, 1991; Smardzewski 1994, 1995) that helped to establish that in a complex state of loads, the greatest tangential stress focuses in the corners of the glue-line.
Mortise and tenon joints and bridle joints are some of the most common in the constructions of furniture frames (Fig. 6.39). Therefore, let us separate corner from the frame (Fig. 6.39a), and let us indicate in its cross sections the forces N, T, M. Reducing these forces into the middle of the glue-line 0, we shall obtain that it is subjected to resultant force:
F = N + T, (6.90)
and the moment:
M = Mi + F • l. (6.91)
The state of stresses in such a node was provided in Haberzak’s (1975) work in a simplified manner, but useful for engineering practices.
Because force F passes through the middle of the glue-line, it causes (arbitrarily) an average state of stresses on the entire surface of the glue-line with a value of:
a, b dimensions of the glue-line, n number of the glue-lines, kk shearing strength of the glue-line.
While moment M, attempting to bend the frames of the joint, and simultaneously twist the glue-line, is counteracted by tangential stresses t2 emerging on the surfaces, different in different points of these surfaces. By assuming that their distribution is the same as in Fig. 6.39b, then we can write:
By continuing to maintain the state of balance between the external moment and the moment originating from the sum of stresses t2, we obtain:
M = m2 J J Vx2 + y2dxdy. (6.94)
Therefore, the value of the maximum stress occurring in the corners of the glue-line can be determined according to the formula:
M maximum bending moment,
J moment of inertia of the cross section of the glue-line in relation to the middle of rotation 0.
2b ¥ 2 2
J = J J (x2 + y2)dxdy = ab(a +b ). (6.96)
Therefore, the greatest static stress is the sum of component stresses x1 i x2 in one of the corners of the glue-line. According to Fig. 6.40, it amounts to:
Smax = Jx + S^x – 2s1s2max cos d < fcf, (6.97)
whereas for glue-line with the dimensions of a < b,
d = 90° + b + У, (6.98)
b = arctg ь,
у = 90° – a, (6.99)
a = arcctg ^,
and for glue-line with proportions a > b,
d = 90° + a + у, (6.100)
у = 90° – b. (6.101)
When designing or checking bridle joints, the number n of glue-lines should be established for its provided dimensions a x b and for an established distribution of internal forces. The solutions proposed above are based on the elementary strength of materials and are completely correct from an engineering point of view. They take into account only a linear variability of tangential stresses caused by the bending moment and the lack of variability of the distribution of stresses originating from loads of axial forces, while these distributions should be nonlinear functions.
On this assumption, Smardzewski (1994) suggested a description of the distribution of tangential stresses in a square glue-line of a joint subject to clean torsion, using the appropriate equations of the theory of elasticity.
However, it is known that in angular connections, the glue-line is in a complex state of loads. Establishing in them the distribution of tangential stresses requires constructing a suitable mathematical model. To this end, let us consider the single bridle joint, loaded by shearing forces and the bending moment (Fig. 6.41). In this joint, the distribution of tangential stresses in the glue-line is caused by the forces T = Fsine i N = Fcos^. Therefore, we can write down the first components of the state of stresses in the form of:
Fig. 6.41 Deformations in a joint subject to torsion
Establishing the distribution of tangential stresses caused by the bending moment M = 2qbh requires the analysis of forms of deformations of connected elements (Fig. 6.42).
The deformations of elementary sections of joint shown in Fig. 6.42, of the dimensions dx, dy, result from both the shear strains of elements and the glue-line. By undertaking to properly describe the state of tangential stresses in the joint, two adjacent elements were selected, with a width dx (Fig. 6.42) and of elastic properties Gj, Et and thickness St (1 in the bottom index concerns the upper element, 2—lower element, k—glue-line). These segments were then separated by planes perpendicular to the surface of joint at a distance of y — dy and y + dy, by also entering the loads substituting the impact of the cut-out parts. An additional assumption was also adopted here that the loads between the elements and the layer of glue are shifted along the edges dx and dy. By writing the equation of balance for the considered parts in the form:
Fig. 6.42 The distribution of internal forces in elementary sections of joint
soxdxdy = ^T1x – dy^ • S1dx – s^dx = ^x – dy^ S2dx – T2xS2dx,
and neglecting the small sizes of the higher orders, we shall obtain the dependence between tangential stresses Tix in the element:
Tox = S1 dT^x and Tox = S2 ~~r~. (6.107)
By further assuming the condition of the continuity of displacements, which requires that the adjacent walls of separated elements were also adjacent after deformation (Fig. 6.43), we can write the equation:
x’o = X2 + X0 – X1, (6.108)
where the displacement of a segment of glue in the upper element:
xO = Goo-Sk; (6.109)
Fig. 6.43 Deformations of elements’ section and the layer of glue-line
the displacement of a segment of glue in the lower element:
Xo = (xox + dy^ G; (6.110)
displacement of the upper element section:
x1 = (s, x+tt dy)s“: • (6111)
displacement of the lower element section:
X2 = (s2x + ds2xdy^ gL. (6.112)
V dy ) Gxy2
By entering the above equations to the previous equation, and neglecting the small sizes of the higher orders, we shall obtain the equation:
dsox Gk I s1x s2x
dy Sk Gxy1 Gxy2
from which by differentiating on both sides in relation to y, we shall obtain the differential equation of the second degree in the form:
we obtain the characteristic equation:
ery(r2 – k2) = 0; (6.116)
that has two specific solutions:
rj = k and r2 = —к.
The general equation of the differential equation of the second order, expressing tangential stresses sox in the glue-line, can therefore be written as follows:
Constants of integration c1 and c2 are determined by assuming the appropriate border conditions. Therefore,
By solving the above system of equations and entering c1 and c2, we shall obtain the general form of the equation describing the distribution of tangential stresses along the edge y of the glue-line in the following form:
For direction x, the distribution of stresses soy can be written in the analogous equation:
M bending moment,
b, h dimensions of the glue-line.
Finally, the tangential stresses in a rectangular glue-line, caused by a complex state of load, can be written in the vector form:
Sxy Sx ^ Sy ^ SOx ^ SOy, (6.122)
or also, respectively, for specific parts of the glue-line (Fig. 6.44) in the form:
Fig. 6.44 The components of vectors of tangential stresses in a glue-line
Fig. 6.45 Orientation of wooden elements in an angular connection with respect to the adhesion of connectors by planes: a tangential with tangential LT-LT, b tangential with radial LT-LR, c radial with radial LR-LR
The correctness of the results obtained with the use of those equations will depend on the correctness of the constant flexibilities of wood and glue-line, determined in the course of experimental studies. It should also be noted that in addition to constant elasticity related to linear deformations in the directions x and y of the local system of coordinates, the values of coefficients of elasticity associated with figural deformations should be determined. In practice, there are a few basic ways of connecting wooden elements in angular and cross-connections (Fig. 6.45).