# Stability of Rear Wall

In addition to torsional loads, the board elements also shift shield loads, that is forces lying in their plane. Hence, for flaccid elements, in this case the rear walls, the possibility of the loss of stability occurs (Smardzewski 1991). Therefore, it is necessary to check the stability of this element for the most unfavourable load. Such a load occurs in the case of supporting the furniture body in four corners (Fig. 7.52).

To determine the value of critical loads in boards, the method proposed by Southwell (1954) is commonly used. This method in laboratory works on wood-based boards was also used by Ozarska-Bergandy (1983). It consists in measuring deflections of the board and the loads corresponding to these deflections. By drawing up a graph of the load-deformation dependencies for the central points of the surface of the shield, the value Fkr can be read from it as an ordinate of asymptote, to which the deformation curve strives (Fig. 7.53).

The nature of work of rear walls, being in a state of clear shearing, requires a broader discussion. The primary task is determining the value of critical loads in which the rear walls, as isotropic and orthotropic boards attached to the body in various ways, lose their stability. The values of loads and critical stresses are calculated on the basis of the linear theory. For boards subjected to shearing like in Fig. 7.54, the differential equation of the bending surface takes the following form:

(7.160)

Fig. 7.52 Shield loads of the rear wall caused by supporting the body in four corners

Fig. 7.54 Scheme of a sheared isotropic board

where

Nxy edge tangential forces,

w function describing the surface of the board bending, D = Eh3/(12(1 – u2)), E linear elasticity module of the board, h thickness of the board and

и Poisson’s ratio.

By applying the energy methods commonly used in the theory of elasticity, critical values of edge contact loads are sought. By determining the work of external forces by AW, the energy of the bent board by AU, we determine the value of critical forces from the equation:

AW = DU

whereby

(7.163)

The orthotropic board is a board of orthogonal anisotropy. This type of anisotropy occurs when the structural elements of the board are mutually perpen­dicular, e. g. the perpendicular-fibre plywood, particle board veneered on both sides or a glued wooden board. And both directions of the constructions are the main axes of elasticity of an orthotropic board. By determining the main directions of anisotropy for a board loaded like in Fig. 7.55, the differential equation of the bent surface takes the form:

where

Ex linear elasticity module in the direction of x, Ey linear elasticity module in the direction of y, G shear modulus and h thickness of the board,

whereby

DxVy = DyVx. (7.168)

Solutions of critical stresses of rectangularly non-unidirectional (orthotropic) boards loaded by contact forces should be sought by integrating the differential equation or on the basis of the previously presented energy methods.

Stability of Board with Articulated Support on Perimeter

The buckling surface for a free support of edges of the isotropic board can be adopted in the form of a double trigonometric series fulfilling all the border conditions:

By using one of the presented solving methods, we obtain an expression for the value of critical edge forces in the following form:

where

b smaller dimension of the rear wall,

k coefficient dependent on the relation c/a, determined on the basis of Table 7.7.

A freely supported sheared orthotropic board is described by Dutko (1976). He demonstrates how to calculate critical loads according to the following equation:

4p2

Nkr = k-b2rV4DxD3y. (7.171)

In this equation, the value of the coefficient k is determined on the basis of chart (Fig. 7.56), for the following characteristics:

Table 7.7 Value of the coefficient k for isotropic boards

 b/a 1 1.2 1.4 1.5 1.6 1.8 2 2.5 3 k 9.34 8 7.3 7.1 7 6.8 6.6 6.1 5.9

stiffness characteristic of the board,

reduced ratio of sides,

where

H, Dx, Dy as previously.

Updated: October 9, 2015 — 5:44 am