Stiffness of Bodies of Furniture

By analysing the next stages of assembling the body of the furniture piece, starting from the side walls and bottom, it is easy to notice that the side walls, due to low stiffness of joints, are not able to shift the side load P (Fig. 7.22). This structure can be improved by adding a rear wall, which will begin to shift side load P in its plane. By subsequently introducing load on free corners of the side walls by forces Q, a clearly curved deformation of boards is caused. And because the torsional stiffness is many times smaller than the shearing stiffness, in order to improve the quality of the structure, load Q should be shifted in the top plane, in this case, shield-loaded. Another loading of the top with force V vertical to its surface causes torsional deformation of the entire body of the furniture piece. In this situation, the only treatment, leading to the elimination of figural deformations of boards, is to

Fig. 7.22 Example of eliminating torsional deformations of the boards of the body of case furniture

mount doors inserted in the body in such a way, that the bottom, top and sides of the wardrobe meet with their outline. Forces V will then be shield-displaced by the doors, and the entire construction of the furniture piece will remain almost perfectly stiff.

Situating board elements in such a manner enables to obtain a high stiffness of the body and strength of joints, only when the doors of the furniture piece are closed. In the construction of case furniture without doors or with lift-off doors, during operational loads, the form of torsional deformation dominates.

The stiffness of furniture’s body depends on the value of displacement APz, measured at the direction of the operation of the force Pz. This displacement is caused by torsional deformation of individual boards. Therefore, without neglecting essential construction requirements of furniture in many studies (Ganowicz et al. 1977, 1978; Ganowicz and Kwiatkowski 1978), an important assumption was adopted that the elements of a wardrobe are connected together articulately and only in the corners. This enables to use energy methods, known in the theory of construction and express the value APz in the course of analytical calculations. Knowing that the work of external forces Pz on external displacement APz must be equal to the sum of the work of internal forces Pi on internal displacements A;, it can be written as

^ 1 .

-PzAPz = ^2 2 PiDi;

where

n number of boards (excluding shelves and doors) forming the body of furniture, P, internal load of the board and а, internal displacement of the board.

As it has already been mentioned, these displacements are the result of torsions that the sides, partitions, rear walls and other fixtures on permanent board elements of furniture are subject to. It should be added here that shelves, as elements that are unrelated to the body of the wardrobe, are not subject to torsions, and thus in no way does their number and location determine the stiffness of the body, that is the values of sought displacements APz.

The angle of torsion of the ith element in the state of torsional loads P,, according to Fig. 7.23, can be determined from equations binding for the rod of a rectangular cross section subjected to torsion. Therefore,

Msl1i Di

pGrfhi = hi ’

torsion moment (N m), internal load (N), dimensions of the ith board (m), thickness of the ith board (m),

coefficient dependent on the relation (l2/d);-, for (l2/d)i > 10в = 1/3 and shear modulus of the ith board.

By solving the system of equations:

we obtain the expression connecting external displacements APz with internal displacements A, in the form as follows:

From the geometry of deformations of the wardrobe (Fig. 7.24) and on the assumption of the perfect stiffness of board elements in their plane, particular relations result between the displacement APz of vertical external board elements and the displacement Ac of horizontal elements and Ab rear walls in the form of the equations:

. byax.

Az = y x ДР;

bc

A* = by£i dp,

bc

Dy = aXCz AP, (7.59)

bc

where

a, b and c appropriate overall dimensions of furniture in the directions x, y and z of the local coordinate system,

ax, by and cz dimensions of board elements corresponding to the directions of overall dimensions of the furniture piece

By marking the displacement of the ith board in the general form as

A; = XiAP (7.60)

where

ai the coefficient determining the geometric dependency of the element and the body of furniture according to the equations:

external deformation APz can be written down as

Pz

n Gidj a2

i=1 3(ll2)i ai

For practical purposes, the concept of stiffness of the furniture’s body is used more often, proving the quality of approval tests based on the value of the stiffness coefficient k and visual inspection of the state of damage. By comparing the obtained equations, it is easy to demonstrate that the global stiffness of any multi-chamber furniture body is the sum of torsional stiffness of its particular elements and is expressed in the form of:

^ Gd 2

k = gwa2:

A natural characteristic of wood-based board, used to construct case furniture, is the module of figural deformation G. For the purposes of designing furniture, this module must be specified, because it is one of the main stiffness parameters of the body. The method of determining the module of transverse elasticity for boards has been shown in Fig. 7.25. Experimental studies are carried out on rhombus and preferably square samples of boards of the dimensional proportion:

25d < l < 40d (7.64)

where

d thickness of the board and l length of the side of the sample

Load value P, in material tests should be adopted in such a way that deformation A, measured in the direction of the force was smaller than the triple thickness of the board d (A, < 3d). And the value of the module is established from the already known equation:

For boards with a complex construction, e. g. veneered or strengthened by socle skirts and fins, in an analogous experiment, only a certain substitute of their stiff­ness Dz can be determined. This stiffness can also be established in a theoretical way, if the geometric and physical characteristics of individual components of the system are known. The substitute torsion stiffness of the board after veneering (Fig. 7.26) is expressed in the form

where

Dz substitute torsion stiffness,

Gz substitute figural deformations module and dz substitute thickness of the board after veneering.

Taking into account the properties of the board constituting the core of the layer system and properties of the veneer, the substitute figural deformations module Gz of furniture board built from n layers can be written by the following equation:

4=1,

number of layers, thickness of veneers,

thickness of the core (of a particle board, MDF, etc.), shear modulus of the veneer and shear modulus of the core,

And for a symmetrical board built from two layers of veneer (Fig. 7.27), the substitute module of figural deformations amounts to

thickness of the system, thickness of the core (particle board), shear modulus of the veneer and shear modulus of the particle board

In order to determine Gz of a veneered board, the shear modules of boards and veneers must be known. By analysing the stiffness model of case furniture, where the body of the furniture piece constitutes an open case, supported by three corners and loaded in such a way that causes its torsional deformation, the doors are only extra mass load and do not improve the global stiffness of the body. Only when they are placed inside the construction in such a way as to fit tightly to the sides, the

bottoms and tops, then they cause that instead of torsional deformations, all com­ponents of the furniture piece shift shield loads.

Sealed single-chamber case furniture, assuming that they are made from linear-elastic boards interconnected articulately, in accordance with the theory of torsion of thin-walled profiles can be treated as a construction of a cohesive closed structure, in which the rear wall and door fulfil the function of membranes closing streams of tangential stresses. Therefore, in the calculation scheme, it was adopted that the construction was torsionally loaded, as it has been shown in Fig. 7.28a. The deformation, which the closed thin-walled case will be subject to under the influ­ence of the torsion moment in relation to the bending centre 00′, is explicitly described by the values of the rotation angle ф and displacements A1P, A2P, A3P (Fig. 7.28b).

Due to the symmetry of the construction of the case subjected to torsion in theoretical calculations, it was decided to assume that the bending centre of the case is located in the geometric centre overlapping with the centre of the gravity of the block. Moments of inertia of the cross section, as it has been shown in Fig. 7.28c, therefore amount to

Ix = — db2(b + 3a), Iy = — da2 {a + 3b).

Determining the streams of tangential intensities qp:

qp = q + qo,

whereby

Fig. 7.28 Scheme of a load, b deformation of the case furniture with doors inserted to the interior of the body, as well as the characteristics of the streams of intensities in the thin-walled profile: c cross-sectional geometry, d streams of intensities of tangents relative to pole B, e the basic state of streams of tangential intensities, f virtual state of streams of tangential intensities

_ PSx

q = ’

x

(7.72)

X = 2ab,

(7.73)

where

Ms torsion moment,

P load,

Sx static moment of the considered part of the cross section,

Ix moment of inertia in relation to x axis,

Iy moment of inertia in relation to y axis,

Q double surface of the profile, h distance of the centre of gravity from the pole, a width of the cabinet, b height of the cabinet, c depth of the cabinet and

d thickness of the board,

and assuming the pole in point B (Fig. 7.28c), for which Mj = 0, the streams of intensities q of values provided in Fig. 7.28d are obtained. For a coherent closed profile, streams of intensities of tangents qp relative to the bending centre in given points of the profile have been provided in Fig. 7.28e, while the graph of streams of intensities of tangents q, at the virtual state of loads have been provided in Fig. 7.28f. The displacement of a closed case in the direction of force impact can be determined from the general equation:

c c

dip = ^> J ^E^ddzds + ^ J qddzds, (7.74)

0 0

where