7.2.2.1 Stiffness of Shelves and Horizontal Elements
Engineering practice requires the selection of the appropriate thickness for boards intended for shelves or partitions of case furniture. The general differential equation of the bending of an elastic thin isotropic board in a rectangular system of coordinates X, Y and Z (Fig. 7.12a) has a known form of Legrange’s equation:
d4 w d4w d4w q(x, y)
~dx4 + dx2dy2 + ~5? _ D~; ( 🙂
where 

q(x, y) D Ed3 
operating surface load, bending stiffness of the board, 
12(1+v2) 

w 
function of bending the board, 
v 
Poisson’s ratio, 
E 
linear elasticity module and 
d 
thickness of the board. 
Expenditures of bending moments Mx, My and torsion moments Mxy, Myx and expenditures of transverse forces Fx, Fy, acting on the sides of the elementary section of the board dA = dxdy (Fig. 7.12b), are determined by the following equations:
Fig. 7.12 Analysis of operation of boards in rectangular coordinates: a external load of the board, b internal forces 
The abovementioned equation is solved, taking into account the border conditions of support of rectangular boards: of a fixed edge, freely supported and free edge. If we place these conditions at the edge of the board y = 0, then they have the form:
for a fixed edge (horizontal partitioning)
H=0= (f)y=0= (7’9)
which means that the deflection and angle of deflection on this edge are equal to zero, for free support (shelf)
(w)y=0= 0; M )y=0= 0; (@w) o= 0; (7’10)
for free edge
(My)y=0 = 0; My=0= 0; (Fy)y=0 = 0′ (7’11)
In the last two cases, we assume that the edges are not loaded by bending moments and transverse forces. The solution to the above equation is obtained by entering a rapidly convergent triple trigonometric series for the bending function. The exception here is the cylindrical bending of shelves and partitions supported or fixed on the two opposite sides (Fig. 7.13a, b). This occurs when a board of the dimensions b < a, supported freely along the edges with a length of x =0 and x = a, is loaded in any way in the direction of x and in a constant way in the direction of y. In such case,
and according to these equations,
Fy = 0
Therefore, eventually, we obtain a differential equation of bending an elastic, thin isotropic board in the form as follows:
Therefore, cutting an element from a board with a width dy = S and calculating the course of the bending moment using known methods:
@2w,
Mx = @W2D (715)
we obtain all the necessary information in order to determine the deflection and the My moment in a simple way. For support and load schemes from Fig. 7.13a, b we write an equation of moments for the left half of the beam as a function of length and insert it to the differential equation of the deflection line. The bending value is calculated by integrating the equation:
and determining constant integrations with the appropriate border conditions.
Solutions for the discussed boards have the form: maximum deflection of a freely supported shelf (Fig. 7.14a)
5 qAa4
w = 384 D – Wp’ Wp = 0’004a’ (717)
maximum deflection of a partition fixed on both sides (Fig. 7.14b)
1 a4
W = 384 Wp; Wp = 0’002a’ (718)
where
qA surface load of the board element according to Table 7.1, a length of the free side of the board,
wp acceptable deflection in the middle of the shelf’s span, which should not exceed 4 mm/m in length, acceptable deflection in the middle of the partition’s span, which should not exceed 2 mm/m in length.
In many structural solutions of furniture, the boards can additionally be supported on a third edge from the side of the rear wall (Fig. 7.15). Strict solutions of these cases come down to the integration of the general equation of a board subject to bending and torsion. A similar solution can also be obtained on the basis of the elementary strength of materials.
A shelf that is additionally supported can be regarded as a rod subject to bending or torsion. External load can then be broken down into two states (Eckelman and Resheidat 1984):
• balanced load of a freely supported shelf (Fig. 7.16a),
• concentrated or continuous load (Fig. 7.16b, c) in the location of support.
Fig. 7.14 Acceptable deflection of board furniture elements: a shelves, b partitions 
Fig. 7.15 The deflections of shelves supported along three edges: a by point from the side of the rear wall, b discreetly from the side of the rear wall 
Fig. 7.16 Load states of the shelf: a board bending, b, c board tension 
Deflection in the first load state is the same as for cylindrical bending wp = (5/384) • (qAa4/D). The load of the second state can be broken down into two components, symmetrical (Fig. 7.17a) and asymmetrical (Fig. 7.17b). Symmetrical load gives cylindrical bending of a maximum ordinate:
Fig. 7.17 The state of edge load of boards: a symmetrical, b asymmetrical 
_ 1 Ra[1] W1 _ 48~Ej ‘ in the case of distributed load along the free edge 
This moment is applied in the middle of the shelf’s span. Taking into account the dependence of calculating the torsion angle ф of the board:
where M(x) X G = 2E(1 + u) Js = bd3/12 
rod torsion moment, torsion length, shear modulus,
moment of inertia of rectangular cross section of the board.
Deflection of the board at midlength x = a/2 loaded by the moment M(05b) = Rb/4 amounts to
where
is the moment causing the torsion of the board. In solving this equation, we obtain the value of deflection of the board caused by torsion in the form of the equation:
Because bending along a supported edge is equal to zero, the value of reaction R and q'(x) can be calculated from the equations:
for a shelf supported at one point
5 qAa 1 Ra3 3 Rab
384~D + 48~EJ 7 8Ed3 = ’
which gives
5 2a2
R = 8 qAab202T3b2
hence,
, , , 5a2 q(x)= «7 57767′
The maximum deflection can be determined from the appropriate sum of component deformations:
Fig. 7.18
A separate example of a furniture board is a partition supported elastically from the side of the rear wall and attached to the side walls using joints of the known stiffness у (Fig. 7.18). According to Eckelman (1967), the maximum deflection of free edges is determined by the equation:
Bending stiffness D, present in the formulas, is a certain substitute stiffness, which value should be determined experimentally in the test of static bending. This size, for multilayer boards (Fig. 7.26), can also be calculated analytically on the basis of tabular data and the equation:
whereby
П 1
^ ^ di di = 1 2 dz.
For a board that is symmetrically veneered on both sides (Fig. 7.27), stiffness is expressed in the form as follows:
where
dz thickness of the board after veneering, dw thickness of the core of the board (particle board), Ew linear elasticity module of the particle board,
Eo linear elasticity module of the veneer, vw Poisson’s ratio of the particle board and vo Poisson’s ratio of the veneer.