Let us assume that the subject of optimisation is a chair with a bar, whose side frame construction is shown in Fig. 6.94.
The cost of manufacturing is considered a natural optimisation criterion. This often leads to adopting the dimensions of cross sections of the construction components as the function of purpose. Achieving the optimal solution should be preceded by a conjunctive fulfilment of a number of restriction conditions. To illustrate the way of discretisation of the decision variables cube and the build of the appropriate restrictions, Fig. 6.95 shows the scheme of internal forces in four nodes of the side frame.
Fig. 6.95 Internal forces in nodes of side frame of a chair
Mathematical model of optimisation of the side frame of a chair includes the following:
(a) Decision variables, for which the cube of variables Kz, have the form:
Kz = {x = (X1. . ,X4) : Xi(min) < Xi < Xi(max) : i = ■■■ 4} ; (6-268)
where
i number of decision variables for one construction node,
x2 = g, x1 = h dimensions of cross sections of vertical elements, x4 = k, x3 = l dimensions of cross sections of horizontal elements.
(b) Parameters, which constitute internal forces in the construction (Fig. 6.96) and border strength of the material and glue-line in joints, that is,
M bending moment in the jth node,
T cutting force in the jth node,
N normal force in the jth node, kg bending strength of the wood, kt shearing strength of the wood, ks shearing strength of the glue-line.
(c) Permissible set u is made of inequality restrictions, ui = (x) > 0, that is,
U = {x =(x1,…,x4) : Ui(x)>0 : i = 1,…, 4}. (6.269)
(d) Strength restrictions, ui = (x) > 0, shearing strength of the tenon in the jth node
where
n safety coefficient,
bending strength of the tenon on the jth joint
6М2П
kg
shearing strength of the element in the cross section of mortise for loads:
(gh – g1k1 + 2) > Tmkaxn, (6.272)
kt
bending strength of the element in the cross section of mortise, centre of gravity of the examined cross section according to Fig. 6.97:
2 g2h – g2k0 (gh – g2k1) ’
Fig. 6.97 Geometry of the cross section at the place of implementation of the mortise and tenon joints
moment of inertia of cross section in relation to xe-axis:
(e) Construction conditions resulting from the provisions of BN-76/7140-02:
• length of the tenon g1 < 3 g,
• depth of the mortise for the tenon g2 = g1 + 2 mm
• thickness of the tenon 1 k < k1 < 7 k,
• height of the tenon Z1 < 11.
(f) Aesthetic conditions for individual horizontal and vertical elements: h > k. Minimisation of the area of cross section of components of the side frame of a chair at the place of their connection has been assumed as the function of purpose. Therefore, the minimum area of cross section of the horizontal element with a tenon is as follows:
A1 = k • l! min, (6.279)
minimum area of cross section of the vertical element with a mortise is as follows:
A2 = g • h! min.
When attempting to optimise a chosen construction of a chair, all the optimisation parameters had to be established. To this end, the geometric parameters discussed previously, and presented in Fig. 6.94, were additionally complemented with the values of internal forces caused by the standard loads (Fig. 6.96). A detailed description of each of the planned static optimisation methods can be found in the rich literature of the subject, and therefore, only the way of determining the number of samplings No and the choice of the optimisation step t and size of the cell T have been discussed below.
Systematic search method (Fig. 6.98) consists in a gradual search of the entire permissible area, counting the function value at each point, choosing the point in which the value of the function of purpose is the smallest. In this case, discretisation points can be selected so that the optimisation step is equal to 0.5 mm. Then, the total number of discretisation points for a single structural node is equal to:
No = Y ^1(mx)~X(max^ + j. (6.281)
The following values can be used as initial data for the process of optimisation:
– internal forces M1, M2, M3, T1, T2, T3, N1, N2, N3, as in Figs. 6.94 and 6.96b-d,
– minimum dimensions—assumed arbitrarily—gmin =15 mm, hmin =15 mm, lmin = 15 mm, kmin = 15 mm,
– maximum dimensions—assumed arbitrarily—gmax = 60 mm, hmax = 60 mm, lmax = 60 mm, kmax = 60 mm,
– bending strength of the pinewood along the fibres kg = 105 MPa,
– shearing strength of the pinewood across the fibres kt =8 MPa,
– shearing strength of the polyvinyl acetate glue ks =12 MPa,
– safety coefficient n = 1.5,
– optimisation step t = 0.5 mm,
– number of samplings No = 68,610.
