Monte Carlo Method

Construction parameters selected by the manufacturer as a result of using optimi­sation must meet the basic criterion, which in this case is the minimum area of cross section of rails. And the given optimum dimensions must meet numerous additional restricting conditions. For example, let us consider the construction of an external wing of a reinforced single window with the dimensions given in Fig. 6.99.

Structural joints of window wings should shift loads of 100 N on the arm 250 mm and of internal wings—330 N on the same arm (Fig. 6.100a). For the wing of the given standard dimensions, the load state corresponds to the distribution of bending moments caused by concentrated force P of 500 N, as it is shown in Fig. 6.100b. Stiffness and strength of such a loaded window wing depend on the strength of the joints and on the bending strength of rails.

Figure 6.100a shows the scheme of a structural node of the external window wing, whose strength will be the main source of restrictions in the process of optimisation. Cross section of rails shown in Fig. 6.100a results from adopting for the analysis only active part of cross section of rails, in which connections have been made. The remaining part of the cross section, situated outside the hatched

Fig. 6.99 Window: a basic state, b distribution of bending moments

outline of bridle joint (Fig. 6.101), meets only specific technological requirements, without the effect on strength of the joint.

Mathematical model of optimisation of construction is as follows:

– selected construction parameters—according to Fig. 6.100a—constitute trans­verse dimensions of rails: vertical 11 • bt and horizontal ht • bt, accepted in the subsequent n-steps of optimisation for i = (1, 2, …, n),

– restricting conditions resulting from the construction of the window wing constitute:

(a) bending strength of the tenon loaded with bending moment Mg1 (Fig. 6.99b), described by the inequality:




kg bending strength of the wood,

(b) shearing strength of the tenon loaded with the force P, expressed as follows: (6.283)


kt shearing strength of the wood,

l‘h2 > Mg’ ,

n1 ■ a ■ smax smax kS;


kS shearing strength of the glue-line, nj number of glue-line,

a coefficient dependent from the relation hi/li, Tmax maximum static stress.


bending strength of the loaded vertical rail, at the point of mounting of hinges, maximum bending moment Mg2 (Fig. 6.99b), which has been illustrated below:

2 >6Mg2,

‘‘" kg ’


as well as construction conditions tested in practice, which have been pre­sented by the following relations:







(c) shearing strength of the glue-line, which has been presented as follows:


A natural optimisation criterion, as previously, is the cost of manufacturing. Assuming that these costs are proportional to the amount of used materials, the cross sections of rails should be minimised. Therefore, we should try to determine:

Fbhmin bi ‘ hi,


Fblmin = bi ‘ li:


Mathematical model of optimisation of construction of a window wing, on the example of the window, therefore includes the following:

– decision variables lt, bt, ht,

– parameters Mg1, Mg2, P, kg, kt, ks, n1, a,

– permissible area determined by restricting conditions,

– function of purpose.

Before solving the model, the 3-D variability cube presented in Fig. 6.102 should be described in detail. Therefore, we have to give minimum—intuitively

Fig. 6.102 Decision variables cube

estimated—values of dimensions of the cross sections of rails Zmin, bmn, hmin, and maximum values of dimensions Zmax, bmax, hmax, which we intend to optimise. In the decision variables cube, these values are described by extreme points D1, D2. In the optimisation process, computer randomly chooses points from the given cube Dj, lt, hi, bj and remembers only those which fulfil all the restricting conditions. The point, which describes the function of purpose best, represents optimal dimensions of cross sections.

For the external wing of the examined window construction with the specified dimensional characteristics and given—according to Fig. 6.99b—distribution of bending moments, optimisation of construction using Monte Carlo method was carried out, according to the algorithm presented in Fig. 6.103.

The following values have been used as initial data for the process of optimi­sation using Monte Carlo method:

– maximum bending moment lowering the joint Mg1 = 10.249 daNm,

– maximum bending moment lowering the vertical rail Mg2 = 11.586 daNm,

– concentrated force (cutting the tenon) P = 50 daN,

– bending strength of the pinewood kg = 87.0 MPa,

– shearing strength of the pinewood kg = 10.0 MPa,

– shearing strength of the urea glue ks = 8.0 MPa,

– minimum dimensions (freely assumed) lmin = 20 mm, hmin = 20 mm, bmin = 10 mm,

– maximum dimensions of the cross sections of rails, according to Fig. 6.101, lmax = 44 mm, hmax = 34 mm, bmax = 33 mm.

Entering the number of samplings No = 50, which the computer should perform in order to find the optimal solution, is followed by generation of random numbers and selection from the cube of decision variables of any points D;(l;, ht, bj). The point that specifies the minimum value of the function of purpose after performing

Ьгтип’Ьтах»Ьтіп, hmax, Mgi, Mg2.Pfkg, kt, Nq,

+ ——————

N=0, S=0, FA=bmaKhm3x, Fв=ЬтахІтзх’





Fig. 6.103 Algorithm of optimisation using Monte Carlo method

No samplings presents the optimal dimensions of cross sections of rails. And this process is considered completed, when the result after a great number of samplings has not improved. For the data presented above, we obtain the following results:

– cross section of the vertical rail L = 26.3 mm, b = 20.6 mm,

– cross section of the horizontal rail h = 32.3 mm, b = 20.6 mm.

The obtained optimal dimensions—as it has been stated earlier—refer to the dimensions of the bridle joints. Therefore, taking into account the necessary tech­nological profiles, according to Fig. 6.100, a new, optimal cross section of rails is obtained with dimensions and surface as shown in Fig. 6.104.