DETERMINATION OF THE OPTIMAL BEAMS STEP FOR THE WOOD FLOOR

Abstract

The article presents the results of optimization of the beams step of the building 
wooden floor according to the criterion of the minimum material consumption. The slab consists of 
parallel wooden beams of rectangular cross-section, on which the boardwalk is laid. 
The  optimization  problem  is  solved  proceeding  from  the  minimization  of  the  total  wood 
consumption for beams and boardwalk. For this, the target function of the total volume of timber for 
flooring and beams per unit floor area is used. It represents the reduced thickness of the slab. The step 
of the beams is taken as an argument to the objective function. Based on the use of the method for 
finding the minimum of a function, the optimal value of the argument is determined. 
It is shown that the objective function depends on the unknown dimensions of the rectangular 
cross-section of the beams and the thickness of the boardwalk. They are determined based on the 
solution of the strength condition by normal stresses for the boardwalk and beams. To simplify the 
solution of the problem, it is assumed that the ratio of the width of the section of the beams to their 
height is a given value that does not depend on the step of the beams. In this case, only the height of 
the beam and the thickness of the boardwalk remain unknown. 
When solving the strength condition for the beam, it became necessary to solve the third degree 
inequality with respect to the unknown beam height. To avoid significant algebraic difficulties, it is 
proposed to solve the problem by a numerical method in the MathCAD computer environment. For 
specific values of the initial data, a graph of the objective function of wood consumption per unit of 
floor area was built and, by tracing, its minimum was determined, which corresponds to the optimal 
step of the beams. 
It was found at what values of the beams step, the rigidity of the structures is ensured for the 
case of equality of the right and left sides of the strength condition by normal stresses from bending. 
The corresponding conditions are obtained, the analysis of which has shown that the structures will 
be  quite  rigid  at  a  relatively  small  value  of  the  wood  strength  in  bending.  On  this  basis,  a 
recommendation has been made to use relatively low strength timber in the design of wooden floors 
with an optimal beam step. The performed theoretical studies and a numerical experiment showed 
that it is possible to determine the beams step of a wooden floor, at which the total volume of wood 
for it will be minimal.