Calculation of wooden beams on the stability of a flat bending shape enhancement

Flat bending stability problem of constant rectangular cross section wooden beam, loaded by a distributed load is considered. Differential equation is provided for the cases when load is located not in the center of gravity. The solution of the equation is performed numerically by the method of finite differences. For the case of applying a load at the center of gravity, the problem reduces to a generalized secular equation. In other cases, the iterative algorithm developed by the authors is implemented, in the Matlab package. A relationship between the value of the critical force and the position of the load application point is obtained. A linear approximating function is selected for this dependence.


Introduction
For reasons of reducing wood consumption in rectangular section wooden beams design they try to increase the ratio of the cross section height to its width.The necessity of flat deformation shape stability test comes in this case [1,2].
The problem of flat bending shape stability of constant rectangular section wooden beam is solved with following differential equation [3,4] where Gshear modulus, t I -polar moment of inertia, z I -axial moment of inertia, y M bending moment, θangle of twist.Equation ( 1) is written for the case when load is applied at cross section center of gravity.Current design standards of wooden structures [5] are based on the solutions of this equation for different options for securing the beam.
In A. Karamysheva dissertation [6] following differential equation was obtained, that considers variable beam stiffness, and loading not only in the center of gravity as well: where acenter of gravity to load location distance.___________________________ Fig. 1.A beam considered in the article A. Karamysheva considered only two cases of a load applying not at the center of gravity: a cantilever beam with concentrated force at the end, and a hinged beam loaded with a concentrated force in the middle of the span.In the paper we consider a beam of constant cross-section loaded by a uniformly distributed load q (Figure 1).The second term of the equation ( 2) equals zero because the beam that we calculate has constant section.

The derivation of the final equation
Bending moment is determined by the following formula: Inserting dimensionless coordinate k z q l GI EI   , the equation can be represented in a form: where Boundary conditions for the equation ( 4):   The solution of equation ( 4) is performed numerically by the method of finite differences [7].Uniform grid with  -step is inserted.The finite differences approximation of equation ( 4) is written as [8]: Constructing equation ( 5) for all internal nodes of the grid, we have a homogeneous system of linear algebraic equations: ( where MATEC Web of Conferences 196, 01003 (2018) https://doi.org/10.1051/matecconf/201819601003XXVII R-S-P Seminar 2018, Theoretical Foundation of Civil Engineering The system (6) has a nonzero solution if its determinant is equal to zero: | In the special case for 0   , i.e. when the load is applied at the center of gravity, equation ( 7) represents a generalized secular equation.The first critical load corresponds to the minimum of the eigenvalues .
 The critical load can be determined by formula:

Method of calculation
For 0   coefficient K equals 28.3, and the eigenvalue 1 800.
  In the case when 0   equation ( 7) is no longer a generalized secular equation.For its solution an iterative process was developed by authors, the essence of which is as follows.
In the first approximation, instead of equation ( 7), we solve the secular equation, which has the form: where 1

[ ] [ ] [ ]
As a result of the equation ( 9) solution we obtain the minimum eigenvalue 1   .In second approximation we substitute 2 The criterion for the iterative process termination is the condition: 100% , where predetermined error.

Results and discussion
The calculation was implemented in Matlab package.The resulting graph of the change in the coefficient K, as a function of  , is shown in As it is seen in the graph, when a load is applied above the center of gravity, its critical value decreases, and if the load is applied under the center of gravity, its critical value increases.The same dependency was observed in the calculation of wooden articulated beam, loaded with the concentrated force [9].
To verify the correctness of the results, a finite-element calculation of the wooden beam in the LIRA-SAPR 2013 program was performed.Initial data for the calculation: b = 1 cm, h = 10 cm, l = 2 m.The orthotropy of the material was considered, the values of the elastic constants were assumed to be equal to: E 1 = 10 4 MPa, E 2 = 400 MPa, G = 500 MPa, ν 12 = 0.018, ν 21 = 0.45.The beam was modeled by plane shell finite elements.For the case when load is applied at the center of gravity of the cross section, the value of critical load was found: q cr =0.133 kN/m (Figure 3), and when the load is above the gravity center q cr =0.123 kN/m (Figure 4).Theoretical values that were received using formula (9), equals 0.132 and 0.123 kN/m.The coincidence of the results indicates their reliability. :

Figure 2 .Fig. 2 .
Fig. 2. Dependency graph of coefficient K as a function of α

Fig. 3 .Fig. 4 .
Fig. 3. Loss of stability for the case when a load is applied at the center of gravity of the cross section