Calculation of beams with corrugated wall on the stability of a flat bending shape

The article presents the method of calculating beams with a corrugated wall on the stability of a flat bending shape. The problem reduces to a differential equation of fourth order with respect to the twist angle. The solution is performed numerically by the finite difference method. A comparison of the results with the software package LIRA is presented.


Introduction
I-beams with corrugated walls are actively used in modern construction. In the literature there are a large number of works on such structures, including [1][2][3][4]. Most of them are devoted to the study of the stress-strain state. Finite element complexes make it possible to model beams with a corrugated wall with regard to their real geometry, but this approach is rather laborious. Therefore, there is a need for an approximate method suitable for engineering calculations.

Methods
We will use the general V.Z. Vlasov theory of thin-walled rods of an open profile [5] to solve the problem of lateral buckling of a beam with a corrugated wall. The relationship between the torque Mt and the twisting angle θ for a thin-walled rod has the form [6]: where Iω is the sectorial moment of inertia, It is the moment of inertia during torsion, E and G are the elastic moduli of the first and second kind, respectively. The cross section of the corrugated beam is shown in fig. 1.
where s i and t i are respectively the length and thickness of the i-th wall; γ is a coefficient that depends on the shape of the cross section. Applied to a corrugated beam, the formula (2) takes the form: The position of the center of gravity of the cross section can be calculated by the formula: The moment of inertia of the section relative to the main central axis z is calculated by the formula: 12 The center of the bending of the section A 0 is located at a distance α y from the wall, the value of which is calculated by the formula: The main sectorial moment of inertia of a section is determined as follows: 2 3 2 22 The geometric characteristics of I z and I ω depend on λ, which in turn is a function of x. The value of I t is not dependent on the x coordinate.
To determine the critical load, we use the static Euler criterion. The beam element at the moment of buckling is shown in fig. 2 (the wall is conventionally shown smooth). At the moment of buckling, a torque M x occurs in the beam. Writing the sum of moments relative to the x axis of the beam in the undeformed state, we get: .
x dM q v a dx     Torque on the x' axis is calculated as: .
Substituting (9) into (11) and given that dQ q dx  and y dM Q dx  , we get: The bending moment relative to the axis z' at the moment of buckling is defined as: We differentiate equality (1) with respect to x: Expressing from (13) the second derivative of the deflection and substituting further the value 2 2 dv dx in (12) and then equating the right-hand sides (14) and (12), we obtain the basic resolving equation: Thus, the problem has been reduced to a fourth-order differential equation with respect to the twist angle.
We will consider the calculation method on the example of a hinged at the ends beam under a uniformly distributed load ( fig. 3). In the calculation, we assume that the rotation of the beam about its x axis is excluded in the reference sections, and also that in the endsection deplanation occurs freely, i.e. they do not have a bimoment B. The corresponding boundary conditions at x = 0 and x = l are written in the form: The solution of equation (15) is performed by the finite difference method. Finally, the problem is reduced to a homogeneous system of linear algebraic equations: , . given here because of their bulkiness.

Fig. 3. Calculation scheme
The critical load is determined from the condition that the determinant of the system (17) is zero:

Summary
A method for calculating corrugated wall beams based on the theory of thin-walled rods of V.Z. Vlasov was developed. For test problems, a good agreement of results was obtained with a solution in a three-dimensional formulation using FEM. Compared with a beam with a smooth wall for a structure with a corrugated wall, the critical load was higher by 30.6%