A new method for calculating volumetric structural elements in the general case of their stress-strain state

. The parametric equations of the strength surface in the space of internal force factors (IFF) are given for 9 forces and 9 moments in homogeneous anisotropic bodies. As special cases, similar equations are given for isotropic bodies that resist tension and compression differently (and equally). A variant of the kinematic method of the theory of limit equilibrium is proposed. A rigid-plastic model of a deformable solid body is used. It is assumed that massive structural elements are destroyed by dividing into parts that deform relatively little (“a bsolutely rigid finite elements”, ARFE) and have 6 degrees of freedom in three -dimensional space. The process of destruction of the material goes along infinitely thin generalized destruction surfaces (GDS), on which the work of all active IFFs is taken into account. To determine the minimum kinematically possible load, the simplex linear programming (LP) method was used.


Introduction
The method for calculating structural elements by limit equilibrium originates from the works of G. Kazinchi [1] and A.A. Gvozdev [2]. As applied to bar systems, this method was developed, in particular, in the works of A.R. Rzhanitsyn [3], A.A. Chiras [4], E.S. Sibgatullin, K.E. Sibgatullin [5][6][7][8], for plates and shells -in the works of E.S. Sibgatullin, K.F. Islamov [9][10][11][12] and others. Here, the application of this method for massive structural elements is proposed. Appropriate algorithms and computer programs have been developed, with the use of which numerical results have been obtained, some of which are presented in this paper.
Equality (24) takes place when the generalized forces and velocities of generalized displacements are related by the associated deformation law [15] ∆v ⃗ = λ̇∂ Here ⃗ , ⃗⃗ is any combination of IFF that satisfies the condition Φ( ⃗ , ⃗⃗ ) = 0, but is not necessarily associated with the associated deformation law with the combination ∆ ( 1 2 ), ∆ ⃗ ⃗ ( 1 2 ).
Let the body be loaded with external surface and volume forces: Here 0 , 1 , 0 , 1 are functions of only spatial coordinates; i,j = X,Y,Z, μ is a monotonically increasing parameter. The method of fixing the body eliminates the possibility of its displacement and rotation as an entirely rigid body. It is necessary to determine the limit value 0 of the external loading parameter when the considered massive element loses the property of geometric invariability. When the destruction of a massive body occurs according to the GDS between neighboring ARFE, the equilibrium equation in the Lagrange form is [15,16]: Here is the area of the kth GDS; m is the number of such surfaces; 0 , 1 are the areas on the body surface, where the forces 0 , 1 act, respectively, 0 , 1 are the volumes in the body composition, where the forces 0 and 1 act, respectively; N is the power of the IFF per unit area; , are the components of the velocity vectors of displacement of the body points, where the forces , are applied, respectively. If external forces act in areas of the body that are separate from each other, then on the right side of (29) the sums of integrals over the corresponding areas are taken with 0 , 0 , 1 , 1 ,l,m,n,p = 1,2,3, ….
Let's represent the problem as a mathematical programming problem: to find min + , where under integral constraint and when equalities of the form (24) are satisfied at all points of all GDSs. When using LP, integration in (30) and (31) is replaced by summation, instead of (24) we use a system of constraints of the form (27), where various combinations of ⃗ and ⃗⃗ correspond to the vertices of a convex limit polyhedron approximating the limit surface Φ( ⃗ , ⃗⃗ ) = 0.

Results and discussion
Below are some of the results obtained using the above method for a lightweight concrete body (Fig. 4). Below are some of the results we obtained for a cube with dimensions of 10x10x10 cm ( Table 1)     The destruction scheme in Figure 5 and the empirical result in Table 3 are taken from [17].

Conclusion
The proposed parametric equations of strength surfaces for massive bodies form the basis of a variant of the kinematic method of the theory of limit equilibrium. The proposed method for determining the safety factor in the IFF space is more realistic than its determination based on dangerous stresses. Our theoretical results have been repeatedly compared with the corresponding experimental data of other authors; some of them are shown in Fig. 5 and in table 2.
The above results were obtained using algorithms and computer programs developed to study the bearing capacity of homogeneous massive bodies in a volumetric stress state, for the most general case. It can be assumed that the methods developed by us will find their application in engineering practice, in particular, in such works as [18][19][20].