A transient dynamic process in a structural nonlinear system “beam – two-parameter foundation”

A method for analytical assessment of dynamic added stress in elastic loaded beam resting on elastic two-parameter Pasternak’s foundation due to sudden destruction a part of foundation is proposed. Equations of static bending, natural and forced oscillations are written in a matrix form using state vectors including deflection, rotational angles, bending moments, and shear forces at arbitrary cross section of a beam and also using the matrices of the initial parameters influence on the stressstrain state in arbitrary cross section. The influence of foundation failure on beam’s stress-strain state, taking into account a relation between the stiffness parameters of foundation, is analyzed. The condition of smallness for the shear stiffness parameter (Pasternak’s parameter) in comparison with the stretching-compressing stiffness parameter (Vinkler’s parameter) is accepted. It is shown that the accounting of Pasternak’s parameter reduces the level of dynamic added stress in a beam when sudden destructing of a foundation. The factor of sudden defect occurrence in the system “beam – foundation” increases considerably the internal forces in a beam in comparison with quasistatic formation of the same defect.


Introduction
In this work, a problem of construction a mathematical model for the dynamical process in a load-bearing beam resting on Pasternak's two-parametrical foundation [1] during sudden occurrence a defect in the form of destruction a part of foundation. Before the defect occurs, the stress-strain state of all the construction is determined by a static influence. The sudden defect appearance leads to reduction in the overall construction stiffness. This reduced stiffness does not already provide with static stability of all the system. The occurring inertial forces cause a dynamical response, the beam begins to move and this results in re-distribution and growth of strain and stress. Due to the dynamical added stress, violations in the established performance or loss of bearing capacity along with progressive destruction are possible. At present, investigations on forced oscillations for the non-linear system "beam-foundation" (i.e. a system that changes a proper calculation scheme under loading) are practically absent in the literature. In several works, manifestations of the structural non-linearity and its after-effects are studied during complete or partial

Results
First, find natural bending oscillations of a free beam partially resting on Pasternak's elastic foundation.

Natural frequencies of a beam partially supported by an elastic foundation (
The equation of natural bending oscillations is as follow: where ( 1 4) j Aj = -integration constants. In congenial works [2][3][4][5][6], an efficiency of the initial parameters method in combination with a vector-matrix representation of an arbitrary beam's cross section state is demonstrated in order to analyze both displacements and stresses when interacting with foundation. An analogous approach is used in the present work.
Replacing the integration constants j A by the internal parameters of the first segment where matrices ( ) 12 1 ;  (8) and the state of arbitrary section where the influence matrix of the first segment on the state of the second segment takes the form are considered below.

Natural frequencies and modes of bending oscillations for a beam with free endpoints partially supported by two-parametrical Pasternak's foundation
In  It means that the following problem is considered: for a given beam which oscillates with unknown frequency  along with a foundation, we bring into correspondence some conditional free (i.e., without a foundation) beam with a natural frequency the same as for the given beam, i.e.

Free beam with unfixed endpoints
As is known [7], a free (i.e. without foundation) beam with unfixed endpoints, besides natural frequencies coinciding with frequencies of a beam with fixed endpoints, has also two zero frequencies corresponding to translational and rotational motion of a beam as a rigid body. That is displacements caused by beam oscillations can be supplemented by movement of a rigid body. This combined motion can be described by the function 12

W C C  =+
In the framework of accepted model for the system "beam-foundation", the presence of an indefinitely small beam's segment of length 0   interacting with foundation excludes the possibility of motion (as a rigid body) for a beam having free endpoints. And calculation of the first natural frequency should be carried out according to the variant (7) and (9) 1  3  2  4  1  4  1  4  3  4   4  2  3  2  3  1  3  1  3  3  3   4  3  3  3  4  2  4  1  4  4  4   4  4 3 3 3 ; . n n n n n n n n n n n n Bending moments in segments (both in free and in resting ones) are determined by the functions ; . nn n n n n n n n n n

 
Further, accept the condition 01 .

 
The states of segments can be expressed by the matrix equations (6) and (9) ( ) ( ) ( ) ( ) ( ) The frequency equation takes the form The oscillation modes and bending moments, after the frequencies have been evaluated from (30), are determined by the functions 1  4  1  1 3  1   2  2  2  4  2  4  3  2  1  2  2  2  1  2  2   2  1  4  2  3  3  2  4  2  2  1  1  2  2 ; ( ) ( ); n n n n n n n n n n n n  Experimental data concerning the second parameter K2 are practically absent. According to recommendation [8][9][10][11][12][13][14], accept the value  Table 1, the first three dimensionless frequencies 1 3   − obtained from (12) and (13) for the two combinations of conditional frequencies 01 02 and  characterizing the general stiffness of the system "beamfoundation" are adduced along with length ν of the indestructed part of foundation after its partial destruction.  Proper modes along with bending moments are adduced in Figure 2 the beam rests, are described by the equation  = Figure 5 demonstrates the fact that the maximum bending moment by quasistatic process of destruction appears to be approximately twice lesser than by sudden destruction.

Conclusion
An analytical solution for the problem on determination of forces, modes, and frequencies (both natural and forced) of transversal oscillations for a beam resting on elastic twoparametrical foundation is obtained. This solution can be used for testing of mathematical models describing static-dynamic and quasi-static deformation of a complex non-linear "erection − foundation" system under special crash impacts associated with sudden destruction of foundation's segments.
This analytical solution can also be applied in numerical analysis of building and erection defense against a progressive destruction when an additional loading of a "band foundation − erection" system is caused by sudden subsidence of foundation in correspondence with a possible scenario of crash impact. For example, it may be actual for objects build on slopes in case of foundation destruction due to displacement of foundation.