PSI - Issue 6

A.D. Lovtsov et al. / Procedia Structural Integrity 6 (2017) 122–127 Author name / Structural Integrity Procedia 00 (2017) 000–000

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2. Mathematical model and computational methods The shelter is exposed to short-term loads which are idealized by instantaneous impulses. Thus, the problem leads to investigating the behavior of the system under the action of its own weight and instantaneous impulses, applied at different times to different masses. We denote this load as ( , ) t x F . Equation of motion can be written as: ( ) ( ) ( ) ( ) ( , ), (0) 0, (0) 0 , t t t t t x      Mz Dz R z F z z z     (1) где ( ) t z , ( ) t z  and ( ) t z  are the nodal displacements, velocities and accelerations; M and D are the diagonal mass and damping matrices; ( ) t  R stiffness matrix at the moment t . Only nonnegative vertical displacements are considered due to unilateral contact between foundation and structure where ( ) t u and ( ) t w are lateral and vertical displacements of masses. Contact region is defined by current length of elastic elements. If current length of the element due to deformation is less than initial length 0 L 0 L L  , (3) then this element is not taken into consideration because it doesn’t affect stiffness of the system. Various time integration strategies can be used to solve equation (1) with the Newmark methods being very popular. Dynamic equilibrium is considered at the end of the time step t  The solution of the equation (4) can be obtained using of a “predictor-corrector technique” (Crisfield, 1997). Equation (5) provide an incremental “predictor step”  z : n    F R z , (5) where   4 2 2 , 1 n n n n n t               F F F M z z Dz    and 4 2 2 n n t t      R R M D , where n  R static tangent stiffness matrix. Having solved (5) for  z , the incremental displacements, velocities and accelerations at step 1 n  are obtained. For iterative “corrector” technique equation (4) can be rewritten as 1 1 1 1    1 n n n n n      Mz  Dz  R z F . (4) ( ) ( ) t t        u w ( ) t , z ( ) 0, t  w (2)