Dynamic analysis of Earthquake
The earthquake problem is solved by the means of dynamic analysis of a continuous body. At each point x and each time instant t the following differential equation is satisfied:
where: | c | - | coefficient of viscous damping |
ρ | - | mass density | |
u | - | displacement |
| - | velocity |
| - | accelerationí |
| - | gradient |
σ | - | stress |
The stresses are provided by:
where: | Dijkl | - | material stiffness tensor |
εkl | - | strain tensor | |
εklpl | - | plastic strain tensor |
The strains are equal to the symmetric part of the displacement gradient:
where: | ui, j | - | derivative of the i-th component of the displacement in the direction of the j-axis. |
Finite element discretization of the equations of motion gives the system of ordinary differential equations in the form:
where: | M | - | mass matrix |
C | - | damping matrix | |
K | - | stiffness matrix | |
F(t) | - | vector of time-dependent loading | |
r(t) | - | vector of nodal displacements |
As for time integration, the user may choose between the Newmark method and the Hilber-Hughes-Taylor Alpha method.
Further details are available in the theoretical manual on our website.
Literature:
Z. Bittnar, P. Řeřicha, Metoda konečných prvků v dynamice konstrukcí, SNTL, 1981.
T. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice Hall, INC., Engelwood Clifts, New Jersey 07632, 1987.
Z. Bittanr, J. Šejnoha, Numerical methods in structural engineering, ASCE Press, 1996.