Implicit Dynamic Solver

Implicit dynamic solver using non-linear Newmark's method

Implicit dynamic solver using non-linear Newmark's method with example file

function Result=Newmark_Nonlinear(Elements,Material,Support,Free,M,C,f,fs,delta)

Input

Elements: a structure containing Elements{i}.DOFs and Elements{i}.Material

where Elements{i}.DOFs=[j k] means element i connect DOF j with k

and Elements{i}.Material=m assign material m to element i

Material: a structure containing material properties for bilinear springs

where Material{m}.k1 is Spring stiffness

Material{m}.x1 is Spring deformation beyond which the stiffness decreases

Material{m}.k2 is Reduced stiffness

Support: a vector of support (Fixed) DOFs of size (nSupport,1)

Free: a vector of free DOFs of size (nFree,1)

M:mass matrix (nFree*nFree)

C:damping matrix (nFree*nFree)

f:external force matrix(nFree,N)

fs: sampling frequency

delta: convergance criterion for residual force

where N is the length of data points of dynamic force

Output:

Result: is a structure consist of

Result.Displacement: Displacement (nFree*N)

Result.Velocity: Velocity (nFree*N)

Result.Acceleration: Acceleration (nFree*N)

Note: Elements are assumed to be springs connecting nodes with bi-linear stiffness (No hysteresis).

References

Chopra, Anil K. "Dynamics of Structures. Theory and Applications to." Earthquake Engineering (2017).