Implicit Dynamic Solver 2

Implicit dynamic solver using non-linear Newmark's method

Implicit dynamic solver using non-linear Newmark's method with two examples file.

function Result=Newmark_Nonlinear(Elements,Material,Support,M,C,f,U_s,dU_s,ddU_s,fs,delta)

Input:

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

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

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

Elements{i}.Material_State.e initial material strain

Elements{i}.Material_State.s initial material stress

Elements{i}.Material_State.k initial material hardening

Material: a structure containing material properties for bilinear springs

where Material{m}.E Stiffness

Material{m}.f_y Stress beyond which the stiffness decreases

Material{m}.Ep Reduced stiffness

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

M: mass matrix (n*n)

C: damping matrix (n*n)

f: external force matrix(n,N)

U_s: support displacement matrix (nSupport,N)

dU_s: support velocity matrix (nSupport,N)

ddU_s: support acceleration matrix (nSupport,N)

fs: sampling frequency

delta: convergance criterion for residual force

where n is the number of DOFs, and N is the length of data points of dynamic force

Output:

Result: is a structure consist of

Result.Displacement: Displacement (n,N)

Result.Velocity: Velocity (n,N)

Result.Acceleration: Acceleration (n,N)

Result.F_s: Support Force (nSupport,N)

References

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