# Iterative Solution Methods for Nonlinear Studies

## Nonlinear Static Studies

In nonlinear static analysis, the basic set of equations to be solved at any “time” step, t+Δt, is:

^{ t+Δt}{R} - ^{ t+Δt}{F} = 0,

where:

^{ t+Δt}{R} = Vector of externally applied nodal loads

^{ t+Δt}{F} = Vector of internally generated nodal forces.

Since the internal nodal forces ^{ t+Δt}{F} depend on nodal displacements at time t+Δt, ^{ t+Δt}{U}, an iterative method must be used. The following equations represent the basic outline of an iterative scheme to solve the equilibrium equations at a certain time step, t+Δt,

{ΔR}^{(i-1)} = ^{ t+Δt}{R} - ^{ t+Δt}{F}^{(i-1)
}

^{ t+Δt}[K]^{(i)} {ΔU}^{(i)} = {ΔR}^{(i-1)}

^{ t+Δt}{U}^{(i)} = ^{ t+Δt}{U}^{(i-1)} + {ΔU}^{(i)}

^{ t+Δt}{U}^{(0)} = ^{t}{U}; ^{ t+Δt}{F}^{(0)} = ^{t}{F}

where:

^{ t+Δt}{R} = Vector of externally applied nodal loads

^{ t+Δt}{F}^{(i-1)} = Vector of internally generated nodal forces at iteration (i)

{ΔR}^{(i-1)} = The out-of-balance load vector at iteration (i)

{ΔU}^{(i)} = Vector of incremental nodal displacements at iteration (i)

^{ t+Δt}{U}^{(i)} = Vector of total displacements at iteration (i)

^{ t+Δt}[K]^{(i)} = The Jacobian (tangent stiffness) matrix at iteration (i).

There are different schemes to perform the above iterations. A brief description of two methods of the Newton type are presented below: