Equilibrium#

This section covers the description of global system equilibrium equations and their linearization. For local (element-based) contributions see refer to Truss Element.

System Equilibrium#

The system vector of nodal equilibrium equations is formulated as the component-wise difference for each DOF of nodal element fixing forces \(\boldsymbol{r}(\boldsymbol{U})\) and prescribed nodal external forces \(\boldsymbol{f} = \lambda~\boldsymbol{f}_0\). Given to the fact that we enforce the stiffness matrix to contain a positive sign for the partial derivative of the system fixing force column vector w.r.t to the nodal displacement system column vector the sign of the system equilibrium \(\boldsymbol{g}\) in equation (1) is shifted.

(1)#\[\begin{split}-&\boldsymbol{g}(\boldsymbol{U},\lambda) &= \boldsymbol{r}(\boldsymbol{U}) - \lambda~\boldsymbol{f}_0 = \boldsymbol{0}\\ ||(-)&\boldsymbol{g}(\boldsymbol{U},\lambda)|| &\le \varepsilon_{tol}\end{split}\]

with

Overview of system parameters#
Symbol	Description
\(\boldsymbol{g}\)	system column vector of (nonlinear) nodal equilibrium equations
\(\boldsymbol{r}\)	system column vector of nodal element fixing forces
\(\boldsymbol{U}\)	system column vector of nodal displacements
\(\lambda\)	load-proportionality-factor (LPF)
\(\boldsymbol{f}_0\)	prescribed external nodal load vector (reference loadcase)
\(\boldsymbol{\varepsilon}_{tol}\)	tolerance vector for allowable numerical violation of the equilibrum state

Linearized System Equlibrium#

The linearized equilibrium equations for a given equlibrium state \(\boldsymbol{g}(\boldsymbol{U},\lambda)\) are approximated with the help of a 1st order taylor - expansion:

(2)#\[\begin{split}-\boldsymbol{g}(\boldsymbol{U}+\boldsymbol{dU}, \lambda+d\lambda) &= \boldsymbol{r}(\boldsymbol{U}+\boldsymbol{dU}) &- (\lambda + d\lambda)\boldsymbol{f}_0 &&\\ &\approx \left(\boldsymbol{r}(\boldsymbol{U}) + \frac{\partial \boldsymbol{r}}{\partial \boldsymbol{U}} \boldsymbol{dU} \right) &- (\lambda + d\lambda)\boldsymbol{f}_0 & &= \boldsymbol{0}\\ &\approx \quad \phantom{-}\boldsymbol{r}(\boldsymbol{U}) - \lambda \boldsymbol{f}_0 &+ \frac{\partial \boldsymbol{r}}{\partial \boldsymbol{U}} \boldsymbol{dU} &- d\lambda~\boldsymbol{f}_0 &= \boldsymbol{0} \\ &\approx \quad-\boldsymbol{g}(\boldsymbol{U}, \lambda) &+ \boldsymbol{K}_T(\boldsymbol{U})~\boldsymbol{dU} &- d\lambda~\boldsymbol{f}_0 &= \boldsymbol{0}\end{split}\]

The linearized equilibrium equations may also be expressed as a simple linear equation system. The right hand side of this equation enforces a self-correction over incremental updates of the displacement vector.

(3)#\[\quad\boldsymbol{K}_T(\boldsymbol{U})~\boldsymbol{dU} - d\lambda~\boldsymbol{f}_0 = \boldsymbol{g}(\boldsymbol{U}, \lambda)\]

Newton-Rhapson Iteration and Update of Displacements#

(4)#\[\quad\boldsymbol{K}_T(\boldsymbol{U})~\boldsymbol{dU} - d\lambda~\boldsymbol{f}_0 = \boldsymbol{g}(\boldsymbol{U}, \lambda)\]

with

(5)#\[\begin{split}\boldsymbol{U} &\leftarrow (\boldsymbol{U} + d\boldsymbol{U}) \\ \lambda &\leftarrow (\lambda + d\lambda)\end{split}\]