Equilibrium Analysis

About

There are generally two methods for solving structural equilibrium analysis. One is solving for force equilibrium, and the other is solving for energy equilibrium. Although force equilibrium analysis is more commonly used, each method has its own advantages and disadvantages. Below is a simple summary:

• For simple mechanical problems (such as springs and pendulums), using potential energy is often easier.
• To determine stability or instability, checking \frac{d^2 U}{dx^2} is sufficient.
• If there are non-conservative forces (such as friction, air resistance, or external forces), force equilibrium is more intuitive and easier to understand.
• For static or structural problems, directly solving for force equilibrium is more convenient.
• The potential energy equation makes handling state derivatives easier, making it useful for optimization problems (e.g., topology optimization).

Force Equilibrium

Force Equilibrium

In the case of a one-dimensional spring problem, it corresponds to a formulation that relates the spring extension to the applied force and the spring constant.

kx=f

Yap, it is simple

KU=F

If we introduce lagrange multiplier \lambda and boundary constraint BU=0,

\begin{align} 
\begin{bmatrix}
K & B^T \\
B & 0
\end{bmatrix}
\begin{bmatrix}
U \\
\lambda
\end{bmatrix}
=
\begin{bmatrix}
F \\
0
\end{bmatrix}
\end{align} 

Energy Equilibrium

It is a force formulation corresponding to the energy stored in the spring.

E=\frac{1}{2}kx^2-fx

We can represent the form as quadratic form

E(U) = \frac{1}{2} U^T K U - F^T U

If we introduce lagrange multiplier \lambda and boundary constraint BU=0,

   \mathcal{L}(U, \lambda) = \frac{1}{2} U^T K U - f^T U + \lambda^T B U