Advanced structural deformation
Introduction
In structural engineering, the Flexibility Method is the classic method of using deformation to calculate forces in members and displacements in structural systems. Its modern version formulated in terms of the flexibility matrix of the members is also known as Force Matrix Method due to the use of forces in the members as those primarily known.
Member Flexibility
Flexibility is the inverse of rigidity. For example, consider a spring that has Q and q as, respectively, its force and deformation:.
• - The spring stiffness ratio is Q = k q, where k is the spring stiffness.
• - Its flexibility ratio is q = f Q, where f is the flexibility of the spring.
• - Therefore, f = 1/k.
The flexibility ratio of a typical member has the following general form:.
where.
For a system composed of many members interconnected at points called nodes, the flexibility relationships of the members can be put together within a single matrix equation, dropping the superscript m:.
where M is the total number of member deformation characteristics or forces in the system.
Unlike the matrix stiffness method, where member stiffness relationships can be easily integrated using nodal equilibrium and compatibility conditions, the present flexibility form of equation (2) presents serious difficulties. With member forces such as the first unknown, the number of nodal equilibrium equations is insufficient for the solution, in general (unless the system is statically indeterminate).
Nodal Balance Equations
To solve this difficulty, we first make use of the nodal equilibrium equations in order to reduce the number of unknown forces in independent members. The nodal equilibrium equations for the system have the form:.
where.
In the case of given systems, the matrix b is square and the solution to can be found immediately (3) as long as the system is stable.