# Structural analysis

## Degrees of Indeterminacy

When analyzing structural models, the first question to ask yourself is whether the system is determinant or indeterminate. If you are looking at a truss, the first step is to count the number of unknown forces. This is the number of members (a single axial force for each member) summed with the number of unknown reaction forces. Second, count the available equations of equilibrium. This number is found by taking the number of joints and multiplying by 2 (two equations of equilibrium per joint). Finally, subtract the number of equilibrium equations from the unknown forces to find the degrees of indeterminacy.

## Degrees of freedom

Degrees of freedom(DoF) In two dimension space, single part of structure have 2 displacements and 1 rotation.

In three dimension space, single part of structure have 3 displacements and 3 rotations.

In general, a rigid body in d dimensions has d(d + 1)/2 degrees of freedom (d translations and d(d −1)/2 rotations), 2 directions combine to 1 rotation, therefore use Binomial coefficient to determine number of rotations.

## Energy method

w:energy method These are methods based on linear elastic behavior and conservation of energy, i.e. the work done by external forces equals the energy stored in the structure under load. Energy U = Fx/2 = F2 /2k where F is the applied force, x is the distance moved in the direction of the force at its point of application and k is the elastic stiffness of the part, again in the direction of the force at its point of application.

## Influence line

Influence line is defined as the function which shows the variation of a structural quantity (shear force, bending moment, deflection etc.) due to the change in the position of unit load.

## Virtual work

Virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement will be different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action, and, therefore, is the one followed by the particle by the principle of least action. The work of a force on a particle along a virtual displacement is known as the virtual work.

Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies, but they have also been developed for the study of the mechanics of deformable bodies.

## Castigliano's method

Virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement will be different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action, and, therefore, is the one followed by the particle by the principle of least action. The work of a force on a particle along a virtual displacement is known as the virtual work.

Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies, but they have also been developed for the study of the mechanics of deformable bodies.

## Flexibility method

In structural engineering, the flexibility method, also called the method of consistent deformations, is the traditional method for computing member forces and displacements in structural systems. Its modern version formulated in terms of the members' flexibility matrices also has the name the matrix force method due to its use of member forces as the primary unknowns

## Stiffness method

Stiffness is defined as the resistance to deformation. stiffness is inversely proportion to flexibility