Analyzing Indeterminate Structures - Equilibrium and Redundancy of a Structure

While analyzing any indeterminate structure using any method, it is necessary that the solution satisfy the following requirements:

1. Equilibrium of the Structure
2. Compatibility of the Structure
3. Force Displacement Requirements

I. Equilibrium of a Structure

Equilibrium of a Structure是满意的fied when the actions (applied loads) and reactions hold the structure at rest. For a finite sized structure, substructure, element or joint, the following six equations must be satisfied:

ΣFx = 0 ΣFy = 0 ΣFz = 0

ΣMx = 0 ΣMy = 0 ΣMz = 0

Number of equations reduces to 3 for a 2D element or structure:

ΣFx = 0 ------------------------(1)

ΣFy = 0 ------------------------(2)

ΣM = 0-------------------------(3)

These equations are known asEquations of Equilibrium. A structure in which all the unknownscan be determinedusing the Equations of Equilibrium is known asStatically Determinate Structure. While a structure in which all the unknownscannot be determinedusing these equations is known asStatically Indeterminate Structure.

II. Compatibility of a Structure

Compatibility of a Structure is satisfied when the various segments of the structure fit together except intentional breaks or overlaps. By compatibility we mean that:

• Members initially connected together remain connected together (the distance between them may have altered due to the deformation)
• Two initially separate points remain separate (do not overlap or move to another common point)
• Cracks or Gaps do not appear as a structure deforms

Consider the indeterminate truss. Each of the members has been elongated. PointAhas movedto a new PointA’.However, by compatibility of displacements, the elongations are such that the three members remain connected even after deformation, which is additional information and helps in developing an extra equation or set of equations.

In the following indeterminate propped beam; we know that at the supports A and B there is no deflection i.e.

dA= 0AlsodB= 0

In addition slope at a point of maximum deflection is zero i.e.

dy/dx = 0

At the fixed end rotation is resisted hence i.e.

ΣA = 0

All these equations are known as compatibility equations.

Redundancy of a Structure

Any constrain in a structure when removed and do not cause instability to the structure is known as redundant. Consider the following simply supported beam. The horizontal and vertical reactions at a hinged supportAand the only vertical reaction at the roller supportBprevent both the translation and rotation of the beam. In other words these supports are sufficient to keep the structurestable.

If a third supportCis provided between these two, it will make the structure more stable,but its absence is not causing any instability. Thus the vertical reaction provided by this support may be regarded as redundant and can be removed. The structure is then known as basic released structure or primary structure.

The choice of redundant is increased with this extra support thus if the support B is removed, the structure is still stable. Thus any of the support B or C may be removed except thehingedone, which if removed will cause a parallel system of reactions and thence causing the instability of a structure.

Examples of Redundancy/Basic Released Structures