## Force Method of Analysis (Structure Indeterminate Structures)

Purpose of choosing statically indeterminate structure as compared to statically determinate structure are as follows-

- For a given loading, the maximum stress and deflection of indeterminate structures are generally smaller than that in statically determinate structures.

- Indeterminate structures have a tendency to redistribute its load to its redundant supports in the case where faulty design or overloading occurs.

As load P is increased, the plastic hinge will form at supports first and hence it will be treated as simply supported structures. The further load can also be resisted. However, in case of simple supports, the hinge will form at the center hence collapse will be early.

Although there will be cost saving in material due to lesser stress in member, the cost of construction of supports and joint may sometime offset the saving in material.

Differential settlement of supports, temperature variation, change in length due to fabrication errors in indeterminate structures will introduce internal stresses in structure.

In statically determinate structures internal stresses will not be introduced because of these factors.

In statically indeterminate structures, it is necessary to satisfy:

- Equilibrium equations
- Compatibility equations
- Force displacement requirement

There are two different ways to satisfy these requirements:

- Force Method
- Displacement Method

## Difference between Force and Displacement Method

S.No. | Force Method | Displacement Method |

1 | Also called- Compatibility Method, Method of Consistent Deformation, and Flexibility Method | Also called- Stiffness Method |

2 | Forces are unknown in this Method (Reaction, BM, SF) | Displacement is unknown in this Method (Δ, θ) |

3 | Force displacement equations are written and solution for unknown forces is obtained from compatibility equations. | Force displacement equations are written and solution for unknown displacement is obtained from equilibrium equations. |

4 | Once unknown forces are known, the reaction forces are found using equilibrium equations. | Once unknown displacements are known, internal forces are found using compatibility and load-displacement equation. |

5 | Example- Castigliano’s TheoremStrain energy Method Virtual Work Method/Unit Load Method Claperon’s three-moment Equations Column analogy Method Flexibility Matrix Method | Example- Slope Deflection MethodMoment Distribution Method Stiffness Matrix Method Kam’s Method |

## Fundamental Assumptions in Structural Analysis

### Principal of Superposition

The total displacement or internal loading (stress) at a structure subjected to several external loading can be determined by adding together the displacements or internal loading (stress) caused by each of the external loads acting separately.

Two requirements for Principal of superposition are-

- Linear elastic response i.e., Hooke’s Law valid. (
**Load****∝ Displacement**) - Small displacement theory applies i.e., geometry of the structure must not undergo significant change when loads are applied.

The above requirements are also the requirements for **Linear 1 ^{st} order analysis**.

## Linear-1^{st} Order Analysis

Linear method stress ∝ strain i.e., **Hooke’s law is valid.**

1^{st}-order analysis means the assumption that, length of ABC = length of A’B’C’ in the following figure-

**The 1 ^{st}-order analysis** is valid for small displacement, as otherwise, for large displacement, axial forces as shown below will also produce moments like

**P-Δ**. Thus

**P-Δ**effect is neglected in 1

^{st}order analysis.

However, the effect of **P-Δ** is considered in 2^{nd}-order analysis.

## Betti’s Theorem

The virtual work done by a **P-force system** is going through the deformation of the Q-force system is equal to the virtual work is done by the Q-force system is going through the **deformation of** the **P-force system**.

Hence per **Betti’s Theorem**–

**P _{1}**

**Δ**

_{P1Q}**+ P**

_{2}**Δ**

_{P2Q}= Q_{2}**Δ**

_{Q1P}+ Q_{2}**Δ**

_{Q2P}## Maxwell’s Reciprocal Theorem

If only two force P & Q are acting and magnitude of P & Q are unity.

**Δ _{PQ}**

**=**

**Δ**

_{QP}Δ_{PQ}= Deflection at the location of **P** due to unit load at the location of **Q**

Δ_{QP}= Deflection at the location of **Q** due to unit load at the location of **P**

*i.e., deflection at the location Q due to unit load at P is equal to deflection at the location of P due to unit load at Q.*

## Castigliano’s Theorem

Castigliano has two theorems-

### First Theorem

### Second Theorem

Castigliano extended the **principal of least work**, later on, to the **self-straining system.**

If **λ** = small displacement in the direction of redundant force R then.

*∂U/∂R= λ*

Self-straining may be caused by the **settlement of support** of a redundant structure by an amount **λ** or by the **initial misfit** of a member by an amount **λ** too short/long.

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