# Muller-Breslau principle

**Muller-Breslau principle** is the most important tool in obtaining influence lines for statically determinate us well u statically indeterminate structures. The method is based on the‘ concept of the influence line as a deflection curve. The **Muller-Breslau principle** may be stated as follows:

If an internal stress component or a reaction ‘ component is considered to act through some small distance and thereby to deflect or displace a structure, the curve of 1 the deflected or displaced structure will be, to some scale, the influence line for the stressor reaction component.

## Muller-Breslau principle is applicable

1. Statically determinate beams

2. Statically indeterminate beams

The **Muller-Breslau principle** influence theorem for ‘ statically determinate beams may be stated as follows:

The influence line for an assigned function of a statically determinate beam may be obtained by removing ‘ the restraint offered by that function and introducing a directly related generalized unit displacement at the location! and in the direction of the function.

## Muller Breslau principle indeterminate structures

#### I.L. for Reaction Ra and Rb

The I.L. for reaction (Ra) at A can be found by lifting the beam of the support by a unit distance, as shown in figure (b). The deflected shape gives the I.L. for Ra. Similarly reaction RB can be found out [figure (c)].

#### I.L. for S.F. at C

We know that S.F. acts to both ‘ the sides of the section and is represented by hence cut the beam at c in two parts AC and CB. The free-body diagram of the two parts is shown in the figure. Let the beam go through rigid body motions ‘ of parts AC and CB, so that the total movement C1C2 = unity. The deflected shape will then give the influence line for the sheer force at C. Values of the ordinates will be as shown in figure (d).

#### LL. for B.M. at C

For obtaining I.L. for Mc introduce a hinge at C, and let the system go through rigid-body motions of AC and C B as shown in the figure. The deflected shape will thus be the influence tine for bending moment at C, various values of different elements are as shown in figure (e).

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