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 For
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
(a) 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)].
(1)) 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).
(c) 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).