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KIC000121

# KIC000121 - Influence Lines using the MOiler Breslau...

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8 c Influence Lines using the MOiler Breslau Principle MOiler Breslau Principle The MUller Breslau Principle is another alternative available to qualitatively develop the influence lines for different functions. The MUller Breslau Principle states that the ordinate value of an influence line for any function on any structure is proportional to the ordinates of the deflected shape that is obtained by removing the restraint corresponding to the function from the structure and introducing a force that causes a unit displacement in the positive direction. s B fj C --+-6'--1 1---16' I 6'--1 ,till For example, to obtain the influence line for the support reaction at A for the beam shown in Figure 1, above, remove the support corresponding to the reaction and apply a force in the positive direction that will cause a unit displacement in the direction of Y The resulting deflected shape will be proportional to the true influence line for this reaction. i.e., for the support reaction at A. The deflected shape due to a unit displacement at A is shown below. Notice that the deflected shape is linear, i.e., the beam rotates as a rigid body without any curvature. This is true only for statically determinate systems. A s Y" 1~ R.~pe s ~-3/8 Similarly, to construct the influence line for the support reaction Y B, remove the support at B and apply a vertical force that induces a unit displacement at B. The resulting deflected shape is the qualitative influence line for the support reaction Y k ~8~C 1, ~ 13/8 : + Rb nn=: ; ~A 8 C

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Once again, notice that the influence line is linear, since the structure is statically determinate. This principle will be now be extended to develop the influence lines for other functions. Shear at s To determine the qualitative influence line for the shear at s, remove the shear resistance of the beam at this section by inserting a roller guide, i.e., a device that does not resist shear, but maintains axial force and bending moment resistance. ~ ~fV; B ..J.b V" ~~s ~ 1-10'-- 6'-1 1----16' I c 6'--1 1I1 ~ n Removing the shear resistance will then allow the ends on each side of the section to move perpendicular to the beam axis of the structure at this section. Next, apply a shear force, i.e., V s .
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