m7l38

m7l38 - Module 7 Influence Lines Version 2 CE IIT Kharagpur...

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Module 7 Influence Lines Version 2 CE IIT, Kharagpur
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Lesson 38 Influence Lines for Beams Version 2 CE IIT, Kharagpur
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Instructional Objectives: The objectives of this lesson are as follows: How to draw qualitative influence lines? Understand the behaviour of the beam under rolling loads Construction of influence line when the beam is loaded with uniformly distributed load having shorter or longer length than the span of the beam. 38.1 Müller Breslau Principle for Qualitative Influence Lines In 1886, Heinrich Müller Breslau proposed a technique to draw influence lines quickly. The Müller 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. Let us say, our objective is to obtain the influence line for the support reaction at A for the beam shown in Figure 38.1. Figure 38.1: Simply supported beam First of all remove the support corresponding to the reaction and apply a force (Figure 38.2) in the positive direction that will cause a unit displacement in the direction of R A . The resulting deflected shape will be proportional to the true influence line (Figure 38.3) for the support reaction at A. Figure 38.2: Deflected shape of beam Version 2 CE IIT, Kharagpur
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Figure 38.3: Influence line for support reaction A The deflected shape due to a unit displacement at A is shown in Figure 38.2 and matches with the actual influence line shape as shown in Figure 38.3. Note 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. Similarly some other examples are given below. Here we are interested to draw the qualitative influence line for shear at section C of overhang beam as shown in Figure 38.4. Figure 38.4: Overhang beam As discussed earlier, introduce a roller at section C so that it gives freedom to the beam in vertical direction as shown in Figure 38.5. Figure 38.5: Deflected shape of beam
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This note was uploaded on 06/30/2009 for the course CE 358 taught by Professor Trifunac during the Fall '07 term at USC.

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m7l38 - Module 7 Influence Lines Version 2 CE IIT Kharagpur...

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