L34_Superposition-fall 10

# L34_Superposition-fall 10 - EAS 209-Fall 2010 Instructors:...

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EAS 209-Fall 2010 Instructors: Christine Human Lecture 34 Superposition We saw last lecture how we could use superposition to solve statically indeterminate beams. In general we can use superposition to determine the slope and deflection for both statically determinate and indeterminate beams. This is one of the most practical methods. Todays’ Lecture: Use the method of superposition to determine beam slope and deflection Today’s Homework: 12/20/2010 1 Lecture 34-Superposition

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EAS 209-Fall 2010 Instructors: Christine Human For the beam shown above, we can determine the slope at A by finding the slope due to a concentrated force P and then finding the slope due to the distributed load w . The total slope is sum of the slopes from both cases. What is so good about this approach? The key is that we are breaking down a complicated problem into a series of simple problems. The solutions to these simple problems are often tabulated in textbooks. See Appendix D, page 762, and hence no integration is required. 12/20/2010 2 Lecture 34-Superposition
EAS 209-Fall 2010 Instructors: Christine Human Table of beam deflections and slopes (pg 762) IMPORTANT: When combining loadings, you cannot just add maximum deflections. Maximum deflections may occur at different point on the beam 12/20/2010 3 Lecture 34-Superposition

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EAS 209-Fall 2010 Instructors: Christine Human Example For the loading shown, find the slope at end A Loading I Loading II Get slopes at A from table for each loading case Both loads cause clockwise rotation 12/20/2010 4 Lecture 34-Superposition ( 29 ( 29 EI PL EIL b L Pb I A 128 7 6 2 2 2 - = - - = θ ( 29 EI wL II A 24 3 - = for b = 3L/4 ( 29 ( 29 EI wL EI PL II A I A A 24 128 7 3 2 + = + = SLOPE SLOPE
EAS 209-Fall 2010 Instructors: Christine Human Since the maximum displacements do not occur at the same location, if we wanted to find y max we would need to: add the elastic curves for the two loading cases set the derivative to zero to find location of the maximum deflection ( x max ) plug x max into total elastic curve to get y max Another Limitation of Method : We can not always express the loading as a combination of the cases found in the tables. For example, in Lecture 33 we solved the problem below using singularity functions Consequently, I would always use singularity functions to solve this problem.

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L34_Superposition-fall 10 - EAS 209-Fall 2010 Instructors:...

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