L32_singularity functions-10 fall

L32_singularity functions-10 fall - EAS 209-Fall 2010...

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EAS 209-Fall 2010 Instructors: Christine Human Lecture 32 Beam Deflection using Singularity Functions We can use singularity functions to solve for beam deflection by writing a single equation for w(x) and integrating four times. Today’s Homework: 12/20/2010 1 Lecture 32-Singularity Functions
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EAS 209-Fall 2010 Instructors: Christine Human In the process of graphical integration we have seen that: Similarly, there will be a jump or “singularity” in the slope diagram at a hinge, due to rotation at the hinge. 12/20/2010 2 Lecture 32-Singularity Functions Concentrated load Jump or “singularity” in shear diagram Kink (change in slope) in moment diagram Concentrated moment Jump or “singularity” in moment diagram No change in shear diagram
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EAS 209-Fall 2010 Instructors: Christine Human Normal integration requires the function to be smooth and continuous between the limits. Singularity functions allow us to handle discontinuities with a single equation for the entire beam, and to integrate across these jumps and kinks (discontinuities). Family of singularity functions STEP FUNCTION RAMP In general n a x - is a singularity function defined as: For x<a 0 = - n a x For x≥a n n a x a x ) ( - = - 12/20/2010 3 Lecture 32-Singularity Functions
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EAS 209-Fall 2010 Instructors: Christine Human The pointed brackets (Macaulay’s brackets) are like ordinary brackets except that they are blind to negative quantities (and negative exponents). n n a x a x a x a x a x - - - - - - - 1 0 1 , , ..... are all singularity functions A useful feature of singularity functions is that the brackets < > can be differentiated and integrated as ordinary parentheses provided n ≥0 Integration - + = - + 0 1 1 1 n a x n dx a x n n Differentiation 0 1 1 - + = - + n a x n a x dx d n n 12/20/2010 4 Lecture 32-Singularity Functions
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EAS 209-Fall 2010 Instructors: Christine Human However, if n <0 the function simply integrates by increasing the exponent by one.
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L32_singularity functions-10 fall - EAS 209-Fall 2010...

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