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singularity functions

singularity functions - LECTURE Third Edition BEAMS...

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• A. J. Clark School of Engineering •Department of Civil and Environmental Engineering Third Edition LECTURE 17 9.5 – 9.6 Chapter BEAMS: DEFORMATION BY SINGULARITY FUNCTIONS by Dr. Ibrahim A. Assakkaf SPRING 2003 ENES 220 – Mechanics of Materials Department of Civil and Environmental Engineering University of Maryland, College Park LECTURE 16. BEAMS: DEFORMATION BY SINGULARITY FUNCTIONS (9.5 – 9.6) Slide No. 1 ENES 220 ©Assakkaf Singularity Functions Introduction – The integration method discussed earlier becomes tedious and time-consuming when several intervals and several sets of matching conditions are needed. – We noticed from solving deflection problems by the integration method that the shear and moment could only rarely described by a single analytical function.
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LECTURE 16. BEAMS: DEFORMATION BY SINGULARITY FUNCTIONS (9.5 – 9.6) Slide No. 2 ENES 220 ©Assakkaf Singularity Functions Introduction – For example the cantilever beam of Figure 9a is a special case where the shear V and bending moment M can be represented by a single analytical function, that is ( ) ( ) ( ) ( ) 2 2 2 and x Lx L w x M x L w x V + = = (15a) (15b) LECTURE 16. BEAMS: DEFORMATION BY SINGULARITY FUNCTIONS (9.5 – 9.6) Slide No. 3 ENES 220 ©Assakkaf Singularity Functions Introduction x y w y x w P Figure 9 (a) (b) L L L/ 2 L/ 4
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LECTURE 16. BEAMS: DEFORMATION BY SINGULARITY FUNCTIONS (9.5 – 9.6) Slide No. 4 ENES 220 ©Assakkaf Singularity Functions Introduction – While for the beam of Figure 9b, the shear V or moment M cannot be expressed in a single analytical function. In fact, they should be represented for the three intervals, namely 0 x L /4, L /4 x L /2, and L /2 x L LECTURE 16. BEAMS: DEFORMATION BY SINGULARITY FUNCTIONS (9.5 – 9.6) Slide No. 5 ENES 220 ©Assakkaf Singularity Functions Introduction – For the three intervals, the shear V and the bending moment M can are given, respectively, by + = L x L L x w wL L x L wL L x wL P x V 2 / for 2 2 2 / /4 for 2 4 / 0 for 2 ) ( y x w P L L/ 2 L/ 4
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LECTURE 16. BEAMS: DEFORMATION BY SINGULARITY FUNCTIONS (9.5 – 9.6) Slide No. 6 ENES 220 ©Assakkaf Singularity Functions Introduction and + + + + = L x L L x w x wL wL L x L x wL wL L x x wL Px wL PL x M 2 / for 2 2 2 8 3 2 / /4 for 2 8 3 4 / 0 for 2 8 3 4 ) ( 2 2 2 2 y x w P L L/ 2 L/ 4 LECTURE 16. BEAMS: DEFORMATION BY SINGULARITY FUNCTIONS (9.5 – 9.6) Slide No. 7 ENES 220 ©Assakkaf Singularity Functions Introduction – We see that even with a cantilever beam subjected to two simple loads, the expressions for the shear and bending moment become complex and more involved. Singularity functions can help reduce this labor by making V or M represented by a single analytical function for the entire length of the beam.
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LECTURE 16. BEAMS: DEFORMATION BY SINGULARITY FUNCTIONS (9.5 – 9.6) Slide No.
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