Explanation Given: The expression for the load on cantilever beam of length L is w ( x ) = 2 w 0 L [ L 2 − x + ( x − L 2 ) U ( x − L 2 ) ] . Concept used: (1) The Laplace transformation of a linear differential equation with constant coefficient is expressed as follows: a n L { d n y d t n } = a n [ s n Y ( s ) − s ( n − 1 ) y ( 0 ) − ⋯ − y ( n − 1 ) ( 0 ) ] (2) The differential equation for the load distribution on cantilever is given as follows: E I d 4 y d x 4 = w ( x ) (3) If F ( s ) = L { f ( t ) } and a > 0 , then L − 1 { e − a s F ( s ) } = f ( t − a ) U ( t − a ) and L − 1 { e − a s s } = U ( t − a ) . (4) If f ( t ) = L − 1 { F ( s ) } and a > 0 , then L { f ( t − a ) U ( t − a ) } = e − a s F ( s ) and L { U ( t − a ) } = e − a s s . (5) The linearity property of the Laplace transformation is stated as, if a , b are any constant and f , g are any function of t , then L { a f ( t ) + b g ( t ) } = a L { f ( t ) } + b L { g ( t ) } Calculation: The expression for the load on cantilever beam of length L is w ( x ) = 2 w 0 L [ L 2 − x + ( x − L 2 ) U ( x − L 2 ) ] . Substitute 2 w 0 L [ L 2 − x + ( x − L 2 ) U ( x − L 2 ) ] for w ( x ) in equation E I d 4 y d x 4 = w ( x ) . E I d 4 y d x 4 = w ( x ) E I d 4 y d x 4 = 2 w 0 L [ L 2 − x + ( x − L 2 ) U ( x − L 2 ) ] Take the Laplace transform on the both side of the above equation. s 4 L { y } − s 3 y ( 0 ) − s 2 y ′ ( 0 ) − s y ″ ( 0 ) − y ‴ ( 0 ) = 2 w 0 E I L [ L 2 s − 1 s 2 + e − L s 2 s 2 ] Substitute 0 for y ( 0 ) , 0 for y ′ ( 0 ) , c 1 for y ″ ( 0 ) and c 2 for y ‴ ( 0 ) in the above equation. s 4 L { y } − s c 1 − c 2 = 2 w 0 E I L [ L 2 s − 1 s 2 + e − L s 2 s 2 ] s 4 L { y } = s c 1 + c 2 + 2 w 0 E I L [ L 2 s − 1 s 2 + e − L s 2 s 2 ] L { y } = c 1 s 3 + c 2 s 4 + 2 w 0 E I L [ L 2 s 5 − 1 s 6 + e − L s 2 s 6 ] Take inverse Laplace on both sides of above equation

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ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 4, Problem 45CR

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The equation of the deflection y(x) for the load w(x)=2w0L[L2−x+(x−L2)U(x−L2)] on cantilever beam of length L.

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Chapter 4, Problem 44CR

Chapter 4, Problem 46CR

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The equation of the deflection y ( x ) for the load w ( x ) = 2 w 0 L [ L 2 − x + ( x − L 2 ) U ( x − L 2 ) ] on cantilever beam of length L .

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The equation of the deflection y(x) for the load w(x)=2w0L[L2−x+(x−L2)U(x−L2)] on cantilever beam of length L.

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Chapter 4 Solutions