HW_CH4_set2

HW_CH4_set2 - Shear function V(x) = R 1 < x-...

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Unformatted text preview: Shear function V(x) = R 1 < x- 0>- w < x- 0> 1 + w < x- a > 1- F < x- b > 0 + R 2 < x- L > Moment function M(x) = R 1 < x- 0> 1- w < x- 0> 2 /2 + w < x- a > 2 /2 - F < x- b > 1 + R 2 < x- L > 1 Modulus of elasticity E 207 GPa = Reactions R 1 264.0 N = R 2 316.0 N = Maximum shear V max 316 N = (negative, from x = b to x = L ) Maximum moment M max 126.4 N m = (at x = b ) 2. Integrate the moment function, multiplying by 1/ EI , to get the slope. ( x ) = [ R 1 < x > 2 /2 - w < x > 3 /6 + w < x- a > 3 /6 - F < x- b > 2 /2 + R 2 < x- L > 2 /2 + C 3 ]/ EI 3. Integrate again to get the deflection. y ( x ) = [ R 1 < x > 3 /6 - w < x > 4 /24 + w < x- a > 4 /24 - F < x- b > 3 /6 + R 2 < x- L > 3 /6 + C 3 x + C 4 ]/ EI 4. Evaluate C 3 and C 4 At x = 0 and x = L , y = 0, therefore, C 4 = 0. R 1 6 L 3 w 24 L 4 - w 24 L a- ( ) 4 + F 6 L b- ( ) 3 - C 3 L + C 3 1 L R 1 6- L 3 w 24 L 4 + w 24 L a- ( ) 4 - F 6 L b- ( ) 3 + = C 3 31.413- N m 2 = 5. Define the range for x x m 0.005 L , L .. = 6. For a Mathcad solution, define a step function S. This function will have a value of zero when x is less than z , and a value of one when it is greater than or equal to z . S x z , ( ) if x z 1 , , ( ) = PROBLEM 4-23a Statement: A beam is supported and loaded as shown in Figure P4-11a. Find the reactions, maximum shear, maximum moment, maximum slope, maximum bending stress, and maximum deflection for the data giv in row a from Table P4-2. Units: N newton = MPa 10 6 Pa = GPa 10 9 Pa = Given: Beam length L 1 m = Distance to distributed load a 0.4 m = Distance to concentrated load b 0.6 m = Distributed load magnitude w 200 N m 1- = Concentrated load F 500 N = Moment of inertia I 2.8510 8- m 4 = FIGURE 4-23A Distance to extreme fiber c 2.0010 2- m = Free Body Diagram for Problem 4-23 Solution: See Figures 4-23 and Mathcad file P0423a. 1. The reactions, maximum shear and maximum moment were all found in Problem 3-23a. Those results are summarized here. Load function q(x) = R 1 < x- 0>-1- w < x- 0> + w < x- a >- F < x- b >-1 + R 2 < x- L >-1 Slope and Deflection Diagrams for Problem 4-23a FIGURE 4-23aB 0.2 0.4 0.6 0.8 1 2 1.5 1 0.5 y x ( ) mm x m 0.2 0.4 0.6 0.8 1 0.01 0.005 0.005 0.01 x ( ) x m DEFLECTION, mm SLOPE, radians y max 1.82- mm = y max y c ( ) = Substituting c into the deflection equation, c 0.523 m = c B- B 2 4 A C - + 2 A = C 33.547- N m 2 = B 16.000 N m = A 92.000 N = C C 3 w 6 a 3 - = B 3 w 6 a 2 = A R 1 2 3 w 6 a - = Solving for c , 1 E I R 1 2 c 2 w 6 c 3 - w 6 c a- ( ) 3 + C 3 + 9. Maximum deflection occurs at x = c , where = 0 and c < b ....
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HW_CH4_set2 - Shear function V(x) = R 1 < x-...

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