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Page  616.Consider Sample Problem 9.3. A. First, solve the Sample Problem 9.2 problem as stated. b. Now let’s replace the triangularly distributed load by a uniformly distributed load w per unit length;and solve the problem. C. Now solve the problem again, but this time you need to simultaneously apply both distributed loads, the triangularly distributed and the uniformly distributed.Remember about superposition

Page  616.Consider Sample Problem 9.3. A. First, solve the Sample Problem 9.2 problem as stated. b. Now let’s replace the triangularly distributed load by a uniformly distributed load w per unit length;and solve the problem. C. Now solve the problem again, but this time you need to simultaneously apply both distributed loads, the triangularly distributed and the uniformly distributed.Remember about superposition

Structural Analysis
Structural Analysis
6th Edition
ISBN:9781337630931
Author:KASSIMALI, Aslam.
Publisher:KASSIMALI, Aslam.
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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Page  616.Consider Sample Problem 9.3.

A. First, solve the Sample Problem 9.2 problem as stated.

b. Now let’s replace the triangularly distributed load by a uniformly distributed load w per unit length;and solve the problem.

C. Now solve the problem again, but this time you need to simultaneously apply both distributed loads, the triangularly distributed and the uniformly distributed.Remember about superposition

Wo B Sample Problem 9.3 For the uniform beam AB (a) determine the reaction at A, (b) derive the equation of the elastic curve, (c) determine the slope at A. (Note that the beam is statically indeterminate to the first degree.) STRATEGY: The beam is statically indeterminate to the first degree. Treating the reaction at A as the redundant, write the bending-moment equation as a function of this redundant reaction and the existing load. After substituting the bending-moment equation into the differential equa- tion of the elastic curve, integrating twice, and applying the boundary conditions, the reaction can be determined. Use the equation for the elastic curve to find the desired slope. MODELING: Using the free body shown in Fig. 1, obtain the bending moment diagram: +)EMD = 0: A RAX - 1/Woxx L 2 - - M = 0 D W = Wo M M = RAX RA Fig. 1 Free-body diagram of portion AD of beam. Wox 6L
Transcribed Image Text:Wo B Sample Problem 9.3 For the uniform beam AB (a) determine the reaction at A, (b) derive the equation of the elastic curve, (c) determine the slope at A. (Note that the beam is statically indeterminate to the first degree.) STRATEGY: The beam is statically indeterminate to the first degree. Treating the reaction at A as the redundant, write the bending-moment equation as a function of this redundant reaction and the existing load. After substituting the bending-moment equation into the differential equa- tion of the elastic curve, integrating twice, and applying the boundary conditions, the reaction can be determined. Use the equation for the elastic curve to find the desired slope. MODELING: Using the free body shown in Fig. 1, obtain the bending moment diagram: +)EMD = 0: A RAX - 1/Woxx L 2 - - M = 0 D W = Wo M M = RAX RA Fig. 1 Free-body diagram of portion AD of beam. Wox 6L
[x = 0, y = 0] em họ lại 1) [x = Ly=0) Fig. 2 Boundary conditions. Fig. 3 Deformed elastic curve showing slope at A. X x ANALYSIS: Differential Equation of the Elastic Curve. Use Eq. (9.4) for WOX 6L El El- = R₁x dx²² Noting that the flexural rigidity El is constant, integrate twice and find dy WOX El 0 = + C₁ (1) dx 24L Wor El y = = R₁x²- 120L [x = 0, y = 0]: C₂=0 [x=L0=0]: 2- W² + C Boundary Conditions. The three boundary conditions that must be satisfied are shown in Fig. 2. Making x = 0, + C₁ = 0 120 aucery mucomma B R₁²_ [x=Ly=0]: ₁² a. Reaction at A. Multiplying Eq. (4) by L, subtracting Eq. (5) member by member from the equation obtained, and noting that C₂ = 0, give + + GL+ C₂ = 0 (₂²³-₂²+ C₁ = 0 b. Equation of the Elastic Curve. into Eq. (2), + C₂x + C₂ El 1y = -=-(-1/wal) x² - Wort - ( 1210 wal² ) x 120L dy Wo 8= dx 120EIL y = 8₁ = R₁L²-L=0 R₁ = twal. 14 The reaction is independent of E and I. Substituting R₁ = w into Eq. (4), Wo 120EIL Wo 120E/ 6 -(-x² + 2L²x²³ - L²x) (2) -(-5x² +6L²x²-L²¹) C₁ = = =₁2²³ Substituting for R₁, C₁, and C₂ c. Slope at A (Fig. 3). Differentiate the equation of the elastic curve with respect to x (3) 0₁ wal 120EI (4) (5)
Transcribed Image Text:[x = 0, y = 0] em họ lại 1) [x = Ly=0) Fig. 2 Boundary conditions. Fig. 3 Deformed elastic curve showing slope at A. X x ANALYSIS: Differential Equation of the Elastic Curve. Use Eq. (9.4) for WOX 6L El El- = R₁x dx²² Noting that the flexural rigidity El is constant, integrate twice and find dy WOX El 0 = + C₁ (1) dx 24L Wor El y = = R₁x²- 120L [x = 0, y = 0]: C₂=0 [x=L0=0]: 2- W² + C Boundary Conditions. The three boundary conditions that must be satisfied are shown in Fig. 2. Making x = 0, + C₁ = 0 120 aucery mucomma B R₁²_ [x=Ly=0]: ₁² a. Reaction at A. Multiplying Eq. (4) by L, subtracting Eq. (5) member by member from the equation obtained, and noting that C₂ = 0, give + + GL+ C₂ = 0 (₂²³-₂²+ C₁ = 0 b. Equation of the Elastic Curve. into Eq. (2), + C₂x + C₂ El 1y = -=-(-1/wal) x² - Wort - ( 1210 wal² ) x 120L dy Wo 8= dx 120EIL y = 8₁ = R₁L²-L=0 R₁ = twal. 14 The reaction is independent of E and I. Substituting R₁ = w into Eq. (4), Wo 120EIL Wo 120E/ 6 -(-x² + 2L²x²³ - L²x) (2) -(-5x² +6L²x²-L²¹) C₁ = = =₁2²³ Substituting for R₁, C₁, and C₂ c. Slope at A (Fig. 3). Differentiate the equation of the elastic curve with respect to x (3) 0₁ wal 120EI (4) (5)
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