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The figure shows a simply supported beam PQ of uniform flexural rigidity EI carrying two moments M and 2M. The slope at P will be (A) 0 (B) ML/(9EI) (C) ML/(6EI) (D) ML/(3EI)?
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The figure shows a simply supported beam PQ of uniform flexural rigidi...
Solution:
Understanding the problem:
- A simply supported beam PQ with flexural rigidity EI is given.
- The beam carries two moments M and 2M.
- We need to find the slope at point P.
Calculating the slope:
- Let the length of the beam be L.
- The support reactions at P and Q are RA and RB respectively.
- By taking moments about point Q, we get RA = 2M/L.
- By taking moments about point P, we get RB = M/L.
- The bending moment equation for the beam is given by Mx = RAx - wx^2/2 for 0 ≤ x ≤ L.
- For the section just to the left of point P, the bending moment equation becomes MP = RAP - M(P/2) = M/2.
- For the section just to the right of point P, the bending moment equation becomes MP = RBP - M(P/2) = M/2.
- The slope at point P is given by θP = d/dx(Mx/EI) at x = P.
- Substituting the value of Mx, we get θP = d/dx[(RAx - wx^2/2)/EI] at x = P.
- Differentiating the above equation with respect to x, we get θP = RA/EI - (wP/EI).
- Substituting the values of RA and wP, we get θP = 2M/(LEI) - (ML)/(EI × L^3).
- Simplifying the above equation, we get θP = ML/(3EI).
Therefore, the slope at point P is ML/(3EI). Answer is (D).
Understanding the problem:
- A simply supported beam PQ with flexural rigidity EI is given.
- The beam carries two moments M and 2M.
- We need to find the slope at point P.
Calculating the slope:
- Let the length of the beam be L.
- The support reactions at P and Q are RA and RB respectively.
- By taking moments about point Q, we get RA = 2M/L.
- By taking moments about point P, we get RB = M/L.
- The bending moment equation for the beam is given by Mx = RAx - wx^2/2 for 0 ≤ x ≤ L.
- For the section just to the left of point P, the bending moment equation becomes MP = RAP - M(P/2) = M/2.
- For the section just to the right of point P, the bending moment equation becomes MP = RBP - M(P/2) = M/2.
- The slope at point P is given by θP = d/dx(Mx/EI) at x = P.
- Substituting the value of Mx, we get θP = d/dx[(RAx - wx^2/2)/EI] at x = P.
- Differentiating the above equation with respect to x, we get θP = RA/EI - (wP/EI).
- Substituting the values of RA and wP, we get θP = 2M/(LEI) - (ML)/(EI × L^3).
- Simplifying the above equation, we get θP = ML/(3EI).
Therefore, the slope at point P is ML/(3EI). Answer is (D).
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The figure shows a simply supported beam PQ of uniform flexural rigidity EI carrying two moments M and 2M. The slope at P will be (A) 0 (B) ML/(9EI) (C) ML/(6EI) (D) ML/(3EI)?
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The figure shows a simply supported beam PQ of uniform flexural rigidity EI carrying two moments M and 2M. The slope at P will be (A) 0 (B) ML/(9EI) (C) ML/(6EI) (D) ML/(3EI)? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about The figure shows a simply supported beam PQ of uniform flexural rigidity EI carrying two moments M and 2M. The slope at P will be (A) 0 (B) ML/(9EI) (C) ML/(6EI) (D) ML/(3EI)? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The figure shows a simply supported beam PQ of uniform flexural rigidity EI carrying two moments M and 2M. The slope at P will be (A) 0 (B) ML/(9EI) (C) ML/(6EI) (D) ML/(3EI)?.
The figure shows a simply supported beam PQ of uniform flexural rigidity EI carrying two moments M and 2M. The slope at P will be (A) 0 (B) ML/(9EI) (C) ML/(6EI) (D) ML/(3EI)? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about The figure shows a simply supported beam PQ of uniform flexural rigidity EI carrying two moments M and 2M. The slope at P will be (A) 0 (B) ML/(9EI) (C) ML/(6EI) (D) ML/(3EI)? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The figure shows a simply supported beam PQ of uniform flexural rigidity EI carrying two moments M and 2M. The slope at P will be (A) 0 (B) ML/(9EI) (C) ML/(6EI) (D) ML/(3EI)?.
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