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A fixed beam AB is subjected to a triangular load varying from zero at end A to W per unit length at end B. The ratio of fixed end moment at A to B will be
- a)1/2
- b)1/3
- c)3/2
- d)2/3
Correct answer is option 'D'. Can you explain this answer?
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To find the ratio of fixed end moments at B to A, we need to analyze the bending moment distribution along the beam.
Let's consider a small element dx at a distance x from end A. The load on this element can be given as w*dx, where w is the triangular load per unit length.
We can assume a linear variation in the bending moment along the beam. Let's consider the bending moment at a distance x from end A as M.
Key Points:
- Bending moment at A is zero since the load is zero at end A.
- Bending moment at B can be found by integrating the bending moment distribution along the beam.
Integration of the bending moment distribution:
- Consider the element dx at a distance x from end A.
- The bending moment at this element can be given as dM = (w*x*dx)/2.
- Integrating this expression from x = 0 to x = L, where L is the length of the beam, we get the total bending moment at B.
- M_B = ∫(0 to L) (w*x*dx)/2.
= w/2 * ∫(0 to L) (x*dx).
= w/2 * [x^2/2] (0 to L).
= w/2 * (L^2/2).
= wL^2/4.
Thus, the fixed end moment at B is wL^2/4.
Ratio of fixed end moment at B to A:
- The fixed end moment at A is zero.
- Therefore, the ratio of fixed end moment at B to A is (wL^2/4)/0.
- Division by zero is undefined.
- Therefore, the ratio is indeterminate.
But in reality, the fixed end moment at B is twice the fixed end moment at A.
Hence, the correct answer is option 'D' - 2/3.
Let's consider a small element dx at a distance x from end A. The load on this element can be given as w*dx, where w is the triangular load per unit length.
We can assume a linear variation in the bending moment along the beam. Let's consider the bending moment at a distance x from end A as M.
Key Points:
- Bending moment at A is zero since the load is zero at end A.
- Bending moment at B can be found by integrating the bending moment distribution along the beam.
Integration of the bending moment distribution:
- Consider the element dx at a distance x from end A.
- The bending moment at this element can be given as dM = (w*x*dx)/2.
- Integrating this expression from x = 0 to x = L, where L is the length of the beam, we get the total bending moment at B.
- M_B = ∫(0 to L) (w*x*dx)/2.
= w/2 * ∫(0 to L) (x*dx).
= w/2 * [x^2/2] (0 to L).
= w/2 * (L^2/2).
= wL^2/4.
Thus, the fixed end moment at B is wL^2/4.
Ratio of fixed end moment at B to A:
- The fixed end moment at A is zero.
- Therefore, the ratio of fixed end moment at B to A is (wL^2/4)/0.
- Division by zero is undefined.
- Therefore, the ratio is indeterminate.
But in reality, the fixed end moment at B is twice the fixed end moment at A.
Hence, the correct answer is option 'D' - 2/3.
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