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A beam having uniform cross-section carries a uniformly distributed load of intensity q per unit length over its entire span, and its mid-span deflection is δ.
The value of mid-span deflection of the same beam when the same load is distributed with intensity varying from 2q unit length at one end to zero at the other end is:
- a)1/3 δ
- b)1/2 δ
- c)2/3 δ
- d)δ
Correct answer is option 'D'. Can you explain this answer?
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A beam having uniform cross-section carries a uniformly distributed lo...
Solution:
Given: The mid-span deflection of the beam carrying a uniformly distributed load of intensity q per unit length over its entire span is δ.
To find: The mid-span deflection of the same beam when the load is distributed with intensity varying from 2q unit length at one end to zero at the other end.
Assumption: The beam is initially straight and the deflection is small.
Let's consider the following steps to solve the problem.
Step 1: Determine the expression for deflection of the beam carrying uniform load
We know that the deflection of a beam carrying a uniformly distributed load of intensity q per unit length over its entire span is given by,
δ = (5qL^4)/(384EI)
where L is the span of the beam, E is the modulus of elasticity, and I is the moment of inertia of the cross-section of the beam.
Step 2: Determine the expression for deflection of the beam carrying load varying from 2q to 0
Let the load per unit length on the beam be q(x/L), where x is the distance from one end of the beam. Given that the load varies from 2q at one end to 0 at the other end, we can write,
q(x/L) = 2q(1 - x/L)
The deflection of the beam carrying load varying from 2q to 0 can be determined by applying the principle of superposition. We can consider the beam to be loaded with a uniformly distributed load of intensity q per unit length over its entire span and an additional load of intensity q(x/L) - q per unit length. The deflection due to the uniformly distributed load is given by the expression derived in Step 1. The deflection due to the additional load can be determined using the moment area method.
The deflection due to the additional load is given by,
δ' = ∫(q(x/L) - q) dx^2/(2EI) from x = 0 to L
On substituting the expression for q(x/L), we get,
δ' = (qL^4)/(24EI) ∫x(1-x/L)^2dx from x = 0 to L
Solving the integral, we get,
δ' = (5qL^4)/(384EI)
Therefore, the total deflection of the beam carrying load varying from 2q to 0 is given by,
δ_total = δ + δ' = (5qL^4)/(192EI)
Step 3: Determine the ratio of deflection
The ratio of deflection of the beam carrying load varying from 2q to 0 to that of the beam carrying uniform load is given by,
δ_total/δ = (5qL^4)/(384EI) / (5qL^4)/(192EI) = 1/2
Therefore, the mid-span deflection of the beam carrying load varying from 2q to 0 is half of the mid-span deflection of the beam carrying uniform load.
Hence, the correct option is (D) 2/3.
Given: The mid-span deflection of the beam carrying a uniformly distributed load of intensity q per unit length over its entire span is δ.
To find: The mid-span deflection of the same beam when the load is distributed with intensity varying from 2q unit length at one end to zero at the other end.
Assumption: The beam is initially straight and the deflection is small.
Let's consider the following steps to solve the problem.
Step 1: Determine the expression for deflection of the beam carrying uniform load
We know that the deflection of a beam carrying a uniformly distributed load of intensity q per unit length over its entire span is given by,
δ = (5qL^4)/(384EI)
where L is the span of the beam, E is the modulus of elasticity, and I is the moment of inertia of the cross-section of the beam.
Step 2: Determine the expression for deflection of the beam carrying load varying from 2q to 0
Let the load per unit length on the beam be q(x/L), where x is the distance from one end of the beam. Given that the load varies from 2q at one end to 0 at the other end, we can write,
q(x/L) = 2q(1 - x/L)
The deflection of the beam carrying load varying from 2q to 0 can be determined by applying the principle of superposition. We can consider the beam to be loaded with a uniformly distributed load of intensity q per unit length over its entire span and an additional load of intensity q(x/L) - q per unit length. The deflection due to the uniformly distributed load is given by the expression derived in Step 1. The deflection due to the additional load can be determined using the moment area method.
The deflection due to the additional load is given by,
δ' = ∫(q(x/L) - q) dx^2/(2EI) from x = 0 to L
On substituting the expression for q(x/L), we get,
δ' = (qL^4)/(24EI) ∫x(1-x/L)^2dx from x = 0 to L
Solving the integral, we get,
δ' = (5qL^4)/(384EI)
Therefore, the total deflection of the beam carrying load varying from 2q to 0 is given by,
δ_total = δ + δ' = (5qL^4)/(192EI)
Step 3: Determine the ratio of deflection
The ratio of deflection of the beam carrying load varying from 2q to 0 to that of the beam carrying uniform load is given by,
δ_total/δ = (5qL^4)/(384EI) / (5qL^4)/(192EI) = 1/2
Therefore, the mid-span deflection of the beam carrying load varying from 2q to 0 is half of the mid-span deflection of the beam carrying uniform load.
Hence, the correct option is (D) 2/3.
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A beam having uniform cross-section carries a uniformly distributed lo...
First things first, the beam we are dealing here with is a simply supported beam because the term 'mid-span deflection' .
Now, we know that the mid-span deflection formula for a simply supported beam carrying a 'udl' of intensity q per unit length is : 5/384× (ql^4/EI)
For the second case, the mid span deflection would be just half of the deflection in the previous case. So, the mid span deflection formula for a simply supported beam carrying a 'uvl' of intensity q per unit length is : 5/768×(ql^4/EI)
Here in this problem, you have to simply take '2q' because in the second case the intensity of loading is 2q/unit length. So 2 and 2 gets cancelled out and the deflection remains same.
Thank you, hope you will understand.
Now, we know that the mid-span deflection formula for a simply supported beam carrying a 'udl' of intensity q per unit length is : 5/384× (ql^4/EI)
For the second case, the mid span deflection would be just half of the deflection in the previous case. So, the mid span deflection formula for a simply supported beam carrying a 'uvl' of intensity q per unit length is : 5/768×(ql^4/EI)
Here in this problem, you have to simply take '2q' because in the second case the intensity of loading is 2q/unit length. So 2 and 2 gets cancelled out and the deflection remains same.
Thank you, hope you will understand.
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A beam having uniform cross-section carries a uniformly distributed load of intensity q per unit length over its entire span, and its mid-span deflection is δ.The value of mid-span deflection of the same beam when the same load is distributed with intensity varying from 2q unit length at one end to zero at the other end is:a)1/3 δb)1/2 δc)2/3 δd)δCorrect answer is option 'D'. Can you explain this answer?
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A beam having uniform cross-section carries a uniformly distributed load of intensity q per unit length over its entire span, and its mid-span deflection is δ.The value of mid-span deflection of the same beam when the same load is distributed with intensity varying from 2q unit length at one end to zero at the other end is:a)1/3 δb)1/2 δc)2/3 δd)δCorrect answer is option 'D'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about A beam having uniform cross-section carries a uniformly distributed load of intensity q per unit length over its entire span, and its mid-span deflection is δ.The value of mid-span deflection of the same beam when the same load is distributed with intensity varying from 2q unit length at one end to zero at the other end is:a)1/3 δb)1/2 δc)2/3 δd)δCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A beam having uniform cross-section carries a uniformly distributed load of intensity q per unit length over its entire span, and its mid-span deflection is δ.The value of mid-span deflection of the same beam when the same load is distributed with intensity varying from 2q unit length at one end to zero at the other end is:a)1/3 δb)1/2 δc)2/3 δd)δCorrect answer is option 'D'. Can you explain this answer?.
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