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Maximum deflection of a cantilever beam of length „l‟ carrying uniformly distributed load w per unit length will be:
[Where E = modulus of elasticity of beam material and I = moment of inertia of beam crosssection]
- a)wl4/ (EI)
- b)wl4/ (4 EI)
- c)wl4/ (8 EI)
- d)wl4/ (384 EI)
Correct answer is option 'C'. Can you explain this answer?
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Maximum deflection of a cantilever beam of length l carrying uniformly...
Maximum deflection of a cantilever beam carrying a uniformly distributed load:
To find the maximum deflection of a cantilever beam carrying a uniformly distributed load, we can use the formula derived from the Euler-Bernoulli beam theory.
The formula for the maximum deflection of a cantilever beam with a uniformly distributed load can be given as:
δmax = (w * l^4) / (8 * E * I)
Where:
δmax = Maximum deflection of the beam
w = Uniformly distributed load per unit length
l = Length of the cantilever beam
E = Modulus of elasticity of the beam material
I = Moment of inertia of the beam cross-section
Explanation:
The Euler-Bernoulli beam theory is a fundamental theory used to analyze the behavior of beams subjected to bending. According to this theory, the deflection of a beam is inversely proportional to the modulus of elasticity (E) and the moment of inertia (I) of the beam. It is directly proportional to the load applied.
The maximum deflection occurs at the free end of the cantilever beam, and it can be calculated using the formula mentioned above.
Derivation:
To derive the formula, we start with the differential equation of the deflection of the beam:
d^2y/dx^2 = -M(x)/(E * I)
Where:
y = Deflection of the beam at a distance x from the fixed end
M(x) = Bending moment at a distance x from the fixed end
For a cantilever beam carrying a uniformly distributed load, the bending moment can be given as:
M(x) = (w * x^2) / 2
Substituting this into the differential equation, we get:
d^2y/dx^2 = -(w * x^2) / (2 * E * I)
Integrating this equation twice and applying the boundary conditions (y(0) = 0 and dy/dx(0) = 0), we can find the deflection equation:
y(x) = (w * x^4) / (8 * E * I)
The maximum deflection occurs when x = l, which gives us the formula for the maximum deflection:
δmax = (w * l^4) / (8 * E * I)
Conclusion:
Therefore, the correct answer is option 'C': Maximum deflection of a cantilever beam of length l carrying a uniformly distributed load w per unit length will be wl^4 / (8 * E * I). This formula is derived from the Euler-Bernoulli beam theory and takes into consideration the load, length, modulus of elasticity, and moment of inertia of the beam.
To find the maximum deflection of a cantilever beam carrying a uniformly distributed load, we can use the formula derived from the Euler-Bernoulli beam theory.
The formula for the maximum deflection of a cantilever beam with a uniformly distributed load can be given as:
δmax = (w * l^4) / (8 * E * I)
Where:
δmax = Maximum deflection of the beam
w = Uniformly distributed load per unit length
l = Length of the cantilever beam
E = Modulus of elasticity of the beam material
I = Moment of inertia of the beam cross-section
Explanation:
The Euler-Bernoulli beam theory is a fundamental theory used to analyze the behavior of beams subjected to bending. According to this theory, the deflection of a beam is inversely proportional to the modulus of elasticity (E) and the moment of inertia (I) of the beam. It is directly proportional to the load applied.
The maximum deflection occurs at the free end of the cantilever beam, and it can be calculated using the formula mentioned above.
Derivation:
To derive the formula, we start with the differential equation of the deflection of the beam:
d^2y/dx^2 = -M(x)/(E * I)
Where:
y = Deflection of the beam at a distance x from the fixed end
M(x) = Bending moment at a distance x from the fixed end
For a cantilever beam carrying a uniformly distributed load, the bending moment can be given as:
M(x) = (w * x^2) / 2
Substituting this into the differential equation, we get:
d^2y/dx^2 = -(w * x^2) / (2 * E * I)
Integrating this equation twice and applying the boundary conditions (y(0) = 0 and dy/dx(0) = 0), we can find the deflection equation:
y(x) = (w * x^4) / (8 * E * I)
The maximum deflection occurs when x = l, which gives us the formula for the maximum deflection:
δmax = (w * l^4) / (8 * E * I)
Conclusion:
Therefore, the correct answer is option 'C': Maximum deflection of a cantilever beam of length l carrying a uniformly distributed load w per unit length will be wl^4 / (8 * E * I). This formula is derived from the Euler-Bernoulli beam theory and takes into consideration the load, length, modulus of elasticity, and moment of inertia of the beam.
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