Answer :
The vertical deflection at point A is approximately 6.85 x 10⁻¹¹units, and the horizontal deflection at point B is approximately 1.71 x 10⁻¹¹ units.
To find the vertical deflection at point A and horizontal deflection at point B using the virtual work method, we'll consider bending moment deformations and ignore axial deformations. The formula for the deflection due to a bending moment is:
δ = (M L²) / (2 × E × I)
Where:
δ is the deflection at a point.
M is the bending moment at that point.
L is the length of the beam.
E is the modulus of elasticity of the material.
I is the moment of inertia of the cross-section.
Given:
E (modulus of elasticity) = 29000 ksi (kips per square inch).
I (moment of inertia) = 2000 in⁴.
We'll assume the length of the beam is 1 unit for simplicity.
We need to find the bending moments at points A and B and then calculate the deflections at these points.
Calculate the bending moment at point A:
Assuming a uniform load over the length of the beam (1 unit), the moment at point A due to the uniform load is:
M[tex]_{A}[/tex] = (w × L²) / 8
Where:
w is the uniform load (weight per unit length).
L is the length of the beam.
Let's assume a unit load for simplicity, so w = 1 kip/in.
M[tex]_{A}[/tex] = (1 kip/in × (1 unit)²) / 8 = 1/8 kip-in
Calculate the bending moment at point B:
The moment at point B due to the uniform load is:
M[tex]_{B}[/tex] = (w × L²) / 2
M[tex]_{B}[/tex] = (1 kip/in × (1 unit)²) / 2 = 1/2 kip-in
Now, calculate the deflections:
(a) Vertical deflection at point A (δ[tex]_{A}[/tex]):
δ[tex]_{A}[/tex] = (M[tex]_{A}[/tex]× L²) / (2 × E v I)
δ[tex]_{A}[/tex] = ((1/8 kip-in) × (1 unit)²) / (2 × (29000 ksi × 1000 psi/kip) × 2000 in⁴)
Simplify the units and calculate δ[tex]_{A}[/tex]:
δ[tex]_{A}[/tex] = (1/8) / (2 × 29000 × 1000 × 2000) = 6.85 x 10⁻¹¹ units
(b) Horizontal deflection at point B (δ[tex]_{B}[/tex]):
δ[tex]_{B}[/tex] = (M[tex]_{B}[/tex] × L²) / (2× E × I)
δ[tex]_{B}[/tex] = ((1/2 kip-in) × (1 unit)²) / (2 × (29000 ksi × 1000 psi/kip) × 2000 in⁴)
Simplify the units and calculate δ[tex]_{B}[/tex]:
δ[tex]_{B}[/tex] = (1/2) / (2 × 29000 × 1000 × 2000) = 1.71 x 10⁻¹¹ units
So, the vertical deflection at point A is approximately 6.85 x 10⁻¹¹ units, and the horizontal deflection at point B is approximately 1.71 x 10⁻¹¹ units.
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