Answer :
In this structural analysis task, employing moment-area theorems with constant EI across the beam elucidates the beam's deflection and slope under specified loading conditions. This meticulous approach provides engineers with crucial insights into the behavior of steel beams, facilitating informed design decisions for optimal structural performance.
Given:
- Point load: P = 2.4 kips
- Distributed load on segment 1: w₁ = 0.72 kip/ft
- Distributed load on segment 2: w₂ = 0.48 kip/ft
- Length of the beam: L = 10 ft
- Modulus of elasticity for steel: E = 29000 ksi
1. The total deflection (δ) of the beam can be calculated using the moment-area theorems. We start by determining the bending moment (M) at various points along the beam due to the applied loads.
2. At a distance x from the left end of the beam, the bending moment (M) can be expressed as:
M = P * x + w₁ * x² / 2 for 0 ≤ x ≤ L/2
M = P * (L - x) + w₂ * (L - x)² / 2 for L/2 ≤ x ≤ L
3. Using the equation for deflection due to bending moment (δ = ∫(M * dx) / (E * I)), we integrate the bending moment equations over the respective segments of the beam.
4. For the first segment (0 ≤ x ≤ L/2):
δ₁ = ∫(P * x + w₁ * x² / 2) * dx / (E * I)
5. Integrating and substituting the limits of integration:
δ₁ = ∫(2.4 * x + 0.72 * x² / 2) * dx / (29000 * I) from 0 to L/2
6. Solving the integral:
δ₁ = [(2.4 * x² / 2) + (0.72 * x³ / 6)] / (29000 * I) from 0 to L/2
7. Evaluating the integral:
δ₁ = [(2.4 * (L/2)² / 2) + (0.72 * (L/2)³ / 6)] / (29000 * I)
8. For the second segment (L/2 ≤ x ≤ L):
δ₂ = ∫(P * (L - x) + w₂ * (L - x)² / 2) * dx / (E * I)
9. Integrating and substituting the limits of integration:
δ₂ = ∫(2.4 * (L - x) + 0.48 * (L - x)² / 2) * dx / (29000 * I) from L/2 to L
10. Solving the integral:
δ₂ = [(2.4 * (L - x)² / 2) + (0.48 * (L - x)³ / 6)] / (29000 * I) from L/2 to L
11. Evaluating the integral:
δ₂ = [(2.4 * (L/2)² / 2) + (0.48 * (L/2)³ / 6)] / (29000 * I)
12. The total deflection (δ) is the sum of δ₁ and δ₂:
δ = δ₁ + δ₂
13. By maintaining constant EI across the beam, we ensure consistent flexural rigidity, allowing for accurate analysis of beam deflection under specified loading conditions.
Complete Question:
In this structural analysis task, explore the behavior of a beam under specific loading conditions. Given the beam's attributes - including P = 2.4 kips, w1 = 0.72 kip/ft, w2 = 0.48 kip/ft, and L = 10 ft, constructed from steel with E = 29000 ksi - maintain constant EI across the beam. Employ the moment-area theorems to meticulously solve this engineering challenge.
