7 0.5 2 m 1.5 Shear Diagram 2 Distance [m] Moment Diagram (2.5,900) 2 m 800 N/m ↓↓ 2.5
7 0.5 2 m 1.5 Shear Diagram 2 Distance [m] Moment Diagram (2.5,900) 2 m 800 N/m ↓↓ 2.5
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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Question
![**Beam Deflection Analysis**
Given the equations for the bending moment \(M(x)\) applied to the shown beam, determine the vertical deflection of the beam at distances of 1 [m], 2 [m], 2.5 [m], and 3 [m] from the left support. Assume an elastic modulus of \(E = 200 \text{ GPa}\) and the cross-sectional moment of inertia is \(I = 4.5 \times 10^{-6} \text{ m}^4\).
### Beam Configuration
The beam is simply supported at both ends and subjected to a uniform distributed load (UDL) of \(800 \text{ N/m}\) over a portion of its span.
- Length of left section: \(2 \text{ m}\)
- Length of loaded section: \(2 \text{ m}\)
- Distributed load: \(800 \text{ N/m}\)
### Shear and Moment Diagrams
**Shear Diagram:**
- The shear force diagram shows the variation of the shear force along the length of the beam.
- At \(0 \, \text{m}\), the shear force starts at 400 \, \text{N}.
- At \(2 \, \text{m}\), shear force transitions from 400 \, \text{N} to -800 \, \text{N}.
- From \(2 \, \text{m}\) to \(4 \, \text{m}\), the linear decrease in shear force reflects the effect of the distributed load.
**Moment Diagram:**
- The bending moment diagram shows the variation of the bending moment along the length of the beam.
- At \(0 \, \text{m}\), the moment starts at 0 Nm.
- The maximum moment of \(900 \text{ Nm}\) occurs at \(2.5 \text{ m}\).
- The moment then decreases back to 0 Nm at \(4 \, \text{m}\).
### Bending Moment Equations
For the given beam, the bending moment \( M(x) \) is defined piecewise as follows:
\[
M(x) =
\begin{cases}
400x & \text{if} \quad x < 2 \\
-400x^2 + 2000x - 1600 & \text{if} \quad 2 \leq x \le](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe374e874-5461-490c-9f50-64cb80000b51%2F6cb80dbd-d648-4313-aed8-6c9842373da8%2F1ot48h9_processed.png&w=3840&q=75)
Transcribed Image Text:**Beam Deflection Analysis**
Given the equations for the bending moment \(M(x)\) applied to the shown beam, determine the vertical deflection of the beam at distances of 1 [m], 2 [m], 2.5 [m], and 3 [m] from the left support. Assume an elastic modulus of \(E = 200 \text{ GPa}\) and the cross-sectional moment of inertia is \(I = 4.5 \times 10^{-6} \text{ m}^4\).
### Beam Configuration
The beam is simply supported at both ends and subjected to a uniform distributed load (UDL) of \(800 \text{ N/m}\) over a portion of its span.
- Length of left section: \(2 \text{ m}\)
- Length of loaded section: \(2 \text{ m}\)
- Distributed load: \(800 \text{ N/m}\)
### Shear and Moment Diagrams
**Shear Diagram:**
- The shear force diagram shows the variation of the shear force along the length of the beam.
- At \(0 \, \text{m}\), the shear force starts at 400 \, \text{N}.
- At \(2 \, \text{m}\), shear force transitions from 400 \, \text{N} to -800 \, \text{N}.
- From \(2 \, \text{m}\) to \(4 \, \text{m}\), the linear decrease in shear force reflects the effect of the distributed load.
**Moment Diagram:**
- The bending moment diagram shows the variation of the bending moment along the length of the beam.
- At \(0 \, \text{m}\), the moment starts at 0 Nm.
- The maximum moment of \(900 \text{ Nm}\) occurs at \(2.5 \text{ m}\).
- The moment then decreases back to 0 Nm at \(4 \, \text{m}\).
### Bending Moment Equations
For the given beam, the bending moment \( M(x) \) is defined piecewise as follows:
\[
M(x) =
\begin{cases}
400x & \text{if} \quad x < 2 \\
-400x^2 + 2000x - 1600 & \text{if} \quad 2 \leq x \le
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