Based on the given 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 GPa and the cross-sectional moment of inertia is 1 = 4.5 x 10-6 m². Shear Force [N] Moment [Nm] 400 200 D -200 400 -600 -800 -1000 -1200 1000 800 0 600 400 200 0.5 0.5 2 m 1.5 1.5 Shear Diagram 2 Distance [m] Moment Diagram (2.5,900) 2 Distance (m) 800 N/m 2.5 2 m 2.6 3.5 3.5
Based on the given 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 GPa and the cross-sectional moment of inertia is 1 = 4.5 x 10-6 m². Shear Force [N] Moment [Nm] 400 200 D -200 400 -600 -800 -1000 -1200 1000 800 0 600 400 200 0.5 0.5 2 m 1.5 1.5 Shear Diagram 2 Distance [m] Moment Diagram (2.5,900) 2 Distance (m) 800 N/m 2.5 2 m 2.6 3.5 3.5
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
Related questions
Topic Video
Question
![## Beam Bending Moment and Deflection Analysis
### Problem Statement:
Based on the given 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 \( E = 200 \) GPa and the cross-sectional moment of inertia is \( I = 4.5 \times 10^{-6} \) m\(^4\).
### Beam Diagram:
The beam setup is described as follows:
- Total length of beam: 4 m
- Left section: 2 m (supported by a hinge support)
- Middle section: 2 m with a uniform distributed load of 800 N/m (supported by a roller support at the end)
### Shear Force Diagram (SFD):
The graph labeled "Shear Diagram" illustrates the shear force distribution along the length of the beam. Key observations from the diagram:
- The shear force starts from 400 N at the left support.
- It linearly decreases due to the distributed load and drops to -1200 N at the roller support.
### Bending Moment Diagram (BMD):
The graph labeled "Moment Diagram" illustrates the moment distribution along the length of the beam:
- Starts from 0 at the left hinge.
- Peaks at 2.5 m with a value of 990 Nm.
- Returns to 0 at the right roller support.
### Bending Moment Equation:
The bending moment is piecewise-linear, represented by the function \( M(x) \) and defined as follows:
\[ M(x) = \begin{cases}
400x & \text{if } 0 \leq x < 2 \\
-400x^2 + 2000x - 1600 & \text{if } 2 \leq x \leq 4
\end{cases} \]
### Analysis Procedure:
Using the given beam properties and boundary conditions, the deflection at specified points (1m, 2m, 2.5m, and 3m) along the beam length can be calculated through deflection formulas and integration methods specific to Euler-Bernoulli beam theory.
### Additional Notes:
The diagrams provided are essential tools in analyzing how loads affect beam structures in engineering. The Shear Force](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff88f37cb-86ab-403b-91ba-83c290e9656c%2Ff0e1d18d-cf63-485e-bf07-d907b8b887d5%2F6crk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Beam Bending Moment and Deflection Analysis
### Problem Statement:
Based on the given 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 \( E = 200 \) GPa and the cross-sectional moment of inertia is \( I = 4.5 \times 10^{-6} \) m\(^4\).
### Beam Diagram:
The beam setup is described as follows:
- Total length of beam: 4 m
- Left section: 2 m (supported by a hinge support)
- Middle section: 2 m with a uniform distributed load of 800 N/m (supported by a roller support at the end)
### Shear Force Diagram (SFD):
The graph labeled "Shear Diagram" illustrates the shear force distribution along the length of the beam. Key observations from the diagram:
- The shear force starts from 400 N at the left support.
- It linearly decreases due to the distributed load and drops to -1200 N at the roller support.
### Bending Moment Diagram (BMD):
The graph labeled "Moment Diagram" illustrates the moment distribution along the length of the beam:
- Starts from 0 at the left hinge.
- Peaks at 2.5 m with a value of 990 Nm.
- Returns to 0 at the right roller support.
### Bending Moment Equation:
The bending moment is piecewise-linear, represented by the function \( M(x) \) and defined as follows:
\[ M(x) = \begin{cases}
400x & \text{if } 0 \leq x < 2 \\
-400x^2 + 2000x - 1600 & \text{if } 2 \leq x \leq 4
\end{cases} \]
### Analysis Procedure:
Using the given beam properties and boundary conditions, the deflection at specified points (1m, 2m, 2.5m, and 3m) along the beam length can be calculated through deflection formulas and integration methods specific to Euler-Bernoulli beam theory.
### Additional Notes:
The diagrams provided are essential tools in analyzing how loads affect beam structures in engineering. The Shear Force
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.Recommended textbooks for you
Elements Of Electromagnetics
Mechanical Engineering
ISBN:
9780190698614
Author:
Sadiku, Matthew N. O.
Publisher:
Oxford University Press
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:
9780134319650
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:
9781259822674
Author:
Yunus A. Cengel Dr., Michael A. Boles
Publisher:
McGraw-Hill Education
Elements Of Electromagnetics
Mechanical Engineering
ISBN:
9780190698614
Author:
Sadiku, Matthew N. O.
Publisher:
Oxford University Press
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:
9780134319650
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:
9781259822674
Author:
Yunus A. Cengel Dr., Michael A. Boles
Publisher:
McGraw-Hill Education
Control Systems Engineering
Mechanical Engineering
ISBN:
9781118170519
Author:
Norman S. Nise
Publisher:
WILEY
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:
9781337093347
Author:
Barry J. Goodno, James M. Gere
Publisher:
Cengage Learning
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:
9781118807330
Author:
James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:
WILEY