Use the moment function from Part A to determine the equation for the elastic curve from A to B. Express vour answer in terms of w. z. and L.
Use the moment function from Part A to determine the equation for the elastic curve from A to B. Express vour answer in terms of w. z. and L.
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
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![### Learning Goal:
To use the method of integration to solve for the maximum deflection of a statically indeterminate beam.
The method of integration can be used to solve for reactions and deflections of a statically indeterminate beam. The elastic curve of the beam is obtained using the equation:
\[ EI \frac{d^2v}{dx^2} = M(x) \]
and two boundary conditions for either the slope or the deflection. For a statically indeterminate beam, the deflection function \( v(x) \) will contain one or more variables due to the unknown reactions at the redundant supports. However, these supports also provide more boundary conditions, equating the unknowns with the number of boundary conditions.
### Problem Description:
A beam is subjected to a uniform load \( w \) and is supported by a pin support at \( A \) and two rollers at \( B \) and \( C \). This configuration is statically indeterminate to the first degree. Use the method of integration to determine the maximum deflection.
A diagram depicts the beam with a uniform load \( w \) across its length with three supports at points \( A \), \( B \), and \( C \). The distance between each support is \( L \).
### Part A - Write the Moment Function
Since the loading and supports are symmetric, the vertical reactions at \( A \) and \( C \) must be the same, and the shape of the elastic curve from \( A \) to \( B \) must be the same as the shape of the curve from \( C \) to \( B \). Write the moment function for the segment of the beam between \( A \) and \( B \). Use the standard sign convention for beams.
**Express your answer in terms of \( A_y \), \( w \), and \( x \).**
Answer:
\[ M(x) = -\frac{w}{2} (x^2) + A_y x \]
Correct.
### Part B - Determine the Elastic Curve
Use the moment function from Part A to determine the equation for the elastic curve from \( A \) to \( B \).
**Express your answer in terms of \( w \), \( x \), and \( L \).**
An input box is provided below to complete this expression:
\[ EI v(x) = \]
**Note:** The determination of \( EI v(x) \) requires further calculation using integration methods](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e366a85-c23b-450a-9bc7-d2ea0be52d3f%2Fb966c9b3-b500-4961-a764-7443122a1b74%2Fcgzr5ij8_processed.png&w=3840&q=75)
Transcribed Image Text:### Learning Goal:
To use the method of integration to solve for the maximum deflection of a statically indeterminate beam.
The method of integration can be used to solve for reactions and deflections of a statically indeterminate beam. The elastic curve of the beam is obtained using the equation:
\[ EI \frac{d^2v}{dx^2} = M(x) \]
and two boundary conditions for either the slope or the deflection. For a statically indeterminate beam, the deflection function \( v(x) \) will contain one or more variables due to the unknown reactions at the redundant supports. However, these supports also provide more boundary conditions, equating the unknowns with the number of boundary conditions.
### Problem Description:
A beam is subjected to a uniform load \( w \) and is supported by a pin support at \( A \) and two rollers at \( B \) and \( C \). This configuration is statically indeterminate to the first degree. Use the method of integration to determine the maximum deflection.
A diagram depicts the beam with a uniform load \( w \) across its length with three supports at points \( A \), \( B \), and \( C \). The distance between each support is \( L \).
### Part A - Write the Moment Function
Since the loading and supports are symmetric, the vertical reactions at \( A \) and \( C \) must be the same, and the shape of the elastic curve from \( A \) to \( B \) must be the same as the shape of the curve from \( C \) to \( B \). Write the moment function for the segment of the beam between \( A \) and \( B \). Use the standard sign convention for beams.
**Express your answer in terms of \( A_y \), \( w \), and \( x \).**
Answer:
\[ M(x) = -\frac{w}{2} (x^2) + A_y x \]
Correct.
### Part B - Determine the Elastic Curve
Use the moment function from Part A to determine the equation for the elastic curve from \( A \) to \( B \).
**Express your answer in terms of \( w \), \( x \), and \( L \).**
An input box is provided below to complete this expression:
\[ EI v(x) = \]
**Note:** The determination of \( EI v(x) \) requires further calculation using integration methods
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