Moment area method: Difference between revisions
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'''Theorem 1:''' The change in slope between any two points on the [[elastic curve]] equals the area of the M/EI diagram between these two points. | '''Theorem 1:''' The change in slope between any two points on the [[elastic curve]] equals the area of the M/EI diagram between these two points. | ||
< | <math> | ||
\theta_{AB} = {\int_A}^B \frac{M}{EI}\;dx | \theta_{AB} = {\int_A}^B \frac{M}{EI}\;dx | ||
</tex> | </tex> | ||
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* M moment | * M moment | ||
* EI flexural rigidity | * EI flexural rigidity | ||
* < | * <math>\theta_{AB}</tex> ... change in slope between points A and B | ||
* A, B ... points on the elastic curve | * A, B ... points on the elastic curve | ||
| Line 22: | Line 22: | ||
[[Image:Moment area method theorem 2.gif|right|frame|Picture illustrating the second theorem]] | [[Image:Moment area method theorem 2.gif|right|frame|Picture illustrating the second theorem]] | ||
'''Theorem 2:''' The deviation of the tangent at point B on the elastic curve with respect to the tangent at point A equals the "moment" of the < | '''Theorem 2:''' The deviation of the tangent at point B on the elastic curve with respect to the tangent at point A equals the "moment" of the <math>M/EI</tex> diagram between points A and B computed about point A (the point on the elastic curve), where the deviation <math>t_{A/B}</tex> is to be determined. | ||
< | <math> | ||
t_{A/B} = {\int_A}^B \frac{M}{EI} \bar{x} \;dx | t_{A/B} = {\int_A}^B \frac{M}{EI} \bar{x} \;dx | ||
</tex> | </tex> | ||
| Line 32: | Line 32: | ||
* M moment | * M moment | ||
* EI flexural rigidity | * EI flexural rigidity | ||
* < | * <math>t_{A/B}</tex> ... deviation of tangent at point B with respect to the tangent at point A | ||
* < | * <math>\bar{x}</tex> ... centroid of M/EI diagram measured horizontally from point A | ||
* A, B ... points on the elastic curve | * A, B ... points on the elastic curve | ||
Revision as of 02:35, 22 October 2024
The method for finding deflections in a framed structure by use of the moment area curve.
First Theorem
.gif)
Picture illustrating the first theorem
Theorem 1: The change in slope between any two points on the elastic curve equals the area of the M/EI diagram between these two points.
<math> \theta_{AB} = {\int_A}^B \frac{M}{EI}\;dx </tex>
where
- M moment
- EI flexural rigidity
- <math>\theta_{AB}</tex> ... change in slope between points A and B
- A, B ... points on the elastic curve
Second Theorem
Picture illustrating the second theorem
Theorem 2: The deviation of the tangent at point B on the elastic curve with respect to the tangent at point A equals the "moment" of the <math>M/EI</tex> diagram between points A and B computed about point A (the point on the elastic curve), where the deviation <math>t_{A/B}</tex> is to be determined.
<math> t_{A/B} = {\int_A}^B \frac{M}{EI} \bar{x} \;dx </tex>
where
- M moment
- EI flexural rigidity
- <math>t_{A/B}</tex> ... deviation of tangent at point B with respect to the tangent at point A
- <math>\bar{x}</tex> ... centroid of M/EI diagram measured horizontally from point A
- A, B ... points on the elastic curve
References
- Russel C. Hibbeler: Structural Analysis, 3rd Edition, Prentice Hall, 1995, chapter 8, p. 354-569, ISBN 0-02-354041-9
External links
- Deflection of Cantilever Beam by Area Moment Method
- Deflection of Simply Supported Beam by Area Moment Method
- Colin Caprani: Structural Analysis III The Moment Area Method – Mohr’s Theorems (lecture notes), 2007
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| See Also | Computational analysis methods |
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