User:Aaronleopold/Moment-Area Theorems

The moment-area theorems were developed by Otto Mohr and later published by Charles Greene in 1873. These methods are used to find the deflection and slope of a beam during bending. The first theorem is used to find the slope at a location of the beam. Simply stated, the change in slope in a member is equal to the change in area of the beam’s moment diagram. The second method is used to find the vertical displacement of the beam. The vertical deviation of the tangent at a point on the elastic curve with respect to the tangent extended from another point equals the "moment" of the area under moment diagram between the two points.

Derivation
A statically determinate beam is first assumed to have an arbitrary loading applied. The radius of curvature can be derived to be the following through Hooke’s law and the flexure formula. These equations are measured from the neutral axis of the member, which for uniform beams, occurs in the middle. Combining these equations results with the following:



\frac{1}{\rho} = \frac{M}{EI}\. $$

Where ρ is the radius of curvature, M is the internal moment where ρ is determined, E is the material’s modulus of elasticity, and I is the beam’s moment of inertia. The moment of inertia is also calculated about the neutral axis. Young's Modulus, or the modulus of elasticity, is a property of the beam determined by its composition and microstructural properties. It then follows from the relationship of curvature and slope that



d\theta = \frac{M}{EI}\,dx. $$

Integrating from point A to B on the elastic curve forms the basis for the first moment-area theorem. It can formally be stated as follows.

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. (Hibbeler)



\theta_{B/A} = \int_A^B \frac{M}{EI}\,dx. $$

The second theorem is derived from the relative deviation of tangents to the elastic curve. The slope of the curve is assumed to be negligible. Therefore, the length of each tangent line can be approximated by x and the arc by dt. The vertical deviation of the tangent at A with respect to the tangent at B can be found by integration, where



t_{B/A} = \int_A^B x\frac{M}{EI}\,dx. $$

From statics, x may be pulled outside of the equation as the centroid of an area. The equation may then be written as



t_{B/A} = \overline{x}\int_A^B \frac{M}{EI}\,dx. $$

Where $$\overline{x}$$ is the distance from the vertical axis through A to the centroid of the area between A and B. From this equation, the second therom can be stated as followed.

'''Theorom 2: The vertical deviation of the tangent at a point (A) on the elastic curve with respect to the tangent extended from another point (B) equals the "moment" of the area under the M/EI diagram between the two points (A and B). This moment is computed about point A (the point on the elastic curve), where the deviation is to be determined.''' (Hibbeler)

Discussion
The term $$\theta_{B/A}$$ is referred to as the angle of the tangent at B measured with respect to the tangent at A. It is measured counterclockwise from tangent A to tangent B if the area of the moment diagram is positive. Alternately, if the area of the M/EI diagram is negative, below the x axis, then the angle $$\theta_{B/A}$$ is measured clockwise from tangent A to tangent B. The resulting angle is measured in degrees. As for deflections from the tangent line, $$t_{B/A}$$ is not the deflection of point A or B. To find a point of deflection on the member, another t value may be found at another point. Using similar triangle geometry and subtraction of the larger t value from the smaller, the deflection may finally be calculated. Again, the deflection is found to have the same similar sign convention.

Applications
This method of analysis is advantageous when solving problems involving beams. It is also very useful for members subjected to vaious loadings and members with different moments of inertia. However, the moment-area theorems only are used to determine the angles or deviations between two tangents on the beam's elastic curve. They do not give the final solution of the member's slope or deflection. Geometric relationships need to be defined, which usually involve similar triangles and endpoints of the beam, to find these actual values. This method is mainly used to understand the fundamentals of beam behavior. Most of these calculations in structural design or analysis are done with computers using fairly simple software.