Moment Area Method Lesson Objectives: 1) Identify the formulation and sign conventions associated with the Moment Area method. 2) Derive the Moment Area method theorems using mechanics and mathematics. 3) Outline procedure on how to apply the Moment Area method. 4) Compute the structural response using the Moment Area method. Background Reading: 1) Read Moment Area Method Overview: 1) The moment area method consists of. 2) These were developed by and and utilize to determine the due to. 3) With these developed theorems, one is able to determine: a. b. Moment Area Derivation: 1) Let s consider a beam as illustrated below in Figure 1 in both and shape. 2) In this diagram, the deformation is induced only due to. a. Therefore the assumption is made that are negligible. b. Reasonable for a beam member? Moment Area Method Notes prepared by: R.L. Wood Page 1 of 12
Figure 1. Undeformed and deformed state diagram of a prismatic beam section. Moment Area Method Notes prepared by: R.L. Wood Page 2 of 12
3) In the figure above, a few takeaways can be noted: a. AB is the of the beam. b. A B is the position of AB when. c. The angle at the center of the arc (at point ) is the. d. is noted to be a very short length of the beam and therefore can be measured as along the curve and along the x-axis. e. is the small angle at the center of the arc,, where this represents the. f. In the bottom diagram, is the average over the portion (between points ). First Moment Area Theorem: 1) To derive the first theorem, note that the angles are and measured in. Therefore a geometric expression can be written as: 2) From mechanics, the states that: 3) Therefore a substitution can be made with the original equation expressed as: Moment Area Method Notes prepared by: R.L. Wood Page 3 of 12
4) Assuming that and small deflections, then: 5) Where substitution yields: 6) Integrating over the entire length of the beam (from points ), the total between A and B can be found. 7) This can be interpreted as the is equal to the : 8) The first theorem can be summarized as: The change in between any two points on the elastic curve for a member subjected to is equal to the area of the between these two points. Second Moment Area Theorem: 1) To derive the second theorem, note that the angles again are and measured in. Therefore another geometric expression can be written as: Moment Area Method Notes prepared by: R.L. Wood Page 4 of 12
2) With an expression for derived in the first moment area theorem, substitution yields: 3) Integrating over the entire length of the beam (from points ), the total between A and B can be found. 4) The second theorem can be summarized as: The at a point (B) on the elastic curve with respect to the tangent extended from the another point (A) equals the of the area under the diagram between the two points (A and B). This moment is computed about, where the is to be computed. 5) Note that the vertical intercept is not, but is referred to as the. 6) Sketch: Moment Area Method Notes prepared by: R.L. Wood Page 5 of 12
Summary of Moment Area Method: 1) Two separate theorems were developed: a. The first quantifies a. b. The second quantifies the. 2) To solve a structure with the Moment Area Method, key steps include: a. The deflection is only due to. b. Find the distribution of over the structure of interest. c. Its critical to correctly draw the. 3) An alternative detailed summary is shown in Figure 2. Figure 2. Detailed summary diagrams of the Moment Area Method (from Leet et al., 2011) Moment Area Method Notes prepared by: R.L. Wood Page 6 of 12
Moment Area Method: Example #1 For a cantilever beam, find the slope and deflection at the beam end. Assume EI is constant throughout the length of the beam. 1) First, draw the moment curve and then divide the ordinates by EI (constant for this example member). Moment Area Method Notes prepared by: R.L. Wood Page 7 of 12
2) Then, draw the deflected (or displaced) shape: 3) Applying the Moment Area Theorem to find : 4) Applying the Moment Area Theorem to find : Moment Area Method Notes prepared by: R.L. Wood Page 8 of 12
Moment Area Method: Example #2 For the two member frame structure below, find the horizontal deflection at point B. Assume EI is constant for all members and assume joint B is rigid. Moment Area Method Notes prepared by: R.L. Wood Page 9 of 12
1) First, draw the moment curve and then divide the ordinates by EI (constant for this example member). 2) Then, draw the deflected (or displaced) shape: Moment Area Method Notes prepared by: R.L. Wood Page 10 of 12
3) Solution: Moment Area Method Notes prepared by: R.L. Wood Page 11 of 12
Moment Area Method Notes prepared by: R.L. Wood Page 12 of 12