Sa 1,moment area theorem

This ppt is all about the introduction of moment area theorem.

1. Vishwakarma Government Engineering College, Chandkheda Name: - Vekaria Darshil(150170106061) Branch: - Civil Sub : -SA-1 Sem: - 4 th Topic :- SOLVING STATICALLY INDETERMINATE STRUCTURES BY MOMENT AREA METHOD .

2. I NTRODUCTION:- The idea of moment area theorem was developed by Otto Mohr and later started formally by Charles E. Greene in 1873.It is just an alternative method for solving deflection problems. In this method we will establish a procedure that utilizes the area of the moment diagrams [actually, the M/EI diagrams] to evaluate the slope or deflection at selected points along the axis of a beam or frame . In numerous engineering applications where deflection of beams must be determine, the loading is complex and cross sectional areas of the beam vary.

3. What is moment area method? This is one of the easiest methods for finding deflection and slope of a beam with t he help of area of bending moment of the beam, which is loaded with some load or any moment at any point or on whole span.

4. Scope of the study :- When the superposition technique of indeterminate beam accelerated according to following reasons restrained and continues beams differ from the simply supported beams mainly by the presence of redundant moment at the supports then moment area method can be used . It is convenient to use this method with great advantage in the following type of problems: Cantilever beams(slope at the fixed end is zero). Simply supported beams carrying symmetrical loading.(slope at mid span is zero) Beams fixed at both ends(slope at each end is zero).

5. DEFLECTION OF BEAMS:- Assumptions for slope and displacement by MOMENT AREA THEOREM are as follows:- Beam is initially straight . Is elastically deformed by the loads, such that the slope and deflection of the elastic curve are very small. Deformations are caused by bending.

6. Theorem 1 :- The change in slope between any two points on the elastic curve equals the area of the M/EI diagram between two points . Figure : Interpretation of small change in an element

7. Theorem 2 :- The vertical deviation of the tangent at a point A on the elastic curve with respect to the tangent extended from another B equals the moment of the area under the M/EI diagram between the two points A and B. Figure : Vertical deviation

8. T heorem(Continue):- This method requires an accurate sketch of the deflected shape, employs above two theorems. THEOREM 1 is used to calculate a change in slope between two points on the elastic curve and THEOREM 2 is used to compute the vertical distance (called a tangential deviation) between a point on the elastic curve and a line tangent to the elastic curve at a second point. Figure : Moment area theorem.

9. PROOF:- ------------(1)

10. For finding slope:-

11. For deflection:-

12. Now We get:- Where M.ds=area of b.m. diagram M.x.ds=moment of area of b.m. diagram

13. Process:- Process to Draw M/EI diagram:-   Determine a redundant reaction, that establish the numerical values for the bending moment diagram. Divided moment diagram by EI. Plot the value and sketch the M/EI Process to Draw E lastic Curve:-   1 Draw an exaggerated view of the beam’s curve. Recall that points of zero slope occur at fixed supports and zero displacement occurs at all fixed, pin and roller supports 2. If it becomes difficult to draw the general shape of the elastic curve, use the M/EI diagram. Realize that when the beam is subjected to a positive moment the beam bends concave up, where negative he negative moments bends the beam concave down. And change in curvature occurs where the moment of the beam is zero.

14. P rocess (continue):- Process to Calculate Deviation:-   Apply theorem 1 to determine the angle between two tangents and theorem 2 to determine vertical deviation between these tangents. Realize that theorem 2 in general will not yield the displacement of a point on the elastic curve. When applied properly it will only give the vertical distance or deviation of a tangent at a point A on the elastic curve from the tangent at B. After applying either theorem 1 or theorem 2 the algebraic sign of the answer can be verified from the angle or deviation as indicated on the elastic curve.

15. Problem:- Find the maximum downward deflection of the small aluminum beam shown in figure due to an applied force P=100N. The beam constant flexure rigidity EI=60N.

16. Problem(Continue):- Solution : The solution of this problem consists of two parts. First a redundant reaction must be determined to establish the numerical values for the bending moment diagram. Then the usual moment-area procedure is applied to find the deflection.

17. P roblem (Continue):- By assuming the beam is released from the redundant end moment, a simple beam-moment diagram is constructed as given here. The moment diagram of known shape due to the unknown redundant moment

18. P roblem (Continue):- ,

19. Problem(Continue ):-

20. Example:-

21. Diagrams:- B.M. Diagram M/EI diagram

22. Solution:-

23. THANK YOU