Macaulay's Method

Macaulay's Method for finding Slope and deflection of various beams.

1. Macaulay’s method Abdikarim yasin (171310140063) Wyman Chisanga (171310140065) Mohamed nur ali (171310140066) Sabba sachi saha (171310140067) Sowrav saha (171310140068) 1

2. Introduction The problems of deflections in beam are bit tedious and laborious, specially when a beam is subjected to point loads or in general discontinuous loads. Mr. W. H Macaulay devised a method, a continuous expression, for bending moment & integration it in such a way that the constants of integration are valid for all section of the beam; even though the law of bending moment varies from section to section. Typically partial uniformly distributed loads and uniformly varying loads over the span and a number of concentrated loads are conveniently handled using this technique. 2

3. Rules for Macaulay's method Always take origin on the extreme left of the beam Take clockwise moment as positive and anticlockwise moment as negative For any term when i.e. negative, the term is neglected. The quantity should be integrated as and not as   3

4. Steps for Macaulay's method 4

5. We have following information from above figure, W1, W2 and W3 = Loads acting on beam AB a1, a2 and a3 = Distance of point load W1, W2 and W3 respectively from support A AB = Position of the beam before loading AFB = Position of the beam after loading θA  = Slope at support A θB  = Slope at support B yC , yD  and yE  = Deflection at point C, D and E respectively Boundary condition  We must be aware with the boundary conditions applicable in such a problem where beam will be simply supported and loaded with multiple point loads. 5

6. Deflection at end supports i.e. at support A and at support B will be zero, while slope will be maximum. Step: 1 First of all we will have to determine the value of reaction forces. Here in this case, we need to secure the value of RAand RB. Step: 2 Now we will have to assume one section XX at a distance x from the left hand support. Here in this case, we will assume one section XX extreme for away from support A and let us consider that section XX is having x distance from support A. 6

7. Step: 3 Now we will have to secure the moment of all the forces about section XX and we can write the moment equation as mentioned here. MX = RA. x – W1 (x- a1) – W2 (x- a2) – W3 (x- a3) We have taken the concept of sign convention to provide the suitable sign for above calculated bending moment about section XX. For more detailed information about the sign convention used for bending moment, we request you to please find the post “Sign conventions for bending moment and shear force”. 7

8. Step: 4 Now we will have to consider Differential equation for elastic curve of a beam and bending moment determined earlier about the section XX and we will have to insert the expression of bending moment in above equation. We will have following equation as displayed here in following figure. 8

9. Step: 5 We will now integrate this equation. After first integration of differential equation, we will have value of slope i.e. dy /dx. Similarly after second integration of differential equation, we will have value of deflection i.e. y. 9

10. Step: 6 We will apply the boundary conditions in order to secure the values of constant of integration i.e. C1 and C2. At x = 0, Deflection (y) = 0    At x = L, Deflection (y) = 0 Step: 7 We will now insert the value of C1 and C2 in slope equation and in deflection equation too in order to secure the final equation for slope and deflection at any section of the loaded beam. Step: 8 We will use the value of x for a considered point and we can easily determine the values of deflection and slope of the beam AB at that respective point. 10

11. Step: 7 We will now insert the value of C1 and C2 in slope equation and in deflection equation too in order to secure the final equation for slope and deflection at any section of the loaded beam. Step: 8 We will use the value of x for a considered point and we can easily determine the values of deflection and slope of the beam AB at that respective point. 11

12. For deriving bending moments For a beam with a number of concentrated loads, separate bending moment equations have had to be written for each part of the beam between adjacent loads. Integration of each expression gives the slope and deflection relationship for each part of the beam. Each slope and deflection relationship includes constants of integration and these have to be determined for each part of the beam. 12

13. For deriving bending moments The constants of integration are then found by equating slopes and deflections given by the expressions on each side of each load. This can be vary laborious if there are many loads. A much less laborious way of tackling the problem is to use the Macaulay’s method Macaulay’s method involves writing just one equation for the bending moment of the entire beam and enables boundary conditions for any part of the beam to be used to obtain the constants of integration. 13

14. Examples 14 Q1.

15. Solution 15

16. Q2. 16

17. 17

18. End Questions? 18