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)
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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.
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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 𝑥 − 𝑎 < 0 i.e. negative, the term is
neglected.
The quantity (𝑥 − 𝑎) should be integrated as
(𝑥−𝑎)2
2
and not as
𝑥2
2
− 𝑎𝑥
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4. Steps for Macaulay's method
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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.
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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.
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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”.
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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.
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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.
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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.
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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.
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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.
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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.
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14. Examples
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Q1.

15. Solution
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16. Q2.
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18. End
Questions?
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