MACAULAY'S METHOD OF FINDING SLOPE AND DEFLECTION OF BEAM

In our previous topics, we have seen some important concepts such as deflection and slope of a simply supported beam with point load, deflection and slope of a simply supported beam carrying uniformly distributed load, deflection and slope of a cantilever beam with point load at free end and deflection and slope of a cantilever beam loaded with uniformly distributed load in our previous post.

Now we will start here, in this post, another important topic i.e. Macaulay’s method to determine the deflection and slope of a loaded beam with the help of this post.

We have already seen terminologies and various terms used in deflection of beam with the help of recent posts and now we will be interested here to discuss the Macaulay’s method and its significance with the help of this post.

Basic concepts

There are basically three important methods by which we can easily determine the deflection and slope at any section of a loaded beam.

Double integration method

Moment area method

Macaulay’s method

Double integration method and Moment area method are basically used to determine deflection and slope at any section of a loaded beam when beam will be loaded with a single load.

While Macaulay’s method is basically used to determine deflection and slope at any section of a loaded beam when beam will be loaded with multiple loads.

Differential equation for elastic curve of a beam will also be used in Macaulay’s method to determine the deflection and slope of the loaded beam and hence we must have to recall here the differential equation for elastic curve of a beam.

Differential equation for elastic curve of a beam

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.

As we have seen above that Macaulay’s method is basically used to determine deflection and slope at any section of a loaded beam when beam will be loaded with multiple loads.

Let us consider one case of a simply supported beam with multiple loading and let us write here the complete steps one by one in order to understand the complete process of Macaulay’s method.

Let us consider a beam AB of length L is simply supported at A and B as displayed in following figure, Let us consider that there are three loads W₁, W₂ and W₃ are acting on the beam AB at point C, D and E respectively and we have displayed these loads in following figure.

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

θ_A = Slope at support A

θ_B = Slope at support B

y_C, y_D and y_E= 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.

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 R_A and R_B.

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.

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.

M_X = R_A. 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”.

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.

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.

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.

We will see another topic in our next post. Please comment your feedback and suggestions in comment box provided at the end of this post.