Using Macaulay's method of deflection on a cantilever beam

• Kyle Grayston
In summary, the slope of the beam 1m from the wall is negative, meaning that the beam is inclined upwards.
Kyle Grayston

Homework Statement

i) Determine the slope of the beam 1m from the wall.
ii) Calculate the deflection at the free end of the beam.

Homework Equations

Macauly's method of deflection: M = -RX + W[X-a]
Integrate once for slope
Integrate twice for deflection

The Attempt at a Solution

[/B]
It is my understanding that a positive deflection is downwards and a negative deflection is upwards.

I am getting a negative answer for the slope which would indicate an upwards slope?

When I use my initial equation to calculate the moment before integrating, I get the correct moment value (27.84kNm) so I feel like that is correct, I am worried that maybe some of the values in my equation should be negative which would change the value of my constants A and B.

Any help would be greatly appreciated.

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Your hand drawn diagram does not match the printed one. You have 0.5m to the first load where the printed diagram has 1m.

Also, shouldn't there be a torque and vertical force from the support in the expression for M?

haruspex said:
Your hand drawn diagram does not match the printed one. You have 0.5m to the first load where the printed diagram has 1m.

Also, shouldn't there be a torque and vertical force from the support in the expression for M?

Hi, sorry I've drawn the beam out mirrored as I was trying to compare it to a similar one done in class. My diagram is shown with the fixed end on the right.

There is 19.2kN reaction force at the fixed end, I am just not sure whether this is incorporated into the equation.

Kyle Grayston said:
My diagram is shown with the fixed end on the right.
Ok, so you are measuring x from the free end, but the question asks for the slope 1m from the wall, and you have substituted x=1.
Also, viewed from the free end, your loads will have clockwise moments, so should have negative torques. That is why you got a negative slope.

Ah yes, I see where I have put the incorrect x value in, thanks!

As for the negative slope, is this what you mean by having a negative slope when looking at the free end?

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Kyle Grayston said:
As for the negative slope,
Ignore what I wrote before... not saying it was wrong, but here's a simpler way..
A downward force left of x leads to an anticlockwise torque at x and makes a negative contribution to y". So if you are taking torques as positive clockwise then your M=... equations are correct but y"=-kM.

Ah I see what you're saying, that makes more sense. Thanks very much for your help, I really appreciate it.

1. What is Macaulay's method of deflection?

Macaulay's method of deflection is a mathematical approach used to calculate the deflection of a cantilever beam under various loading conditions. It takes into account the beam's geometry, material properties, and applied loads to determine the deflection at any given point along the beam.

2. How does Macaulay's method work?

The method involves dividing the beam into segments and writing an equation for the deflection of each segment. These equations are then combined and solved for the unknown deflection coefficients using boundary conditions. The final equation can then be used to calculate the deflection at any point along the beam.

3. What are the advantages of using Macaulay's method?

Macaulay's method is relatively simple and efficient compared to other methods of calculating beam deflection. It also allows for easy incorporation of different load types and boundary conditions, making it a versatile tool for analyzing cantilever beams.

4. Are there any limitations to using Macaulay's method?

Macaulay's method is limited to analyzing linear elastic beams and cannot be used for beams with nonlinear material properties. It also assumes that the beam is homogeneous and isotropic, meaning that the material properties are constant throughout the beam.

5. How can I apply Macaulay's method to a cantilever beam?

To use Macaulay's method, you will need to have a thorough understanding of the beam's geometry and material properties, as well as the applied loads and boundary conditions. You can then follow the steps outlined in the method to calculate the deflection at any point along the beam. Alternatively, there are many software programs available that can perform the calculations for you using Macaulay's method.

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