# Deflection equation using Macaulay's

In summary, the equation for the deflection of a beam loaded by the reaction R2 located at x = 5 m is incorrect, and the deflections calculated are far beyond the elastic limit for steel.

## Homework Statement

I need to calculate the deflection equation.
R1=33kN
R2=32kN
E=210GPa
calculated I=5.4*10^-7 m^4

## The Attempt at a Solution

Is the equation correct? I'm not sure if I got the u.d.l. right.

## Homework Statement

View attachment 83515

I need to calculate the deflection equation.
R1=33kN
R2=32kN
E=210GPa
calculated I=5.4*10^-7 m^4

## The Attempt at a Solution

View attachment 83516

Is the equation correct? I'm not sure if I got the u.d.l. right.
Indeed, there is a problem with the UDL. The UDL starts at x = 0, whereas your BM equation has it starting at x = 2 m.

Also, for completeness, I would add a term for the reaction R2 located at x = 5 m, even though this would not affect any values calculated by the BM expression.

Sorry for the late reply. My current solution is attached.

#### Attachments

• equation.docx
15.7 KB · Views: 231
Sorry for the late reply. My current solution is attached.
I noticed several things which need fixing.

1. In your expression for the BM, you have EI y" = 103[Mess].
When you integrate to find the slope, you wind up with EI y' = 103[Mess + A] and EI y = 103[Mess + Ax + B].
There's really no need to bring the constants of integration inside the brackets where the other terms are multiplied by 1000 due to the loading of the beam.

2. When you integrated the terms for the UDL, the expression -5x2 became -1.67x3 and then -0.56x4. I think -1.67 / 4 ≠ -0.56.
This same error pops up in the term which cuts off the UDL.

You should look at your deflections again, particularly checking the calculation of A and B. The magnitudes of your calculated deflections (one the order of 0.5 m) seem rather large given the length of this beam. Make sure you haven't misplaced a decimal somewhere.

Thanks, in that case one more time! I've corrected the 0.56 to 0.4175 and the value of A is now A=-75.878. Those changes however made the deflection even greater. I've checked everything a couple of times now and I can't see where I when wrong. I've attached the full calcs again. I'm a bit stuck now.

#### Attachments

• equation.docx
71.8 KB · Views: 251
Thanks, in that case one more time! I've corrected the 0.56 to 0.4175 and the value of A is now A=-75.878. Those changes however made the deflection even greater. I've checked everything a couple of times now and I can't see where I when wrong. I've attached the full calcs again. I'm a bit stuck now.

I double checked your deflection calculations, and they are correct, but for a few round off errors. What you have as 0.4175 should be 0.4167, and I get A = -76.00.

Now that you have included the dimensions of the cross section of the beam, I can see why the deflections are so large. The max. bending stresses for the beam are way beyond the elastic limit for steel (whose E = 210 GPa) and are the same order of magnitude as E. This suggests that a real beam made of steel loaded and supported as shown in the OP would have failed completely.

In order for the equation M/EI = y" to be valid, the slope of the beam must be small, such that θ << 1, which is not the case here. While this beam is a good exercise for showing how to calculate deflection using McCauley's Method, it is a terrible choice because its deflections are much too large to satisfy the small slope approximation on which elastic beam theory is based.

Thanks for your help on that, it took a wee while. I shall round up all calcs to 4 d.p. for a better accuracy.

## What is Macaulay's method for deflection equations?

Macaulay's method is a mathematical technique used to calculate the deflection of a beam under a load. It involves dividing the beam into segments and using a set of equations to determine the deflection at each point.

## How does Macaulay's method differ from other methods for calculating deflection?

Unlike other methods, Macaulay's method takes into account the discontinuity in slope at the point where the load is applied. This allows for a more accurate calculation of deflection.

## What are the assumptions made when using Macaulay's method?

Macaulay's method assumes the beam is linearly elastic, the load is a point load, and the beam is subjected to small deflections. It also assumes that the beam is homogeneous and isotropic, meaning it has the same properties throughout and in all directions.

## What are the limitations of Macaulay's method?

Macaulay's method is limited to simple beam structures and cannot be applied to more complex structures such as frames or trusses. It also does not take into account the effect of shear and axial forces on the deflection of the beam.

## How is Macaulay's method used in real-world applications?

Macaulay's method is commonly used in engineering and construction to design and analyze beams and other structural elements. It allows engineers to determine the maximum deflection and bending moments in a beam, which is important for ensuring the safety and stability of a structure.

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