Constructing a Bending Moment Function Using McCauley's Method

[James20]

I have A beam of 1.2 m long, supports at 0m and 0.8 m. forces of 10 N at 0.4 m and 5N at 1.2 m I need to find the deflection equation for this situation. Can someone have a look and see if they can come up with the equation. As I have tried but my results do not match what I am expecting

 

[SteamKing]

I have A beam of 1.2 m long, supports at 0m and 0.8 m. forces of 10 N at 0.4 m and 5N at 1.2 m I need to find the deflection equation for this situation. Can someone have a look and see if they can come up with the equation. As I have tried but my results do not match what I am expecting

Why don't you post your calculations? You may have made some mistakes in calculating deflections for this beam.

 

[James20]

I split the beam into 3 sections and found the moments acting at each section so section on 2.5 x second section -7.5x and the last section 5x I then integrated twice for each section and got 5/12x^3 +c1x + c2
5/4x^3 +c3x +c4 and 5/6x^3 + c5x + c6 all equal to yEI. Does what I have done so far seem right to you because all I did next was find the constants and if this part isn't right then if my constants are wrong it won't matter as it's all wrong
Cheers

 

[SteamKing]

I split the beam into 3 sections and found the moments acting at each section so section on 2.5 x second section -7.5x and the last section 5x

You have found some of the parts of the moment expression, but these parts act over limited portions of the beam.

For instance, starting at the first support, M(x) = 2.5 x, for 0 ≤ x ≤ 0.4 m
Similarly, the next part, M(x) = -7.5 x for 0.4 < x ≤ 0.8 m
Finally, M(x) = 5 x for 0.8 < x ≤ 1.2 m

You've got to write the bending moment function is such a way that all of the segments of the bending moment curve are treated together. This is difficult to do with regular algebra, but there are special ways to construct a bending moment function in a piece-wise manner. McCauley's method is one such procedure:

http://www.codecogs.com/library/engineering/materials/beams/macaulay-method.php

I then integrated twice for each section and got 5/12x^3 +c1x + c2
5/4x^3 +c3x +c4 and 5/6x^3 + c5x + c6 all equal to yEI.

You've wound up with too many constants of integration. When you integrate the moment equation to find the slope, you'll generate one constant of integration. When you integrate the slope equation to find the deflection, you'll generate another constant of integration as follows:

M(x) = bending moment as a function of position x

Θ(x) = slope as a function of position x

y(x) = deflection as a function of position x

Θ(x) = ∫ M(x) dx = Θ(x) + C₁

y(x) = ∫ Θ(x) dx = ∫ [Θ(x) + C₁] dx = ∫∫ M(x) dx dx + C₁ * x + C₂

The boundary conditions for this beam are that the deflection = 0 at the two supports, so you can have at most two constants of integration.

Does what I have done so far seem right to you because all I did next was find the constants and if this part isn't right then if my constants are wrong it won't matter as it's all wrong
Cheers

You're on the right track. You just have to make some adjustments in how you construct the bending moment function and then integrate it.

 

[James20]

You have found some of the parts of the moment expression, but these parts act over limited portions of the beam.

For instance, starting at the first support, M(x) = 2.5 x, for 0 ≤ x ≤ 0.4 m
Similarly, the next part, M(x) = -7.5 x for 0.4 < x ≤ 0.8 m
Finally, M(x) = 5 x for 0.8 < x ≤ 1.2 m

You've got to write the bending moment function is such a way that all of the segments of the bending moment curve are treated together. This is difficult to do with regular algebra, but there are special ways to construct a bending moment function in a piece-wise manner. McCauley's method is one such procedure:

http://www.codecogs.com/library/engineering/materials/beams/macaulay-method.php
You've wound up with too many constants of integration. When you integrate the moment equation to find the slope, you'll generate one constant of integration. When you integrate the slope equation to find the deflection, you'll generate another constant of integration as follows:

M(x) = bending moment as a function of position x

Θ(x) = slope as a function of position x

y(x) = deflection as a function of position x

Θ(x) = ∫ M(x) dx = Θ(x) + C₁

y(x) = ∫ Θ(x) dx = ∫ [Θ(x) + C₁] dx = ∫∫ M(x) dx dx + C₁ * x + C₂

The boundary conditions for this beam are that the deflection = 0 at the two supports, so you can have at most two constants of integration.
You're on the right track. You just have to make some adjustments in how you construct the bending moment function and then integrate it.

Thank you i will give it another go

 

[James20]

You have found some of the parts of the moment expression, but these parts act over limited portions of the beam.

For instance, starting at the first support, M(x) = 2.5 x, for 0 ≤ x ≤ 0.4 m
Similarly, the next part, M(x) = -7.5 x for 0.4 < x ≤ 0.8 m
Finally, M(x) = 5 x for 0.8 < x ≤ 1.2 m

You've got to write the bending moment function is such a way that all of the segments of the bending moment curve are treated together. This is difficult to do with regular algebra, but there are special ways to construct a bending moment function in a piece-wise manner. McCauley's method is one such procedure:

http://www.codecogs.com/library/engineering/materials/beams/macaulay-method.php
You've wound up with too many constants of integration. When you integrate the moment equation to find the slope, you'll generate one constant of integration. When you integrate the slope equation to find the deflection, you'll generate another constant of integration as follows:

M(x) = bending moment as a function of position x

Θ(x) = slope as a function of position x

y(x) = deflection as a function of position x

Θ(x) = ∫ M(x) dx = Θ(x) + C₁

y(x) = ∫ Θ(x) dx = ∫ [Θ(x) + C₁] dx = ∫∫ M(x) dx dx + C₁ * x + C₂

The boundary conditions for this beam are that the deflection = 0 at the two supports, so you can have at most two constants of integration.
You're on the right track. You just have to make some adjustments in how you construct the bending moment function and then integrate it.

Thank managed to get the answer thanks to yoir help