# How to Find the First Buckling Load from the Deflection Equation?

• Engineering
• Master1022
In summary, the conversation discusses a problem about calculating buckling loads for a column with specific geometry and boundary conditions. The approach to finding a solution is using a cut and moments, leading to a complex equation with the parameter Beta. The conversation also mentions a link with more information on multiload column buckling. The conversation ends with a hint from the professor on how to find the first buckling load using the displacement at x = L/2 and solving a trigonometric equation.
Master1022
Homework Statement
Use the derived deflection equation to calculate the first buckling load?
Relevant Equations
Derived equation
Hi,

I was working through a problem about calculating buckling loads. The problem had the following geometry (I apologise for the poor drawing skills):
and the total length is ## L ##.

The boundary conditions are therefore:
1. ## y(0) = 0 ##
2. ## y(L) = 0 ##

My approach:
After taking a cut and taking moments we can eventually get the following solution (which agrees with the answer):

$$y(x) = \frac{-wEI}{P^2} cos(\beta x) + \frac{wEI}{P^2} \left(\frac{cos(\beta L) - 1}{sin(\beta L)} \right) sin(\beta x) - \frac{w}{2P} x^2 + \frac{wL}{2P}x + \frac{wEI}{P^2}$$

where ## \beta^2 = \frac{P}{EI} ##

but I really have no clue how to proceed by using this equation to find the first buckling load. Usually we find expressions for ## \beta ## that arise from the boundary conditions...

Any help is greatly appreciated. Thanks

sully_01
We could consider this case as a buckling column, axially loaded by P and laterally loaded by a uniformly distributed load W.
Note that the column is pivoted in both ends, which, together with the lateral load, reduces the magnitude of critical P.

This Math is too complicated for me, but it may help you:

Master1022
Lnewqban said:
We could consider this case as a buckling column, axially loaded by P and laterally loaded by a uniformly distributed load W.
Note that the column is pivoted in both ends, which, together with the lateral load, reduces the magnitude of critical P.

This Math is too complicated for me, but it may help you:

Thank you very much for sharing this @Lnewqban - I will try to make my way through this!

Although, I feel that this method, whilst correct, might slightly be a bit too long for 2 marks of working. I have managed to get a hint from the professor that perhaps we can use the displacement at ## x = L/2 ## but I don't really see how that information is useful. Are you able to see how that could be useful?

Your professors hint is correct. The first buckling mode is likely to be when the maximum deflection occurs at the centre of the beam. Hence dy/dx evaluated at x = L/2 will be 0, and you can solve for the corresponding Beta. (The math simplifies down quite a bit and you'll end up having to solve a simple trigonometric equation!).

Last edited:

## 1. What is the deflection equation used to find the first buckling load?

The deflection equation used to find the first buckling load is the Euler's formula, which is given by Pcr = π^2EI/L^2, where Pcr is the critical buckling load, E is the elastic modulus, I is the moment of inertia, and L is the length of the column.

## 2. How do I determine the elastic modulus (E) for the deflection equation?

The elastic modulus (E) can be determined by consulting material properties tables or by conducting experiments to measure the material's stiffness and elasticity. It is an intrinsic property of the material and varies depending on the type of material being used.

## 3. What is the moment of inertia (I) and how do I calculate it?

The moment of inertia (I) is a measure of an object's resistance to changes in its rotational motion. It is calculated by integrating the square of the distance from the axis of rotation of each element of the object. For a simple column, the moment of inertia can be calculated using the formula I = (π/64) x d^4, where d is the diameter of the column.

## 4. Can the deflection equation be used for any type of column or beam?

The deflection equation can be used for slender columns or beams that are subjected to axial compressive loads. It is not applicable for columns or beams that are subjected to bending or torsional loads.

## 5. How do I interpret the results of the deflection equation to determine the first buckling load?

The first buckling load is the critical load at which the column or beam will buckle and fail under compression. To interpret the results of the deflection equation, compare the calculated critical load (Pcr) to the applied load on the column. If the applied load is less than the critical load, the column is considered stable and will not buckle. If the applied load is greater than the critical load, the column will buckle and fail.

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