# Question involving Slenderness Ratio and E.S.R.

• Gundam404
In summary, the column will buckle at the length determined in a) if the ESR is below the S.R., and it will crush instead of buckle when it's length is shorter than the minimum buckling length determined at the yield stress of the steel.

## Homework Statement

A column has the dimensions shown in the diagram below. (a) What is the minimum length of the column at which buckling is likely to occur? (b) If the column is the length determined in (a), (i) What will be the mode of failure? (ii) At what load would you expect failure to occur?

The specific problem I am having with this question is for b)i) how to determine what the mode of failure will be from the E.S.R. and the S.R. Admittedly this question has been asked many times and people seem to say it will buckle at the length determined in a). Now reading my course material, it states that if your E.S.R is below your S.R. then it will be a crushing mode of failure and above will be buckling. Now I have an E.S.R of 118.741 and a S.R. of 237.52. Am I correct in thinking that this will be a crushing mode of failure with the length of 5.938 m as we have just worked out that the minimum length at which buckling is likely to occur is 5.938 m, so how can this length be the minimum for bucking when the slenderness ratio is 237.52. I would like to make sense of this and I feel like I am chasing my own tail at the moment. Thank you very much for any assistance.

## Homework Equations

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Euler's equation. Fc= ((pi^2)*E*I)/Le^2

E.S.R. = Le/k

S.R. = L/K

## The Attempt at a Solution

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Effective Length = 2.969 m
Length = 5.938 m
k = 0.025
Fc = 306 kN
I = 13.7*10^-7
E.S.R. = 118.741
S.R. = 237.52

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The ESR is
kL/r, where k is a function of the boundary conditions and equal to 0.5 for the fixed-fixed ends, and r is the radius of gyration which is equal to sq rt of (I/A). I don’t know where you are getting your formulas. Also, the SR L/r has no place here. The column will crush instead of buckle when it’s length is shorter than the minimum buckling length determined at the yield stress of the steel.

Thank you for the reply and clarification. I have taken a screen shot of a couple of pages from the learning materials which can show you where I get some formulas from and the part about slenderness. In particular the paragraph below the E.S.R. formula on the euler's equation page. This is the one talking about slenderness ratio of a material, i.e. steel, so if I don't know the material being used, does that mean the way it fails will change with different material choices?

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Yes. As long as the geometric properties of the column are the same, the material used does not change the ESR. However , since bucking depends on E, and type of failures depends on the yield stress of the material, then it makes a big difference when using other materials. Steel for example has an E modulus about 20 times greater than E for wood, so for a given long length its critical buckling load is 20 times greater than wood. And in terms of the minimum length for buckling at yield stress, the wood , which has a breaking stress about 10 percent that of steel, will buckle at about a half the min length of the steel. Except for short columns, the wood will always buckle before it crushes.

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Thank you very much PhanthomJay. I feel far more confident with my understanding of this now.

Best regards.

## 1. What is the definition of Slenderness Ratio?

The Slenderness Ratio (SR) is a measure of the stability of a structural member, typically used for columns, and is calculated by dividing the length of the member by its least radius of gyration.

## 2. What is the significance of Slenderness Ratio in structural design?

The Slenderness Ratio is an important factor in structural design as it helps determine the buckling behavior of a member. A high SR indicates a higher risk of buckling, while a low SR indicates a more stable member.

## 3. How is the Effective Slenderness Ratio (E.S.R.) different from the Slenderness Ratio?

The Effective Slenderness Ratio (E.S.R.) takes into account the end conditions of the member, such as whether it is fixed or pinned. It is a more accurate measure of stability and is used in design calculations for columns.

## 4. What is the formula for calculating E.S.R.?

The formula for calculating E.S.R. is E.S.R. = SR / (K x C), where SR is the Slenderness Ratio, K is a factor that depends on the end conditions of the member, and C is a factor that depends on the material properties of the member.

## 5. How does the E.S.R. affect the design of a structural member?

The E.S.R. is a crucial factor in determining the maximum load a structural member can withstand before buckling. A higher E.S.R. requires a larger cross-sectional area for the member to be considered stable, while a lower E.S.R. allows for a smaller cross-sectional area, resulting in more efficient design.