Correct Answer - Option 3 : A column with both ends fixed has minimum equivalent or effective length
Explanation:
According to Euler's column theory, the crippling load for a column of length L,
\({P_{cr}}= \;\frac{{{\pi ^2}EI}}{{{{\left( {L_e} \right)}^2}}}\)
Where Leq is the effective length of the column.
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Support Condition
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Effective Length
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Both ends hinged
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\({L_e} = L\)
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Both ends fixed
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\({L_e} = \frac{L}{2}\)
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| One end fixed other end free |
\({L_e} = 2L\)
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| One end fixed other end hinged |
\({L_e} = \frac{L}{\sqrt{2}}\)
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From the above table, we can see that,
A column with both ends fixed has an effective length equal to \(\frac{L}{2}\) which we minimum among all 4 cases. ..(Option 3 is correct)
The equivalent length of a column with one end fixed and other end hinged is \(\frac {1}{\sqrt 2}\) times of actual length. ...(Option 4 is wrong)
Slenderness ratio:
- It is the ratio of the length of a column and the least radius of gyration of its cross-section. It is used extensively for finding out the design load as well as in classifying various columns in short/intermediate/long.
- Cast iron column with a slenderness ratio not greater than 80 and steel columns with a slenderness ratio not greater than 100 are considered short column.
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Long column is those with a slenderness ratio greater than 100 for ductile materials and greater than 80 for cast iron. ...(Option 2 is wrong)
The following assumptions are made in Euler's column theory:
- The column is initially straight and load is applied axially
- The cross-section of the column is uniform throughout its length
- The column material is perfectly elastic, homogeneous and isotropic and obeys Hooke’s law
- The length of the column is very large as compared to its lateral dimensions.
- The direct stress is very small as compared to the bending stress
- The column will fail by buckling alone
- The self-weight of the column is negligible.
Hence, Euler's theory can only be applicable to the long column. ...(Option 1 is wrong)
