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UPRVUNL JE ME 2016 Official Paper Shift 1

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.**

Support Condition |
Effective Length |

Both ends hinged |
\({L_e} = L\) |

Both ends fixed |
\({L_e} = \frac{L}{2}\) |

One end fixed other end free | \({L_e} = 2L\) |

One end fixed other end hinged | \({L_e} = \frac{L}{\sqrt{2}}\) |

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.
**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)