Euler's Formula

Q. For the case of a slender column of length $l$, and flexural rigidity $EI$ built in at its base and free at the top, the Euler's critical buckling load is

1. $\frac{4{\pi }^{2}EI}{l^2}$
2. $\frac{2{\pi }^{2}EI}{l^2}$
3. $\frac{{\pi }^{2}EI}{l^2}$
4. $\frac{{\pi }^{2}EI}{4l^2}$

Q. If the length of a column is doubled, the critical load becomes

1. 1/2 of the original value
2. 1/4 of the original value
3. 1/8 of the original value
4. 1/16 of the original value

Q. Assertion (A): The buckling load for a column of specified material, cross-section and end conditions calculated as per Euler's formula varies inversely with the column length.
Reason (R): Euler's formula takes into account the end conditions in determining the effective length of column.

1. both A and R are true and R is the correct explanation of A
2. both A and R are true but R is not a correct explanation of A
3. A is true but R is false
4. A is false but R is true

Q. A column of height h with a rectangular cross-section of size a x 2a has a buckling load of P. If the cross-section is changed to 0.5a x 3a and its height changed to 1.5h, the buckling load of the redesigned column will be

1. P/12
2. P/4
3. P/2
4. 3P/4

Q. Two steel columns P (Length L and yield strength $f_y$=250 MPa) and Q (length 2L and yield strength $f_y$=500 MPa) have the same cross-sections and end-conditions. The ratio of buckling load of column P to that of column Q is:

1. 0.5
2. 1.0
3. 2.0
4. 4.0

Q. Consider two axially loaded columns namely 1 and 2, made of linear elastic material with Young's modulus $2\times {10}^5$MPa, square cross-section with side 10 mm and length 1 m. For column 1, one end is fixed and the other end is free. For column 2, one end is fixed and the other end is pinned. Based on the Euler's theory the ratio (up to one decimal place) of the buckling load of column 2 to the buckling load of column 1 is .

1. 8