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Compression members always tend to buckle in the direction of the
Euler’s formula gives 5 to 10% error in crippling load as compared to experimental results in practice because
Euler's formula to calculate crippling load in the column
\({P_{cr}} = \frac{{{\pi ^2}EI_{mini}}}{{L_e^2}}\)
Where P = Crippling load, E = young's modulus, I_{mini} = Minimum area moment of inertia of column, L_{e} = Effective length of a column
The assumption made while developing Euler Formula for the crippling load of a column does not meet in the reallife practice due to which therefore Euler's formula gives 5 to 10 % error compared to experimental results or true value.
The following assumptions are made in Euler's column theory :
Match ListI (Long column) with Listll (Critical Load) and select the correct answer using the codes given below the lists:
ListI
A. Both ends hinged
B. One end fixed and other end free
C. Both ends fixed
D. One end fixed and other end hinged
ListII
1. π^{2} EI/4L^{2}
2. 4π^{2} EI/4L^{2}
3. 2π^{2} EI/4L^{2}
4. π^{2} EI/L^{2}
Codes:
A B C D
(a) 2 1 4 3
(b) 4 1 2 3
(c) 2 3 4 1
(d) 4 3 2 1
(i) Both ends hinged
(ii) One end fixed and other end free
(iii) Both ends fixed
(iv) One end fixed and other end hinged
Given that
P_{E} = the crippling load given by Euler
P_{c }= the load at failure due to direct compression
P_{R} = the load in accordance with the Rankine’s criterion of failure
Then P_{R} is given by
If the crushing stress in the material of a mild steel column is 3300 kg/cm^{2}. Euler’s formula for crippling load is applicable for slenderness ratio equal to/greater than
For Euler’s formula to be applicable the critical stress must not exceed the proportional limit.
Now crushing stress in mild steel
= 3300 kg/cm^{2}
= 330 N/mm^{2}
But stress at proportional limit in mild steel
= 250 M/mm^{2}
Euler’s buckling stress
λ = 88.55 ≈ 89
Thus slenderness ratio should be more than or equal to 89 ideally. Option (d) is the most close one.
For a circular column having its ends hinged, the slenderness ratio is 160. The L/d ratio of the column is
Slenderness ratio,
where r is the least radius of gyration and L_{eff} is effective length
∴
When a column is subjected to an eccentric load, the stress induced in the column will be
An aluminium column of square crosssection (10 mm x 10 mm) and length 300 mm has both ends pinned. This is replaced by a circular crosssection (of diameter 10 mm) column of the same length and made of the same material with the same end conditions. The ratio of critical stresses for these two columns corresponding to Euler ’s critical load (σ_{critical}) square : (σ_{critical})circular is
For same material, same end condition and same length
If one end of a hinged column is made fixed and the other free, how much is the critical load compared to the original value?
Original load =
when one end of hinged column is fixed and other free. New L_{θ} = 2L
∴ New load
A short column of external diameter D and internal diameter d carries and eccentric load W. The greatest eccentricity which the load can be applied without producing tension on the crosssection of the column would be
Direct stress
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