analysis of column

1. Column
A structural member carrying an axial
compression load is called column

2. Failure of column
• Direct compression
• Buckling
• Combination of compression & buckling

3. Direct compression
• Crushing failure occurs when the material of
the column is unable to withstand the
compressive forces and collapses due to
material yielding or rupture. This type of
failure is common in short, columns.

4. Buckling
• Buckling failure, on the other hand, occurs
when the column undergoes sudden large
deflections or lateral displacements, leading
to instability and collapse. This type of failure
is more common in long, slender columns

5. Expression for crippling load
• Both ends of column are hinged

6. Expression for crippling load
• Both ends of column are Fixed

7. Expression for crippling load
• When one end is fixed & other is hinged

8. Expression for crippling load
• When one end is fixed & other is free

9. Assumption made in Eulers Column
Theory
• Initially straight column: The column is assumed
to be perfectly straight before any load is applied.
• Uniform cross-section: The cross-sectional shape
and dimensions of the column remain constant
along its length.
• Axial load through centroid: The applied load is
assumed to be axial and passes through the
centroid of the column's cross-section.
• Neglecting self-weight: The weight of the column
itself is neglected in calculations, assuming it is
relatively small compared to the applied loads.

10. Assumption made in Eulers Column
Theory
• Homogeneous and isotropic material: The column is
composed of a material that is homogeneous (uniform
composition) material and isotropic (has the same
properties in all directions).
• Elastic limit of stresses: The stresses within the column
are within the elastic limit of the material, ensuring
that the column remains in the elastic range.
• Length dominance: The length of the column is
significantly larger than its cross-sectional dimensions.
• Buckling as the only failure mode: The failure of the
column is by buckling, disregarding other failure
mechanisms.

11. Problem
• A bar of length 4m when used as a simply
supported beam subjected beam & subjected
to a uniformly distributed load of 30KN/m
over the whole span deflects 15mm at the
centre . Determine crippling loads when it is
used as a column with the following condition
• Both ends hinged
• One fixed other hinged
• Both fixed

15. Torsion of circular shaft
• Analysis of circular shaft subjected to Torsion

16. Analysis of circular shaft subjected to
Torsion
• The shaft is considered to be in pure torsion if
it is subject to two opposing turning
moments. This causes the shaft to shear off at
every perpendicular cross-section. This
moment or torsion will bend the circular
shafts without affecting the shaft's cross-
section.

17. Assumption in the theory of pure
torsion
• The material is homogeneous and isotropic
• Hook's law is obeyed by the material.
• The shaft is circular in section.
• The cross-section of the shaft remains uniform
throughout.
• The shaft is subjected to pure torque only.
• The shaft is not subjected to any initial torque.
• The transverse sections which were plane before
application of torque remain plane even after
application of torque.

18. Maximum Torsion transmitted by solid
circular shaft

19. Maximum Torsion transmitted by
hollow circular shaft

20. Power transmitted by shaft

21. Problem
• Find the power transmitted by a shaft 50mm in
diameter at 2.5rev/sec if the max permissible shear
stress is 80N/mm2?

25. Problem
• A solid shaft 80mm in diameter transmit 80KN ot 180rev/min.
Calculate the max shear induced & the angle of twist in
degree for a length of 8m consider G = 80000N/mm2

29. Problem
• A solid shaft has to transmit 60KW at 10/3 hz. The max torque
transmitted in each revolution exceeds the mean by 30%. If
the shear stress is not to exceed 80N/mm2. find a suitable
diameter for the shaft?

33. Shaft in series
• When two shaft are connected so as to remain continuous lengthwise
they are said to be in series

35. Shaft in parallel
• When two shaft are connected so as to remain
surrounded in two shaft are said to be in parallel

37. Problem
• A solid steel shaft 50mm in diameter is fixed rigidly & concentrically inside
a bronze sleeve 75mm external diameter. Find the angle of twist in a 1.5m
length of the composite shaft under the action of a torque of 8Nm
modulus of rigidity of steel is 8x10 4 N/mm2 & that of bronze is 4x10 4
N/mm2?