A structural member carrying an axial compression load is called column

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

12. Torsion of circular shaft Analysis of circular shaft subjected to Torsion

13. 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.

14. 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.

15. Maximum Torsion transmitted by solid circular shaft

16. Maximum Torsion transmitted by hollow circular shaft

17. Power transmitted by shaft

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

19. 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

20. 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?

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

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

23. 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?