Strength of Materials : Thin Cylinders & Buckling of column

(i) on the basis of ratio of diameter to its thickness

where, D is the inner diameter of the shell & t is the thickness of the shell.

(ii) On the basis of shape of the pressure vessel

However, Spherical pressure vessels are better, but due to fabrication difficulty, cylindrical pressure vessels are most commonly used.

Common examples of pressure vessels are steam boilers, reservoirs, tanks, working chambers of engines, gas cylinders etc. 

THIN CYLINDRICAL SHELL SUBJECT TO INTERNAL PRESSURE

Consider a thin cylinder of internal diameter d and wall thickness t, subject to internal gauge pressure P. The following stresses are induced in the cylinder-
(a) Circumferential tensile stress (or hoop stress) σ_H.
(b) Longitudinal (or axial) tensile stress σ_L.
(c) Radial compressive stress σ_R which varies from a value at the inner surface equal to the atmosphere pressure at the outside surface.

Assumptions followed in thin pressure vessels

• Stresses are assumed to be distributed uniformly
• Area is calculated considering the pressure vessel as thin
• Radial stresses are neglected
• Biaxial state of stress is assumed to be applicable

(a) Circumferential stress or Hoop stress, σ_H
There are normal stresses which act in the direction of circumference. Due to internal fluid pressure these are tensile in nature. In thin pressure vessels, hoop stresses are assumed to be uniform across thickness.

In the figure we have shown a one half of the cylinder. This cylinder is subjected to an internal pressure P.

Pressure force by fluid ≤ Resisting force owing to hoop stresses σ_H
P x Projected Area ≤ 2.σ_h.L.t 

P.d.L ≤ 2.σ_h.L.t 
 

In η_L is the efficiency of the Longitudinal riveted joint, 

Similarly,

(b) Longitudinal stress (or axial stress) σ_L

Pressure force by fluid ≤ Resisting force owing to longitudinal stresses σ_L

In η_L is the efficiency of the circumferential riveted joint, 

Thus, the magnitude of the longitudinal stress is one half of the circumferential stress, both the stresses being of tensile nature.

Hoop strain or Circumferential strain -

Longitudinal Strain or axial strain

Ratio of Hoop Strain to Longitudinal Strain

Volumetric Strain or Change in the Internal Volume

THIN SPHERICAL SHELLS
Figure shows a thin spherical shell of internal diameter ‘d’ and thickness ‘t’ and subjected to an internal fluid pressure ‘P’.

Hoop stress/longitudinal stress

Pressure force by fluid ≤ Resisting force owing to Hoop/Longitudinal stresses

Hoop stress/longitudinal strain

Volumetric strain of sphere

Columns and Struts:

• A structural member subjected to an axial compressive force is called strut. As per definition strut may be horizontal, inclined or even vertical.
• The vertical strut is called a column.

Euler’s Column Theory

Assumptions of Euler's theory:

Euler's theory is based on the following assumptions:

(i). Axis of the column is perfectly straight when unloaded.

(ii). The line of thrust coincides exactly with the unstrained axis of the column.

(iii). Flexural rigidity El is uniform.

(iv) Material is isotropic and homogeneous.

Limitation of Euler’s Formula

• There is always crookedness in the column and the load may not be exactly axial.
• This formula does not take into account the axial stress and the buckling load is given by this formula may be much more than the actual buckling load.

Euler’s Buckling (or crippling load)

• The maximum load at which the column tends to have lateral displacement or tends to buckle is known as buckling or crippling load. Load columns can be analysed with the Euler’s column formulas can be given as  

where, E = Modulus of elasticity, L_e= Effective Length of column, and I = Moment of inertia of column section.

  For both end hinged:

in case of Column hinged at both end L_e = L

 For one end fixed and other free:

in case of column one end fixed and other free: L_e = 2L

 For both end fixed:

in case of Column with both end Fixed L_e = L/2

For one end fixed and other hinged:

in case of Column with one end fixed and other hinged L_e = L/√2

Effective Length for different End conditions

Slenderness Ratio ( S)

The slenderness ratio of a compression member is defined as the ratio of its effective length to least radius of gyration.

Modes of failure of Columns

 Rankine’s Formula:

Rankine proposed an empirical formula for columns which coven all Lasts ranging from very short to very long struts. He proposed the relation

   P_c = σ_C. A = ultimate load for a strut

Eulerian crippling load for the standard case

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