Strength of Materials : Columns and springs & Shear Centre and Pressure Vessels Notes

By Deepanshu Rastogi|Updated : December 8th, 2021



Shear Centre and Pressure Vessels

Shear Centre

The shear centre is defined as the point about which the external load has to be applied so that it produces no twisting moment.

  • What happens when loads act in a plane that is not a plane of symmetry?
    Loads must be applied at particular point in the cross section, called shear center, if the beam is to bend without twisting.

Shear Stress distribution

  • Constrained by the shape of the cross section
  • Its resultant acts at the shear center
  • Not necessarily the centroid



Shear Center

  • A lateral load acting on a beam will produce bending without twisting only if it acts through the shear center
    For a doubly symmetric section Shear centre  and Centroid coincide




Locating Shear Centre


Unsymmetric Loading of Thin-Walled Members

Beam loaded in a vertical plane of symmetry deforms in the symmetry plane without twisting.



Beam without a vertical plane of symmetry bends and twists under loading.


Unsymmetric Loading of Thin-Walled Members


If the shear load is applied such that the beam does not twist, then the shear stress distribution satisfies



F and F’ indicate a couple Fh and the need for the application of a torque as well as the shear load.



When the force P is applied at a distance e to the left of the web centerline, the member bends in a vertical plane without twisting.

Shear stress distribution strategy

  • Determine the location of centroid and Iyy, Izz and Iyz as needed - (symmetric sections subject to Vy needs only Izz)
  • Divide section into elements according to geometry (change in slope)
  • Start with a vector s following element centre line from a free end
  • Calculate first moment of area(s). This determines the shear flow distribution
  • Negative shear value indicate the direction of shear flow opposite to assumed vector s
  • Calculate the first moment of area(s). This determines the shear flow distribution

For symmetric sections subject to bending about one axis

· Elements parallel to bending axis-Linear distribution

· Elements normal to bending axis-Parabolic distribution

For unsymmetric sections shear flow in all elements is parabolic

When moving from one element to another the end value of shear in one element equals the initial value for the subsequent element (from equilibrium)

Shear Centres for Some Other Sections




Shear Center

How to locate Shear Center?

Doubly symmetric cross sections- Coincides with centroid

Singly symmetric cross sections- Lies on the axis of symmetry

· Unsymmetric Cross sections

Thin-walled open sections

Opposite side of open part

· Doubly or singly symmetric section



Pressure Vessels

Types of Pressure Vessels Pressure vessels are mainly of two type:

  • Thin shells

If the thickness of the wall of the shell is less than 1/10 to 1/15 of its diameter, then shell is called Thin shells.


  • Thick shells

If the thickness of the wall of the shell is greater than 1/10 to 1/15 of its diameter, then shell is called Thick shells.


where Nature of stress in thin cylindrical shell subjected to internal pressure

  1. Hoop stress/circumferential stress will be tensile in nature.
  2. Longitudinal stress/axial stress will be tensile in nature
  3. Radial stress will be compressive in nature.

Stresses in Thin Cylindrical Shell s3

  • Circumferential Stress /Hoop Stress


Where, p = Intensity of internal pressure

d = Diameter of the shell

t = Thickness of shell

η = Efficiency of joint

  • Longitudinal Stress


  • Hoop Strain


  • Longitudinal Strain


  • Ratio of Hoop Strain to Longitudinal Strain


  • Volumetric Strain of Cylinder

s11 Stresses in Thin Spherical Shell

  • Hoop stress/longitudinal stress



  • Hoop stress/longitudinal strain


  • Volumetric strain of sphere


Lame’s Theory/Analysis of Thick Cylinders Lame’s theory is based on the following assumptions



  1. Homogeneous, isotropic and linearly elastic material.
  2. Plane section of cylinder, perpendicular to longitudinal axis remains plane.
  • Hoop stress at any section image014
  • Radial pressure image015

Subjected to Internal Pressure (p)

  • At image016
  • At image017
Subjected to External Pressure (p)
  • At image018
  • At image019

Note: Radial and hoop stresses vary hyperbolically.


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