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
  • The shear center
  • Is a property of the cross section like the centroid
  • It lies on an axis of symmetry
  • For a doubly symmetric section S and C coincide


Why a property of the cross section?


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

  1. Determine location of centroid and Iyy, Izz and Iyz as needed – (symmetric sections subject to Vy needs only Izz)
  1. Divide section into elements according to geometry (change in slope)
  2. Start with a vector s following element center line from a free end
  3. Calculate first moment of area(s). This determines the shear flow distribution

– Negative shear value indicate direction of shear flow opposite to assumed vector s

  1. Calculate first moment of area(s). This determines the shear flow distribution
    • For symmetric sections subject to bending about one axis
      • Elements parallel to bending axisLinear distribution
      • Elements normal to bending axisParabolic distribution
    • For unsymmetric sections shear flow in all elements is parabolic
    1. 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 sectionsCoincides with centroid
      • Singly symmetric cross sectionsLies on the axis of symmetry
    • Unsymmetric Cross sections
      • Thin-walled open sections
      • Opposite side of open part
    • Doubly or singly symmetric section
    • 10.JPG

      Also see Shear Center with Examples