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
to the left of the web centerline, the member bends in a vertical plane without twisting.*e*

**Shear stress distribution strategy **

- Determine location of centroid and
*I*_{yy}*,**I*and_{zz}*I*as needed – (symmetric sections subject to_{yz}*V*needs only_{y}*I*)_{zz}

- Divide section into elements according to geometry (change in slope)
- Start with a vector
*s*following element center line from a free end - 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*

- 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 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

- Thin-walled open sections

- Opposite side of open part

**Doubly or singly symmetric section**