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Shear Flow in Thin-walled Members (continued)

*Example: For the closed, thin-walled box section shown below find the minimum

and maximum shear stress in the web of the box section. All dimensions are in mm.

Take the shear force at the section to be V = 100 kN.

Strategy: Apply the equation for shear stress given by:

τ = VQ/It

where       V is the value of the shear force at the section
                Q is the first moment of the area between the location where the shear stress
                      is being calculated and the location where the shear stress is zero about
                      the neutral (centroidal) axis
                  I is the moment of inertia of the entire x-section about the neutral axis
                   t is the wall thickness

The key step is in the correct evaluation of the first moment of area, Q, about the neutral
axis shown above by the x-hatched areas in the two figures on the right.

For τ_min the area between zero shear stress and the location for minimum shear stress
in the web is as follows:

Q = (Area)(y_bar for area about n.a.) = 100(20)(140) = 2.8 x 10⁵ mm³

The moment of inertia of the entire x-section about the neutral axis is:

I = (1/12)(200)(300)³ - (1/12)(160)(260)³ = 2.157 x 10⁸ mm⁴

So the minimum shear stress in the web is (100)( 2.8 x 10⁵ ) / (2.157 x 10⁸)(20) = 6.49 MPa

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