Question 16.1: Determine the shear flow distribution in the thin-walled Z s...

The origin for our system of reference axes coincides with the centroid of the section at the midpoint of the web. From antisymmetry, we also deduce by inspection that the shear center occupies the same position. Since S_{y} is applied through the shear center, no torsion exists and the shear flow distribution is given by Eq. (16.14), in which S_{x}=0, that is,

q_{s}=-\left(\frac{S_{x} I_{x x}-S_{y} I_{x y}}{I_{x x} I_{y y}-I_{x y}^{2}}\right) \int_{0}^{s} t x d s-\left(\frac{S_{y} I_{y y}-S_{x} I_{x y}}{I_{x x} I_{y y}-I_{x y}^{2}}\right) \int_{0}^{s} t y d s  (16.14)

or

q_{s}=\frac{S_{y}}{I_{x x} I_{y y}-I_{x y}^{2}}\left(I_{x y} \int_{0}^{s} t x d s-I_{y y} \int_{0}^{s} t y d s\right)  (i)

The second moments of area of the section were determined in Example 15.14 and are

Substituting these values in Eq. (i) we obtain

q_{s}=\frac{S_{y}}{h^{3}} \int_{0}^{s}(10.32 x-6.84 y) d s  (ii)

On the bottom flange 12, y=-h / 2 and x=-h / 2+s_{1}, where 0 \leq s_{1} \leq h / 2. Therefore,

giving

q_{12}=\frac{S_{y}}{h^{3}}\left(5.16 s_{1}^{2}-1.74 h s_{1}\right)  (iii)

Hence, at 1,\left(s_{1}=0\right), q_{1}=0 and, at 2, \left(s_{1}=h / 2\right), q_{2}=0.42 S_{y} / h . Further examination of Eq. (iii) shows that the shear flow distribution on the bottom flange is parabolic with a change of sign \text { (i.e., direction) } at s_{1}=0.336 h. For values of s_{1}<0.336 h, q_{12} is negative and therefore in the opposite direction to s_{1}. In the web 23, y=-h / 2+s_{2} \text {, } where 0 \leq s_{2} \leq h where 0 \leq s_{2} \leq h and x=0. Then,

q_{23}=\frac{S_{y}}{h^{3}} \int_{0}^{s_{2}}\left(3.42 h-6.84 s_{2}\right) d s_{2}+q_{2}  (iv)

We note, in Eq. (iv), that the shear flow is not zero when s_{2}=0 but equal to the value obtained by inserting s_{1}=h / 2 in Eq. (iii), that is, q_{2}=0.42 S_{y} / h. Integration of Eq. (iv) yields

q_{23}=\frac{S_{y}}{h^{3}}\left(0.42 h^{2}+3.42 h s_{2}-3.42 s_{2}^{2}\right)  (v)

This distribution is symmetrical about Cx with a maximum value at s_{2}=h / 2(y=0) and the shear flow is positive at all points in the web. The shear flow distribution in the upper flange may be deduced from antisymmetry, so that the complete distribution is of the form shown in Fig. 16.7.