## 9 Non-Symmetric Section
### Shear Flow Under \(F_{y}{}^{\prime}\) and \(F_{x}{}^{\prime}\)  
Equation (2b,c) showed that the second moments \(I_{x}\), \(I_{y}\) and \(I_{xy}\) provide equivalent shear forces \(F_{x}{}^{\prime}\) and \(F_{y}{}^{\prime}\) that refer to this section's principal axes \(u\) and \(v\) in Figure 11b. Conveniently, the net flexural shear flow retains axes \(x\) and \(y\) with these equivalent forces \(F_{x}{}^{\prime}\) and \(F_{y}{}^{\prime}\). This enables a further progression to \(q_{net}\) using general symbols for the section geometry as  
\[q_{net}=F_{y}{}^{\prime}\ D_{x}/I_{x}+F_{x}{}^{\prime}\ D_{y}/I_{y} \tag{18}\]  
where \(F_{y}{}^{\prime}\) and \(F_{x}{}^{\prime}\) are defined in Equation (2a,b). The first term in Equation (18) is shear flow \(q_{x}\) due to \(F_{y}{}^{\prime}\) within which \(D_{x}\) is the first moment of area about the \(x\)-axis, which is established for median points 1, 2, 3...7 (see Figure 12a) as follows:1-3 : \(D_{x}=t\int_{s}y\;\mathrm{d}s=t\int_{0}^{s}(d-\gamma^{\prime})\mathrm{d}s=t(d- \gamma^{\prime})s\)  
at 1: \(s=0\), \(D_{x1}=0\)  
at 2 (\(y\)-axis) \(s=a-X^{\prime}\); \(D_{x2}=t(d-Y^{\prime})(a-X^{\prime})\)  
at 3: \(s=a\), \(D_{x3}=t(d-Y^{\prime})a\)  
3-5: \(D_{x}=t\int_{s}y\;\mathrm{d}s=t\int_{s}(d-Y^{\prime}-s)\mathrm{d}s=t[(d-Y^{ \prime})s-s^{2}/2]+D_{x3}\)  
at 3: \(s=0\), \(D_{x}=D_{x3}=t(d-Y^{\prime})a\)  
at 4: (\(x\)-axis) \(s=d-Y^{\prime}\); \(D_{x4}=t(d-Y^{\prime})^{2}/2+t(d-Y^{\prime})a\)  
at 5: \(s=d\), \(D_{x5}=t[(d-Y^{\prime})d-d^{2}/2]+t(d-Y^{\prime})a\)  
5-7: \(D_{x}=t\int_{s}y\;\mathrm{d}s=-tY^{\prime}s+D_{x5}\)  
at 5: \(s=0\), \(D_{x}=D_{x5}\)  
at 6: \(s=X^{\prime}\), (\(y\)-axis); \(D_{x6}=-tY^{\prime}X^{\prime}+D_{x5}\)  
at 7: \(s=b\), \(D_{x7}=-tY^{\prime}\mathrm{b}+D_{x5}=0\)  
The \(D_{x}\) shear flow distribution (Figure 12b) is linear for points within the flanges (1-3) and (5-7) and is parabolic in the web (3-5), attaining its maximum at point 4 (the neutral \(x\)-axis). The second term in Equation (18) is the shear flow due to \(F_{x}{}^{\prime}\), which is established within the first moment of area \(D_{y}\) for median points 1, 2, 3...7 in Figure 13a as follows:  
1-3: \(D_{y}=t\int_{s}x\;\mathrm{d}s=t\int_{0}^{s}(a-X^{\prime}-s)\;\mathrm{d}s=t[(a- X^{\prime})s-s^{2}/2]\)  
at 1: \(s=0\), \(D_{y1}=0\)  
at 2 (\(y\)-axis) \(s=a-X^{\prime}\); \(D_{y2}=t(a-X^{\prime})^{2}/2\)  
at 3: \(s=a\), \(D_{y3}=t[(a-X^{\prime})a-a^{2}/2]=ta(a/2-X^{\prime})\)  
3-5: \(D_{y}=t\int_{s}x\;\mathrm{d}s=-tX^{\prime}s+D_{y3}\)  
at 3: \(s=0\), \(D_{y}=D_{y3}=ta(a/2-X^{\prime})\)  
at 4: (\(x\)-axis) \(s=d-Y^{\prime}\); \(D_{y4}=-tX^{\prime}(d-Y^{\prime})+ta(a/2-X^{\prime})\)  
at 5: \(s=d\), \(D_{y5}=-tX^{\prime}d+D_{y3}=-tX^{\prime}d+ta(a/2-X^{\prime})\)  
5-7: \(D_{y}=t\int_{s}x\;\mathrm{d}s=t\int_{s}(s-X^{\prime})\;\mathrm{d}s+D_{y5}=t( s^{2}/2-X^{\prime}s)-tX^{\prime}d+ta(a/2-X^{\prime})\)  
at 5: \(s=0\), \(D_{y}=D_{y5}=-tX^{\prime}d+ta(a/2-X^{\prime})\)  
at 6: (\(y\)-axis) \(s=X^{\prime}\); \(D_{y6}=-tX^{\prime 2}/2-tX^{\prime}d+ta(a/2-X^{\prime})\)  
at 7: \(s=b\), \(D_{y7}=t(b^{2}/2-X^{\prime}b)-tX^{\prime}d+ta(a/2-X^{\prime})=0\)  
Figure 12: (a) Channel dimensions showing (b) shear flow due to \(F_{y}{}^{\prime}\). The \(D_{y}\) shear flow distribution (Figure 13b) is linear in the web (3-5) and parabolic for points within the flanges (1-3) and (5-7), attaining therein their maxima at point 2 and 6 (the neutral \(y\)-axis). To establish the \(D_{x}\) and \(D_{y}\) distributions, the shear flows given above \(q\)'s are written in normalised form \(D/d^{3}\) within Tables 10 and 11 and evaluated for a non-symmetric channel with dimension ratios \(a/d=1/3\), \(b/d=2/3\) and \(t/d=1/40\). Equation (17c,d) give the centroidal co-ordinates \(X^{\prime}/d=5/36\) and \(Y^{\prime}/d=5/12\). Note 1: The conversion from \(D_{x}\) and \(D_{y}\) given in Table 10, to net shear flow in Equation (18) follows from the multiplication involving equivalent forces and second moments of area in Equation (2a,b).