Question 15.12: Determine the horizontal and vertical components of the tip ...

From Eqs. (15.28),

\left\{\begin{array}{l}u^{\prime \prime} \\v^{\prime \prime}\end{array}\right\}=\frac{-1}{E\left(I_{x x} I_{y y}-I_{x y}^{2}\right)}\left[\begin{array}{cc}-I_{x y} & I_{x x} \\I_{y y} & -_{xy}\end{array}\right]\left\{\begin{array}{l}M_{x} \\M_{y}\end{array}\right\}  (15.28)

u^{\prime \prime}=\frac{M_{x} I_{x y}-M_{y} I_{x x}}{E\left(I_{x x} I_{y y}-I_{x y}^{2}\right)}  (i)

In this case, M_{x}=W(L- z ), M_{y}=0, so that Eq. (i) simplifies to

u^{\prime \prime}=\frac{W I_{x y}}{E\left(I_{x x} I_{y y}-I_{x y}^{2}\right)}(L-z)  (ii)

Integrating Eq. (ii) with respect to z,

u^{\prime}=\frac{W I_{x y}}{E\left(I_{x x} I_{y y}-I_{x y}^{2}\right)}\left(L z-\frac{z^{2}}{2}+A\right)  (iii)

and

u=\frac{W I_{x y}}{E\left(I_{x x} I_{y y}-I_{x y}^{2}\right)}\left(L \frac{z^{2}}{2}-\frac{z^{3}}{6}+A z+B\right)  (iv)

in which u^{\prime} denotes du/dz and the constants of integration A and B are found from the boundary conditions, namely, u^{\prime}=0 and u = 0 when z = 0. From the first of these and Eq. (iii), A = 0, while from the second and Eq. (iv), B = 0. Hence, the deflected shape of the beam in the xz plane is given by

u=\frac{W I_{x y}}{E\left(I_{x x} I_{y y}-I_{x y}^{2}\right)}\left(L \frac{z^{2}}{2}-\frac{z^{3}}{6}\right)  (v)

At the free end of the cantilever (z = L), the horizontal component of deflection is

u_{ f . e .}=\frac{W I_{x y} L^{3}}{3 E\left(I_{x y} I_{y y}-I_{x y}^{2}\right)}  (vi)

Similarly, the vertical component of the deflection at the free end of the cantilever is

v_{ f . e .}=\frac{-W I_{y y} L^{3}}{3 E\left(I_{x x} I_{y y}-I_{x y}^{2}\right)}  (vii)

The actual deflection \delta_{\text {f.e. }} at the free end is then given by

at an angle of \tan ^{-1} u_{\text {f.e. }} / v_{\text {f.e. }} to the vertical. Note that, if either Cx or Cy were an axis of symmetry, I_{x y}=0 and Eqs. (vi) and (vii) reduce to

the well-known results for the bending of a cantilever having a symmetrical cross-section and carrying a concentrated vertical load at its free end (see Example 15.5).