$$V(X) = -w (X) + \frac{w L}{2}$$

Where:

$X = x-x_0$
$x$ - is relative to the end of the beam
$x_0$ - is the absolute position

$$V(X) = -w X + \frac{w L}{2}$$$$M(X) = -\frac{w}{2} X^2 + \frac{w L}{2} X$$$$M(X) = \frac{w}{2}\left(-X^2 + L X\right)$$$$\theta(X) = \frac{-w}{2 E I}\left(-\frac{X^3}{3} +L \frac{X^2}{2} - C\right)$$$$\theta(X) = \frac{w}{2 E I}\left(\frac{X^3}{3} - L \frac{X^2}{2} + C\right)$$$$\Delta(X) = \frac{w}{2 E I}\left(\frac{X^4}{12} - L \frac{X^3}{6} +C X + D\right)$$$$\Delta(0) = \frac{w}{2 E I}\left(\frac{0^4}{12} - L \frac{0^3}{6} +C\cdot 0 + D\right) = y_0$$$$\frac{w}{2 E I} D = y_0$$$$\text{therefore:}\quad D = \frac{ y_0 2 E I}{w}$$$$\Delta(L) = \frac{w}{2 E I}\left(\frac{L^4}{12} - L \frac{L^3}{6} +C \cdot L + \frac{2 y_0 E I}{w}\right) = y_0$$$$\frac{w}{2 E I}\left(\frac{L^4}{12} - L \frac{L^3}{6} +C \cdot L\right) + y_0 = y_0$$$$ \frac{w}{2 E I}\left(\frac{L^4}{12} - L \frac{L^3}{6} +C \cdot L \right) = 0$$$$ \frac{L^4}{12} - L \frac{L^3}{6} +C \cdot L = 0$$$$ \frac{L^3}{12} - \frac{L^3}{6} +C = 0$$$$C = \frac{L^3}{6} - \frac{L^3}{12}$$$$C = \frac{L^3}{12}$$

$$\Delta(X) = \frac{w}{2 E I}\left(\frac{X^4}{12} - L \frac{X^3}{6} +\frac{L^3}{12} X + \frac{2 y_0 E I}{w}\right)$$$$X = x-x_0$$$$\Delta(x) = \frac{w}{2 E I}\left(\frac{(x-x_0)^4}{12} - L \frac{(x-x_0)^3}{6} +\frac{L^3}{12} (x-x_0) + \frac{2 y_0 E I}{w}\right)$$



In [11]:

    
from sympy import symbols, collect, expand, latex, simplify
D, x, x_0, E, I, w, y_0, L, y, t = symbols('Delta x x_0 E I w y_0 L y t')
from sympy import init_printing 
init_printing(use_unicode=True)

D = w/(2*E*I)*((x-x_0)**4/12-L*(x-x_0)**3/6+L**3/12*(x-x_0)+2*y_0*E*I/w)
D









    Out[11]:





$$\frac{w}{2 E I} \left(\frac{2 E}{w} I y_{0} + \frac{L^{3}}{12} \left(x - x_{0}\right) - \frac{L}{6} \left(x - x_{0}\right)^{3} + \frac{1}{12} \left(x - x_{0}\right)^{4}\right)$$



In [2]:

    
E = expand(D)
E









    Out[2]:





$$y_{0} + \frac{L^{3} w x}{24 E I} - \frac{L^{3} w x_{0}}{24 E I} - \frac{L w x^{3}}{12 E I} + \frac{L w x^{2} x_{0}}{4 E I} - \frac{L w x x_{0}^{2}}{4 E I} + \frac{L w x_{0}^{3}}{12 E I} + \frac{w x^{4}}{24 E I} - \frac{w x^{3} x_{0}}{6 E I} + \frac{w x^{2} x_{0}^{2}}{4 E I} - \frac{w x x_{0}^{3}}{6 E I} + \frac{w x_{0}^{4}}{24 E I}$$



In [3]:

    
F = collect(E,x)
F









    Out[3]:





$$x^{3} \left(- \frac{L w}{12 E I} - \frac{w x_{0}}{6 E I}\right) + x^{2} \left(\frac{L w x_{0}}{4 E I} + \frac{w x_{0}^{2}}{4 E I}\right) + x \left(\frac{L^{3} w}{24 E I} - \frac{L w x_{0}^{2}}{4 E I} - \frac{w x_{0}^{3}}{6 E I}\right) + y_{0} - \frac{L^{3} w x_{0}}{24 E I} + \frac{L w x_{0}^{3}}{12 E I} + \frac{w x^{4}}{24 E I} + \frac{w x_{0}^{4}}{24 E I}$$



In [4]:

    
latex(F)









    Out[4]:





'x^{3} \\left(- \\frac{L w}{12 E I} - \\frac{w x_{0}}{6 E I}\\right) + x^{2} \\left(\\frac{L w x_{0}}{4 E I} + \\frac{w x_{0}^{2}}{4 E I}\\right) + x \\left(\\frac{L^{3} w}{24 E I} - \\frac{L w x_{0}^{2}}{4 E I} - \\frac{w x_{0}^{3}}{6 E I}\\right) + y_{0} - \\frac{L^{3} w x_{0}}{24 E I} + \\frac{L w x_{0}^{3}}{12 E I} + \\frac{w x^{4}}{24 E I} + \\frac{w x_{0}^{4}}{24 E I}'

$$\Delta(x) = x^{3} \left(- \frac{L w}{12 E I} - \frac{w x_{0}}{6 E I}\right) + x^{2} \left(\frac{L w x_{0}}{4 E I} + \frac{w x_{0}^{2}}{4 E I}\right) + x \left(\frac{L^{3} w}{24 E I} - \frac{L w x_{0}^{2}}{4 E I} - \frac{w x_{0}^{3}}{6 E I}\right) + y_{0} - \frac{L^{3} w x_{0}}{24 E I} + \frac{L w x_{0}^{3}}{12 E I} + \frac{w x^{4}}{24 E I} + \frac{w x_{0}^{4}}{24 E I}$$

$$\Delta(x) = \frac{w}{2 EI}\left(\frac{x^{4}}{12} - \left(\frac{L}{6 } + \frac{x_{0}}{3}\right)x^{3} + \left(\frac{L x_{0}}{2} + \frac{x_{0}^{2}}{2}\right) x^{2} + \left(\frac{L^{3}}{12} - \frac{L x_{0}^{2}}{2} - \frac{x_{0}^{3}}{3}\right) x \right) +\frac{w}{2 E I}\left(- \frac{L^{3} x_{0}}{12} + \frac{L x_{0}^{3}}{6} + \frac{ x_{0}^{4}}{12}\right)+ y_{0}$$

$$\Delta(x) = \frac{w}{2 EI}\left(\frac{x^{4}}{12} - \frac{1}{3}\left(\frac{L}{2} + x_{0}\right)x^{3} + \frac{1}{2}\left(L x_{0} + x_{0}^{2}\right) x^{2} + \left(\frac{L^{3}}{12} - \frac{L x_{0}^{2}}{2} - \frac{x_{0}^{3}}{3}\right) x \right) +\frac{w}{2 E I}\frac{1}{6}\left(- \frac{L^{3} x_{0}}{2} + L x_{0}^{3} + \frac{ x_{0}^{4}}{2}\right)+ y_{0}$$

The following was not used due to another approach being found.

The following calculations were done in an effort to determine what the loacation of the supports needed to be if they did not move in the drawing when the beam deflects. This is necessary due to the simplifiaction made where the ends of the beam centerline do not move when the bem deflects.



In [8]:

    
G = simplify(F)
G









    Out[8]:





$$\frac{1}{24 E I} \left(24 E I y_{0} - L^{3} w x_{0} + 2 L w x_{0}^{3} + w x^{4} - 2 w x^{3} \left(L + 2 x_{0}\right) + 6 w x^{2} x_{0} \left(L + x_{0}\right) - w x \left(- L^{3} + 6 L x_{0}^{2} + 4 x_{0}^{3}\right) + w x_{0}^{4}\right)$$

$$\Delta(x)=\frac{1}{24 E I} \left(24 E I y_{0} - L^{3} w x_{0} + 2 L w x_{0}^{3} + w x^{4} - 2 w x^{3} \left(L + 2 x_{0}\right) + 6 w x^{2} x_{0} \left(L + x_{0}\right) - w x \left(- L^{3} + 6 L x_{0}^{2} + 4 x_{0}^{3}\right) + w x_{0}^{4}\right)$$$$\theta(x) =\frac{1}{24 E I} \left(4 w x^{3} - 6 w x^{2} \left(L + 2 x_{0}\right) + 12 w x x_{0} \left(L + x_{0}\right) - w \left(- L^{3} + 6 L x_{0}^{2} + 4 x_{0}^{3}\right) \right)$$

$$\theta(x) = \frac{1}{24 E I} \left(4 w x^{3} - 6 w x^{2} \left(L + 2 x_{0}\right) + 12 w x x_{0} \left(L + x_{0}\right) - w \left(- L^{3} + 6 L x_{0}^{2} + 4 x_{0}^{3}\right) \right)$$$$y_s = y_0 +(y - y_0) +\sqrt{\frac{t^2}{4} - (x-x_0)^2}$$$$y_s = \frac{-1}{\theta(x)}(x_0-x) + y$$$$ 0 = y_0 +(y - y_0) +\sqrt{\frac{t^2}{4} - (x-x_0)^2} - \left( \frac{-1}{\theta(x)}(x_0-x) + y \right)$$$$ 0 = y_0 +y - y_0 +\sqrt{\frac{t^2}{4} - (x-x_0)^2} + \frac{1}{\theta(x)}(x_0-x) - y $$$$ 0 = \sqrt{\frac{t^2}{4} - (x-x_0)^2} + \frac{x_0-x}{\theta(x)} $$



In [33]:

    
from sympy import diff,sqrt, roots
T = diff(G,x)
T









    Out[33]:





$$\frac{1}{24 E I} \left(4 w x^{3} - 6 w x^{2} \left(L + 2 x_{0}\right) + 12 w x x_{0} \left(L + x_{0}\right) - w \left(- L^{3} + 6 L x_{0}^{2} + 4 x_{0}^{3}\right)\right)$$



In [26]:

    
H = sqrt(t**2/4-(x-x_0)**2)*T #+ (x-x_0)
J = H**2



In [32]:

    
K = collect(expand((J - (x_0-x)**2)/(w**2/(12*E**2*I**2))),x)
K









    Out[32]:





$$- \frac{12 E^{2}}{w^{2}} I^{2} x_{0}^{2} + \frac{L^{6} t^{2}}{192} - \frac{L^{6} x_{0}^{2}}{48} - \frac{L^{4} t^{2}}{16} x_{0}^{2} + \frac{L^{4} x_{0}^{4}}{4} - \frac{L^{3} t^{2}}{24} x_{0}^{3} + \frac{L^{3} x_{0}^{5}}{6} + \frac{3 L^{2}}{16} t^{2} x_{0}^{4} - \frac{3 L^{2}}{4} x_{0}^{6} + \frac{L t^{2}}{4} x_{0}^{5} - L x_{0}^{7} + \frac{t^{2} x_{0}^{6}}{12} - \frac{x^{8}}{3} + x^{7} \left(L + \frac{8 x_{0}}{3}\right) + x^{6} \left(- \frac{3 L^{2}}{4} - 7 L x_{0} + \frac{t^{2}}{12} - \frac{28 x_{0}^{2}}{3}\right) + x^{5} \left(- \frac{L^{3}}{6} + \frac{9 x_{0}}{2} L^{2} - \frac{L t^{2}}{4} + 21 L x_{0}^{2} - \frac{t^{2} x_{0}}{2} + \frac{56 x_{0}^{3}}{3}\right) + x^{4} \left(\frac{L^{4}}{4} + \frac{5 x_{0}}{6} L^{3} + \frac{3 L^{2}}{16} t^{2} - \frac{45 L^{2}}{4} x_{0}^{2} + \frac{5 L}{4} t^{2} x_{0} - 35 L x_{0}^{3} + \frac{5 t^{2}}{4} x_{0}^{2} - \frac{70 x_{0}^{4}}{3}\right) + x^{3} \left(- L^{4} x_{0} + \frac{L^{3} t^{2}}{24} - \frac{5 L^{3}}{3} x_{0}^{2} - \frac{3 x_{0}}{4} L^{2} t^{2} + 15 L^{2} x_{0}^{3} - \frac{5 L}{2} t^{2} x_{0}^{2} + 35 L x_{0}^{4} - \frac{5 t^{2}}{3} x_{0}^{3} + \frac{56 x_{0}^{5}}{3}\right) + x^{2} \left(- \frac{12 E^{2}}{w^{2}} I^{2} - \frac{L^{6}}{48} - \frac{L^{4} t^{2}}{16} + \frac{3 L^{4}}{2} x_{0}^{2} - \frac{L^{3} x_{0}}{8} t^{2} + \frac{5 L^{3}}{3} x_{0}^{3} + \frac{9 L^{2}}{8} t^{2} x_{0}^{2} - \frac{45 L^{2}}{4} x_{0}^{4} + \frac{5 L}{2} t^{2} x_{0}^{3} - 21 L x_{0}^{5} + \frac{5 t^{2}}{4} x_{0}^{4} - \frac{28 x_{0}^{6}}{3}\right) + x \left(\frac{24 x_{0}}{w^{2}} E^{2} I^{2} + \frac{L^{6} x_{0}}{24} + \frac{L^{4} x_{0}}{8} t^{2} - L^{4} x_{0}^{3} + \frac{L^{3} t^{2}}{8} x_{0}^{2} - \frac{5 L^{3}}{6} x_{0}^{4} - \frac{3 L^{2}}{4} t^{2} x_{0}^{3} + \frac{9 L^{2}}{2} x_{0}^{5} - \frac{5 L}{4} t^{2} x_{0}^{4} + 7 L x_{0}^{6} - \frac{t^{2} x_{0}^{5}}{2} + \frac{8 x_{0}^{7}}{3}\right) - \frac{x_{0}^{8}}{3}$$



In [43]:

    
c = -12*E**2/w**2*I**2*x_0**2 + L**6*t**2/192 - L**6*x_0**2/48 -L**4*t**2/16*x_0**2\
    +L**4*x_0**4/4 - L**3*t**2/24*x_0**3 + L**3*x_0**5/6 + 3*L**2/16*t**2*x_0**4 \
    -3*L**2/4*x_0**6 + L*t**2/4*x_0**5 - L*x_0**7 + t**2*x_0**6/12 - x_0**8/3
K-c









    Out[43]:





$$- \frac{x^{8}}{3} + x^{7} \left(L + \frac{8 x_{0}}{3}\right) + x^{6} \left(- \frac{3 L^{2}}{4} - 7 L x_{0} + \frac{t^{2}}{12} - \frac{28 x_{0}^{2}}{3}\right) + x^{5} \left(- \frac{L^{3}}{6} + \frac{9 x_{0}}{2} L^{2} - \frac{L t^{2}}{4} + 21 L x_{0}^{2} - \frac{t^{2} x_{0}}{2} + \frac{56 x_{0}^{3}}{3}\right) + x^{4} \left(\frac{L^{4}}{4} + \frac{5 x_{0}}{6} L^{3} + \frac{3 L^{2}}{16} t^{2} - \frac{45 L^{2}}{4} x_{0}^{2} + \frac{5 L}{4} t^{2} x_{0} - 35 L x_{0}^{3} + \frac{5 t^{2}}{4} x_{0}^{2} - \frac{70 x_{0}^{4}}{3}\right) + x^{3} \left(- L^{4} x_{0} + \frac{L^{3} t^{2}}{24} - \frac{5 L^{3}}{3} x_{0}^{2} - \frac{3 x_{0}}{4} L^{2} t^{2} + 15 L^{2} x_{0}^{3} - \frac{5 L}{2} t^{2} x_{0}^{2} + 35 L x_{0}^{4} - \frac{5 t^{2}}{3} x_{0}^{3} + \frac{56 x_{0}^{5}}{3}\right) + x^{2} \left(- \frac{12 E^{2}}{w^{2}} I^{2} - \frac{L^{6}}{48} - \frac{L^{4} t^{2}}{16} + \frac{3 L^{4}}{2} x_{0}^{2} - \frac{L^{3} x_{0}}{8} t^{2} + \frac{5 L^{3}}{3} x_{0}^{3} + \frac{9 L^{2}}{8} t^{2} x_{0}^{2} - \frac{45 L^{2}}{4} x_{0}^{4} + \frac{5 L}{2} t^{2} x_{0}^{3} - 21 L x_{0}^{5} + \frac{5 t^{2}}{4} x_{0}^{4} - \frac{28 x_{0}^{6}}{3}\right) + x \left(\frac{24 x_{0}}{w^{2}} E^{2} I^{2} + \frac{L^{6} x_{0}}{24} + \frac{L^{4} x_{0}}{8} t^{2} - L^{4} x_{0}^{3} + \frac{L^{3} t^{2}}{8} x_{0}^{2} - \frac{5 L^{3}}{6} x_{0}^{4} - \frac{3 L^{2}}{4} t^{2} x_{0}^{3} + \frac{9 L^{2}}{2} x_{0}^{5} - \frac{5 L}{4} t^{2} x_{0}^{4} + 7 L x_{0}^{6} - \frac{t^{2} x_{0}^{5}}{2} + \frac{8 x_{0}^{7}}{3}\right)$$