# Deflection of partially loaded cantilever beam with non-homogeneous EI

• elepolli
elepolli
TL;DR Summary: I have a cantilever beam with fixed end, known rectangular cross section and total length h. A uniform load is applied on the beam from a distance L from the fixed end, to the free end. The E modulus and inertia I are known, and they are two different constant values for 0<x<L and L<x<0.
I want to know the deflection w(h) of the beam at the free end.

This is my approach, what do you think?

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For ## 0 < x_1 \leq L_1## you solve

$$\frac{d^2y}{dx_1^2} = \frac{M_1(x_1)}{E_1I_1}$$

Then, for ## L_1 < x_2 \leq L_2## you can solve

$$\frac{d^2y}{dx_2^2} = \frac{M_2(x_2)}{E_2I_2}$$

Where the constants of integration (slope, deflection) at ##x_2 = 0 ## come from the end condition of the first section.

erobz said:

For ## 0 < x_1 \leq L_1## you solve

$$\frac{d^2y}{dx_1^2} = \frac{M_1(x_1)}{E_1I_1}$$

Then, for ## L_1 < x_2 \leq L_2## you can solve

$$\frac{d^2y}{dx_2^2} = \frac{M_2(x_2)}{E_2I_2}$$

Where the constants of integration (slope, deflection) at ##x_2 = 0 ## come from the end condition of the first section.
How should I calculate $M_1$ and $M_2$?

elepolli said:
How should I calculate $M_1$ and $M_2$?
Find the reactions at point A( force and moment). Then you write the internal moment at the end of the section as function of x in the typical way(by inspection, or integrating the shear).

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