# Homework Help: Tapered Cantilever beam

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1. Apr 15, 2015

### Timisoarian

Hey all,
So I have this beam problem and I honestly am lost with it!
The beam is I-section cantilever, tapers in depth along its length (See picture attached) with a point load at the free end. Im trying to figure out the deflection formula and how to derive it! Also, when calculating its bending stress and shear stress, is it the same way as a normal beam?
Please guys, any help is appreciated it as im completely lost!!!

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2. Apr 15, 2015

### SteamKing

Staff Emeritus
There probably won't be a simple deflection formula. Since the moment of inertia of the beam varies with length, deflections will probably best be calculated using the double integration method:

$θ(x) = \int^x_0 \frac{M(x)}{EI(x)} dx + C_1$

$δ(x) = \int^x_0 θ(x) dx + C_2$

where:

θ(x) - slope of the beam
δ(x) - deflection of the beam
I(x) - moment of inertia of the beam, as a function of the length
E - Young's modulus for the beam material

C1 and C2 - constants of integration; determined by applying the boundary conditions at the fixed end, i.e. θ(0) = δ(0) = 0.

Even if you can determine the moment of inertia I(x) as a function of x, you probably won't get simple functions for M(x)/I(x) to integrate. You may have to use a numerical integration method to obtain θ and δ.

The shear force and bending moment diagrams are calculated based on the loading of the beam only. The taper does not come into play.

The regular formulas for bending stress and shear stress still apply ... you do have to calculate the section properties of the beam at the location where you want to determine the stresses. Unlike a prismatic beam, if you change the location of where the stresses are calculated, you must re-calculate the section properties at that new location.

3. Apr 15, 2015

### PhanthomJay

For the statically determinate beam, bending moments and shear forces at any point along the beam are the same as a non tapered beam, but bending and shear stresses will depend upon the geometric properties of the beam cross section at the point in question. For deflection, since I is non-uniform, you'll have to do the calculus using one of those deflection equations. Like $\int{( mM/EI) }dx$ .

4. Apr 15, 2015

### Staff: Mentor

Two threads merged, and moved to homework.