Analytical solution for bending stiffness of tapered tube

[Sud89]

How to derive the formula to find the bending stiffness of an isotropic tapered tube which is cantilevered with a point load applied at the free end?

 

[SteamKing]

I'm not sure that there is an analytical formula.

You can always go back to first principles and the Euler-Bernoulli beam equation.

If w(x) = the deflection of the beam at a distance x from the fixed end, then

M(x) = -E I(x) d²w(x)/dx²

where M(x) is the bending moment,
E is the modulus of elasticity, and
I(x) is the second moment of area of the beam cross section at a distance x from the fixed end

For this beam, at the fixed end both the slope and deflection will be zero.

If you can write I(x) as a function of x, you might be able to integrate the moment equation twice and apply the initial conditions to determine the constants of integration. There's no guarantee that the resulting integral can be determined analytically, although a numerical solution would probably be more practical.

 

[Sud89]

I'm not sure that there is an analytical formula.

You can always go back to first principles and the Euler-Bernoulli beam equation.

If w(x) = the deflection of the beam at a distance x from the fixed end, then

M(x) = -E I(x) d²w(x)/dx²

where M(x) is the bending moment,
E is the modulus of elasticity, and
I(x) is the second moment of area of the beam cross section at a distance x from the fixed end

For this beam, at the fixed end both the slope and deflection will be zero.

If you can write I(x) as a function of x, you might be able to integrate the moment equation twice and apply the initial conditions to determine the constants of integration. There's no guarantee that the resulting integral can be determined analytically, although a numerical solution would probably be more practical.

This is the model of the tube that I have. I need to derive a common formula in order to find the bending stiffness at different distances. say x=1,2,3..

 

[SteamKing]

At x = 0, let the external radius of the tube be RO and the internal radius be RI

For the values of RO and RI at a distance x from the fixed end then,

RO(x) = RO - x * tan α
RI (x) = RI - x * tan α

The second moment of area of the tube is then

I(x) = (π/4)*[RO(x)⁴ - RI(x)⁴]

and the bending stiffness = E I(x)

If you want to substitute the first expressions for RO(x) and RI(x) into I(x), well, it's probably a lot of algebra to clean up.