User:EML5526.S11.Team3.vnarayanan/Homework4

Problem statement
Given the strong form for the circular bar in torsion:

$$\frac{\mathrm{d} }{\mathrm{d} x}\left [ JG \frac{\mathrm{d}\phi }{\mathrm{d} x} \right ] + m(x) = 0\;\;\;\; 0\leq x\leq 1,$$



Natural Boundary Condition :

$$\;\;\; M(x=1) = \left ( JG \frac{\mathrm{d} \phi }{\mathrm{d} x} \right )_{l} = \bar{M}$$

Essential Boundary Condition :

$$\phi(x=0) = \bar{\phi}$$

where $$m(x)$$ is a distributed moment per unit length, M is the torsion moment, $$\phi$$ is the angle of rotation, G is the shear modulus and J is the polar moment of inertia given by $$J = \frac{\pi C^{4}}{2}$$, where C is the radius of the circular shaft.

Replacing the boundary conditions with the ones stated in problem G1DM1.0 / D1 in Mtg 9-2 the exact solution can be evaluated.

Modified Natural Boundary Condition :

Modified Essential Boundary Condition :

a. Construct the weak form for the circular bar in torsion.

b. Assume that $$m(x) = 0$$ and integrate the differential equation given above. Find the integration constants using boundary conditions.

Solution (a)
As the weight function must vanish on the essential boundaries, we consider all smooth weight functions $$w(x)$$ such that $$w(0)= 0$$.

Multiplying the given differential equation and the natural boundary condition over the domain specified, by an arbitrary weight function:

$$\int_{0}^{1}w(x)\left[ \frac{\mathrm{d} }{\mathrm{d} x}\left ( JG \frac{\mathrm{d} \phi }{\mathrm{d} x} \right ) + m(x) \right]= 0$$

$$w\left[ \left ( JG \frac{\mathrm{d} \phi }{\mathrm{d} x} \right )- \bar{M} \right]_{x=1}= 0$$

Replacing the coefficients and substituting the modified boundary conditions:

$$\int_{0}^{1}w(x)\left[ \frac{\mathrm{d} }{\mathrm{d} x}\left ( (3x + 2) \frac{\mathrm{d} \phi }{\mathrm{d} x} \right ) + 5x \right]dx= 0$$

Now we integrate the Equation 4.4.3 by parts,

Integration by parts:

We have constructed the weight functions so that w(0) =0 therefore, the first term on the RHS of the above vanishes at x = 0. Substituting Equation 4.4.5 in Equation 4.4.1,

Substituting Equation 4.4.4 in Equation 4.4.6,

Solution (b)
Assuming m(x) = 0, reduces the given strong form as follows:

Substituting for the coefficient,

Integrating with respect to x,

From Equation 4.4.4 ,

Hence,

Integrating Equation 4.4.9 with respect to x,

$$\phi = 4log(3x+2)+ C_{2}$$

From Equation 4.4.2,

$$\phi(x=0) = \bar{\phi} = 4$$

$$\therefore\;\;C_{2} =4 - 4log(2) = 1.228$$

Hence,