User:Eml5526.s11.team2.brenner/hwk5

=Problem 5.3: Torsion of a Bar = Modified version of HW 4.4 defined on page 2 of Lecture 21 from Fish & Belytschko Page 73 Problem 3.7

Given
Making use of the data model G1DM1.0/D1b defined on page 2 of Lecture 26 which is a modified version of data model G1DM1.0/D1 defined on page 2 of Lecture 9 where the following conditions are provided.

Where the Weak Form for a bar in torsion can be defined as

Make use of the Lagrange Interpolation Basis Function (LIBF) with uniform discretization (equidistant nodes) using $$ \displaystyle m = 4,6,8   $$ where the LIBF can defined as follows

Find
a) Explain how Langrange Interpolation Basis Functions (LIBF) are used as Constraint Breaking Solutions

b) Plot all LIBF used

c) Use matlab quad function or WolframAlpha.com to integrate

d) Plot approximate vs exact solution and convergence error versus equidistant nodes (m)

(a)
Langrange Interpolation Basis Functions satisfy the condition of the constraint breaking solution by assuming a value of zero at every node except for the node where the LIBF is defined there it assumes a value of 1. It acts like the Kronecker Delta

(b)
The following is the LIBF for the $$ \displaystyle m = 4 $$

The following is the LIBF for the $$ \displaystyle m = 6 $$

The following is the LIBF for the $$ \displaystyle m = 8 $$

Matlab Code
=Problem 4.4: Torsion of a Bar Fish & Belytschko Page 73 Problem 3.7 =

Given
Given the strong form for the circular bar in torsion (Fig. 1):

Where the natural boundary condition is

Where the essential boundary condition is



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

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

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

(a)
Multiplying (4.4.1) by an arbitrary weighting function $$ \displaystyle w(x) $$ and integrating with respect to x across the domain leads to the following form

Making use of Integration by Parts on the first term of (4.4.4) leads to the following equation.

Using the requirement of the essential boundary condition on the weighting function satisfying the essential boundary location implies that $$ \displaystyle w(0) = 0 $$ and also using the natural boundary condition (4.4.2) $$ JG \frac{\partial \phi}{\partial x}_{x=l} = M $$ leads to the following

Therefore the weak form can be represented finding a $$ \displaystyle \phi(x) $$ such that for all smooth $$ \displaystyle \phi(x) $$ with $$ \displaystyle \phi(0) = 0 $$ (4.4.7) is satisfied.

(b)
Applying equation (4.4.1) and with $$ \displaystyle m(x) = 0 $$ leads to the following form.

Integrating (4.4.8) with respect to x generates a constant of integration and provides the following form.

Applying the natural boundary condition from (4.4.2) leads to the following

Solving (4.4.9) for $$ \displaystyle \frac{\partial \phi}{\partial x} $$ and substituting in $$ \displaystyle C_1 = -M $$ leads to the following

Integrating (4.4.11) with respect to x generates a constant of integration and provides the following form, which is the well known equation for angle of twist of a bar.

Applying the essential boundary condition from (4.4.3) to (4.4.12) leads to the following

Therefore the constants and the angle of twist are as follows.