User:Eml5526.s11.team3.sahin/Homework 310

=Problem3.10 Quadratic Trial Solution of Weak Form =

Problem Statement
Refer to lecture notes 17.2

This problem is extracted Problem 3.4 from the textbook of Fish J., and Belytschko T.,

Given
Problem 3.1 given in Fish and Belytschko

The strong form is

The weak form is

Find
Consider a trial (candidate) solution of the form $$u\left(x\right)=\alpha_0+\alpha_1(x-3)+\alpha_2(x-3)^2$$ and a weight function of the same form.

1) Obtain a solution to the weak form in Problem 3.1.

2) Check the equilibrium equation in the strong form in Problem 3.1; is it satisfied?

3) Check the natural boundary condition; is it satisfied?

1)
The trial solutions $$u\left(x\right)$$ must satisfy the essential boundary condition $$u\left(x=3\right)= 0.001$$ so Therefore, it is weight function $$w\left(x=3\right)= 0$$ as the weight function must vanish on the essential boundaries

To find a trial solution, we need to put Eq 10.3 and Eq 10.4 into the weak form Eq 10.2

Evaluating the integrals with the assumption of cross area $$ \begin{align} & \qquad A \end{align} $$ and Young’s modulus $$ \begin{align} & \qquad E \end{align} $$ are constants

LHS was integrated by WolframMathematica and $$\begin{align} \qquad \quad \quad \quad \quad \end{align} $$ RHS was integrated by WolframMathematica

and factoring out $$ \begin{align} & \qquad \beta_1 \end{align} $$ and $$ \begin{align} & \qquad \beta_2 \end{align} $$  gives As it must hold for all $$ \begin{align} & \qquad \beta_1\end{align} $$ and $$ \begin{align} & \qquad \beta_2\end{align} $$ because of weight functions, the term in the parentheses must vanish, so So we obtain linear algebraic equation in $$ \begin{align} & \qquad \alpha_1 \end{align} $$ and $$ \begin{align} & \qquad \alpha_2 \end{align} $$:

Matlab Code: The solution of linear algebraic equation has been calculated above codes in Matlab,

With rearranging the both equations, then we get $$ \begin{align} & \qquad \alpha_1 \end{align} $$ and $$ \begin{align} & \qquad \alpha_2 \end{align} $$

Substituting this result into Eq 10.3 gives quadratic trial solution of the weak solution,

2)
We know that To check the equilibrium equation in the strong form in Problem 3.1, we need to substitute Eq 10.14 into Eq 10.1a

3)
We know that the natural boundary condition is

To check the natural boundary condition; we need to substitute Eq 10.14 into Eq 10.1b

Problem solved by sahin and Ushnish