User:Eml4500.f08.delta 6.krueger/Notes Monday October 27th

Principle of Virtual Work (PVW)

First the weighting coefficient needs to be defined, or reinforced, and an example needs to be given to emphasize its use and importance. We will let w equal the weighting coefficient.

w = weighting coefficient

Now an Example:

Lets look at the calculation of a final course grade. Where: Exam 1 = 82 and is worth 15 % of final grade Exam 2 = 74 and is worth 20 % of final grade Exam 3 = 92 and is worth 25 % of final grade Hw grade = 87 and is worth 40 % of final grade

The formula to calculate the final grade is given as:

$$ finalgrade = w_0(Hw grade) + \sum_{i=1}^{3}w_iExam_i$$

Where $$ w_i$$ and $$ w_0$$ are the exam weighting coefficients and the homework(Hw) weighting coefficient. Now plugging into the above formula:

$$ finalgrade = .4(87) + .15(82) + .2(74) + .25(92) = 84.9$$

Now to emphasize how the weighting coefficients will change the final value, we'll let the Hw grade and all the exam grades remain the same but change the worth to the following: Exam 1 worth 30 % Exam 2 worth 30 % Exam 3 worth 30 % Hw grade worth 10 %

The final grade now becomes:

$$ finalgrade = .1(87) + .3(82) + .3(74) + .3(92) = 83.1$$

This shows the importance of the weighting coefficients. The importance of a value, within an equation, can be changed by changing the weighting coefficient of that value. The final value will also be changed, as shown above, by changing the weighting coefficients.

Now we will return to the Principle of Virtual Work. Recall: $$K^{(e)}= T^{(e)^(T)} * \hat{K}^{(e)} * T^{(e)}$$

Where $$K^{(e)}$$ is a 4x4 matrix, $$T^{(e)^(T)}$$ is a 4x2 matrix, $$\hat{K}^{(e)}$$ is a 2x2 matrix and $$T^{(e)}$$ is a 2x4 matrix.

Question: How do we use the Principle of Virtual Work?

First Recall the Force Displacement Relationship (FD Rel) with respect to the axial degrees of freedom ($$q^{(e)}$$):

Eqn (1)$$\hat{K}^{(e)}q^{(e)} = p^{(e)}$$

Take note that the matrix p is lower case. This implies it is in reference to the element not the global coordinate system.

We then move the p matrix to the left hand side of the equation. Equation (1) is now in the same form as the FD Rel $$Kd-F=0$$.

$$\hat{K}^{(e)}q^{(e)} - p^{(e)}= 0$$

The equation is now multiplied by the weighting coefficient, which is a 2x2 matrix, to give the Principle of Virtual Work:

Eqn (2)$$w\cdot(\hat{K}^{(e)}q^{(e)} - p^{(e)})= 0$$

This holds true for all weighting coefficient matrices. In previous notes and home work we have shown that $$ Eqn(1) \Leftrightarrow Eqn(2)$$. We can ow exploit the Principle of Virtual Work in order to get expressions in the global coordinate system. Recall: Eqn (3)$$q^{(e)}=T^{(e)}d^{(e)}$$ Where $$q^{(e)}$$ is the real displacement and is a 2x1 matrix. $$T^{(e)}$$ is a 2x4 matrix and $$d^{(e)}$$ is a 4x1 matrix. $$w$$ corresponds to the axial degrees of freedom. Since we are using virtual work the axial weighting coefficient needs to be converted into the virtual weighting coefficient. This is done using Eqn (4) below: Eqn (4)$$\hat{w}=T^{(e)}w$$ Where $$\hat{w}$$ is a 2x1 matrix, $$T^{(e)}$$ is a 2x4 matrix and $$w$$ is a 4x1 matrix.