User:Eml4500.f08.delta 6.krueger/Notes Monday November 10th

Continuous Principle of Virtual Work to Discrete Principle of Virtual Work (continued) Now three motivations will be given for the equation being used in the form $$N_{i}(x)+N_{i+1}(x)$$. 1st Motivation: It can be observed from earlier discussion that $$N_{i}(x)+N_{i+1}(x)$$ are linear or represented by straight lines. Therefore the expression for U(x) is linear. This is true because the combination of $$N_{i}(x)$$ and $$N_{i+1}(x)$$ is linear. $$N_{i}(x)$$ and $$N_{i+1}(x)$$ can be written in the typical linear format:

$$N_{i}(x)=\alpha_{i}+\beta_{i}x$$

$$N_{i+1}(x)=\alpha_{i+1}+\beta_{i+1}x$$

where $$\alpha_{i}$$ is the y-intercept of the line and $$\beta_{i}$$ is the slope of the line and they both represent real numbers.

So the linear combination of $$N_{i}(x)$$ and $$N_{i+1}(x)$$ is:

$$N_{i}d_{i}+N_{i+1}d_{i+1}=(\alpha_{i}+\beta_{i}x)d_{i}+(\alpha_{i+1}+\beta_{i+1}x)d_{i+1}$$

Now the like terms are combined to give:

$$U_{x}=(\alpha_{i}d_{i}+\alpha_{i+1}d_{i+1})+(\beta_{i}d_{i}+\beta_{i+1}d_{i+1})x$$

Where $$d_{i}$$ is the weighting coefficient. It can clearly be seen that this is a linear function of x.

2nd Motivation:

Recall the equation for the interpolation of U(x):$$U(x)=N_{i}(x)d_{i}+N_{i+1}(x)d_{i+1}$$

Now replace x with $$x_{i}$$ and the equation becomes:$$U(x)=N_{i}(x_{i})d_{i}+N_{i+1}(x_{i})d_{i+1}$$

In the drawing above it was shown that $$N_{i}(x_{i})d_{i}=1$$ and that $$N_{i+1}(x_{i})d_{i+1}=0$$. The eqaution for $$U(x_{i})$$ becomes:

$$U(x)=d_{i}$$

The equation shows that $$U(x_{i})$$ is equal to the weighting coefficient $$d_{i}$$.