User:Eml4500.f08.team.guan/hw61110

Continuous PVW to Discrete PVW
To continue examining the principle of virtual work along with the continuous finite element method, we will examine the motivation for the form of $$\displaystyle N_i(x)$$ and $$\displaystyle N_{i+1}(x)$$. The first reason for this motivation is as follows. 1) $$\displaystyle N_i(x)$$ and $$\displaystyle N_{i+1}(x)$$ are linear (straight line representations), thus any linear combination of $$\displaystyle N_i(x)$$ and $$\displaystyle N_{i+1}(x)$$ is also linear and in particular the expression for u(x) (31.3). The N functions are shape or interpolation functions. They are functions multiplied to the axial displacements at the local nodes of the element to describe the varying axial displacement along the length of the element. For this simple element, the varying displacement can be linear. Thus, we have the following shape functions. $$\displaystyle N_i(x)=\alpha_i+\beta_ix$$ where $$\displaystyle \alpha_i$$ and $$\displaystyle \beta_i$$ are numbers. Likewise, we have the shaping function of the following node as... $$\displaystyle N_{i+1}(x)=\alpha_{i+1}+\beta_{i+1}x$$ where $$\displaystyle \alpha_{i+1}$$ and $$\displaystyle \beta_{i+1}$$ are numbers. The linear combination of $$\displaystyle N_i(x)$$ and $$\displaystyle N_{i+1}(x)$$is also a linear function as the sum of linear functions result in linear functions. We have the following. $$\displaystyle N_id_i+N_{i+1}d_{i+1}=(\alpha_i+\beta_ix)d_i+(\alpha_{i+1}+\beta_{i+1}x)d_{i+1}=(\alpha_id_i+\alpha{i+1}d_{i+1})+(\beta_id_i+\beta_{i+1}d{i+1})x $$ It is now obvious the resulting equation is a linear one. The $$d_i$$ and $$d_{i+1}$$ are essentially weighting coefficients. The second reason for the motivation is described next 2) Recall the equation for u(x) (interpolation of u(x)) (31.3): $$\displaystyle u(x)=N_i(x)d_i+N_{i+1}d_{i+1}$$ Recall the figure from 31.4 and replacing x with x_i, we have...  $$\displaystyle u(x_i)=N_i(x_i)d_i+N_{i+1}(x_i)d_{i+1}=(1)d_i+(0)d_{i+1}=d_i$$ $$\displaystyle u(x_i)=d_i$$ That is what we set out to do.