User:Eml4500.f08.a-team.melvin/Lecture 38

PVW for beams:  $$ \int_{0}^{b} w(\underbrace{x}_{\tilde{x}}) [-\frac{\delta^2}{\delta x^2} ((EI) \frac{\delta^2 v}{\delta x^2}) +ft-m \ddot{v}] \  dx = 0$$ for all possible w(x)

Integration by parts of 1st term

$$ \alpha = \int_{0}^{L} \underbrace{w(x)}_{g(x)} \frac{\delta^2}{\delta x^2}{(EI) \frac{\delta^2 v}{\delta x^2}} \ dx $$ $$ \underbrace{\frac{\delta}{\delta x} (\underbrace{\frac{\delta}{\delta x}(EI) \frac{\delta^2 v}{\delta x^2}}_{r(x)})}_{r'(x)} $$

$$ \alpha = [\underbrace{w \frac{\delta}{\delta x} {(EI) \frac{\delta^2 v}{\delta x^2}}}_{\beta_{1}}]^{2}_{0} - \int_{0}^{L} \underbrace{\frac{dw}{dx}}_{g'(x)} \underbrace{\frac{\delta}{\delta x} {(EI) \frac{\delta^2 v}{\delta x^2}}}_{r(x)} \ dx$$

$$ \alpha = \beta_{1} - [\underbrace{w \frac{\delta}{\delta x} {(EI) \frac{\delta^2 v}{\delta x^2}}}_{\beta_{2}}]^{2}_{0} + \underbrace{\int_{0}^{L} \frac{\delta^2 w}{\delta x^2} (EI) \frac{\delta^2 v}{\delta x^2} \ dx}_{\gamma}$$

Note : symmetric

Eq. (1) becomes: $$ - \beta_{1} + \beta_{2} - \gamma + \int_{0}^{L} wft \ dx - \int_{0}^{L} wm \ddot{v} \ dx = 0 $$ for all possible w(x)

Let's focus on the stiffness term $$ \gamma $$ for now to derive the beam stiffness matrix and to identify the beam shape functions.



$$ v( \tilde{x}) = N_{2} (\tilde{x})\tilde{d}_{2} + N_{3} (\tilde{x})\tilde{d}_{3} + N_{5} (\tilde{x})\tilde{d}_{5} + N_{6} (\tilde{x})\tilde{d}_{6} $$

Recall : $$ u(\tilde{x}) = N_{1} (\tilde{x})\tilde{d}_{1} + N_{4} (\tilde{x})\tilde{d}_{4} $$



p.246:

$$ N_{2} (\tilde{x}) = 1 - \frac{3 \tilde{x}^2}{L^2} + \frac{2 \tilde{x}^3}{L^3} \ \tilde{d}_{2} $$

$$ N_{3} (\tilde{x}) = \tilde{x} - \frac{2 \tilde{x}^2}{L} + \frac{\tilde{x}^3}{L^3} \ \tilde{d}_{3} $$

$$ N_{5} (\tilde{x}) = \frac{3 \tilde{X}^2}{L^2} - \frac{2 \tilde{x}^3}{L^3} \ \tilde{d}_{5} $$

$$ N_{6} (\tilde{x}) = - \frac{\tilde{x}^2}{L} + \frac{\tilde{x}^3}{L^2} $$