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Master element and Jacobian

All elements are computed on the same master element $\xi\in[-1, 1]$ and mapped to the physical element. The Jacobian $J$ is how much the element is stretched. Integrals get the factor $J$ , derivatives are divided by $J$ .

\varphi_1 = \frac{1 - \xi}{2},\quad \varphi_2 = \frac{1 + \xi}{2}

linear shape functions

x(\xi) = x_1\varphi_1 + x_2\varphi_2,\quad J = \frac{h}{2}

mapping and Jacobian

B^e_{ij} = \int_{-1}^{1} EA\,\frac{d\varphi_i}{d\xi}\frac{d\varphi_j}{d\xi}\frac{1}{J}\,d\xi

element matrix

F^e_i = \int_{-1}^{1} F_x\,\varphi_i\,J\,d\xi

element vector

Symbols

$\xi$	coordinate on the master element
$J$	Jacobian $dx/d\xi$	m
$h$	element length	m

Example

$h = 2$ m, $EA = 600$ kN: $J = 1$ , $\frac{d\varphi_1}{d\xi} = -\tfrac12$ , so $B^e_{11} = \int_{-1}^{1} 600\cdot\tfrac14\,d\xi = 300$ kN/m $= EA/h$ .

Chain rule:

\frac{d\varphi}{dx} = \frac1J\frac{d\varphi}{d\xi}

. Forget

J

and the stiffness is off by a factor

h/2

Practise weak form and shape functions for free →

← The weak form for the bar · Shape functions →

Part of Finite Element Method: Weak form and shape functions.