By Andreas Öchsner
This ebook introduces readers to trendy computational mechanics in response to the finite point approach. It is helping scholars achieve mechanics classes through exhibiting them how you can observe the elemental wisdom they won within the first years in their engineering schooling to extra complex topics.
In order to deepen readers’ figuring out of the derived equations and theories, each one bankruptcy additionally comprises supplementary difficulties. those difficulties commence with primary wisdom questions about the speculation offered within the bankruptcy, by way of calculation difficulties. In overall over eighty such calculation difficulties are supplied, in addition to short suggestions for each.
This publication is principally designed to satisfy the wishes of Australian scholars, reviewing the math lined of their first years at college. The 13-week path includes 3 hours of lectures and hours of tutorials in line with week.
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Extra resources for Computational Statics and Dynamics: An Introduction Based on the Finite Element Method
21) Nn where δu i represents the so-called arbitrary or virtual displacements. It will be shown in the following that the virtual displacements occur on both sides of Eq. 19) and can be eliminated. Thus, these virtual displacements do not need a deeper consideration at this point of the derivation. 19) requires the derivatives of u(x) and W (x) which can be written on the element level as: du e (x) = dx dW (x) = dx d dN T (x) up , N T(x) up = dx dx d dN(x) δuTp N(x) = δuTp . e. not a function of x, and are therefore not affected by the differential operator.
46 2 Rods and Trusses (a) (b) Fig. 16 Free body diagram of the cantilevered rod structure shown in Fig. 14 Based on the indicated reaction forces, the global (non-reduced) system of equations can be stated for the configuration in Fig. 128) L ⎣ 0 −1 2 −1 ⎦ ⎣u 3X ⎦ ⎣ 0 ⎦ u 4X F0 0 0 −1 1 or for Fig. 16b as: ⎤ ⎡ ⎤ ⎤⎡ −R1 1 −1 0 0 u 1X E A ⎢ −1 2 −1 0 ⎥ ⎢u 2X ⎥ ⎢ 0 ⎥ ⎥=⎢ ⎥. 129) Knowing all nodal displacements, the support reaction R1 can be obtained for both cases by evaluating the first equation of the linear system as: R1 = − EA (u 1X − u 2X ) .
2 Derivation of the Governing Differential Equation 13 Fig. 1 Fundamental governing equations of a rod for deformation along the x-axis Expression Equation Kinematics εx (x) = dN x (x) = − px (x) dx σx (x) = Eεx (x) Equilibrium Constitution dσx (x) px (x) + = 0. 9) immediately gives Eq. 6). 1 must be combined. 3) gives: σx (x) = E du x . 10) Considering that a normal stress in the last equation is defined as an acting force N x over a cross-sectional area A: du x Nx =E . 6) to obtain in the general case: d dx E(x)A(x) du x (x) = − px (x).