## MAE533 Finite Element Method课程简介

This course will provide a general preparation in finite element methods with an emphasis on linear finite-elements and material behavior. The course is intended for graduate engineering, science, and mathematics students who will pursue further work and research in specialized areas such as nonlinear continuum mechanics structural mechanics, elasticity, plasticity, fracture mechanics, mechanical design, heat transfer, and numerical analysis.

## PREREQUISITES

Students will develop the mathematical and solid mechanics background to solve and analyze linear stress and analysis problems with the appropriate finite-element methodologies and numerical approximations. Applications of real-world problems with a focus on solid mechanics and elasticity.
Course Requirements
HOMEWORK: 6-8 assignments.
EXAMINATIONS: Midterm and Final.
SOFTWARE REQUIREMENTS: ANSYS will be used for different class projects. Other similar commercial FEM codes can be used. Knowledge of Fortran, C, C++, Matlab, or other programming languages will be useful
Textbook
Most of the material will be based on class notes and lectures. Students will be provided copies of the notes and the lectures.

## MAE533 Finite Element Method HELP（EXAM HELP， ONLINE TUTOR）

Example 1.1 Equation of motion of a solid bar
a) Derive the equation of motion of an elastic bar in terms of its deflection $u(x, t)$. Initially, assume that the bar has a variable cross-sectional area $A(x)$ and that it is subjected to distributed axial load $q(x, t)$ and a concentrated force $F$ at its free end as shown in Fig. 1.2. Also assume small deflections, linear elastic material behavior with constant elastic modulus $E$, and constant mass density $\rho$.
b) Obtain the steady state solution for the case of constant cross-section and zero distributed force.

Solution 1.1a: The solution domain $\Omega$ for this problem spans $0<x<L$. The boundaries $\Gamma$ of the solution domain are located at $x=0$ and $x=L$. Internal forces develop in the bar in response to external loading. The internal normal force $N(x)$ at the cross-section $x$ can be defined as follows:
$$N(x)=\bar{\sigma}(x) A(x)$$
where the average normal stress $\bar{\sigma}$ is defined as follows:
$$\bar{\sigma}(x)=\frac{1}{A(x)} \int_{A(x)} \sigma d A$$
and where $\sigma$ is the internal normal stress, $A$ is the cross-sectional area of the bar. The equation of motion of the bar can be obtained by using Newton’s second law on a small segment of the bar (Fig. 1.2). The balance of internal and inertial forces gives,

\begin{aligned}
& \sum F_x=\rho A d x \frac{\partial^2 u}{\partial t^2} \
& -N+q d x+\left(N+\frac{\partial N}{\partial x} d x\right)=\rho A d x \frac{\partial^2 u}{\partial t^2} \
& \frac{\partial N}{\partial x}=-q+\rho A \frac{\partial^2 u}{\partial t^2}
\end{aligned}

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

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