The Finite Element Method For Solid And Structural Mechanics Pdf

[Monica Okane]

Thefinite element method (FEM) is a powerful technique originally developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems. In the FEM, the structural system is modeled by a set of appropriate finite elements interconnected at discrete points called nodes. Elements may have physical properties such as thickness, coefficient of thermal expansion, density, Young's modulus, shear modulus and Poisson's ratio.

Straight or curved one-dimensional elements with physical properties such as axial, bending, and torsional stiffnesses. This type of element is suitable for modeling cables, braces, trusses, beams, stiffeners, grids and frames. Straight elements usually have two nodes, one at each end, while curved elements will need at least three nodes including the end-nodes. The elements are positioned at the centroidal axis of the actual members.

The elements are interconnected only at the exterior nodes, and altogether they should cover the entire domain as accurately as possible. Nodes will have nodal (vector) displacements or degrees of freedom which may include translations, rotations, and for special applications, higher order derivatives of displacements. When the nodes displace, they will drag the elements along in a certain manner dictated by the element formulation. In other words, displacements of any points in the element will be interpolated from the nodal displacements, and this is the main reason for the approximate nature of the solution.

Large scale commercial software packages often provide facilities for generating the mesh, and the graphical display of input and output, which greatly facilitate the verification of both input data and interpretation of the results.

While the theory of FEM can be presented in different perspectives or emphases, its development for structural analysis follows the more traditional approach via the virtual work principle or the minimum total potential energy principle. The virtual work principle approach is more general as it is applicable to both linear and non-linear material behaviors. The virtual work method is an expression of conservation of energy: for conservative systems, the work added to the system by a set of applied forces is equal to the energy stored in the system in the form of strain energy of the structure's components.

The virtual internal work in the right-hand-side of the above equation may be found by summing the virtual work done on the individual elements. The latter requires that force-displacement functions be used that describe the response for each individual element. Hence, the displacement of the structure is described by the response of individual (discrete) elements collectively. The equations are written only for the small domain of individual elements of the structure rather than a single equation that describes the response of the system as a whole (a continuum). The latter would result in an intractable problem, hence the utility of the finite element method. As shown in the subsequent sections, Eq.(1) leads to the following governing equilibrium equation for the system:

By applying the virtual work equation (1) to the system, we can establish the element matrices B \displaystyle \mathbf B , k e \displaystyle \mathbf k ^e as well as the technique of assembling the system matrices R o \displaystyle \mathbf R ^o and K \displaystyle \mathbf K . Other matrices such as ϵ o \displaystyle \mathbf \epsilon ^o , σ o \displaystyle \mathbf \sigma ^o , R \displaystyle \mathbf R and E \displaystyle \mathbf E are known values and can be directly set up from data input.

Let q \displaystyle \mathbf q be the vector of nodal displacements of a typical element. The displacements at any other point of the element may be found by the use of interpolation functions as, symbolically:

Again, numerical integration is convenient for their evaluation. A similar replacement of q in (17a) with r gives, after rearranging and expanding the vectors Q t e , Q f e \displaystyle \mathbf Q ^te,\mathbf Q ^fe :

In practice, the element matrices are neither expanded nor rearranged. Instead, the system stiffness matrix K \displaystyle \mathbf K is assembled by adding individual coefficients k i j e \displaystyle k_ij^e to K k l \displaystyle K_kl where the subscripts ij, kl mean that the element's nodal displacements q i e , q j e \displaystyle q_i^e,q_j^e match respectively with the system's nodal displacements r k , r l \displaystyle r_k,r_l . Similarly, R o \displaystyle \mathbf R ^o is assembled by adding individual coefficients Q i e \displaystyle Q_i^e to R k o \displaystyle R_k^o where q i e \displaystyle q_i^e matches r k \displaystyle r_k . This direct addition of k i j e \displaystyle k_ij^e into K k l \displaystyle K_kl gives the procedure the name Direct Stiffness Method.

All undergraduate students enrolled in structural engineering courses or admitted into the structural engineering program are expected to meet prerequisite and performance standards. Additional details are given under the various program outlines, course descriptions, and admission procedures for the School of Engineering in this catalog. The department expects that students will adhere to these policies on their own volition and enroll in courses accordingly. Students are advised that they may be dropped at any time from course rosters if prerequisites and/or performance standards have not been met.

While some courses may be offered more than once each year, most SE courses are taught only once per year, and courses are scheduled to be consistent with the curricula as shown in the tables. When possible, SE does offer selected large-enrollment courses more than once each year. A tentative schedule of course offerings is available from the department each spring for the following academic year.

Introduction to probability theory and random processes. Dynamic analysis of linear structural systems subjected to stationary and nonstationary random excitations. Reliability studies related to first excursion and fatigue failures. Applications in earthquake engineering, offshore engineering, wind engineering, and aerospace engineering. Use of computer resources. Recommended preparation: basic knowledge of probability theory (SE 125 or equivalent). Prerequisites: SE 203, graduate standing.

Provides background and tools to apply machine learning to solve problems in computational mechanics and engineering. An overview of the basic principles of machine learning will be provided, including supervised and unsupervised learning, regression, classification, and generative algorithms versus discriminative algorithms. Focus will be given to deep neural networks, convolutional neural networks, recurrent neural networks, physics-informed machine learning and implementation in Python. Recommended preparation: knowledge of computer programming, probability theory, linear algebra, and solid mechanics. Prerequisites: graduate standing or consent of instructor.

Cross-listed with MAE 235. Practical application of the finite element method to problems in solid mechanics. Elements of theory are presented as needed. Covered are static and dynamic heat transfer and stress analysis. Basic processing, solution methods, and postprocessing are practiced with commercial finite element software. Students may not receive credit for SE 233 and MAE 235.

CEE 200. Introduction to Civil and Environmental Engineering

Prerequisite: None; mandatory pass/fail. (1 credit)

An introduction to the nature and scope of the civil and environmental engineering disciplines and specialty programs. Includes case studies from practice and information about academic and professional opportunities for CEE students. CourseProfile (ATLAS)

CEE 265. Sustainable Engineering Principles

Advisory Prerequisite: CHEM 130, MATH 116. (Credit for only one: CEE 265 or MECHENG 489.) (3 credits)

Sustainable engineering principles include calculations of environmental emissions and resource consumption. Mass and energy balance calculations in context of pollution generation and prevention, resource recovery and life-cycle assessment. Economic aspects of sustainable engineering decision-making. Social impacts of technology system design decisions including ethical frameworks, government legislation and health risks. CourseProfile (ATLAS)

CEE 307 (Environ 407). Sustainable Cities

Advisory Prerequisite: Junior or Senior Standing and two environmental science classes. (3 credits)

As economic and ecological pressures increase, it has become increasingly important that greater efforts be expended to have more sustainable urban environments. Specifically, it is essential that the future operation of cities become more sustainable in terms of energy and resource use, while also safeguarding the health and well-being of local citizens. This course will discuss how multiple disciplines can be integrated to identify and discuss this broad goal. A combination of individual and team assignments will be given, culminating in a team term project that provides alternative strategies for consideration by a panel of experts. CourseProfile (ATLAS)

CEE 312. Structural Engineering

Prerequisite: CEE 212 or equivalent. (4 credits)

Introduction to the field of structural engineering. Discussion of structural analysis techniques and concepts such as virtual work, flexibility method, stiffness method, influence lines and matrix structural analysis. Training in AutoCAD and exposure to commonly used structural analysis computer program(s). Discussion of basic design concepts and principles. CourseProfile (ATLAS)

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