### Fundamentals of Finite Element Analysis

Important concepts for finite element computations, such as isoparametric elements for multi-dimensional analysis and Gaussian quadrature for numerical evaluation of integrals, are presented and explained. Practical aspects of FEA and advanced topics, such as reduced integration procedures, mixed finite elements and verification and validation of the FEM are also discussed.

Fundamentals of Finite Element Analysis: Linear Finite Element Analysis is an ideal text for undergraduate and graduate students in civil, aerospace and mechanical engineering, finite element software vendors, as well as practicing engineers and anybody with an interest in linear finite element analysis. View Student Companion Site. His research primarily focuses on the analytical simulation of structural components and systems under extreme events, with an emphasis on reinforced concrete, masonry and steel structures under earthquake loading.

He has authored and co-authored research papers and reports on finite element analysis element formulations, constitutive models, verification and validation of modeling schemes. Permissions Request permission to reuse content from this site. Undetected location. NO YES. Selected type: Paperback. Added to Your Shopping Cart. Evaluation Copy Request an Evaluation Copy. This is a dummy description. Provides detailed derivations of finite element equations for a variety of problems. Incorporates quantitative examples on one-dimensional and multi-dimensional FEA.

Therefore, approximate solutions based on numerical techniques and digital computation are most often obtained in engineering analyses of complex problems. Finite element analysis is a powerful technique for obtaining such approximate solutions with good accuracy. As we treat this example as a two- dimensional problem, it is assumed that the thickness in the z direction is con- stant and z dependency is not indicated in the differential equation. The vertices of the triangular element are numbered to indicate that these points are nodes. Nodes that do not lie on element boundaries are interior nodes and cannot be connected to any other element.

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The triangular element of Figure 1. The interpola- tion functions are most often polynomial forms of the independent variables, derived to satisfy certain required conditions at the nodes. These conditions are discussed in detail in subsequent chapters. The triangular element described by Equation 1. If the domain of Figure 1.

In the latter case, the three-node triangular element has 6 degrees of freedom. How does this element-based approach work over the entire domain of in- terest? As depicted in Figure 1. This feature avoids the physically unacceptable possibility of gaps or voids oc- curring in the domain. In structural problems, such gaps would represent physi- cal separation of the material. This is a critical observa- tion.

However, the magnitudes of discontinuities of derivatives can be used to assess solution accuracy and convergence as the number of elements is increased, as is illustrated by the following example. As most of the commonly used element geometries have straight sides, it is gen- erally impossible to include the entire physical domain in the element mesh if the domain includes curved boundaries.

Such a situation is shown in Figure 1. Triangular elements could approximate the boundaries even better. To illustrate convergence, we consider a relatively simple problem that has a known solution. Figure 1.

Stippled areas are not included in the model. A total of 41 elements is shown. A total of elements is shown. The solution is obtained using the strength of materials theory. Next, we model the tapered cylinder as two uniform bars in series, as in Figure 1.

In the two- element model, each element is of length equal to half the total length of the cylinder and has a cross-sectional area equal to the average area of the corre- sponding half-length of the cylinder. Actual tapered cylinder is shown as dashed lines. The area of each element is average over the respective tapered cylinder length. We note that the displacement variation in each element is a linear approximation to the true nonlinear solution. The linear variation is directly attributable to the fact that the interpolation functions for a bar element are linear.

The previous paragraph discussed convergence of the displacement of the tapered cylinder. Hence, strains and stresses are referred to as derived variables. We also note that, as the number of elements increases, the jump discontinuities in stress decrease in magnitude. The answer to this question is not simple.

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If we did not have the dashed line in Figure 1. Does the heat output balance with the heat input? Details of the technique are discussed in Chapter 7 in the context of transient heat The method is also illustrated in Chapter 10 for transient dynamic analy- sis of structures. Equation 1. To illustrate, Figure 1. The manner of variation There are also certain similarities between the two methods.

The variable of interest is explicitly evaluated at such points.

## Mechanical Engineering at IIT Madras » Finite Element Analysis

And in both cases, differential equations are reduced to algebraic equations. The steps are described as follows. The computed values are then used by back substitution to For static, linear problems, a wave front solver, based on Gauss elimination Appendix C , is commonly used. Examples of opera- tions that can be accomplished include Sort element stresses in order of magnitude.

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Check equilibrium. Calculate factors of safety.

Plot deformed structural shape. Animate dynamic model behavior. Produce color-coded temperature plots.

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While solution data can be manipulated many ways in postprocessing, the most important objective is to apply sound engineering judgment in determining whether the solution results are physically reasonable. Approximate methods for solving differential equations using trial solu- tions are even older in origin.

Lord Rayleigh  and Ritz  used trial functions in our context, interpolation functions to approximate solutions of differential equations. Galerkin  used the same concept for solution. In the late s, aircraft engineers were dealing with the invention of the jet engine and the needs for more sophisticated analysis of airframe structures to withstand larger loads associated with higher speeds.

In this text, we adhere exclusively to the displacement method. In the early years, applications were performed using mainframe computers, which, at the time, were considered to be very pow- erful, high-speed tools for use in engineering analysis.