section 1: introduction · 2014-08-25 · section 1: introduction washkewicz college of engineering...

Section 1: INTRODUCTIONWashkewicz College of Engineering

Course – Ground Rules1

• Lecture notes will be posted to my web page (http://academic.csuohio.edu/duffy_s/duffy.html)

• Homework assignments are due on the following Monday from the day they are assigned.

• Collaboration with other students on the homework assignments is permitted. Since this is a matrix intensive class you are encourage to use MATLAB.

• Homework assignments will be returned one week after they are collected.

• Neatness counts. Homework assignments should be stapled, and pages numbered.

• There will be two exams and numerous home work assignments. The second exam is not comprehensive.

1 a sports rule adopted to modify play on a particular field, court, or course


Course Progression

From the civil engineering student’s perspective:

Structural Analysis (CVE 311) dealt with “classical” methods of analysis that were also performed by hand, covering relatively simple truss, beam, and frame structures.

In CVE 511 we revisited elements of CVE 311, then concentrated on modern “matrix” methods of analysis of certain classes of structural elements.

In CVE 513 (Advanced Strength of Materials) we learn to analyze buckling, torsion and failure theories – all problems we can solve by hand.

In CVE 512 we delve into the mathematical concepts that support structural analysis using finite element analysis. In the process the student develops analytical skills that encompass a wide range of structural component.


From the mechanical engineering student’s perspective:

In MCE 367 (Machine Design) the elements of ESC 211 are revisited. In this course we learn how to design machine elements under static and fatigue loading. We design gears and use force analysis of spur, helical, bevel and worm gears. In addition we learn about the use of keys, pins, and splines to attach gears to shafts.

In MCE 465 (Advanced Machine Analysis) we learn about the analysis of stresses and deflections in complex mechanical systems under static and dynamic loading. Modeling techniques are integrated with 2D- and 3D-CAD systems. Comparisons of finite element results with theoretical and empirical results are made.

In MCE 580 we delve into the mathematical concepts that support structural analysis using finite element analysis. In the process the student develops analytical skills that encompass a wide range of structural component.


Importance of Topic by Technical Sequence (Duffy)

The following list depicts the relative importance of analytical (not design) topics based on building technical competence in a sequential fashion.

Statically determinate (by hand)Statically indeterminate (by hand)

BucklingTorsion

Elastic analysis Vibrations

Plastic AnalysisMatrix AnalysisPlates and shells

Finite Element analysisNon-linear Elasticity

Non-linear Finite Element Analysis


Momentarily focusing on categorizing the different fields of numerical analysis for mechanics a moment, the topic can be subdivided as follows:

TheoreticalMechanics Applied

Experimental NanomechanicsComputational Micromechanics

Continuum mechanicsSystems

Focusing on continuum mechanics

GasContinuum mechanics Fluids

Solids & Structures DynamicsStatics Nonlinear

Linear

The study of mechanics encompasses a wide range of technical content. We will study how to derive and implement numerical solution strategies within a narrow span of this field.


Finite Element Method Defined

• Many problems in engineering and applied science are governed by differential or integral equations (strong formulations).

• The solutions to these field equations would provide an exact, closed-form expressions to the particular problem posed.

• However, complexities in the geometry, properties and in the boundary conditions that are seen in most real-world problems usually means that an exact solution cannot be obtained, or obtained in a reasonable amount of time.

• Current product design cycle times demand that engineers must obtain design solutions in a ‘short’ amount of time.

• Industry is content to obtain approximate solutions that can be readily obtained in a reasonable time frame, and with reasonable effort. The FEM is one such approximate solution technique. But the new graduate engineer must use this tool with caution

• The FEM is a numerical procedure for obtaining approximate solutions to many of the problems encountered in engineering analysis.


Course Objectives

• You should gain an understanding of the theory and development of methods to analyze linear elastic finite elements using the knowledge of mathematics (linear algebra) engineering (mechanics and material behavior), and computer science (MATLAB, COMSOL, ABAQUS, NASTRAN)

• You will start (or continue) to develop foundation of experience for how, in general, structural components behave under load through an understanding of the flexibility (force) and then displacement methods of solution. Industry refers to this as understanding patterns of component behavior (building experience).

• You will develop knowledge of the effective use of the general purpose finite element method and the diverse types of elements available for analysis.

• This course provides a preliminary basis for the theoretical underpinnings of applying finite element analysis to other areas of engineering, e.g., heat transfer, electromagnetics, fluid flow, etc.


Course Outcomes

You will be expected to show proficiency in both the theory and practical application of structural analysis via finite elements. You will be able to use MATLAB to analyze simple structural components and structures. This will require developing several skills and abilities, including:

1. Choosing correct modeling elements, and choosing correct load representation,

2. Developing appropriate geometry and boundary conditions,

3. Apply software to attain displacements and stresses,

4. Interpret and validate results using simplified models and hand calculations.


Course Assessment

This course will measure progress in meeting the previously mentioned objectives by requiring students to:

• Understand the theory and application of finite-element methods;

• Model and analyze stress and strain fields for two- and three-dimensional structural components in engineering.

This assessment is made via

Exams – 80%

Quizzes – 10%

Homework – 10%


Brief History of Discrete Analysis Methods (also see first reading assignment, Felippa 2000)

1850-75 Interaction concepts introduced by Castigliano, Maxwell, and Mohr

1875-1920 No significant progress due to obvious limitations in solving large numbers of equations.

1920 Truss analysis utilizing joint displacements as solution unknowns is introduced (Maney, Ostenfeld) .

1932 Moment distribution by Hardy-Cross increases solvability of larger structures by an order of magnitude. Method helps in “visualizing” the interaction of members.

1930’s Aeroelastic research at England’s National Physical Laboratory (NPL) extends previous efforts.

1950’s Computers become available.


Brief History of Discrete Analysis Methods (continued)

1956 The landmark contribution of Turner, Clough, Martin and Topp finally succeeded in deriving the stiffness of a triangular plate. Clough observes later that this paper represents earlier work at Boeing and is one of the two intellectual sources of present day finite element analysis.

1960 Paper by Argyris and Kelsey introduces the concept that the displacement-based method of structural analysis is a dual methodology to the force-based of structural analysis.

1965-1969 In 1965 NASA’s RFP to develop the NASTRAN finite element software called for the simultaneous development of displacement-based and force-based versions. Two separate contracts were awarded to MSC and Martin. The contract for the force-based version was cancelled in 1969. The following year may be taken as the end of the force-based methods as a serious contender for general purpose finite element programs.


The finite element method involves modeling a structural component using small interconnected elements, i.e., finite elements. A displacement function is associated with each finite element. Every interconnected element is linked directly, or indirectly, to every other element through common interfaces that include nodes, boundary lines and boundary surfaces. By using stress-strain relationships (constitutive relationships) one can determine the behavior of a given node in terms of the properties of every other element in the structural component.

Finite Element Analysis


• With finite element analysis a complex region defining a continuum is discretized into simple geometric shapes called elements.

• The properties and the governing relationships are assumed for these elements and expressed mathematically in terms of unknown values at specific points in the elements called nodes.

• An assembly process is used to link the individual elements to the given system. When the effects of loads and boundary conditions are considered, a set of linear or nonlinear algebraic equations is usually obtained.

• Solution of these equations gives the approximate behavior of the continuum or system (solve via a “weak formulation” of the problem).

• The continuum has an infinite number of degrees-of-freedom (DOF), while the discretized model has a finite number of DOF. This is the origin of the name, finite element analysis.

• The number of equations is usually rather large for most real-world applications of the FEM, and requires the computational power of modern computers.


First principle stress (psi) from a cold start on Ingersoll-Rand’s ceramic microturbine vane

Finite element mesh used for the analysis of a single vane (axisymmetric model)


Two features of the finite element method are worth noting.

• The piecewise approximation used by finite element analysis of the displacement fields within a continuum provides good precision even with simple approximating functions. Simply increasing the number of elements can achieve increasing precision.

• The approximation leads to sparse equation systems for a discretized problem. This helps to ease the solution of problems having very large numbers of nodal unknowns. It is not uncommon today to solve systems containing a million primary unknowns.


Applications of Finite Element Analysis

Finite element analysis can be used to analyze both structural and non-structural problems. Typical structural applications include:

• Stress analysis

• Buckling

• Vibrational analysis

Non-structural analyses include:

• Heat transfer

• Fluid flow (computational fluid dynamics – CFD)

• Distribution of electric or magnetic potential (“electromag”)


Advantages of Finite Element Analysis

Finite element analysis has the capability to:

• model irregularly shaped bodies quite easily (Civil Engineering – structural connections);

• allow for general load conditions without difficulty;

• model components composed of several different materials (reinforced concrete);

• utilize various types and sizes of elements, e.g., densify meshes in “hot spots” to increase accuracy;

• alter models quickly;

• readily include dynamic effects; and

• allow for nonlinear behavior (large deformations of material nonlinearity, e.g., plasticity);

Finite element analysis allows the engineer to evaluate multiple design options prior to fabrication or construction.


Analytical Steps – Executive Summary

In a structural analysis displacements and stresses are determined throughout the structural component. There are two general approaches to do this with FEA:

• Force, or flexibility method

• Displacement, or stiffness method

The flexibility method uses internal forces as unknowns and equilibrium equations as well as compatibility equations to establish a system of equations to determine the unknown forces.

In the stiffness method nodal displacements serve as unknowns and again equilibrium as well as relationships between forces and displacements are use to establish a system of equations to solve for the unknown displacements.


The displacement method dominates the commercial finite element software. The reason evolves from the fact that:

• Once displacements are determined strains can be computed via the strain displacement relationships studied in Elasticity and Continuum Mechanics

• With strains and a multiaxial constitutive relationships, e.g., Hookes law, stresses can be computed.

Reversing this two step process is not impossible, i.e., find forces, stresses, strains and then displacements, but has not found commercial application. So we focus on displacement based approaches in this course. Finding forces, then stresses, then strains and finally displacements is the “flexibility method.”


Obtaining Answers

Structural analyses of complex structures involve five key ingredients:

1. Basic mechanics relationships (stress-strain, compatibility, equilibrium)

2. Use of discrete elements (exact vs. numerical)

3. Equation Formulation (algebraic system)

4. Equation Solution (algorithms to solve simultaneous equations)

5. Solution Interpretation (“answers”)


Advanced Topics

Finite Element analysis can be extended to include more complex structural behavior including:

1. Geometric nonlinearity (large displacements, e.g., tip of an airplane wing)

2. Material nonlinearity • Plasticity• Creep• Viscous elasticity

3. Time dependent dynamic analysis including all of above.

In this course we will focus on SMALL STRAIN, LINEAR ELASTIC, STATIC analysis.

section 1: introduction · 2014-08-25 · section 1: introduction washkewicz college of engineering...

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