ce 257 discrete methods of structural analysis€¦ · j. n. reddy, an introduction to the finite...

CE 257 Discrete Methods of Structural Analysis

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This lecture material is not for sharing/distribution outside of CE257 and CE297 class.

Last meeting

• Timoshenko Beam Theory

• Plane Frames

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For this meeting

• Eigenvalue Problems

Reference for this lecture (notes and figures):

J. N. Reddy, An Introduction to the Finite Element Method (Chapter 6)

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CE28 Analytical and Computational Methods in CEIII

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Eigenvalue Problems: Introduction

Introduction

• An eigenvalue problem is defined to be one inwhich we seek the values of the parameter 𝜆 suchthat the equation

𝐴(𝑢) = 𝜆 𝐵(𝑢)

• is satisfied for nontrivial values of 𝑢. Here 𝐴 and 𝐵denote either matrix operators or differentialoperators, and values of 𝜆 for which the equationabove is satisfied are called eigenvalues. For eachvalue of 𝜆 there is a vector 𝑢 , called aneigenvector or eigenfunction.

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Introduction

• For example, the equation

• which arises in connection with natural axialvibrations of a bar or the transverse vibration of acable, constitutes an eigenvalue problem. Here𝜆 denotes the square of the frequency of vibration𝜔.

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Introduction

• In general, the determination of the eigenvalues isof engineering as well as mathematicalimportance.

• In structural problems, the eigenvalues denoteeither natural frequencies or buckling loads.

• In fluid mechanics, and heat transfer, eigenvalueproblems arise in connection with thedetermination of the homogeneous parts of thetransient solution.

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Introduction

• In these cases, eigenvalues often denoteamplitudes of the Fourier components making upthe solution.

• Eigenvalues are also useful in determining thestability characteristics of temporal schemes.

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Introduction

• We develop finite element models of eigenvalueproblems described by differential equations(DEs).

• Differential eigenvalue problems are reduced toalgebraic eigenvalue problems

(i.e. [𝐴]{𝑋} = 𝜆[𝐵][𝑋])

by means of the finite element approximation. Themethods of solution of algebraic eigenvalueproblems are then used to solve for the eigenvaluesand eigenvectors.

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Eigenvalue Problems: Formulation

Formulation of Eigenvalue Problems

Parabolic Equation:

• Consider the PDE:

• Assumed homogeneous solution (by method ofseparation-of-variables):

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Parabolic Equation:

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Parabolic Equation:

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EQ 1:

EQ 2:


Parabolic Equation:

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solutionEQ 1:


Parabolic Equation:

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solution

EQ 2:


Parabolic Equation:

• Try BCs:

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Parabolic Equation:

• IC:

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Parabolic Equation:

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Hyperbolic Equation:

• The axial motion of a bar is described by theequation

• where 𝑢 denotes the axial displacement, 𝐸 themodulus of elasticity, 𝐴 the cross sectional area, 𝜌the density, and 𝑓 the axial force per unit length.

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• The solution of the equation consists of two parts:homogeneous solution 𝑢ℎ (i.e. when 𝑓 = 0) andparticular solution 𝑢𝑝.

• The homogeneous part is (again) determined bythe separation-of-variables technique.

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• Hyperbolic Equation:

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D.E.

EQ 1:

EQ 2:



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EQ 1:

EQ 2:



• Now, let’s try the solution:

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• Similarity

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EQ 2:


• Another eigenvalue problem that arises directlyfrom the governing equilibrium equation is that ofbuckling of beam-columns.

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• In summary, eigenvalue problems associated withparabolic equations are obtained from thecorresponding equations of motion by assumingsolution of the form

• whereas those associated with hyperbolicequations are obtained by assuming solution ofthe form

• where 𝜆 denotes the eigenvalues.

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Eigenvalue Problems: Finite Element Formulation

Finite Element Formulation

Heat Transfer and Bar-like Problems

• Consider the problem of solving the equation

• for 𝜆 and 𝑈(𝑥). Here 𝑎, 𝑐, and 𝑐𝑜 are knownquantities that depend on the physical problem(i.e. data), 𝜆 is the eigenvalue, and 𝑈 is theeigenfunction,

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Heat Transfer and Bar-like Problems

• Model D.E.:

• Cases:

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• Over a typical element Ω𝑒 , we seek a finiteelement approximation of 𝑈 in the form

• The weak form of

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• where 𝑤 is the weight function, and 𝑄1𝑒 and

𝑄𝑛𝑒 are the secondary variables at node 1 and

node n, respectively.

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• Substitution of the finite element approximationinto the weak form gives the finite element modelof the eigenvalue equation

• where

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• The assembly of element equations andimposition of boundary conditions on theassembled equations remain the same as in staticproblems.

• However, the solution of the condensed equationsfor the unknown primary nodal variables isreduced to an algebraic eigenvalue problem inwhich the determinant of the coefficient matrix isset to zero to determine the values of 𝜆 andsubsequently the nodal values of theeigenfunction 𝑈(𝑥).

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Finite Element Formulation of Eigenvalue Problems: Example


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BC SET 1/ IC:

BC SET 1 / IC:

Example 6.1.1

DE:


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• We note that the eigenvalue problem can also beinterpreted as those arising in connection with theaxial vibration of a constant cross sectionalmember. In this case, 𝑈 denotes the axialdisplacement [𝑢 𝑥, 𝑡 = 𝑈 𝑥 𝑒−𝑖𝑤𝑡]. and Eq. 𝜆 =𝜔2𝜌/𝐸 , 𝜔 being the frequency of naturalvibration.

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• For example, the BCs can be interpreted as thoseof a bar fixed at the left end and the right endbeing connected to a linear elastic spring. Theconstant 𝐻 is equal to 𝑘/𝐸𝐴, 𝑘 being the springconstant.

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Natural Vibration of Beams


Euler Bernoulli Beam Theory

• The equation of motion is of the form

• where 𝜌 denotes the mass density per unit length,𝐴 the area of cross section, 𝐸 the modulus, and 𝐼the second moment of area. The expressioninvolving 𝜌𝐼 is called the rotary inertia term.

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D.E.:


• To formulate an eigenvalue problem in the interestof finding the frequency of natural vibration, weassume periodic motion

• where 𝜔 is the frequency of natural transversemotion and 𝑊(𝑥) is the mode shape of thetransverse motion.

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• Substitution of 𝑤(𝑥, 𝑡) to the D.E.:

• where 𝜆 = 𝜔2.

• The weak form is given by

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• where 𝑣 is the weight function

• Note that the rotary inertia term contributes tothe shear force term, giving rise to an effectiveshear force that must be known at a boundarypoint when the deflection is unknown at thepoint.

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• To obtain the finite element model, we assumefinite element approximation of the form

• where 𝜙𝑗𝑒 are the Hermite cubic polynomials,

• We get the finite element model:

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• where

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• For constant values of 𝐸𝐼 and 𝜌𝐴, the stiffnessmatrix [𝐾𝑒] and mass matrix [𝑀𝑒] are

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• When rotary inertia is neglected, we omit thesecond part of the mass matrix.

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Timoshenko Beam Theory

• The equations of motion of the Timoshenko beamtheory are

• where 𝐺 is the shear modulus and 𝐾𝑠 is the shearcorrection factor. Note that second equationcontains the rotary inertia term.

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• We assume periodic motion and write

• and obtain the eigenvalue problem

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• For equal interpolation of 𝑊(𝑥) and 𝑆(𝑥):

• where 𝜓𝑗𝑒 are the (𝑛 − 1) order Lagrange

polynomials, the finite element model is given by

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• where [𝐾𝑒] is the stiffness matrix and [𝑀𝑒 ] is themass matrix

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• Timoshenko Beam Theory

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• For the choice of linear interpolation functions,the F.E. model has the explicit form

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EBT TBT


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• In closing the discussion on natural vibration, it isnoted that when the symmetry of the system isused to model the problem, only symmetricmodes are predicted. It is necessary to model thewhole system in order to obtain all the modes ofvibration.

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Stability (Buckling) of Beams


• EBT

• The study of buckling of beam-columns also leadsto an eigenvalue problem. For example, theequation governing onset of buckling of a columnsubjected to an axial compressive force 𝑁0

according to EBT is

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• EBT

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• EBT

• which describes an eigenvalue problem with 𝜆 =𝑁0 . The smallest value of 𝑁0 is called the criticalbuckling load.

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• The finite element model of the equation is

• where { Δ𝑒 } and { 𝑄𝑒 } are the columns ofgeneralized displacement and force degrees offreedom at the two ends of the Euler-Bernoullibeam element:

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• where the subscripts 1 and 2 refer to the elementnodes 1 and 2 (at 𝑥 = 𝑥𝑎 and 𝑥 = 𝑥𝑏 ,respectively).

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• The coefficients of the stiffness matrix [𝐾𝑒] andthe stability matrix [𝐺𝑒] are

• where 𝜙𝑖𝑒 are the Hermite cubic interpolation

functions.

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• The explicit form of [𝐺𝑒] is

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• TBT

• for TBT, the equations governing buckling ofbeams are

• Here 𝑊(𝑥) and 𝑆(𝑥) denote the transversedeflection and rotation, respectively, at the onsetof buckling.

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• TBT

• The finite element model with equal interpolationof 𝑊 and 𝑆, is

• where

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• TBT

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• TBT

• For the linear RIE, the stiffness matrix [𝐾𝑒] is givenby

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• TBT

• The stability matrix is given by

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Eigenvalue Problem of Frame Structures

• The transformed element equations:

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Homework

• Derive the finite element equations:

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D.E.

Use:

Derive the weak form:

Use:

Derive the finite element equations:

Where:

Recall: Formulation of Eigenvalue Problems


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D.E.

EQ 1:

EQ 2:

Recall: Formulation of Eigenvalue Problems


• Assumed solution

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Finite Element Method

References:

1. J. Fish, T. Belytschko, A First Course in Finite Elements

2. J. N. Reddy, An Introduction to the Finite Element Method

3. O.C. Zienkiewicz, R.L. Taylor, J.Z. Zhu, The Finite Element Method, Its Basis & Fundamentals

4. K-J. Bathe, Finite Element Procedures

5. K. Leet, C-M., Uang, A.M. Gilbert, Fundamentals of Structural Analysis

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This lecture

• Eigenvalue Problems

Reference for this lecture:

J. N. Reddy, An Introduction to the Finite Element Method (Chapter 5)

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This lecture material is not for sharing/distribution outside of CE257 and CE297 class.


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ce 257 discrete methods of structural analysis€¦ · j. n. reddy, an introduction to the finite...

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