Ch#3 Truss Equations (1)

Optimization

Chapter 3Development ofTruss Equations

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Ch. 3 Truss Equations11

3.1. Definition of the Stiffness MatrixWe will consider now the derivation of the stiffness matrix for the linear-elastic, constant-cross-sectional area (prismatic) bar element shown in Figure 3-1.

Figure 3-1 Bar subjected to tensile forces T; positive nodal displacements and forces

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The bar element is assumed to have constant cross-sectional area A, modulus elasticity E, and initial length L. The nodal d.o.f are local axial displacements (longitudinal displacements directed along the length of the bar).

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Ch. 3 Truss EquationsThe following assumptions are used in deriving the bar elements stiffness matrix:

The steps previously outlined in Chapter 1 are now used to derive the stiffness matrix for bar element.

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Step 1 Select Element TypeRepresent the bar by labeling nodes at each end and in general by labeling the element number (see Figure 3-1).Step 2 Select a Displacement FunctionsAssume a linear displacement variation along the local axis of the bar because a linear function with specified endpointshas a unique path.

with the total number of coefficients ai always equal to the total number of d.o.f associated with the element. Using the same procedure as in Section 2.2 for the spring element, we express Eq. (3.1.1) as

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Ch. 3 Truss Equations

The linear displacement function plotted over the length of the bar element os shown in Figure 3-2. The bar is shown with the same orientation as in Figure 3-1.

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Ch. 3 Truss EquationsFigure 3-2 Linear displacement plotted over the length of the element

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Ch. 3 Truss EquationsStep 3 Define the Strain-displacement and Stress-strain RelationshipsThe strain-displacement relationship is.

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Ch. 3 Truss EquationsStep 4 Derive the Element Matrix and Equations

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Ch. 3 Truss EquationsStep 5 Assemble the Element Equations to Obtain the Total/Global EquationsAssemble the global stiffness and forces vectors and global equations using the direct stiffness method described in Chapter 2 can still be adopted in this case.The method applies for structures composedd of more than one element such that

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Ch. 3 Truss EquationsStep 6 Solve for the Nodal DisplacementsDetermine the displacement by imposing boundary conditions and simultaneously solving a system of equations, F = K d.

Step 7 Solve for the Element ForcesFinaly, determine the strains and stress in each element by back-substitution of the displacement into equations similar to Eqs. (3.1.5) and (3.1.6).

Example 3.1

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Ch. 3 Truss Equations3.2. Selecting Approximation Functions for DisplacementConsider the following guidelines, as they relate to the one-dimensional bar element, when selecting a displacement function. Further discussion will be provided in Chapter 4 (for the beam element).Common approximation functions (AF) are usually polynomials such as that given by Eq. (3.1.1) or equivalently by Eq. (3.1.3), where the function is expressed in terms of the shape functions.The AF should be continuous within the bar element. The simple linear function of Eq. (3.1.1) certainly is continuous within the element.The AF should provide interelement continuity for all d.o.f at each node for discrete line elements, and along common boundary lines and surfaces for two- and three-dimensional elements.

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Ch. 3 Truss EquationsFor the bar element, we must ensure that nodes common to two or more elements remain common to the these elements upon deformation and thus prevent overlaps or voids between elements. For the two-bar structure (Figure 3-3), the linear function for displacement within each element will ensure that elements 1 and 2 remain connected; that is, the displacement at node 2 for element 1 will equal the displacement at the same node 2 for element 2. The linear function is then called a conforming (or compatible) function for the bar element because it ensure both the satisfaction of continuity between adjacent elements and of continuity within the element.Figure 3-3 Interelement continuity of a two-bar structure

L1L2312

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Ch. 3 Truss EquationsThe approximation function should allow for rigid-body motion displacement and for a state of constant strain within the element. The 1D displacement function, Eq. (3.1.1), satisfies these criteria because a1 term allows for rigid-body motion and the term allows for constant strain since is a constant. (This state of constant strain in the element can, if fact, occur if elements are chosen small enough).The simple polynomial Eq. (3.1.1) satisfying this fourth guidelines is said to be complete for the bar element. The completeness of a function is a necessary condition for convergence to the exact answer, for instant, for displacement and stresses.The idea that the interpolation (approximation) function must allow for a rigid-body displacement means that the function must be capable of yielding a constant value (say, a1), because such a value can, in fact, occur.

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Ch. 3 Truss Equations3.3 Transformation of Vectors in Two DimensionIn many problem it is convenient to introduce both local and global coordinates. Local coordinates are always chosen to conveniently represent the individual element. Global coordinates are chosen to be convenient for the whole structures.Given the nodal displacement of an element, represented by the vector d in Figure 3-4, we want to relate the components of this vector in one coordinate system to components in another.Figure 3-4General displacement vector d

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Ch. 3 Truss EquationsFor general purposes, we will assume that d is not coincident with neither the local nor the global axes. In this case, we want to relate global displacement components to local ones.In doing that, we will develop a transformation matrix that will subsequently be used to develop the global stiffness matrix for a bar element.

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Ch. 3 Truss Equations3.4. Element Stiffness Matrix in Global CoordinatesWe will now use the transformation relationship Eq. (3.3.16) to obtain the global stiffness matrix for a bar element. We need the global stiffness matrix of each element to assemble the global stiffness matrix of the structure. We have shown in Eq. (3.1.13) that for a bar element in the local coordinate system,

We now want to relate the global element nodal forces f to the global nodal displacement d for a bar element arbitrarily oriented with respect to the global axes as was shown in Figure 3-1.

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We observe from Eq. (3.4.3) that a total of four components of force and four of displacement arise when global coordinates are used (see figure 3.5).

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Figure 3-5 Bar element in global coordintase

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Ch. 3 Truss Equations3.5. Computational of Stress for a Bar in x-y PlaneWe will now consider the determination of the stress in a bar element. For a bar, the local forces are related to the local displacement by Eq. (3.1.13). This equation is repeated here for convenience.

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Figure 3-6 Basic bar element with positive nodal forces

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Ch. 3 Truss Equations3.6. Solution of a Plane TrussWe will now illustrate the use of equations developed in Section 3.4 qnd 3.5, along with the direct stiffness method of assembling the total matrix and equatons, to solve the following plane truss example problem.A plane truss is a structure composed of bar elements all lying in a common plane that connected together by frictionless pins. The plane truss also must have loads acting only in common plane.EXAMPLE 3.5

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Ch. 3 Truss Equations3.7.Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional SpaceWe will now derive the tranformation matrix for a bar element in 3-D space as shown in Figure 3-7.

Figure 3-7 Bar in 3-D space3.7.Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space

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Ch. 3 Truss Equations3.8.Potential Energy ApproachWe now present the principle of minimum potential energy (POMPE) to derive the bar element equations. Recall from Section 2.6 that the total PE, p was defined as the sum of the internal strain energy U and the potential energy of the external forces as

To evaluate the strain energy for a bar, we consider only the work done by the internal forces during deformation. Because we are dealing with a 1-D bar, the internal force doing work is given in Figure 3-8 as x(y)(z), due only to normal stress x. The displacement of the x face of element is x(x); the displacement of x + x face is x(x + dx). The change in displacement is then xdx, where dx is differential change in strain occuring over element x.

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Figure 3-8 Internal force in a 1-D bar

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Ch. 3 Truss EquationsNow, for a linear-elastic (Hookes law) material as shown in Figure 3-9, we see that x = Ex. Hence, substituting this relationship into Eq. (3.8.4), integrating with respect to x, and then substituting x for Ex, we have

Figure 3-9 Linear-elastic (Hookes law) material

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Ch. 3 Truss EquationsFigure 3-10General forces actingon a 1-D bar

In Eq. (3.8.5) and (3.8.6), V is the volume of the body and S1 is the part of the surface S on which surface loading acts. For a bar element with two nodes and one d.o.f per node, M = 2.We are now ready to describe the FE formulation of the bar element equations using th POMPE.The FE process seeks a minimum in the PE within the constraint of an assumed displacement pattern within each element. The greater the number of d.o.f. associated with the element, the more closely will the solution approximate the true one and ensure complete equilibrium. An approximate FE solution using the stiffness method will always provide an approximation value of PE greater than or equal to the correct one.

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Ch. 3 Truss EquationsThe method also results in a structure behavior that is predicted to be physically stiffer than, or at best to have the same stiffness as, the actual one. This is explained by the fact that structure model is allowed to displaced only into shapes defined by the terms of the assumed displacement field within each element of the structure. The correct shape is ussually only approximated by the assumed field, although the correct shape can be the same as the assumed field. The assumed field effectively constraints the structure from deforming in its natural manner, This constraint effect stiffens the predicted behavior of the structure.Apply the following steps when using the POMPE to derive the FE equations.Formulate an expression for the total PE.Assume the displacement pattern to vary with a finite set of undetermined parameters (nodal displacements).Obtain a set of simultaneous equations minimizing the total PE with respect to these nodal displacements.

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Ch. 3 Truss EquationsThe resulting equations are the approximate (or possibly exact) equilibrium equations whose solution for the nodal parameters seeks to minimize the PE when back-substitued into the PE expression. The proceeding three steps will now followed to derive the bar element equations and stiffness matrix.Consider the bar element of length L, with constant cross-sectional area A, shown in Figure 3-10. Using Eqs. (3.8.5) and (3.8.6), the total PE, Eq. (3.8.1), becomes.

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Ch. 3 Truss Equations3.9.Galerkins Residual MethodWe have develop the bar FE equations by the direct method in Section 3.1 and by the PE method (one of number of variation methods) in Section 3.8.In fields other than structural/solids mechanics, it is quite probable that a variational principle, analogous to the principle of minimum PE, for instance, may not be known or even exist. In some flow problems in fluid mechanics and in mass transport problems, we often have the differential equations and BC available.However, the FE method can still be applied.The weighted residual method (WRM) applied directly to the differential equation can be used to develop the FE equations.In this section, we describe Galerkins residual method (GRM) in general and apply it to the bar element.

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Ch. 3 Truss EquationsThis development provides the basis for later applications of GRM to the beam element in Chapter 4 and to the non-structural problems.There are a number of other WRM. Among these are collocation, sub-domain method, least square, and least square collocation. (For more on these methods, see Reference [4]). However, since GRM is more well known than the other WRM, it is the only one described in this text.In WRM, a trial or approximate function is chosen to approximate the independent variable, such as a displacement or a temperature, in a problem defined by a differential equation. This trail function will not, in general, satisfy the governing differential equation. Thus, the substitution of the trail function into the differential equation result in a residual over the whole region of the problem as follows

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Ch. 3 Truss EquationsIn the WRM, we require that a weighted value of the residual be a minimum over the whole region. The weighting functions allow the weighted integral of residuals to go to zero. Denoting the weighting function by W, the general form of the weighted residual integral is

Using GRM, we chose the interpolation function, such as Eq. (3.1.3), in terms of Ni shape functions for the independent variable in the differential equation. In general, this substitution yields the residual R0.By the Galerkin criterion, the shape functions are chosen to play the role of the weighting functions W. Thus, for each i we have.

Eq. (3.9.3) results in total of n equations.

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Ch. 3 Truss EquationsEquation (3.9.3) applies to points within the region of a body without reference to BC such as specified applied loads or displacement.To obtain BC, we apply integration by parts to Eq. (3.9.3), which yields integrals applicable for the region and its boundary.Bar Element FormulationWe now use GRM to formulate the bar element stiffness equations. We begin with the basic element differential equation, without distributed load, derived in Section 3.1 (Eq. (d) as

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Ch. 3 Truss EquationsThe residual R is now defined to be Eq. (3.9.4). Applying Galerkins criterion, Eq. (3.9.3), to Eq. (3.9.4), we have

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Ch. 3 Truss EquationsAE(du/dx = AEepsilon = A x Sigma = gaya57

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Ch. 3 Truss EquationsReference:Logan, D.L., 1992, A First Course in the Finite Element Method, PWS-KENT Publishing Co., Boston.Imbert, J.F.,1984, Analyse des Structures par Elements Finis, 2nd Ed., Cepadues.Zienkiewics, O.C., 1977, The Finite Eelement Method, 3rd ed., McGraw-Hill, London.Finlayson, B.A., 1972, The Method of Weighted Residuals and Variational Principles, Academic Press, New York.

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Ch#3 Truss Equations (1)

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