Hutton - Fundamentals of Finite Element Analysis, страница 12

(b) If k = 50 lb./in., F1 = 20 lb., and F2 = 15 lb., computethe displacement of each trolley and the force in each spring.F12kF2k2kkFigure P2.72.8Use Castigliano’s first theorem to obtain the matrix equilibrium equations for thesystem of springs shown in Figure P2.8.F21F32k1k23k3F44k45Figure P2.82.92.10In Problem 2.8, let k 1 = k 2 = k 3 = k 4 = 10 N/mm, F2 = 20 N, F3 = 25 N,F4 = 40 N and solve for (a) the nodal displacements, (b) the reaction forces atnodes 1 and 5, and (c) the force in each spring.A steel rod subjected to compression is modeled by two bar elements, as shownin Figure P2.10. Determine the nodal displacements and the axial stress in eachelement. What other concerns should be examined?0.5 m10.5 m12 kN2E 207 GPaFigure P2.103A 500mm249Hutton: Fundamentals ofFinite Element Analysis502.

Stiffness Matrices,Spring and Bar ElementsCHAPTER 22.11Text© The McGraw−HillCompanies, 2004Stiffness Matrices, Spring and Bar ElementsFigure P2.11 depicts an assembly of two bar elements made of differentmaterials. Determine the nodal displacements, element stresses, and thereaction force.A1, E1, L1A2, E2, L2120,000 lb.32A1 4 in.2E1 15 106 lb./in.2L1 20 in.A2 2.25 in.2E2 10 106 lb./in.2L2 20 in.Figure P2.112.12Obtain a four-element solution for the tapered bar of Example 2.4. Plot elementstresses versus the exact solution. Use the following numerical values:E = 10 × 10 6 lb./in.22.132.14A0 = 4 in.2L = 20 in.P = 4000 lb.A weight W is suspended in a vertical plane by a linear spring having springconstant k. Show that the equilibrium position corresponds to minimum totalpotential energy.For a bar element, it is proposed to discretize the displacement function asu(x ) = N 1 (x )u 1 + N 2 (x )u 2with interpolation functions2.15N 1 (x ) = cosx2LN 2 (x ) = sinx2LAre these valid interpolation functions? (Hint: Consider strain and stressvariations.)The torsional element shown in Figure P2.15 has a solid circular cross sectionand behaves elastically.

The nodal displacements are rotations 1 and 2 and theassociated nodal loads are applied torques T1 and T2. Use the potential energyprinciple to derive the element equations in matrix form.␪1, T1RLFigure P2.15␪2, T2Hutton: Fundamentals ofFinite Element Analysis3.

Truss Structures: TheDirect Stiffness MethodText© The McGraw−HillCompanies, 2004C H A P T E R3Truss Structures:The Direct StiffnessMethod3.1 INTRODUCTIONThe simple line elements discussed in Chapter 2 introduced the concepts ofnodes, nodal displacements, and element stiffness matrices. In this chapter, creation of a finite element model of a mechanical system composed of any numberof elements is considered. The discussion is limited to truss structures, which wedefine as structures composed of straight elastic members subjected to axialforces only. Satisfaction of this restriction requires that all members of the trussbe bar elements and that the elements be connected by pin joints such that eachelement is free to rotate about the joint. Although the bar element is inherentlyone dimensional, it is quite effectively used in analyzing both two- and threedimensional trusses, as is shown.The global coordinate system is the reference frame in which displacements of the structure are expressed and usually chosen by convenience in consideration of overall geometry.

Considering the simple cantilever truss shown inFigure 3.1a, it is logical to select the global XY axes as parallel to the predominant geometric “axes” of the truss as shown. If we examine the circled joint, forexample, redrawn in Figure 3.1b, we observe that five element nodes are physically connected at one global node and the element x axes do not coincide withthe global X axis. The physical connection and varying geometric orientationof the elements lead to the following premises inherent to the finite elementmethod:1. The element nodal displacement of each connected element must be thesame as the displacement of the connection node in the global coordinatesystem; the mathematical formulation, as will be seen, enforces thisrequirement (displacement compatibility).51Hutton: Fundamentals ofFinite Element Analysis3.

Truss Structures: TheDirect Stiffness Method52CHAPTER 3Text© The McGraw−HillCompanies, 2004Truss Structures: The Direct Stiffness Method733Y2146810295(a)746X(b)Figure 3.1(a) Two-dimensional truss composed of ten elements. (b) Truss joint connecting fiveelements.2. The physical characteristics (in this case, the stiffness matrix and elementforce) of each element must be transformed, mathematically, to the globalcoordinate system to represent the structural properties in the global systemin a consistent mathematical frame of reference.3. The individual element parameters of concern (for the bar element, axialstress) are determined after solution of the problem in the global coordinatesystem by transformation of results back to the element reference frame(postprocessing).Why are we basing the formulation on displacements? Generally, a designengineer is more interested in the stress to which each truss member is subjected,to compare the stress value to a known material property, such as the yieldstrength of the material.

Comparison of computed stress values to material properties may lead to changes in material or geometric properties of individual elements (in the case of the bar element, the cross-sectional area). The answer to thequestion lies in the nature of physical problems. It is much easier to predict theloading (forces and moments) to which a structure is subjected than the deflections of such a structure. If the external loads are specified, the relations betweenloads and displacements are formulated in terms of the stiffness matrix and wesolve for displacements. Back-substitution of displacements into individual element equations then gives us the strains and stresses in each element as desired.This is the stiffness method and is used exclusively in this text.

In the alternateprocedure, known as the flexibility method [1], displacements are taken as theknown quantities and the problem is formulated such that the forces (more generally, the stress components) are the unknown variables. Similar discussion applies to nonstructural problems.

In a heat transfer situation, the engineer is mostoften interested in the rate of heat flow into, or out of, a particular device. Whiletemperature is certainly of concern, temperature is not the primary variable ofinterest. Nevertheless, heat transfer problems are generally formulated such thattemperature is the primary dependent variable and heat flow is a secondary,computed variable in analogy with strain and stress in structural problems.Returning to consideration of Figure 3.1b, where multiple elements are connected at a global node, the geometry of the connection determines the relationsHutton: Fundamentals ofFinite Element Analysis3.

Truss Structures: TheDirect Stiffness MethodText© The McGraw−HillCompanies, 20043.2Nodal Equilibrium Equationsbetween element displacements and global displacements as well as the contributions of individual elements to overall structural stiffness. In the direct stiffnessmethod, the stiffness matrix of each element is transformed from the elementcoordinate system to the global coordinate system. The individual terms of eachtransformed element stiffness matrix are then added directly to the global stiffnessmatrix as determined by element connectivity (as noted, the connectivity relationsensure compatibility of displacements at joints and nodes where elements areconnected).

For example and simply by intuition at this point, elements 3 and 7 inFigure 3.1b should contribute stiffness only in the global X direction; elements 2and 6 should contribute stiffness in both X and Y global directions; element 4should contribute stiffness only in the global Y direction. The element transformation and stiffness matrix assembly procedures to be developed in this chapterindeed verify the intuitive arguments just made.The direct stiffness assembly procedure, subsequently described, results inexactly the same system of equations as would be obtained by a formal equilibrium approach. By a formal equilibrium approach, we mean that the equilibriumequations for each joint (node) in the structure are explicitly expressed, includingdeformation effects.

This should not be confused with the method of joints [2],which results in computation of forces only and does not take displacement intoaccount. Certainly, if the force in each member is known, the physical propertiesof the member can be used to compute displacement. However, enforcing compatibility of displacements at connections (global nodes) is algebraically tedious.Hence, we have another argument for the stiffness method: Displacement compatibility is assured via the formulation procedure. Granted that we have to“backtrack” to obtain the information of true interest (strain, stress), but the backtracking is algebraic and straightforward, as will be illustrated.3.2 NODAL EQUILIBRIUM EQUATIONSTo illustrate the required conversion of element properties to a global coordinatesystem, we consider the one-dimensional bar element as a structural member of atwo-dimensional truss.

Via this relatively simple example, the assembly procedureof essentially any finite element problem formulation is illustrated. We choosethe element type (in this case we have only one selection, the bar element); specify the geometry of the problem (element connectivity); formulate the algebraicequations governing the problem (in this case, static equilibrium); specify theboundary conditions (known displacements and applied external forces); solvethe system of equations for the global displacements; and back-substitute displacement values to obtain secondary variables, including strain, stress, and reaction forces at constrained locations (boundary conditions). The reader is advisedto note that we use the term secondary variable only in the mathematical sense;strain and stress are secondary only in the sense that the values are computed afterthe general solution for displacements.

The strain and stress values are of primaryimportance in design.53Hutton: Fundamentals ofFinite Element Analysis3. Truss Structures: TheDirect Stiffness Method54CHAPTER 3Text© The McGraw−HillCompanies, 2004Truss Structures: The Direct Stiffness MethodU6U4U5F3Y2U6U3223F3XU5␪211YU2␪11XU1(a)(b)Figure 3.2(a) A two-element truss with node and element numbers. (b) Global displacement notation.Conversion of element equations from element coordinates to global coordinates and assembly of the global equilibrium equations are described first in thetwo-dimensional case with reference to Figure 3.2a. The figure depicts a simpletwo-dimensional truss composed of two structural members joined by pin connections and subjected to applied external forces.

The pin connections are takenas the nodes of two bar elements as shown; node and element numbers, as wellas the selected global coordinate system are also shown. The correspondingglobal displacements are shown in Figure 3.2b. The convention used here forglobal displacements is that U2i−1 is displacement in the global X direction ofnode i and U2i is displacement of node i in the global Y direction. The conventionis by no means restrictive; the convention is selected such that displacements inthe direction of the global X axis are odd numbered and displacements in thedirection of the global Y axis are even numbered. (In using FEM software, thereader will find that displacements are denoted in various fashions, UX, UY, UZ,etc.) Orientation angle ␪ for each element is measured as positive from the globalX axis to the element x axis, as shown.