김경종
365 lines
16 KiB
Markdown
365 lines
16 KiB
Markdown
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Galerkin’s method, along with the collocation, the least squares, and the subdomain weighted residual methods are introduced in Chapter 3. To illustrate each method, they will all be used to solve a one-dimensional bar problem for which a known exact solution exists for comparison. As the more easily adapted residual method, Galerkin’s method will also be used to derive the bar element equations in Chapter 3 and the beam element equations in Chapter 4 and to solve the combined heat-conduction/convection/mass transport problem in Chapter 13. For more information on the use of the methods of weighted residuals, see Reference [36]; for additional applications to the finite element method, consult References [37] and [38].
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Using any of the methods just outlined will produce the equations to describe the behavior of an element. These equations are written conveniently in matrix form as
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$$
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\left\{ \begin{array}{l} f _ {1} \\ f _ {2} \\ f _ {3} \\ \vdots \\ f _ {n} \end{array} \right\} = \left[ \begin{array}{c c c c c} k _ {1 1} & k _ {1 2} & k _ {1 3} & \dots & k _ {1 n} \\ k _ {2 1} & k _ {2 2} & k _ {2 3} & \dots & k _ {2 n} \\ k _ {3 1} & k _ {3 2} & k _ {3 3} & \dots & k _ {3 n} \\ \vdots & & & & \vdots \\ k _ {n 1} & & & \dots & k _ {n n} \end{array} \right] \left\{ \begin{array}{l} d _ {1} \\ d _ {2} \\ d _ {3} \\ \vdots \\ d _ {n} \end{array} \right\} \tag {1.4.4}
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$$
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or in compact matrix form as
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$$
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\{f \} = [ k ] \{d \} \tag {1.4.5}
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$$
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where f f g is the vector of element nodal forces, ½k is the element stiffness matrix (normally square and symmetric), and fdg is the vector of unknown element nodal degrees of freedom or generalized displacements, n. Here generalized displacements may include such quantities as actual displacements, slopes, or even curvatures. The matrices in Eq. (1.4.5) will be developed and described in detail in subsequent chapters for specific element types, such as those in Figure 1–1.
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# Step 5 Assemble the Element Equations to Obtain the Global or Total Equations and Introduce Boundary Conditions
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In this step the individual element nodal equilibrium equations generated in step 4 are assembled into the global nodal equilibrium equations. Section 2.3 illustrates this concept for a two-spring assemblage. Another more direct method of superposition (called the direct stiffness method ), whose basis is nodal force equilibrium, can be used to obtain the global equations for the whole structure. This direct method is illustrated in Section 2.4 for a spring assemblage. Implicit in the direct stiffness method is the concept of continuity, or compatibility, which requires that the structure remain together and that no tears occur anywhere within the structure.
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The final assembled or global equation written in matrix form is
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$$
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\{F \} = [ K ] \{d \} \tag {1.4.6}
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$$
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where fF g is the vector of global nodal forces, ½K is the structure global or total stiffness matrix, (for most problems, the global stiffness matrix is square and symmetric) and fdg is now the vector of known and unknown structure nodal degrees of freedom or generalized displacements. It can be shown that at this stage, the global stiffness matrix ½K is a singular matrix because its determinant is equal to zero. To remove this singularity problem, we must invoke certain boundary conditions (or constraints or supports) so that the structure remains in place instead of moving as a rigid body. Further details and methods of invoking boundary conditions are given in subsequent chapters. At this time it is sufficient to note that invoking boundary or support conditions results in a modification of the global Eq. (1.4.6). We also emphasize that the applied known loads have been accounted for in the global force matrix fF g.
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# Step 6 Solve for the Unknown Degrees of Freedom (or Generalized Displacements)
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Equation (1.4.6), modified to account for the boundary conditions, is a set of simultaneous algebraic equations that can be written in expanded matrix form as
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$$
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\left\{ \begin{array}{c} F _ {1} \\ F _ {2} \\ \vdots \\ F _ {n} \end{array} \right\} = \left[ \begin{array}{c c c c} K _ {1 1} & K _ {1 2} & \dots & K _ {1 n} \\ K _ {2 1} & K _ {2 2} & \dots & K _ {2 n} \\ \vdots & & & \vdots \\ K _ {n 1} & K _ {n 2} & \dots & K _ {n n} \end{array} \right] \left\{ \begin{array}{c} d _ {1} \\ d _ {2} \\ \vdots \\ d _ {n} \end{array} \right\} \tag {1.4.7}
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$$
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where now n is the structure total number of unknown nodal degrees of freedom. These equations can be solved for the ds by using an elimination method (such as Gauss’s method) or an iterative method (such as the Gauss–Seidel method). These two methods are discussed in Appendix B. The ds are called the primary unknowns, because they are the first quantities determined using the stiffness (or displacement) finite element method.
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# Step 7 Solve for the Element Strains and Stresses
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For the structural stress-analysis problem, important secondary quantities of strain and stress (or moment and shear force) can be obtained because they can be directly expressed in terms of the displacements determined in step 6. Typical relationships between strain and displacement and between stress and strain—such as Eqs. (1.4.1) and (1.4.2) for one-dimensional stress given in step 3—can be used.
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# Step 8 Interpret the Results
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The final goal is to interpret and analyze the results for use in the design/analysis process. Determination of locations in the structure where large deformations and large stresses occur is generally important in making design/analysis decisions. Postprocessor computer programs help the user to interpret the results by displaying them in graphical form.
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# 1.5 Applications of the Finite Element Method
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The finite element method can be used to analyze both structural and nonstructural problems. Typical structural areas include
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1. Stress analysis, including truss and frame analysis, and stress concentration problems typically associated with holes, fillets, or other changes in geometry in a body
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2. Buckling
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3. Vibration analysis
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Nonstructural problems include
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1. Heat transfer
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2. Fluid flow, including seepage through porous media
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3. Distribution of electric or magnetic potential
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Finally, some biomechanical engineering problems (which may include stress analysis) typically include analyses of human spine, skull, hip joints, jaw/gum tooth implants, heart, and eye.
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We now present some typical applications of the finite element method. These applications will illustrate the variety, size, and complexity of problems that can be solved using the method and the typical discretization process and kinds of elements used.
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Figure 1–2 illustrates a control tower for a railroad. The tower is a threedimensional frame comprising a series of beam-type elements. The 48 elements are labeled by the circled numbers, whereas the 28 nodes are indicated by the uncircled numbers. Each node has three rotation and three displacement components associated with it. The rotations (ys) and displacements (ds) are called the degrees of freedom. Because of the loading conditions to which the tower structure is subjected, we have used a three-dimensional model.
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The finite element method used for this frame enables the designer/analyst quickly to obtain displacements and stresses in the tower for typical load cases, as required by design codes. Before the development of the finite element method and the computer, even this relatively simple problem took many hours to solve.
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The next illustration of the application of the finite element method to problem solving is the determination of displacements and stresses in an underground box culvert subjected to ground shock loading from a bomb explosion. Figure 1–3 shows the discretized model, which included a total of 369 nodes, 40 one-dimensional bar or truss elements used to model the steel reinforcement in the box culvert, and 333 plane strain two-dimensional triangular and rectangular elements used to model the surrounding soil and concrete box culvert. With an assumption of symmetry, only half of the box culvert need be analyzed. This problem requires the solution of nearly 700 unknown nodal displacements. It illustrates that different kinds of elements (here bar and plane strain) can often be used in one finite element model.
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Another problem, that of the hydraulic cylinder rod end shown in Figure 1–4, was modeled by 120 nodes and 297 plane strain triangular elements. Symmetry was also applied to the whole rod end so that only half of the rod end had to be analyzed,
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<details>
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<summary>text_image</summary>
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</details>
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Figure 1–2 Discretized railroad control tower (28 nodes, 48 beam elements) with typical degrees of freedom shown at node 1, for example (By D. L. Logan)
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as shown. The purpose of this analysis was to locate areas of high stress concentration in the rod end.
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Figure 1–5 shows a chimney stack section that is four form heights high (or a total of 32 ft high). In this illustration, 584 beam elements were used to model the vertical and horizontal stiffeners making up the formwork, and 252 flat-plate elements were used to model the inner wooden form and the concrete shell. Because of the irregular loading pattern on the structure, a three-dimensional model was necessary. Displacements and stresses in the concrete were of prime concern in this problem.
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<details>
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<summary>text_image</summary>
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y
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Soil
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Box
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culvert
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Fixed nodes along
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this edge
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Fixed nodes along bottom
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</details>
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Figure 1–3 Discretized model of an underground box culvert (369 nodes, 40 bar elements, and 333 plane strain elements) [39]
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Figure 1–6 shows the finite element discretized model of a proposed steel die used in a plastic film-making process. The irregular geometry and associated potential stress concentrations necessitated use of the finite element method to obtain a reasonable solution. Here 240 axisymmetric elements were used to model the threedimensional die.
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Figure 1–7 illustrates the use of a three-dimensional solid element to model a swing casting for a backhoe frame. The three-dimensional hexahedral elements are
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<details>
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<summary>text_image</summary>
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Applied loads
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y
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Fixed nodes
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</details>
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Figure 1–4 Two-dimensional analysis of a hydraulic cylinder rod end (120 nodes, 297 plane strain triangular elements)
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<details>
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<summary>text_image</summary>
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Concrete shell (plate elements)
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Inner form (plate elements)
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Vertical stiffener (beam elements)
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Rigid rods
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Adjustable rod
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Angle ring
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Sling cable
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Whaler
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(beam
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elements)
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Concrete shell
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(plate elements)
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</details>
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Figure 1–5 Finite element model of a chimney stack section (end view rotated 45
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) (584 beam and 252 flat-plate elements) (By D. L. Logan)
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<details>
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<summary>text_image</summary>
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Fixed edge
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Axis of
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symmetry
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8.5 in.
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</details>
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Figure 1–6 Model of a high-strength steel die (240 axisymmetric elements) used in the plastic film industry [40]
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necessary to model the irregularly shaped three-dimensional casting. Two-dimensional models certainly would not yield accurate engineering solutions to this problem.
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Figure 1–8 illustrates a two-dimensional heat-transfer model used to determine the temperature distribution in earth subjected to a heat source—a buried pipeline transporting a hot gas.
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Figure 1–9 shows a three-dimensional finite element model of a pelvis bone with an implant, used to study stresses in the bone and the cement layer between bone and implant.
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Finally, Figure 1–10 shows a three-dimensional model of a 710G bucket, used to study stresses throughout the bucket.
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These illustrations suggest the kinds of problems that can be solved by the finite element method. Additional guidelines concerning modeling techniques will be provided in Chapter 7.
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# 1.6 Advantages of the Finite Element Method
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As previously indicated, the finite element method has been applied to numerous problems, both structural and nonstructural. This method has a number of advantages that have made it very popular. They include the ability to
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1. Model irregularly shaped bodies quite easily
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2. Handle general load conditions without difficulty
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<details>
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<summary>natural_image</summary>
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3D wireframe model of a mechanical bracket or bracket structure (no text or symbols)
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</details>
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Figure 1–7 Three-dimensional solid element model of a swing casting for a backhoe frame
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3. Model bodies composed of several different materials because the element equations are evaluated individually
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4. Handle unlimited numbers and kinds of boundary conditions
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5. Vary the size of the elements to make it possible to use small elements where necessary
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6. Alter the finite element model relatively easily and cheaply
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7. Include dynamic effects
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8. Handle nonlinear behavior existing with large deformations and nonlinear materials
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The finite element method of structural analysis enables the designer to detect stress, vibration, and thermal problems during the design process and to evaluate design changes before the construction of a possible prototype. Thus confidence in the acceptability of the prototype is enhanced. Moreover, if used properly, the method can reduce the number of prototypes that need to be built.
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Even though the finite element method was initially used for structural analysis, it has since been adapted to many other disciplines in engineering and mathematical physics, such as fluid flow, heat transfer, electromagnetic potentials, soil mechanics, and acoustics [22–24, 27, 42–44].
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<details>
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<summary>text_image</summary>
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50°F
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15 ft
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160°F
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Pipeline
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Earth
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</details>
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<details>
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<summary>text_image</summary>
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20 ft
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20 ft
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15 ft
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15 ft
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</details>
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Figure 1–8 Finite element model for a two-dimensional temperature distribution in the earth
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<details>
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<summary>natural_image</summary>
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3D wireframe model of a human head and neck, rendered in mesh with no text or symbols
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</details>
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Figure 1–9 Finite element model of a pelvis bone with an implant (over 5000 solid elements were used in the model) (> Thomas Hansen/Courtesy of Harrington Arthritis Research Center, Phoenix, Arizona) [41]
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<details>
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<summary>text_image</summary>
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Taper Beams, The Loader Lift Arm
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Parabolic Beam, The Loader Guide Link
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Linear Beams, The Loader Power Link
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Linear Beams, The Lift Arm Cylinders
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The Bucket
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The Loader Coupler
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</details>
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Fig u re 1 – 1 0 F i n ite e l em ent m od el of a 7 1 0G b ucket with 1 69,5 95 e l e ments a n d 1 85,026 n od es u sed (i n cl ud i n g 78,566 th i n s h e l l l i n ea r q uad ri l ate ra l e l e m e nts fo r t h e b u cket a n d co u p l e r, 83, 1 04 so l id l i n ea r b ric k e l e m e nts to m od e l th e bosses, a n d 2 1 2 bea m e l e m e n ts to m od e l l ift a r m s, l ift a r m cyl i n d e rs, a n d g u i d e l i n ks) (Co u rtesy of Yo u s if O m e r, St r u ct u ra l Des i g n E n g i n ee r, Co n st ru ct i o n a n d Forestry Divisio n, Jo h n Deere Du buq ue Works)
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