Hutton - Fundamentals of Finite Element Analysis, страница 88

The instructor may choose to change loading,material properties, or geometry for any of these problems at his or her discretion.E.1 CHAPTER 3Problems E3.1–E3.7 involve two-dimensional trusses to be modeled using thebar element (in some analysis software this may be called a bar, link, spar, ortruss element). In each problem, determine the magnitude and location of themaximum deflection, the stress in each member, and the reaction forces. Use thecomputed reaction forces to check the equilibrium.

Node numbers, where included, are for reference only and can be changed at the analyst’s discretion.E3.1 Each member is steel having E = 30(10 6 ) psi and the cross-sectional area is 1.2 in.21000 lb71000 lb1000 lb951000 lb1000 lb6 ft113500 lb1Problem E3.1476423 ft3 ft63 ft83 ft103 ft500 lb123 ftHutton: Fundamentals ofFinite Element AnalysisBack MatterAppendix E: Problems forComputer Solution© The McGraw−HillCompanies, 2004E.1 Chapter 3E3.2 All members are hollow circular tubing having outside diameter 100 mm andwall thickness 10 mm. The modulus of elasticity is 207 GPa.5 kN9 kN59 kN73m5 kN936484m104m4m9m123m3mProblem E3.2E3.3 All truss members are 2 × 4 lumber having E = 3(106) psi.600 lb12 ft12 ft12 ft9 ft9 ft6700 lb400 lb4700 lb718 ft31218 ft8518 ft18 ftProblem E3.3E3.4 The truss members are square tubular aluminum members having 2.5 outsidedimension and 0.25 wall thickness.

The modulus of elasticity is 107 psi.E3.5 All members are identical with cross-sectional area of 1.6 in.2 and modulus ofelasticity 15(106) psi.E3.6 The horizontal members are solid, square steel bars having basic dimension30 mm; all other members are flat steel sheet stock 6 mm thick by 50 mmwide. Use E = 207 GPa.

Are the computed stresses reasonable for structuralsteel?477Hutton: Fundamentals ofFinite Element Analysis478Back MatterAppendix E: Problems forComputer SolutionAPPENDIX E© The McGraw−HillCompanies, 2004Problems for Computer Solution5000 lb3000 lb3000 lb73 ft953 ft1000 lb1000 lb633 ft11843 ft1023 ft1126 ft6 ft6 ft6 ft6 ft6 ftProblem E3.4800 lb3400 lb250 lb200 lb1242 ft14 ft52 ftProblem E3.55 kN5 kN22m2m5 kN2m583 kN2m104 kN113m479 3 kN3m1363 kNProblem E3.65 ftHutton: Fundamentals ofFinite Element AnalysisBack MatterAppendix E: Problems forComputer Solution© The McGraw−HillCompanies, 2004E.2 Chapter 4E3.7 The truss is composed of solid circular steel members 2 in diameter.

Themodulus of elasticity is 30(106) psi, Other than deflection and stress, whatconcerns should be considered with this truss? Does your answer relate to therelatively low computed stress values?2000 lb41000 lb1200 lb4 ft1500 lb368 ft6 ft212 ft56 ft12 ft17Problem E3.7E.2 CHAPTER 4The problems in this section deal with frame structures (that is, structures inwhich the joints are fixed and transmit bending moment, unlike the pin jointassumption of trusses).Solve problems E3.1–E3.7, respectively, assuming that the joints are fixedas in welded or riveted joints.

Additional required information is as follows.For E3.1,Iz = 0.4 in.4For E3.5,Iz = 0.53 in.4E4.8 The figure shows a basic model of a bicycle frame. All members are 1 diametercircular tubing having wall thickness 0.1 and are made of titanium, which has amodulus of elasticity of 15(106) psi. Determine the maximum deflection and thestress in each member. The nodal coordinates (in inches) are as follows:E4.1–E4.71234567xy01811927293600141818140479Hutton: Fundamentals ofFinite Element Analysis480Back MatterAppendix E: Problems forComputer SolutionAPPENDIX E© The McGraw−HillCompanies, 2004Problems for Computer Solution300 lb50 lb4563172Problem E4.8E4.9Determine the maximum deflection and maximum stress in the frame structureshown if the structural members are 1 diameter, solid aluminum tubes forwhich E = 10(10 6 ) psi.1800 ft · lb200 lb/ft2500 lb38 ft148 ftProblem E4.9E4.10The figure shows an arch that is the main support structure for a footbridge.The arch is constructed of standard AISC 6I17.5 I-beams (height = 6 in.;A = 5.02 in.2; I z = 26.0 in.4).

Use straight beam elements to model thisbridge and examine convergence of solution as the number of elementsis increased from 6 to 12 to 18. In examining convergence, look at bothdeflection and stress. Also note that, owing to the direction of loading, axialeffects must be included. E = 30(10 6 ) psi.5 ft5 ft1500 lb5 ft5 ft2000 lb5 ft1500 lb1000 lb1000 lb6 ft15 ftProblem E4.105 ft15 ftHutton: Fundamentals ofFinite Element AnalysisBack MatterAppendix E: Problems forComputer Solution© The McGraw−HillCompanies, 2004E.3 Chapter 7E4.11The frame structure shown is composed of 10 mm × 10 mm solid squaremembers having E = 100 GPa.

Determine the maximum deflection, maximumslope, and maximum stress.500 N/m3m2.5 m1.5 mProblem E4.11E4.12The structure shown is a model of the support for a freeway light post.For uniformity in wind loading, the structural members are circular. Theoutside diameter of each member is 3.0 and wall thickness is 0.25 .Compute the deflection at each structural joint and determine maximumstresses. Examine the effects on your solution of using more elements(i.e., refine the mesh). E = 10 7 psi.3.75 ft2 ft800 lb8 ft4 ft1500 lbProblem E4.12E.3 CHAPTER 7E7.1E7.2A tapered circular heat transfer pin (known as a pin fin) is insulated all aroundits circumference, as shown.

The large end (D = 12 mm) is maintained at aconstant temperature of 90 ◦ C , while the smaller end (D = 6 mm) is at 30 ◦ C .Determine the steady-state heat flow through the pin using a mesh of straightelements. Thermal conductivity of the material is k = 200 W/m- ◦ C .A rectangular duct in a home heating system has dimensions 12 × 18 asshown. The duct is insulated with a uniform layer of fiberglass 1 thick. The481Hutton: Fundamentals ofFinite Element Analysis482Back MatterAPPENDIX EAppendix E: Problems forComputer Solution© The McGraw−HillCompanies, 2004Problems for Computer SolutionInsulated12 mm6 mm150 mmProblem E7.1duct (steel sheet metal) is maintained at a constant temperature of 115 ◦ F.

Theambient air temperature around the duct is 55 ◦ F.Ta ⫽ 55⬚F115⬚F12 in.18 in.Problem E7.2(a) Calculate the temperature distribution in the insulation and the heat loss perunit length to the surrounding air. Thermal conductivity of the insulationis uniform in all directions and has value k = 0.025 Btu/hr-ft-◦ F; theconvection coefficient to the ambient air is h = 5 Btu/hr-ft2-◦ F.(b) Repeat the calculations for an insulation thickness of 2 in.E7.3 The figure represents a cross section of a long bar insulated on the upper andlower surfaces; hence, the problem is to be treated as two dimensional on a perunit length basis.

The left edge is maintained at constant temperature of 100 ◦ Cand the right edge is maintained at 26 ◦ C. The material has uniform conductivityk = 35 W/m-◦ C. Determine the temperature distribution and the steady-stateheat flow rate. What element should you use? (Triangular, square? Perform theanalysis with different elements to observe differences in the solutions.) Refinethe mesh and examine convergence.100⬚C45 cm26⬚C100 cmProblem E7.3E7.4 A thin copper tube (12 mm diameter) containing water at an average temperatureof 95 ◦ C is imbedded in a long slender solid slab, as shown. The vertical edgesHutton: Fundamentals ofFinite Element AnalysisBack MatterAppendix E: Problems forComputer Solution© The McGraw−HillCompanies, 2004E.3 Chapter 7are insulated.

The horizontal edges are exposed to an ambient temperature of20 ◦ C and the associated convection coefficient is h = 20 W/m 2 -◦ C . The materialhas uniform conductivity k = 200 W/m-◦ C. Compute the net steady-state heattransfer rate and the temperature distribution in the cross section.Ta ⫽ 20⬚C25 mm95⬚C50 mmProblem E7.4E7.5 The figure shows a horizontal cross section of a chimney exhausting the gasesgenerated by a wood stove. The flue is insulated with firebrick 6 thick and havinguniform conductivity k = 2.5 Btu/hr-ft-◦ F.

The chimney is surrounded by air atambient temperature 40 ◦ F and the convection coefficient is 5 Btu/hr-ft2-◦ F. Deter-mine the temperature distribution in the firebrick and the heat loss per unit length.6 in.1200⬚F12 in.Ta ⫽ 40⬚FProblem E7.5E7.6 The heat transfer fin shown is attached to a pipe conveying a fluid at averagetemperature 120 ◦ C.

The thickness of the fin is 3 mm. The fin is surroundedby air at temperature 30 ◦ C and subject to convection on all surfaces withh = 20 W/m2-◦ C. The fin material has uniform conductivity k = 50 W/m-◦ C.Determine the heat transfer rate from the fin and the temperature distributionin the fin.10⬚35 mm25 mm120⬚C100 mmProblem E7.6483Hutton: Fundamentals ofFinite Element Analysis484Back MatterAPPENDIX EAppendix E: Problems forComputer Solution© The McGraw−HillCompanies, 2004Problems for Computer SolutionE7.7 The cross section shown is of a campus footbridge, having embedded heat cablesto prevent ice accumulation.

The vertical edges are insulated and the horizontalsurfaces are at the steady temperatures shown. The material has uniform conductivity k = 0.6 Btu/hr-ft-◦ F and the cables have source strength 200 Btu/hr-in.Compute the net heat transfer rate and the temperature distribution.32⬚F2 in.12 in.15⬚F8 in.3 in.Problem E7.7E.4 CHAPTER 9E9.1 The cantilever beam shown is subjected to a concentrated load F applied at theend.

Model this beam using three-dimensional brick elements and compare thefinite element solution to elementary beam theory. How do you apply theconcentrated load in the FE model?LhFbProblem E9.1E9.2 Refer to a standard mechanical design text and obtain the geometric parametersof a standard involute gear tooth profile. Assuming a tooth to be fixed at the rootdiameter, determine the stress distribution in a gear tooth when the load acts at(a) at the tip of the tooth and (b) the pitch diameter.