Thumb for Mechanical Engineers 2011 Part 12 pptx

320 Rules of Thumb for Mechanical Engineers CREEP RUPTURE Creep is plastic deformation which increases over time under sustained loading at generally elevated temperatures. Stress rupture is the continuation of creep to the point where failure takes place. Metallic and nonmetallic materials vary in their susceptibility to creep, but most common structural materials exhibit creep at stress levels below the propor- tional limit at elevated temperatures which exceed one- third to one-half of the melting temperature. A few metals, such as lead and tin, will creep at ordinary temperatures. The typical strain-time diagram in Figure 39 for a material subject to creep illustrates the three stages of creep behavior. After the initial elastic deformation, the material exhibits a relatively short period of primary creep (stage l), where the plastic strain rises rapidly at first. Then the strain versus time curve flattens out. The flatter portion of the curve is referred to as the secondary or steady state creep (stage 2). This is the stage of most importance to the en- I 0 lime Figure 39. Three stages of creep behavior. gineer in the design process. The final or tertiary creep stage (stage 3) is characterized by an acceleration of the creep rate, which leads to rupture in a relatively short period of time. High stresses and high temperatures have comparable ef- fects. Quantitatively, as a function of temperature, a loga- rithmic relationship exists between stress and the creep rate. A number of empirical procedures are available to correlate stress, temperature, and time for creep in commercial alloys. The Larson-Miller parameter (Km) is an example of one of these procedures. The general form of the Larson-Miller equation is: KLM = (O.OOl)(T + 46O)(lOg ct + 20) where T = temperature in OF. t = time (hours) c = empirical parameter relating test specimens to design Creep strength is specified as the stress corresponding to a given amount of creep deformation over a defined period of time at a specified temperature, i.e., 0.5% creep in 10,000 hours at 1,200"F. The degree of creep that can be tolerated is a function of the application. In gas turbine en- gines, the creep deformation of turbine rotating components must be limited such that contact with the static structure does not occur. In such high temperature applications, stress rupture can occur if the combination of temperature and stress is too high and leads to fracture. As little as a 20" to 30°F increase in temperature or a 10% increase in stress can halve the creep rupture life. Murk's Hurzdbook [16] pro- vides some creep rate information for steels. component FINITE ELEMENT ANALYSIS Over the last 25 years, the finite element method (FEM) has become a standard tool for structural analysis. Ad- vances in computer technology and improvements in finite element analysis (FEA) software have made FEA both af- fordable and relatively easy to implement. Engineers have access to FEA codes on computers ranging hm mainfram es to personal computers. However, while FEA aids engineering judgment by providing a wealth of information, it is not a substitute. Stress and Strain 321 Overview FEM has its origins in civil engineering, but the method first matured and reached a higher state of development in the aerospace industry. The basis of FEM is the representation of a structure by an assemblage of subdivisions, each of which has a standardized shape with a finite number of degrees of freedom. These subdivisions are finite elements. Thus the continuum of the structure with an infinite number of degrees of freedom is approximated by a number of finite elements. The elements are connected at nodes, which are where the solutions to the problem are calculated. FEM proceeds to a solution through the use of stress and strain equations to caldate the deflections in each element pro- duced by the system of fom coming from adjacent elements through the nodal points. From the deflections of the nodal points, the strains and stresses are calculated. This procedure is complicated by the fact that the force at each node is de- pendent on the forces at every other node. The elements, like a system of springs, deflect until all the forces balance. The solution to the problem requires that a large number of simultaneous equations be solved, hence the need for ma- trix solutions and the computer. Each FEA program has its own library of one-, two-, and three-dimensional elements. The elements selected for an analysis should be capable of simulating the deformations to which the actual structure will be subjected, such as bending, shear, or torsion. One-dimensional Elements The term one-dimensional does not refer to the spatial location of the element, but rather indicates that the element will only respond in one dimension with respect to its local coordinate system. A mss element is an example of a one-dimensional element which can only support axial loads. See Figure 40. Figure 40. One-dimensional element. Two-dimensional Elements A general two-dimensional element can also span threedimensional space, but displacements and forces are limited to two of the three dimensions in its local coordinate system. "bo dimensional elements are categorized as plane stress, plane strain, or axisymmetric. Plane stress problems assume a small dimension in the longitudinal direction such as a thin circular plate loaded in the radial direction. As a result the shear and normal stresses in the longitudinal direction are zero. Plane strain problems pertain to situations where the longitudinal dimension is long and displacements and loads are not a function of this dimension. The shear and normal strains in the longitudinal direction are equal to zero. Axisymmetric elements are used to model components which are symmetric about their central axes, i.e., a volume of revolution. Cylinders with uniform internal or external pressures and turbine disks are examples of axisymmetric problems. Symmetry permits the assumption that there is no variation in stress or strain in the circumferential direction. Two-dimensional elements may be triangular or quadrilateral in shape. Lower order linear elements have only cor- ner nodes while higher order isoparametric elements may have one or two midsides per edge. The additional edge nodes allow the element sides to conform to curved boundaries in addition to providing a more accurate higher order displacement function. See Figure 41. Three-dimensional Elements Three-dimensional solid elements are used to model structures where forces and deflections act in all three di- rections or when a component has a complex geometry that does not permit two-dimensional analysis. Three-dimensional elements may be shell, hexahedra (bricks), or tetra- hedra; and depending upon the order may have one or two midside nodes per edge. See Figure 42. 322 Rules of Thumb for Mechanical Engineers U El Quadrilateral Elements Tiangular Elements Hexahedral elements @licks) Telrahedmt elements Figure 41. Two-dimensional elements. Figure 42. Three-dimensional elements. Modeling Techniques The choice of elements, element mesh density, bound- ary conditions, and constraints are critical to the ability of a model to provide an accurate representation of the phys- ical part under operating conditions. Element mesh density is a compromise between making the mesh coarse enough to minimize the compu- tation time and fine enough to provide for convergence of the numerical solution. Until a “feel” is developed for the number of elements necessary to adequately predict stresses, it is often necessary to modify the mesh density and make additional runs until solution convergence is achieved. Reduction of solution convergence error achieved by reducing element size without changing element order is known as h-convergence. Models intended for stress prediction require more elements than those used for thermal or dynamic analy- ses. Mesh density should be increased near areas of stress concentration, such as fillets and holes (Figure 43). Abrupt changes in element size should be avoided, as the mesh density transitions away from the stress concentration feature. Compared to linear comer noded elements, fewer higher order isoparametric elements are required to model a structure. In general, lower order 2D triangular elements and 3D tetrahedral solid elements are not ade- quate for structural analysis. Some finite element codes use an automated convergence analysis technique Figure 45. Increase mesh density near stress concen- trations. known as the p-convergence method. This method maintains the same number of elements while increasing the order of the elements until solution convergence is achieved OT the maximum available element order is reached. Convergence of the maximum principal stress is a much better indicator than the maximum Von Mises equivalent stress. The equivalent stress is a local mea- sure and does not converge as smoothly as the maximum principal stress. Stressandstrain 323 Elements with large aspect ratios should be avoided. For two-dimensional elements, the aspect ratio is the ratio of the larger dimension to the smaller dimension. While an aspect ratio of one would be ideal, the maximum allow- able element aspect ratio is really a function of the stress field in the component. Larger aspect ratios with a value of 10 may be acceptable for models of components such as cylinders subjected only to an axial load. Generally, the largest aspect ratio should be on the order of 5. Highly distorted elements should be avoided. Two-dimensional quadrilateral and threedimensional brick elements should have comers which are approximately right angled and resemble rectangles and cubes re- spectively as much as possible, particularly in regions of high stress gradient. The angle between adjacent edges of an element should not exceed 150" or be less than 30". Many current finite element modeling codes have built-in options which permit identification of elements with sufficient distortion to affect the model's accuracy. Symmetry in a component's geometry and loading should be considered when constructing a model. Often, only the repeated portion of the component need be modeled. A section of a shaft contains three equally spaced holes. A solid model of the shaft con- taining one hole or even onehalf of a hole (Figure a), if the holes are loaded in a symmetric manner, must be modeled to perform the analysis. Appropriate constraints which define the hoop continuity of the shaft Figure 44. Sector model of a shaft cross-section con- taining three holes. must be applied to the nodes on the circumferential boundaries of the model. A number of FEA modeling codes have automated meshing features which, once the solid geometry is defined, will create a mesh at the punch of a button. This greatly speeds the production of a model, but it cannot be assumed that the model that is created will be free of distorted elements. Auto mesh programs are prone to creating an excessive number of elements in areas where the stress field is fairly uniform and such mesh density is unwarranted. The analyst must use available mesh controls and diagnostic tools to minimize these potential problems. Advantages and limitations of FEM Generally, the finer the element mesh, the more accurate the analysis. However, this also assumes that the model is loaded appropriately to mimic the load conditions to which the part is exposed. It is always advisable to ground the analysis with actual test results. Once an initial correlation between the model and test is established, then subsequent modifications can be implemented in the model with relative confidence. In many instances, the FEA results predict relative changes in deflection and stress between design iterations much better than they predict absolute deflections and stresses. 324 Rules of Thumb for Mechanical Engineers CENTROIDS AND MOMENTS OF INERTIA FOR COMMON SHAPES Key to table notation: A = area (in.2); II = moment of inertia about axis 1-1 (in."; J, = polar moment of inertia (in.4); c-denotes centroid location; a and p are measured in radians. Rectangle k Circle A=d d I, =- 4 7lP J =- "2 1 Semicircle 1-4 LW 2 d A=- 2 d I, =- 8 I, =0.1098R4 Hollow Circle A = n(%* - 42) J, =-(44-44) A I, =:(la4 e4) 2 Triangle h N3 bh A=- 3 bh3 36 I, =- Trapezoid Circular Sector 2 A=& I, = fa+sinacosa R" -16sinZa/9a] 4 R1 1 I, =-[a -sinacoscr] 4 yi =R[i-~sina/3aa] 1 Radius=R 1 y2 = 2ltsina/3a Solid Ellipse k h fl dU A= 4 1rbu3 I, =- 64 Hollow Ellipse 1 A = -(bu IT -6,u,) 1 (6u3 x -4~~') 4 1-64 Thin Annulus t 2 A = 2pRI Radius=R 2 BEAMS: SHEAR, MOMENT, AND DEFLECTION FORMULAS FOR COMMON END CONDITIONS Key to table notation: P = concentrated load (lb.); W = uniform load (lbhn.); M = moment (in, lb.); V = shear (lb); R = reaction (lb.); y = deflection (in.); 0 = end slope (radians); E = modulus (psi); I = moment of inertia (in?). Loads are positive upward. Moments which produce com- pression in the upper surface of beam are positive. 1. Cantilever - End LMd I I c- 3. Cantilever - Uniform had yt RA =P v= -P M= P(x- L) M, -PL. (st A) -PLZ e=- 2EI M=-P(u-x), (A~oB) y, =-($-)3u2L-a3), (atB) RA=P V=-P, (AtoB) M=O, (BtoC) v=o, (BtoC) M,=-Pa, (atA) @=- I @to0 -Pa' 2H RA =WL W(LZ+x2) -& M=tPLr- Y, == , (at B) 2 t 4. Cantilever - End Moment 5. End Supports - Intermediate Load t 6. End Supports - Uniform Load .f w I RA=O v=o MEMO M,, = M, M Lz 2EI y,, =o @=%, (atB) ET -Pb(Lz -b')y2 Y,, = 96LEi ' Pb Pa Pb L L L M=-x, (AtoB) RA =- , R, =- -Pb( L' - b') V=-, Pb (AtoB) M=-(L-XI, Pa (B toc) 0, = L L 6LEI -Pa M, =- Pub P~(L' -a') V ,(BtoC) 9 (atB) 0, = 6LET L Y, =- (atL/2) 2 384~1' WL3 0, =- WL3 0 = WL2 M,, =-, (atL/2) 8 A 24ET ' 24El 7. One End Supported and One End Fixed - Intermediate Load P 3b2L-b’ RA =- - 2[ L3 1 M =-( L2 Rc = P-R, V=RA, (AtoB) V=RA-P, (BtcC) P b’ +2bL2 - 3b2L “2 M = RAx (A to B) ivi=R,rtP(u-n), @toe) (+)M,, = RA(a), (at B when a =.366L) (-)M- = -M, , (when b =.4227L) p13 y, = 0098- EI’ (atB when b =.586L). 8. One End Supported and One End Fixed - Uniform Load WL‘ EI’ M=m y, = 0054- 3wL SWL R =-; R A 8 B-8 9WL2 , (atx=3L/8) (for x =.4215L). (+)M,, = - 8t 128 FEZ -FEZ -m3 0 (at A) ’-48ET’ MB =T (-)Mm = -, (at B) 8 Ifa>b - Pub2 M=- +RAx, (A to B) Pbz 9. Both Ends Fmed - Intermediate Load RA = -(3a t b) 4 =-(3b+a) V=RA, (AtoB) -Pab2 3a+b (+M, = 7 + RAa, (at B) L’ L2 Pa2 L3 LZ -Pub2 M=- +R,x-P(x-u), (BtOC) 2a1 (at x = -). Ifa<b V=RA-P, (BtoC) 10. Both Ends Fired - Uniform Load PL 8 Pub’ = (fora=L/2) Pa2b M, =- MA =- L’ L2 2b1 (-)Mmy = MA = 1481PL, (for a = L/ 3) (-)M, =M, = 1481PL, (fora=2L/3) (atx=L-3b+a). WL R,, = RB =- 2 -w14 (for x = L / 2) wLz ym - 384EI ’ (+)M, =-, (forx=L/2) 24 WL2 -wL2 B- 12 (-)Mm&? = 12 , (at A and B) M,=M I 328 Rules of Thumb for Mechanical Engineers 1. Dept. of Defense and Federal Aviation Administra- tion, Mil-Hdbk-5D, Metallic Materials and Elements for Aerospace Vehicle Structures, Vol. 1-2, Philadelphia: Naval Publications and Forms Center, 1983. 2. Aerospace Structural Metals Handbook Vol. 1-5,1994 ed. W. Brown Jr., H. Mindlin, and C. Y Ho @Is.). West Lafayette, IN: CINDAS / USAF CRDA Handbooks Op- erations Fkudue University. 3. Wang, C., Applied Elasticity. New York: McGraw-Hill Book Co., 1953, pp. 3&3 1. 4. Young, W. C., Roark 's Formulas for Stress and Strain, 6th Ed. New York: McGraw-Hill Book Co., 1989. 5. Hsu, T. H., Stress and Strain Data Han&ook Houston: Gulf Publishing Co., 1986, pp. 364-366. 6. Seely, F. B. and Smith, J. O., Advanced Mechanics of Materials, 2nd Ed. New York John Wiley & Sons, Inc., 1952, p. 415. 7. Peterson, R. E., Stress Concentration Factors. New York John Wiley & Sons, Inc., 1974. 8. Hsu, T. H., Stuctural Engineering &Applied Mechan- ics Data Handbook, Volume I: Beam. Houston: Gulf Publishing Co., 1988. 9. Higdon, A., Ohlsen, E. H., Stiles, W. B., and Weese, J. A., Mechanics of Materials, 2nd Ed. New York John Wiley & Sons, Inc., 1967, p. 236. 10. Perry, D. J. and Azar, J. J., Aircraft Structures, 2nd Ed. New York McGraw-Hill Book Co., 1982, p. 313. 11. Shigley, J. E., Mechanical Engineering Design, 3rd Ed. New York McGraw-Hill Book Co., 1977, p. 208. 12. Gleason Works, Gleason Curvic@ Coupling Design Manual. Rochester, Ny: Gleason Works, 1973. 13. Machine Design 1993 Basics of Design Engineering Reference Volme, Vol. 65, No. 13, June 1993, p. 271. 14. Dann, R. T. 'Wow Much Preload for Fasteners?" Ma- chine Design, Aug. 21, 1975, pp. 66-69. 15. Franm, I? R. "Are Your Fasteners Really Reliable?" Ma- chine Design, Dec. 10, 1992, pp.66-70. 16. MacGregor, C. W. and Symonds, J. "Mechanical Prop- erties of Materials" in Marks ' Standard Handbook for Mechanical Engineers, 8th ed. T. Baumeister, E. A. Avallone, and T. Bameister 111 (Eds.). New Yak: Mc- Graw-Hill Book Co., 1978, pp, 5-11. Fatigue J . Edward Pope. Ph.D., Senior Project Engineer. Allison Advanced Development Company Introduction 330 Design Approaches to Fatigue 331 Residual Stresses 332 Notches 332 Real World Loadings 335 Temperature Interpolation 337 Material Scatter 338 Estimating Fatigue Properties 338 339 Stages of Fatigue 330 Crack Initiation Analysis 331 Crack Propagation Analysis 338 K-The Stress Intensity Factor Crack Propagation Calculations 342 Creep Crack Growth 344 Inspection Techniques 345 Fluorescent Penetrant Inspection (PI) 345 Magnetic Particle Inspection (MPI) 345 Radiography 345 Ultrasonic Inspection 346 Eddy-Current Znspection 347 Evaluation of Failed Parts 347 Nonmetallic Materials 348 Fatigue T~ng 349 Liabhty Issues 350 References 350 329 [...]... single large crack For cases of uniform stress, the tendency is towards a single crack 348 Rules of Thumb for Mechanical Engineers NONMETAUIC MATERIALS This chapter has focused on metals, which are widely used for structural components A few comments will be made about plastics, composites, and ceramics as well The Properties of plastics vary greatly, but the same methodology that is used for metals can... of Thumb for Mechanical Engineers Material Scatter Since designers want to e s that only a very small fracnm tion of their components fail, material scatter must be taken into account This is generally done by specifying that -30 material properties should be used Only one out of about 800 specimens should have fatigue properties below this level Three rules of thumb are commonly used to account for. .. than the frachn ture toughness, stable crack growth under cyclic loading occurs For the English system, this is typically (hi) (inches).j For the metric system, it is typically (mpa) meter^).^ The conversion factor from the metric to the English system is: 1 (mpa) (meters).j= 91 (hi) (iizches).j Rules o Thumb for Mechanical Engineers f 340 Fast Fracture As stated earlier, fast fracture will occur when... John Wiley & Sons, Inc.) Figure 22 Common crack types 344 Rules of Thumb for Mechanical Engineers Computer codes which calculate crack growth use two approaches: 1 The cycle-by-cycle approach is the simplest, but it can be excessively time-consuming for slow-growing cracks With this method, the stress intensity factors and crack growth for one cycle is calculated The crack length is then increased by... the crack initiation life When the sum of the damage for all applied cycles equals one, crack initiation is assumed to occur For example: 10- During each mission, a component is subjected to: Figure 8 Evaluating cyclic content of a mission by the range-pair method 1 major cycle of 0-100 ksi 15 minor cycles of 0-85 ksi 336 Rules of Thumb for Mechanical Engineers If the crack initiation life is: -1 10,000... diameter-to-plate Figure 4 (Continued) I 0.4 I 0.5 I 0.6 width ratio, c/w Figure 5 Stress concentrationfactors with and without auxiliary holes [16] (Reprinted with permission of Society for ExperimentalMechanics.) 334 Rules o Thumb for Mechanical Engineers f The notch sensitivity factor “q” is a material property which varies with temperature Its value ranges from 1 (fully notch sensitive) to 0 (notch insensitive)...330 Rules of Thumb for Mechanical Engineers INTRODUCTION Fatigue is the failure of a component due to repeated applications of load, which are referred to as cycles An example of fatigue failure can be generated using a paper clip Bending it back and forth will cause failure in only a few cycles It has been estimated that up to 90%of... at the surface (Figure 2 ) 4 X-Rays 1 Film + More radiation is received here because there was less material along the path to abwrb the x-rays Figure 2 Radiographic inspection for cracks 3 346 Rules of Thumb for Mechanical Engineers Bad Weld Good Weld Figure 24 Radiography is often used to inspect welds Ultrasonic Inspection Ultrasonic inspection utilizes high-frequency sound waves which are reflected... inspections will be done in volume The negative aspects are t a special maht chinery and reference standards (showing the signals for cracked and uncracked parts) are necessary, making it impractical for one-of-a-kind inspections Evaluation of Failed Parts If a failure occurs, all relevant information about the event should be recorded as quickly as possible Seemingly minor details may help pinpoint the cause... factor For a given alternating stress, increasing the R ratio will decrease the crack initiation life For example, a component with stresses varying from 50 to 100ksi will have a lower life than a component with stresses varying from 0 to 50 ksi 5 10’ 105 10’ N Figure 2 Typical S-N (log stress versus log life)plot [14] (Reprinted with permission of John Wiley & Sons, lnc.) 332 Rules of Thumb for Mechanical . -w14 (for x = L / 2) wLz ym - 384EI ’ (+)M, =-, (forx=L/2) 24 WL2 -wL2 B- 12 (-)Mm&? = 12 , (at A and B) M,=M I 328 Rules of Thumb for Mechanical Engineers. they predict absolute deflections and stresses. 324 Rules of Thumb for Mechanical Engineers CENTROIDS AND MOMENTS OF INERTIA FOR COMMON SHAPES Key to table notation: A = area (in.2);. 320 Rules of Thumb for Mechanical Engineers CREEP RUPTURE Creep is plastic deformation which increases over time under sustained loading

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