# MECH 344/X Machine Element Design

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1 1 MECH 344/X Machine Element Design Time: M 14:45-17:30 Lecture 2

2 Contents of today's lecture Introduction to Static Stresses Axial, Shear and Torsional Loading Bending in Straight and Curved Beams Transverse Shear in Beam Mohr Circle Combined Stresses Stress Concentration Residual Stresses

3 3 After we identify the external loads on a member, we need to see what the resulting stresses on the member due to those loads are. Body stresses, existing within the member as a whole Surface or contact stresses in localized regions where external loads are applied. Stresses resulting from static loading, as opposed to stresses caused by impact or fatigue loading. Convention capital letter S for material strength (S u, S y for ultimate, yield strength) and using Greek letters and for normal and shear stress. This chapter is going to be a review of subjects you learned in mechanics of materials

4 Figure illustrates a case of simple tension. P for tension and P for compression, in both cases axial. Small block E represents an arbitrarily located small element of material It is important to remember that the stresses are acting on faces perpendicular to the paper. This is made clear by the isometric view in Figure 4.1b.

5 Figure illustrates equilibrium of the left portion of the link From this we have perhaps the simplest formula in all of engineering: It is important to remember that although this formula is always correct as an expression for the average stress in any cross section, disastrous errors can be made by naively assuming that it also gives the correct value of maximum stress in the section. Unless several important requirements are fulfilled, the maximum stress will be several times greater than P/A

6 Figure illustrates lines of force on the link Maximum stress = P/A only if load distribution is uniform 1. The section being considered is well removed from the loaded ends. Uniform distribution is reached at points about three d from the end 2. The load is applied exactly along the centroidal axis. If, there is eccentricity, bending moment will come into play 3. The bar is a perfect straight cylinder, with no holes, notches, threads, internal imperfections giving rise to stress concentration

7 Figure illustrates lines of force on the link Maximum stress = P/A only if load distribution is uniform 4. The bar is free of stress when the external loads are removed. - No residual stresses due to manufacture and past mechanical/thermal loading 5. The bar comes to stable equilibrium when loaded. If, the bar is in compression, or if it long, buckling occurs, and elastically unstable. 6. The bar is homogeneous. Not a composite material (where 2 different Es make a material so that one with larger E gets stressed more)

8 Figure shows where unexpected failure can result from assuming P is 600 N and 6 identical welds are used. The average load 100 N per weld 6 welds represent redundant force paths of different stiffnesses. The paths to welds 1 and 2 are much stiffer; they may carry nearly all the load. A more uniform distribution could be obtained by adding two side plates One might despair of ever using P/A as an acceptable value of maximum stress for relating to the strength properties of the material. The student should acquire increasing insight for making engineering judgments relative to these factors as his or her study progresses and experience grows.

9 Direct shear loading involves equal and opposite forces, collinear that the material between them experiences shear stress, with negligible bending. In figure neglecting interface friction direct shear with If the nut in Figure is tightened to produce an initial bolt tension of P, the direct shear stresses at the root of the bolt threads (area 2 ), and at the root of the nut threads (area 3), have average values P/A. The thread root areas involved are cylinders of a height equal to the nut thickness. (for V threads) If the shear stress is excessive, shearing or stripping of the threads occurs in the bolt or nut, whichever is weaker.

10 Figure shows a hinge pin loaded in double shear, where the load P is carried in shear through two areas; hence, the area A used in Eq. is twice the CSA of pin Direct shear loading does not produce pure shear (as does torsional loading), and the actual stress distribution is complex. It involves fits between the mating members and relative stiffnesses. The maximum shear stress will always be somewhat in excess of the P/A value In the design of machine and structural members, however, it is commonly used in conjunction with appropriately conservative values of working shear stress. Furthermore, to produce total shear fracture of a ductile member, the load must simultaneously overcome the shear strength in every element of material in the shear plane. Thus, for total fracture the equation will work if being set equal to the ultimate shear strength, S us.

11 Figure illustrates torsional loading Direction of T determines that the left face of element E is subjected to a shear stress, and the right face to an stress, making a CCW couple, balanced by a corresponding CW couple with shear stresses on top/bottom The state of stress shown on element E is pure shear. In axial force + for tension and for compression; Compression can cause buckling but tension cannot, a chain can withstand tension but not compression, concrete is strong in compression but weak in tension. The sign convention for shear loading serves no similar function; + and - shear are basically the same; and the sign convention is purely arbitrary. In this book CCW ve and CW +ve

12 For round bar in torsion stresses vary from 0 at center to max at surface = Tr/J where r is the radius and J polar moment Max shear stress will be Important assumptions for this equation are 1. The bar must be straight and round (either solid or hollow), and the torque must be applied about the longitudinal axis. 2. The material must be homogeneous and perfectly elastic within the stress range involved. 3. The cross section considered must be sufficiently remote from points of load application and from stress raisers (i.e., holes, notches, keyways, surface gouges, etc.).

13 For bars of nonround cross section, the analysis gives erroneous results. This can be demonstrated with an ordinary rubber eraser with small square elements 1, 2, and 3 as shown in Figure. When the eraser is twisted, Eq. implies highest shear stress would be at element 2 farthest from the neutral axis. Lowest surface stress at element 1 closest to the axis. Observation of the twisted eraser shows exactly the opposite; element 2 does not distort at all, while element 1 experiences the greatest distortion In Eq. the basic assumption that what are transverse planes before twisting remain planes after twisting. If such a plane is represented by drawing line A on the eraser, obvious distortion occurs upon twisting; therefore, the assumption is not valid for a rectangular section. The equilibrium requirement of corner element 2 makes it clear that this element must have zero shear stress: (1) the free top and front surfaces do not contact anything that could apply shear stresses;

14 (2) this being so, equilibrium requirements prevent any of the other four surfaces from having shear. Hence, there is zero shear stress along all edges of the eraser. Torsional stress equations for nonround sections are summarized in references

15 15 Pure Bending it is rare for beam to be loaded in pure bending. It is useful though to understand the situation Mostly shear loading and bending moments act together Applying point loads P equidistant at the simply supported beam, absence of shear loading makes this pure bending Assumptions used for analysis

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17 17 N-N along the neutral axis, no change in length A-A shortents (in compression) and B-B lengthens (in tension Bending stress is 0 at N-N and is linearly proportional to distance y away from N-N Where M is the bending moment and I is the area moment of Inertia of the beam cross section at the neutral plane, y the distance from N-N The max stress at outer plane is Where c is distance from neutral plane (it should be same both sections if the beam is symmetrical about the neutral axis C is taken as +ve initially and proper sign applied based on loading compression ve and tension +ve

18 Figure shows bending load applied to beam of CS having 2 axes of symmetry cutting-plane stresses max are obtained from Eq. 4.6 by substituting c for y Section modulus Z (I/c) is used, giving max as

19 Figure shows bending load applied to beam of CS having 1 axis of symmetry The properties for calculating max for these beams are given in Appendix B

20 20 Machines have curved beams like in C clamps, hooks etc, with a ROC) the first 6 assumptions still apply If it has a significant curvature the neutral axis will not be coincident with the centroidal axis

21 21 If it has a significant curvature the neutral axis will not be coincident with the centroidal axis and the shift e is found from Where A is area, r c is ROC of centroidal axis and r is the ROC of the differential area da For a rectangular beam this can be e = r c -(r o -r i )/LN(r o / r i )

22 22 Stress distribution is not linear but hyperbolic. Sign convention is +ve moment straightens the beam (tension inside and compression outside) For pure bending loads stresses at inner and outer surface is And if a force F on the CSA A is applied on the curved beam then the stresses will be

23 Stress values given by Eq differ from the straight-beam Mc/I value by a curvature factor, K. Thus, using subscripts i and o to denote inside and outside fibers, respectively, we have Values of K in Figure illustrates a common rule of thumb: If r is at least ten times inner fiber stresses are usually not more than 10 percent above the Mc/I value. Values of Ko, Ki, and e are tabulated for several cross sections in references.

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25 A rectangular beam has an initial curvature equal to the section depth h, as shown in Figure. How do its extreme-fiber-bending stresses compare with those of an identical straight beam? Known: A straight beam and a curved beam of given cross section and initial curvature are loaded in bending. Find: Compare the bending stresses between the straight beam and the curved beam.

26 Assumptions: 1. The straight bar must initially be straight. 2. The beams are loaded in a plane of symmetry. 3. The material is homogeneous, and all stresses are within the elastic range. 4. The sections where stresses are calculated are not too close to significant stress raisers or to regions where external loads are applied. 5. Initial plane sections remain plane after loading. 6. The bending moment is positive; that is, it tends to straighten an initially curved beam.

27 e = r c -(r o -r i )/LN(r o / r i ) r o = 1.5h and r i =.5h and r c = h

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29 Shear due to transverse loading 29 Common case is both shear and bending moment on the beam. Fig shows a point loaded beam shear and moment diagram Cutting our segment P of width dx around A, cut from outer side at c upto depth y 1 it is seen that the M(x 1 ) to left < M(x 2 ) to right and the difference being dm Similarly for stresses (can be seen in fig b) since the stresses are proportion to moment

30 Shear due to transverse loading 30 Similarly for stresses (can be seen in fig b) since the stresses are proportion to moment. This stress imbalance is countered by shear stress component Stress acting on left hand side of p at a distance y from neutral axis is stress times the differential area da at that point The total force acting on the left hand side will then be Similarly for right hand side Shear force on the top face is Where bdx is the area of the top face of the element

31 Shear due to transverse loading 31 For equilibrium, the forces acting on p is 0 Gives an expression for shear stress as a function of change in momentum wrt x Since slope of dm/dx is the magnitude of the shear function V Assigning the integral as Q, then Shear stresses vary across y And becomes 0 when c=y 1 And maximum at neutral axis A common rule of thumb is the shear stress due to transverse loading in a beam will be small enough to ignore if the length to depth ratio of the beam is 10 or more. Short beams below that ratio should be investigated for both transverse shear and bending.

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34 Determine the shear stress distribution for the beam and loading shown in Figure Compare this with the maximum bending stress.

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40 X Z Y Along axis direction Opposite axis direction

41 Axis of the stresses is arbitrarily chosen for convenience. Normal and shear stresses at one point will vary with direction along the coordinate system. There will be planes where the shear stress is 0, and the normal stresses acting on these planes are principal stresses, and planes as principal planes There will be planes where the shear stress components are 0, and the normal stresses acting on these planes are principal stresses, and planes as principal planes Direction of surface normals to the planes is principal axes. And the normal stresses acting in these directions are principal normal stresses The principal shear stresses, act on planes at 45 to the planes of principal normal stresses 41

42 The principal shear stresses from principal normal stresses can be calculated as 43 Since usually 1 > 2 > 3, max is 13 and the direction of principal shear stresses are 45 to the principal normal planes and are mutually orthogonal

43 The Mohr s Circle Mohr circle is a graphical method find the principal stresses 44

44 The Mohr s Circle 45

45 The Mohr s Circle 46

46 The Mohr s Circle 47

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48 The shaft is subjected to torsion, bending, and transverse shear. Torsional stresses are shaft surface. Bending stresses are a maximum at points A and B Transverse shear stresses are relatively small compared to bending stresses, and equal to zero at points A and B 49

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54 Since the largest of the three Mohr circles always represents the maximum shear stress as well as the two extreme values of normal stress, Mohr called this the principal circle.

55 A common example in which the maximum shear stress would be missed if we failed to include the zero principal stress in the Mohr plot is the outer surface of a pressurized cylinder. Here, the axial and tangential stresses are tensile principal stresses, and the unloaded outer surface ensures that the third principal stress is zero. Figure illustrates both the correct value of maximum shear stress and the incorrect value obtained from a simple two-dimensional analysis. The same situation exists at the inner surface of the cylinder, except that the third principal stress (which acts on the surface) is not zero but a negative value numerically equal to the internal fluid pressure.

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60 Figure indicates lines of force flow through a tensile link. Uniform distribution of these lines exist in regions away from the ends. At the ends, the force flow lines indicate stress concentration near outer surface. We need to evaluate the stress concentration associated with various geometric configurations to determine maximum stresses existing in a part The first mathematical treatments of stress concentration were published after 1900 with experimental methods for measuring highly localized stresses In recent years, FEM studies have also been employed. The results of many of these studies are available in the form of published graphs giving values of the theoretical stress concentration factor, K t (based on a theoretical elastic, homogeneous, isotropic material), for use in the equations

61 For eg, the maximum stress for axial loading would be P/A * appropriate K t. Note that the stress concentration graphs are plotted on the basis of dimensionless ratios, indicating that only part shape (not size) is involved. Also note that stress concentration factors are different for axial, bending, and torsional loading. In many situations involving notched parts in tension or bending, the notch not only increases the primary stress but also causes one or both of the other principal stresses to take on nonzero values. This is referred to as the biaxial or triaxial effect of stress raisers ( stress raiser is a general term applied to notches, holes, threads, etc.). Consider, for example, a soft rubber model of the grooved shaft in tension illustrated in Figure 4.36b. As the tensile load is increased, there will be a tendency for the outer surface to pull into a smooth cylinder. This will involve an increase in the diameter and circumference of the section in the plane of the notch. The increased circumference gives rise to a tangential stress, which is a maximum at the surface. The increase in diameter is associated with the creation of radial stresses. (Remember, though, that this radial stress must be zero at the surface because there are no external radial forces acting there.)

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77 K t factors given in the graphs are theoretical (hence, the subscript t) or geometric factors based on a theoretical homogeneous, isotropic, & elastic mat l Real materials have microscopic irregularities causing a certain nonuniformity of microscopic stress distribution, even in notch-free parts. Hence, the introduction of a stress raiser may not cause as much additional damage as indicated by the theoretical factor. Moreover, real parts even if free of stress raisers have surface irregularities (from processing and use) that can be considered as extremely small notches. The extent to which we must take stress concentration into account depends on (1) the extent to which the real material deviates from the theoretical and (2) whether the loading is static, or fatigue For materials permeated with internal discontinuities, such as gray cast iron, stress raisers usually have little effect, regardless of the nature of loading, as geometric irregularities affect less than the internal irregularities.

78 For fatigue & impact loading of engineering materials, K t must be considered For the case of static loading, K t is important only with unusual materials that are both brittle and relatively homogeneous; When tearing a package wrapped in clear plastic film, a sharp notch in the edge is most helpful! or for normally ductile materials that, under special conditions, behave in a brittle manner For the usual engineering materials having some ductility (and under conditions such that they behave as ductile), it is customary to ignore K t for static loads.

79 Figure show two flat tensile bars each having a minimum CSA of A, each a ductile material having the idealized stress strain curve shown in e. The load on the unnotched bar can be increased to the product of area times yield strength before failure occurs as shown in c. Since the grooved bar in b has a K t of 2, yielding will begin at only half the load, as shown in d. This is repeated as curve a of f. As the load is increased, the stress distribution (shown in f) becomes b, c, and finally d.

80 These curves reflect a continuous deepening of local yielding, which began at the root; but gross yielding involving the entire CS begins only if d is reached. Note that the load associated with curve d is identical to the unnotched load capacity Also note that curve d can be achieved without significant stretching of the part. The part as a whole cannot be significantly elongated without yielding the entire CS. So, for practical purposes, the grooved bar will carry the same static load as the ungrooved bar.

81 When a part is yielded nonuniformly throughout a CS, residual stresses remain in this CS after the external load is removed. For eg, the 4 levels of loading of the notched tensile bar shown in f. This same bar and the 4 levels of loading are represented in the left column Note that only slight yielding is involved not major yielding such as often occurs in processing. The middle column shows the change in stress when the load is removed.

82 Except for a, where the did not cause yielding at the notch root, the stress change when the load is removed does not exactly cancel the stresses by applying the load. Hence, residual stresses remain after the load is removed. These are shown in the right column In each case, the stress change caused by removing the load is elastic. It must be remembered, too, that this development of residual stress curves was based on assuming that the material conforms to the idealized stress strain curve So the residual stress curves can only be good approximations.

83 Figure illustrates residual stresses caused by the bending of an unnotched 25 * 50-mm rectangular beam. made of steel having an idealized stress strain curve with Sy = 300 MPa. Unknown moment M 1 produces the stress distribution shown in Figure, with yielding to a depth of 10 mm. Let us first determine the magnitude of moment M1 If the distributed stress pattern is replaced with concentrated forces F1 and F2 at the centroids of the rectangular and triangular portions of the pattern, respectively, M1 is equal to the sum of the couples produced by F1 and F2. The magnitude of F1 is equal to the product of the average stress (300 MPa) times the area over which it acts (10 mm * 25 mm).

84 Similarly, F2 is equal to an average stress of 150 MPa times an area of 15 mm * 25 mm. The moment arms of the couples are 40 mm and 20 mm, respectively After M 1 is removed Note that at this point the beam is slightly bent Figure 4.44c shows that the desired center portion stress-free condition requires superimposing a load that develops a compressive stress of 62 MPa, 10 mm below the surface

85 With this load in place, total stresses are as shown. Since center portion stresses are zero, the beam is indeed straight. Stress 396 MPa is due to moment of 4125 Nm. By simple proportion, a stress of 104 MPa requires a moment of 1083 N # m. Let us now determine the elastic bending moment capacity of the beam after the residual stresses have been established. A moment in the same direction as M 1 can be added that superimposes a surface stress of +396 MPa without yielding. It is of 4125 Nm. The release of original moment M 1 = 4125 Nm caused no yielding; so, it can be reapplied.

86 Figure e shows that in the direction opposite the original moment M 1, a moment giving a surface stress of 204 MPa is all that can be elastically withstood. This corresponds to moment of 2125Nm. An overload causing yielding produces residual stresses that are favorable to future loads in the same direction and unfavorable to future loads in the opposite direction. Furthermore, on the basis of the idealized stress strain curve, the increase in load capacity in one direction is exactly equal to the decrease in load capacity in the opposite direction.

87 We ve seen stresses caused by external loads. Stresses can also be caused by expansion and contraction due either to temperature changes or to a material phase change. It is important to become familiar with the basic principles. When the temperature of an unrestrained homogeneous, isotropic body is uniformly changed, it expands (or contracts) uniformly in all directions, according to the relationship where is the strain, is the thermal expansion coef and T is the temperature change. Values of for several common metals are given in Appendix C-1. If restraints are placed on the member during the temperature change, the resulting stresses can be determined by (1) computing the dimensional changes that would take place in the absence of constraints, (2) determining the restraining loads necessary to enforce the restrained dimensional changes, and (3) computing the stresses associated with these restraining loads.

88 We ve seen stresses caused by external loads. Stresses can also be caused by expansion and contraction due either to temperature changes or to a material phase change. A 10-in. L steel tube (E = 30 X 10 6 psi and = 7 X 10-6 per F) having a CSA of 1 in 2 is installed with fixed ends so that it is stress-free at 80 F. In operation, the tube is heated throughout to a uniform 480 F. Careful measurements indicate that the fixed ends separate by in. What loads are exerted on the ends of the tube, and what are the resultant stresses? Known: A given length of steel tubing with a known CSA expands in. from a stress-free condition at 80 F when the tube is heated to a uniform 480 F

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90 If stresses caused by temperature change are undesirably large, the best solution is often to reduce the constraint. - using expansion joints, loops, or telescopic joints Thermal stresses also result due to temperature gradients - if a thick metal plate is heated in the center of one face with a torch, the hot surface is restrained from expanding by the cooler surrounding material; it is in a state of compression. Then the remote cooler metal is forced to expand, causing tensile stresses. If the forces and moments do not balance for the original geometry, it will distort or warp to bring about internal equilibrium. If stresses are within the elastic limit, the part will revert to its original geometry when the initial temperature conditions are restored. If some portion of the part yields, this portion will not tend to revert to the initial geometry, and there will be warpage and internal (residual) stresses when initial temperature conditions are restored. This must be taken into account in the design

91 Residual stresses are added to any subsequent load stresses in order to obtain the total stresses. If a part with residual stresses is machined, the removal of residually stressed material causes the part to warp or distort. As this upsets the internal equilibrium. A common (destructive) method for determining the residual stress in a particular zone of a part is to remove very carefully material from the zone and then to make a precision measurement of the resulting change in geometry. (Hole drilling method) Residual stresses are often removed by annealing. The unrestrained part is uniformly heated (to a sufficiently high temperature and for a sufficiently long period of time) to cause virtually complete relief of the internal stresses by localized yielding. The subsequent slow cooling operation introduces no yielding. Hence, the part reaches room temperature in a virtually stress-free state.

93 The Liberty Bell, cast in 1753, has residual tensile stresses in the outer surface because the casting cooled most rapidly from the inside surface. After 75 years of satisfactory service, the bell cracked, probably as a result of fatigue from superimposed vibratory stresses caused by ringing the bell. Holes were drilled at the ends to keep the crack from growing, but the crack subsequently extended itself. Almen and Black cite this as proof that residual stresses are still present in the bell.

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