failure prediction for loading static

7/24/2019 FAILURE PREDICTION FOR LOADING STATIC

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Fundamentals of Machine Elements, 3rd

ed.Schmid, Hamrock and Jacobson

201 !"! #ress

Chapter 6: Failure Predictionfor Static Loading

The concept of failure is central to thedesign process, and it is by thinkingin terms of obviating failure that

successful designs are achieved.

Henry Petroski,Design Paradigms

The liberty bell, a classic case of brittlefracture. ( R-F Website/Corbis)


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ed.Schmid, Hamrock and Jacobson

201 !"! #ress

Plate with Hole inTension

Figure 6.1: Rectangular plate with hole subjectedto axial load. (a) Plate with cross-sectional plane.(b) Half of plate showing stress distribution. (c)Plate with elliptical hole subjected to axial load.

Stress concentration factor:

Hole in plate:

Sharp crack:


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ed.Schmid, Hamrock and Jacobson 201 !"! #ress

Rectangular Plate with Hole

Figure 6.2: Stress concentration factors for rectangular plate with central hole. (a)Axial load and pin-loaded hole.


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Rectangular Plate with Hole

Figure 6.2: Stress concentration factors for rectangular plate with central hole. (b)bending.


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Rectangular Plate with Fillet

Figure 6.3: Stress concentration factors for rectangular plate with fillet. (a) Axialload


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Rectangular Plate with Fillet

Figure 6.3: Stress concentration factors for rectangular plate with fillet. (b) bending.


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Rectangular Plate with Groove

Figure 6.4: Stress concentration factors for rectangular plate with groove. (a) Axialload.


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Rectangular Plate with Groove

Figure 6.4: Stress concentration factors for rectangular plate with groove. (a) Axialload; (b) bending.


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Round Bar with Fillet

Figure 6.5: Stress concentrationfactors for round bar with fillet.(a) Axial load; (b) bending; (c)torsion.


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Round Bar with Groove

Figure 6.6: Stress concentration factors forround bar with groove. (a) Axial load; (b)

bending; (c) torsion.


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Round Bar with Flat Groove

Figure 6.7: Stress concentration factors for round bar with a flat groove. (a)Bending;


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Round Bar with Flat Groove

Figure 6.7: Stress concentration factors for round bar with a flat groove. (b)torsion;


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Round Bar with Hole

Figure 6.8: Stress concentration factors for round bar with hole.


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Stress Contours

Figure 6.9: Axially loaded flat plate with fillet showing stress contours: (a) squarecorners, (b) rounded corners, (c) small grooves, and (d) small holes.


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Modes of Crack Displacement

Figure 6.10: Three modes of crackdisplacement. (a) Mode I, opening; (b)Mode II, sliding, (c) Mode III, tearing.

Three forms of crack growth:1.Mode I opening.The opening (or tensile) mode,shown in Fig. 6.10a, is the most often encounteredmode of crack propagation. The crack faces separatesymmetrically with respect to the crack plane.

2.Mode II sliding.The sliding (or in-plane shearing)

mode occurs when the crack faces slide relative toeach other symmetrically with respect to the normalto the crack plane but asymmetrically with respectto the crack plane, as shown in Fig. 6.10b.

3.Mode III tearing.The tearing (or antiplane) modeoccurs when the crack faces slide asymmetrically

with respect to both the crack plane and its normal,as illustrated in Fig. 6.10c.


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Yield Strength and Fracture Toughnessfor Various MaterialsYield Strength,

Sy Fracturetoughness,

KIc

iskaPMisklairetaM in MPamMetals

Aluminum alloys9262054564T-41024313093753T-4202

2024-T351 47 325 33 367075-T651 73 505 26 297079-T651 68 470 30 33

Steels4340temperedat260C 238 1640 45.8 50.0

4340temperedat425C 206 1420 80.0 87.4D6AC,temperedat540C 217 1495 93 102

1110012271052835ATitanium alloys

Ti-6Al-4V 119 820 96 106Ti-13V-11Cr-3Al 164 1130 25 27Ti-6Al-6V-2S 157 1080 34 37Ti-6Al-2Sn-4Z-6Mo 171 1180 24 26

CeramicsAluminum oxide 2.7-4.8 3.0-5.3Siliconnitride 3.5-7 4-8

Siliconcarbide 1.8-4.5 2-5Soda-limeglass 0.64-0.73 0.7-0.84.1-2.072.1-81.0etercnoC

PolymersPolymethylmethacrylate 3-7 20-50 0.9-2.7 1-3Polystyrene 4.5-11.5 30-80 0.9-1.8 1-2Polycarbonate 8.5-10 60-70 2.3-2.7 2.5-3Polyvinylchloride 5.8-7 40-50 1.8-2.7 2-3

Table 6.1:Yield stress andfracture toughness data forselected engineeringmaterials at roomtemperature.Source:FromASM International [1989]and Bowman [2004].

Fracture toughness:


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Design Procedure 6.1: FractureMechanics Applied to Design

1.Given a candidate material, obtain its fracture toughness. See Table 6.1 forselected materials, or else find the value in the technical literature or fromexperiments.

2.The dimensionless correction factor for the part geometry,Y, can be obtainedfrom Appendix C for common design situations.

3.Equation (6.6) allows calculation of allowable stress as a function of semi-cracklength,a; similarly, the largest allowable crack (with length 2a) can be determinedfrom the required stress.

4.If design criteria cannot be met, the following alternatives can be pursued:a.Increasing the part thickness will reduce the nominal stress,nom.

b.A different material with a higher fracture toughness can be selected.

c.Local reinforcement of critical areas can be pursued, such as locallyincreasing thickness.d.The manufacturing process can have a significant impact on the initial flawsize. The class of operations (casting versus forging, compression moldingversus extrusion, etc.), quality control procedures, and quality of incomingmaterial are all important factors.


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Yield Locus

Figure 6.11: Three-dimensional yield locus for MSST and DET.


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Maximum Shear Stress Theory

Figure 6.12: Graphical representation ofmaximum-shear-stress theory (MSST) for

biaxial stress state (z= 0).

Mathematical statement:


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Distortion-Energy Theory

Figure 6.13: Graphical representation ofdistortion-energy theory (DET) for biaxialstress state (z= 0).

Statement:

Equivalent or von Mises stress:

In biaxial stress state:


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Example 6.6

Figure 6.14: Rear wheelsuspension used in Example 6.6.


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Example 6.7

Figure 6.15: Cantilevered, round bar with torsion applied to free end (used inExample 6.7). (a) Bar with coordinates and load; (b) stresses acting on an element; (c)Mohr's circle representation of stresses.


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Example 6.8

Figure 6.16: Cantilevered, round bar with torsion and transverse force applied to freeend (used in Example 6.8). (a) Bar with coordinates and loads; (b) stresses acting onelement at top of bar and at wall; (c) Mohr's circle representation of stresses.


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Maximum Normal Stress Theory

Figure 6.17: Graphical representationof maximum-normal-stress theory(MNST) for a biaxial stress state (z=

0).

Failure will occur if:

or:


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Internal Friction and Modified MohrTheories

Figure 6.18: Internal frictiontheory and modified Mohr theoryfor failure prediction of brittlematerials.

Internal Friction Theory:If1> 0 and3< 0,

If3> 0,

And if1< 0,


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Fundamentals of Machine Elements, 3

rd


Experimental Verification

Figure 6.19: Experimental verification of yield and fracture criteria for severalmaterials. (a) Brittle fracture. (b) Ductile yielding.Source:From Dowling [1993] andMurphy [1964].


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Example 6.10

Part Criterion Equation used Safety factor

a) MSST (6.7) 1.5DET (6.10) 1.73

b) MSST (6.7) 1.28

DET (6.10) 1.33c) IFT (6.14) 1.61

MMT (6.17) 1.69

Table 6.2: Safety factors from using different criteria for three different materialsused in Example 6.10.


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Case Study 6.2: Selecting Failure Criteria

Given a material, where the tensile and compressive yield, ultimate and/orfracture stresses are known, the following steps can be used to help select afailure criterion:1.For a ductile metal, where the strength is the same in tension andcompression, use either the MSST or DET. These criteria are fairly close,

with the largest difference of 15% occurring for pure shear in a plane stressloading. The MSST is more conservative; that is, it predicts yielding at alower stress level than DET.

2.If a ductile metal has a different strength in compression than in tension,such as with certain magnesium alloys, the IFT or MMT are reasonableoptions.

3.Brittle materials are difficult to analyze using failure criteria, and confidencein strength values is difficult to obtain. However, the IFT leads to goodresults without the mathematical complication of the MMT.

4.For circumstances where improved performance is required, MMT may bejustified over the IFT.


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Artificial Hip Stress Concentration

Figure 6.20: (a) Schematic illustration of a portion of a total hip replacement withselected dimensions; (b) idealized geometry used to estimate the stress concentrationfactor at the fillet.Source:Courtesy of T. Hershberger, Biomet, Inc.


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Stress Concentration by Finite ElementAnalysis

Figure 6.21: Example of a finite element mesh to capture the value of a stressconcentration corresponding to Fig. 6.2. Only one-fourth of the problem has beendiscretized to take advantage of symmetry. Note the large number of elements locatednear the stress raiser. Boundary conditions and applied loads have been added for imageclarity.


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Fundamentals of Machine Elements, 3rded.Schmid Hamrock and Jacobson 201 !"! #ress

Photoelastic Stress Visualization

Figure 6.22: Photoelastic comparison of threaded fastener profiles comparing loaddistribution. The left image shows a conventional profile where load per tooth varieswidely, and the right shows a Spiralockprofile with more uniform stresses on eachtooth.Source:Courtesy of Spiralock Corp.

failure prediction for loading static

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