3/25/20 10 1 Chapter 14 Stress Concentrations • Abrupt changes in geometry can give rise to stress concentrations – e.g. screw threads, stepped shafts and holes • Other sources for stress discontinuities – Contact points – Material discontinuities, e.g. an inclusion – Residual stresses, e.g. § • Can calculate stress concentration factors – Analytically using Theory of Elasticity – Numerically using Finite Elements Methods – Experimentally using • Very small strain gages • Brittle coating • Photoelasticity • Moiré • In this chapter will consider stress concentrations – Holes – Notches
Transcript
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Chapter 14Stress Concentrations• Abrupt changes in geometry can give rise
to stress concentrations– e.g. screw threads, stepped shafts and holes
• Other sources for stress discontinuities– Contact points– Material discontinuities, e.g. an inclusion– Residual stresses, e.g. welding and cold
working– Cracks!
• Important because local stress increase can result in an unexpected failure
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• Can calculate stress concentration factors– Analytically using Theory of Elasticity– Numerically using Finite Elements Methods– Experimentally using
• Very small strain gages• Brittle coating• Photoelasticity• Moiré
• In this chapter will consider stress concentrations– Holes– Notches– Cracks
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Presence of a stress concentrationcauses a ―stress gradient‖
High stresses isolated to a local area on the structure
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1. Introduce some basic concepts of Stress Concentrations
2. Theory of ElasticityStress Concentration Factors
14.1 Nature of a Stress Concentration Problem and the Stress Concentration Factor
In a uniaxial tension test, the stress is assumed to be uniformly distributed over the cross section at locations away from the load applicationThe nominal stress is n=P/AThe stress concentration factor is calculated as
(14.1)
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Fig. 14.1 Stress concentrations and stress trajectories8
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•
•
Lines of force, i.e. the stresstrajectories whose paths must lie in the materialCluster together in passing around a geometric hole or discontinuityThe average spacing between the lines of force is reducedResults in more lines of force
squeezed into the same area
Consider a crackThe lines of force may be considered to be transmitted from one row of atoms to another rowThe transmission of force around the tip of the crack entails high straining of the bonds (AB, CD, AC, etc.)
Fig. 14.2Atomic model of a crack in a solid
In real problems, stress state is 3D—not simply 2D9
14.2 Stress Concentration Factors: Theory of Elasticity
14.2.1 Circular Hole in an Infinite Plate under Uniaxial Tension
(14.3)
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14.2.2 Elliptical Hole in an Infinite Plate Stressed in aDirection Perpendicular to the Major Axis of the Hole
(14.6)
(14.7)
•
•
Fig. 14.5Elliptical hole in an infinite plate
In the limit, as 00, theelliptical hole becomes asharp crack of length 2aDimensions a and b are very small compared to the length and width of the plate
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For uniaxial tension stress, perpendicular to the major axis of the elliptical hole, the sum of the stress components and is given by
(14.8)
Because the stress =0 at the hole (= 0 —FreeSurface), Eq. 14.8 yields the stress at the hole
(14.9)
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Fig. 14.6Elliptical coordinates
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The max value of increases w/o bound as b/a0 (a/band the elliptical hole becomes a crack)
(14.11)
For a=b, Eq. 14.11 agrees w/ the circular hole equations
Fig. 14.7Circumferential stress distribution around an edge of a circular hole in an infinite plate
Fig. 14.8Distribution of around an elliptical hole in an infinite plate loadedperpendicular to the major axis
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By geometry, the radius of curvature of an ellipse at the end of the major axis is
(14.12)
(14.13)
By Eq. 14.9, the minimum value of =0 ( @ the ends of the minor axes, where =/2 , -/2)
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14.2.4 Crack in a Plate
• As b0, the elliptical hole in an infinite plate becomes very flat and approaches the shape of a through- thickness crack
• The maximum value of may become quite large compared to the applied stress for nonzero values of b as b0, depending on the nature of the load
• e.g., Fig. 14.5, Eq. 14.11 w/ a/b=100 gives ()max=201• e.g., Eq. 14.15 w/ a/b=100 gives ()max=1.02• The case b=0 leads to a special study of
stress singularities• The practical significance of very large stress
concentrations will be addressed in Chapter 15
Elliptical Hole in an Infinite Plate Stressed in aDirection Perpendicular to the Minor Axis of the Hole
The value of @ any point on the perimeter around the hole
(14.14)
• For =/2 , -/2, attains the maximum value @ the ends of the minor axis
(14.15)
•
Fig. 14.9
For =or attains theminimum value @ the ends of the major axis of = -Distribution of around an elliptical hole in an infinite
plate loadedperpendicular to the minor axis
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Ellipsoidal Cavity
Prolate spheroid (football shape)The semimajor axis a of the ellipsoid (the axis of revolution) is oriented s.t. it is perpendicular to the direction of the axial pull in the memberThe semiminor axis b always lies in a plane parallel to the axial pullDimensions a and b are small compared to the dimensions of the
axial memberThe max stress occurs at the end of the semimajor axis ae.g., A long narrow stringlike internal flaw (b/a=0) oriented in a direction perpendicular to the load is 2.83 n
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Oblate spheroid (doorknob shape)Has its semiminor axis, b (the axis of revolution) oriented in the direction of the uniaxial pull in the memberThe semimajor axis a always lies in a plane perpendicular to the loadThe max stress occurs at the end of the semimajor axis ae.g., An internal flaw or cavity of spheroidal shape such as a gas bubble in a composite resin raises the stress from n to 2.05 n
e.g., a very flat, round cavity oriented s.t. the flat plane is perpendicular to the load gives 13.5 n
If the material remains elastic, then this value is comparable to the value for a narrow elliptical hole as given by Eq. 14.11
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14.2.6 Grooves and Holes
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Fig. 14.10Neuber’s diagram (nomograph) for a calculated stress concentration factor at the root of a notch
Consider the construction of Fig. 14.10Assume a member contains the groove as shown in Fig. A of Table 14.3Subject to an axial load PLet the calculated stress concentration factor be Scs when the
groove is very shallow– Then from Neuber (1958) (14.16)
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Fig. 14.10Neuber’s diagram (nomograph) for a calculated stress concentrationfactor at the root of a notch
Let the calculated stress concentration factor be Scd when the groove is very deepThen from Neuber (1958)
(14.17)
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Fig. 14.10Neuber’s diagram (nomograph) for a calculated stress concentration factor at the root of a notch
Let Scc represent the calculated stress concentration for any depth of grooveThen according to Neuber, an approximate (and usually quite accurate) value of Scc is given by
(14.18)
When the groove is very shallow, Eq. 14.18 reduces to Scc =Scs
When the groove is very deep, Eq. 14.18 reduces to Scc =Scd24
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Fig. 14.10Neuber’s diagram (nomograph) for a calculated stress concentrationfactor at the root of a notch
Curve #1 was plotted using Eqs. 14.16-14.18The rest of the curves were obtained in a similar mannerHow is Fig. 14.10 used?Assume for Fig. A of Table 14.3
––––
=6.35 mmt=38.0 mmb=241.0 mmBar is subjected to a bending moment M
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=6.35 mmt=38.0 mm b=241.0 mm
From these values–
Scale f applies forCurve 2 applies forEnter Fig 14.10 withProceed vertically upward to Curve 2Move horizontally to the left to the axis of ordinatesBy a straight line, join this point to the pointThis line is tangent to the circle corresponding to the approximate Scc=4.25 26
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Some values of Scc for bending obtained from Neuber’sdiagram (Fig. 14.10) as found are given in Fig. 14.11
Fig. 14.11Scc for semicircular grooves in a cylindrical member subjected to a bending only obtained for Neuber’s diagram 27
Fig. 14.10
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Fig. P14.730
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Stress Concentration Factors: Combined Loads
AssumingLinear superposition is applicableIsotropic material
14.3.1 Infinite Plate with a Circular Hole
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14.3.2 Elliptical Hole in an Infinite Plate Uniformly Stressed in Directions of Major and Minor Axes of the Hole
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Pure Shear Parallel to Major and Minor Axes of theElliptical Hole
Let an infinite plate be subject to a uniform shear stress The stress state resulting from this case of pure shear parallel to the (x,y) axes may be found by superposition of the two cases for uniform tension
• (=) @ =/4 and –(=-) @ =3/4See Figs. 14.5 and 14.6 and Eqs. 14.9 and 14.14
(14.22)
Fig. 14.12Distribution of around an elliptical hole in an infinite plate loaded in pure shear 35
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Elliptical Hole in an Infinite Plate with Different Loadsin Two Perpendicular Directions
Let the plate be subjected to uniformly distributed stresses 1 > 2 along straight line edges far removed from the holeLet the major axis be at an angle w.r.t. the edge on which 1 actsThe solution to this problem can be found by superimposing the loadings of Figs. 14.8,14.9 and 14.12
Fig. 14.8 Fig. 14.9 Fig. 14.12 37
Fig. 14.13 Infinite plate withinclined elliptical hole and uniformly distributed stresses1 > 2 Fig. 14.14 Infinite plate with inclined
elliptical hole and uniformly distributed stresses 1 > 2
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Stress Concentration at a Groove in a Circular Shaft
Assume the shaft is subject toAxial force PBending moment MTwisting moment TShear force V (small effect therefore neglect)
Want to calculate the max principal stress at the root of the grooveMax prin occurs at Point AStress components are zz and zx
By Eq. 2.37, the max prin is
(14.40) Fig. 14.15Circumferential groove in acircular shaft
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The axial stress zz is due to P and M (14.41)
The Scc values are determined from Curves 6 and 7 in Fig. 14.10The stress zx is given by
(14.42)
The Scc is determined from Curve 9 in Fig. 14.10
Fig. 14.15Circumferential groove in a circular shaft
Fig. 14.16 Stress concentration factors obtained by use of the photoelastic method45
Fig. 14.17Stress distribution at notches found by the photoelastic method 46
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14.4.2 Strain-Gage Method
Fig. 14.18Calculated stress concentration factors for a shaft in bending with a transverse hole as found by the elastic strain method 47
Fig. 14.19The effect of bearing pressure of a load at the center of a beam on the longitudinal strains in the beam 48
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14.4.3 Elastic Torsional Stress Concentration at a Fillet in a Shaft
Fig. 14.20 Torsional shear-stress concentration at a fillet in a shaft of two diameters49
Elastic Membrane Method: Torsional Stress ConcentrationFound using soap film analogy
Fig. 14.21 Factors of torsional shear-stress concentration at keyway in a hollow shaft50
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Torsional Stress in a Fillet at an Angle Section
Fig. 14.22Angle section tested with soap film method (1917)
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14.4.5 Beams with Rectangular Cross Sections
Fig. 14.23 Stress concentrations at the fillet in a rectangular beam cross section
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If the beam is made of aductile material and is not subject to a large number of repeated loads (fatigue), the effects of stress concentrations are usually disregarded
Implies just use flexure formula=My/I
Respectfully disagree w/ book on this one
If beam is made of a brittle material, then effect of stress concentration must be considered in the design
Fig. 14.24Stress concentration factors for fillets in rectangular section beams
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Fig. 14.25 Stress concentration factors for grooves in rectangular section beams
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14.5 Effective Stress Concentration Factors14.5.1 Definition of Effective Stress Concentration
• Scc (calculated stress concentration factors)– Apply mainly to ideal, elastic materials– Depend mainly on the geometry (or form) of the abrupt change in
section– Often called form factors
• In applications involving real materials,– The significance of the stress concentration factor is
not indicated satisfactorily by the calculated value– Rather, it is found through experience
• The significant or effective stress value that indicates impending failure depends on– The characteristics of the material– Nature of the load– Geometry of the stress riser
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• When possible in practice, use the significant or effective stress concentration factor
• Usually, Sce < Scc
• Sce is found by testing two or more samples or sets of test specimens– One set (or sample) w/o the discontinuity– One set (or sample) w/ the discontinuity– Failure is assumed to occur at same stress level in both sets
• Because Sce < Scc (usually), the Sce may be defined in terms of an effective stress quantity e
where c > e > n
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Consider the case where c > e > nTheory implies the value of c
Experiment gives the value of e
Theory can be used to calculate n
Then e may be expressed as the n plus some proportion q of the increase in the calculated stress caused by the stress concentration
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• q is called the notch sensitivity index– q=0 implies no notch sensitivity– q=1 implies fully notch sensitive
• q is a function of the ability of the material and the member to make adjustments that reduce the effects of the localized high stress, e.g., Local yielding
• Ability of the material to make adjustments depends on– Type of loading (static, cyclic (aka repeated) or impact)– Residual stresses– Material structure– Temperature– Surface finish– Stress gradient
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Fig. 14.26 The influence of the groove radius on the notch sensitivity index
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14.5.2 Static LoadsDuctile Materials
••••
•
Material able to yield at high local stressq is usually between 0 and 0.1
If local deformation is restricted, then may have q 1 At high temperature, may have creep
and may have q 1Brittle Materials
q is usually between 0.5 and 1 for an abrupt change in cross section subjected to static loads– Except for materials that contain many internal stress risers
inherent in the internal structure of the material, e.g. graphite flakes in gray cast iron. In such a material, the value of q is relatively small because the internal stress risers are as important as the abrupt change on the surface.
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Repeated Loads
Mode of failure is of progressive fracture (fatigue)Ability of the material to make adjustments by local yielding is reduced
– Leads to a large value of q (usually between 0.5 and 1.0)
Fig. 14.27 Fatigue cracks in a herringbone gear66
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14.5.4 Residual Stresses
• If the member is made of ductile metal and subjected to static loads, localized yielding relieves the effects of residual stresses– and q is not altered by the residual stresses
• In contrast for a brittle material, the residual stresses can add or subtract from the effects of the stresses– Consequently q may increase or decrease
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14.5.5 Very Abrupt Changes in Section: Stress Gradient
• Let the change in section of a member be very abrupt; i.e., let the hole, fillet or groove have a very small characteristic dimension compared to the dimensions of the section, s.t. the calculated stress gradient is steep in the region of the stress concentration
• The value of Scc is large• However, the Sce found from tests of such members
is relatively low lower q than for less abrupt changes
• i.e., a small scratch is not as critical as a ―big‖ groove
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Significance of Stress Gradient
For notches such as holes, fillets, and grooves, the stress gradient is given by
(14.45)
As S increases, the amount of material that needs to adjust to the stress goes down
– As a consequence q goes down and the notch sensitivity decreases as decreases
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Impact or Energy Loading
If a member is required to absorb energy delivered to it by a relatively high-velocity impact, localized stress is a large influence in decreasing the load-carrying capacity of the memberFor a uniaxial state of stress Uo=2/2EJustification for putting a v-notch in a fracture specimen to initiate point of failureCharpyIzod
Want to find the maximum stress and max strain at the roots of the notchesAssume the stress distribution as shown in Fig. 14.28(a)
max = Scen (14.46) in terms of the significant concentration factorLet n denote the nominal strain as shown by Point A in Fig. 14.28(b)Let C’D’ denote the strain distribution
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