15-stressconcentrations

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Stress Concentrations

INTRODUCTION ...........................................................................................................................3

Stress Concentration – General Information....................................................................................3

CONTROLLING STRESS CONCENTRATIONS.........................................................................6

GEOMETRIC STRESS CONCENTRATION FACTORS.............................................................9

Sheet and Plate.................................................................................................................................9

Mike Mohaghegh Design and Analysis of Aircraft Structures 15-111/26/2008


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15-2 Design and Analysis of Aircraft Structures Mike Mohaghegh11/26/2008


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Introduction

This section presents information on stress concentrations and provides qualitative assistance

to the designer in selecting design details that will result in optimum life and static strength.

No technique is included to calculate fatigue life, but approximate methods are presented tocalculate the effect on static strength. If a structure is subjected to fatigue loads, use the

following guidelines.

Stress Concentration – General Information

This section contains stress concentration factors k t for several common geometric shapes. k t is

defined as the ratio of the peak elastic stress to a reference stress, usually the maximum local

stress that would exist without the specified stress concentration. The factors are independentof material in that their derivation assumes that the material is infinitely elastic, making the

factors independent of modulus of elasticity. However, the significance of the stress

concentration factors to a particular structure does depend on the material characteristics.

The factors are especially important to static strength of structure made from materials thathave little or no ductility, as is common with ceramics and other “brittle” materials. Staticstrength of structures made from highly ductile materials are less affected by stress

concentrations because yielding and plastic flow, in the zone of stress concentration, delay

rupture until the strain in the area of peak stress reaches the ultimate elongation for the material.For highly ductile materials, the net area stresses outside the zone of peak strain approach

ultimate before rupture is initiated. However, the magnitude of stress concentration coupled

with the operating stresses (load intensities) is probably the most important single factordetermining service life for a given material.

The methods presented here for dealing with structural details that cause stress concentrations

minimize the concentration factors and promote the use of improved design details.

Methods for predicting the effect of stress concentration on static strength and for selecting

materials for members that are critical for static strength are also presented.

NOTE: Before calculating stress concentration factors, calculate the stress based on the netarea. If the value exceeds the ultimate stress for the material, failure would be

unavoidable, and no further calculation for stress concentration is needed.

Stress and strain concentrations are increased in local stress and strain in a member. They

result from irregularities or discontinuities in the geometry of the member. Frequentlyencountered discontinuities include notches, holes, threads, keyways, grooves, and scratches.

See Figure 1.



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Hole inplate

Notch in plate

Filletedbar

Weld

Shoulder fillet

Hole insolid rod

Profiled holein tube

Profiled keyway

Semicircular groove

Figure 1. Examples of Geometric Discontinuities

Several distinctly different stress concentration factors can be employed, and the differencesmust be clearly identified. Significant differences relate to load intensity, load type (static or

cyclic), member material geometry, notch geometry, and method of analysis. The most

commonly used is k t , where:

nom

t f

f k max= (See Figure 2)

Actual stress

distribution atnotchedsection.

MM

C

h

f max

f max =

f max

f max

h

dD

Computed frombending

formula

kin

Mc

I

Figure 2. Stress concentration Induced by a Notch

The subscript “t ” indicates “theoretical” because it relates to linear elastic behavior only and is

therefore independent of specific material characteristics and is usually derived analytically.

However, it can also be determined by various test methods in the elastic range.



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The nominal stress can be defined using either the gross cross sectional area or the net cross

sectional area of a notched specimen. Stress concentration factor k t is designated as being based on gross or net area by adding the subscript g or n, as k tg or k tn.

Note that k tg accounts for two effects: (1) increased stress from loss of cross sectional area and(2) increased stress from geometry. k tn accounts for only one effect, increased stress from

geometry.

In contrast, stress concentration factors k r and k f signify rupture and fatigue, respectively, and

always relate to specified materials and are usually determined by test.

The k t values for several geometries with various types of discontinuities are presented in § 3.

Other geometries are readily available in handbooks and structures manuals.

NOTE: Caution is advised in obtaining k t from reference sources. A net or gross k t asdefined above must be specified clearly in the reference and used consistently

throughout the analysis.

The effects of stress concentrations on ductile materials subjected to static loading and uniaxial

stresses are small. If a part is subjected to shock, vibration, or large temperature variations, aductile material may need to be treated as a brittle material for analysis purposes.

The effects of stress concentration on ductile or brittle materials subjected to alternating loadsare of great importance since cracks usually originate at the point of the stress concentration.

The combination of stress concentration and alternating loads is the main cause of fatigue

failure.

When the load on a structure is such that the stresses are in the plastic region, the stressconcentration has a lesser effect on the peak stress. Plastic yielding has the effect of relieving

the stress concentration. The relief in the stress concentration for a given material is seen bycomparing k r to k t for a specific notch. The factor k r , called the stress concentration at rupture,

is used to define the reduction in static strength for a specimen with a given notch.Mathematically, it is the ratio of the stress at rupture for a plain specimen to the stress at

rupture for a notched specimen, and is expressed by the following equation:

notched

plain

r f

f k =

Where

WT

P f plain

1=

WT

P f notched

2=

and, P1 and P2 are test ultimate loads for the plain and notched specimens, respectively.



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Controlling Stress Concentrations

Control of stress concentrations or their effects can usually be accomplished by the following

general methods.

Design the parts to eliminate or minimize the stress concentration factors or restrict them

to lightly stressed areas. If the stress concentration factor cannot be adequately reduced in the design, add

sufficient material to reduce the nominal stresses and overcome the problem of stress

concentration. If the approach in the preceding item is not possible, selective removal of material might

be an acceptable alternative. Examples g, h, and i in the following list illustrate

techniques to control stress concentrations by removing material. Note that these

examples reflect parts that appear to be heavier than parts designed for adequate strength per the two preceding items.

Avoid multiple stress concentrations at a single location because the stress concentration

factor for this location will be the product of the individual stress concentration factors.When stress concentrations are unavoidable, they should be separated.

The following list presents more specific methods and examples of techniques for controlling

stress concentrations.

a. Position material grain direction parallel to the applied load direction whenever possible.

b. Locate all holes in low–stressed regions. Whenever feasible, bush holes if wear could cause

enlargement.

c. Steel stamping of part numbers should be called out on areas of low stress or on raised

bosses. See BAC5307 for additional information. Consult process specifications formaterials with high notch sensitivity, where steel stamping is not allowed.

d. Use high–quality surface finishes for highly stressed members.

e. Shot–peen parts made from ductile material such as aluminum, steel, or titanium toincrease resistance to stress corrosion cracking and improve fatigue life.

f. Avoid knife edges; they cause high stress concentration and are susceptible to additionalnicks and cracks. Examples are overlapping spot faces and countersunk thin sheets. The

depth of countersink should not exceed two–thirds of the sheet thickness.

g. Use a thread relief at the terminus on threaded fittings by undercutting below the thread

root. See Figure 3.

Thread

relief

Figure 3. Thread Relief at the Terminus



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h. Use generous fillet radii or faired lines to avoid abrupt changes in member sections. See

Figure 4.

Figure 4. Relief for Change in Section

i. Reduce the stress concentration at B by removing material at locations A. See Figure 5.

Figure 5. Adding Grooves to Reduce Stress Concentrations

j. Relieve the stress concentration at B, the small–radius shoulder, by adding a groove at A.

See Figure 6.

Figure 6. Adding a Groove to Relieve Stress Concentrationsk. When a shoulder stop is required for another member with a small corner radius, add a

sunken fillet to provide a larger radius at the shoulder. See Figure 7.

Figure 7. Sunken relief for a Shoulder Stop

l. Avoid sharp–cornered discontinuities or notches in a part subjected to repeated loading.

Use a large radius faired into the contour. Use durability methods to determine the life of amember subjected to repeated loading cycles. The type of surface irregularities, material,

and notch sensitivity must also be considered in determining a suitable configuration. See

Figure 8.



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Figure 8. Large Radius Faired into the Contour

m. In some instances, application of a proof load can reduce the maximum stresses resulting

from stress concentrations and increase fatigue life. The approach is applicable to

structures that are loaded primarily in tension, such as pressure vessels. In the case of a pressure vessel, the mechanics involved consist of applying a proof pressure somewhat

above the maximum expected operating pressure and sufficient to exceed the tensile yield

stress in the areas of maximum stress concentrations, resulting in decreased maximum

tension stresses during subsequent pressure cycles. With appropriate materialcharacterization and analyses, the technique can also be used to establish the minimum

number of pressure cycles remaining prior to failure. However, for more complexstructures, the process may be self–defeating because yielding in areas of compressionstresses will lock in tension stresses, and this may reduce rather than increase the fatigue

life. Anyone considering the technique is advised to consult specialists experienced in

fatigue and fracture mechanics.

n. Always control locations of machining mismatches and avoid sharp edges.

o. Avoid eccentricities where cross sections vary or where load directions change.

p. Avoid threaded or tapped holes.

q. Minimize the number of joints and splices, and locate them in low–stress areas whenever possible.

r. Avoid open or unfilled holes; fill tool holes, and reinforce holes for system provisions ordrainage.

s. Whenever feasible in tension applications, cold–work a hole to reduce the peak stress that

results from the stress concentration effect of the hole.

t. When a change in the cross section occurs, provide a gentle taper for the transition.

u. Avoid materials sensitive to stress corrosion cracking where high stress concentration or

residual stresses exist. Stress corrosion cracking is primarily a function of the highestsustained tensile stress. The combination of a high stress concentration in a material

sensitive to stress corrosion cracking could result in fracture of the member at low nominal

stresses.

NOTE: Even when designing with ductile materials, employ details that tend tominimize the stress concentration factor, and, most importantly, avoid details

that place two or more stress concentrations at the same location. When two or

more stress concentrations coincide, each pair of factors multiplies. Majorcracks can develop at a fraction of the limit load, and failure can occur at

fewer than a half dozen cycles of the load that developed the crack.



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Inferior design is indicated if any stress concentration factor significantly reduces the static

strength or fatigue life and requires significant increase to the weight of the member to sustainthe load.

Local increases in stress caused by k t severely limit the load capability of structuralcomponents and can lead to premature failure. Such designs usually result in added excessive

material when detailed for fatigue life. The actual structure is seldom exactly represented by

the discontinuities represented in the analysis, and exact stress concentration factors aredifficult to obtain. The difference between the calculated and actual stress concentration

factors is more significant at higher stress levels and stress concentration factors. Therefore,

design improvement action should be considered if the stress concentration factor approachesfour or more. Even with lower stress concentration factors, designs that reduce the stress

concentrations are desirable.

For areas of high stresses, the designer should consider detailed finite–element analysis to

obtain the most accurate predictions of maximum stress.

Geometric Stress Concentration Factors

Data in Figure 3–1 through Figure 3–41 provide numerical values of the stress concentration

factor k t for a variety of common geometric discontinuities. Additional information and cases

can be found in the book “Stress Concentration Factors,” by R. E. Peterson. For geometriesand loadings not explicitly covered in available references, consider using appropriate detailed

finite element analysis.

The k t factor is given based on either the nominal stress for the gross area, k tg, or the nominal

stress for the net area, k tn. The gross area is the cross sectional area that would exist at the

location of the stress concentration if the material for the stress concentration were not

removed. The net area is the actual cross sectional area that exists at the location of the stressconcentration.

For two discontinuities that are superimposed, it is generally conservative to use the product ofthe individual stress concentration factors. Note that this can be overly conservative in some

cases and may warrant a more accurate analysis, particularly if the stress concentrations peak

at different locations.

Round holes, either singly or in multiple arrays with various patterns, are a primary source of

stress concentrations. Therefore, several charts are presented in 3.1 to support joint analysisconsidering round holes in various patterns in sheets and plates.

Sheet and Plate

Figure 3–1 through Figure 3–26 provides stress concentration factors for various cases of

plates with holes, notches, and steps. Geometric features shown by sketch or equation at an

unspecified distance from the edge are assumed to be in the center of an infinite sheet or plate,

which in practical terms is equivalent to 10 or more hole diameters. Proximity to an edgeresults in further increases in the stress concentration.



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Figure 3-1. Plate with Single Row of Circular Holes, Uniaxial Tension

Figure 3-2. Plate with Single Row of Circular Holes, Biaxial Tension



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Figure 3-3. Plate with Circular Hole, Uniaxial Tension

Figure 3-4. Plate with Single Row of Circular Holes, Uniaxial Tension



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Figure 3-5. Plate with Two Unequal Circular Holes, Transverse Uniaxial Tension

Figure 3-6. Plate with Two Unequal Circular Holes, Oblique Uniaxial Tension



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Figure 3-7. Plate with Unequal Circular Holes, Shear

Figure 3-8. Plate with Double Row of Staggered Circular Holes, Uniaxial Tension



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W d

f ref peak

(Ref stress is gross area stress.)

f k tg=

f ref

3

4

7

6

5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

k tg

d/W

Figure 3-9. Plate with Central Circular Hole, Uniaxial Tension



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Figure 3-10. Plate with Central Symmetrically Padded Circular Hole, Uniaxial Tension

Figure 3-11. Plate with Central Circular Hole, Shear



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Figure 3-12. Plate with Central Symmetrically Padded Circular Hole, Shear

Figure 3-13. Plate with Central Circular Hole, Bending About Shallow Axis



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Figure 3-14. Plate with Central Circular Hole, Bending About Deep Axis

Figure 3-15. Plate with Square Hole, Uniaxial Tension



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Figure 3-16. Plate with Elliptical or Circular Hole, Biaxial Stresses

Figure 3-17. Plate with Unsymmetrical Edge Notch, Tension



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Figure 3-18. Plate with Unsymmetrical Edge Notch, Bending

Figure 3-19. L-Section Plate, Bending



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Figure 3-20. Plate with Unsymmetrical Edge Notch, Bending or Tension – Influence of

Shoulder Slope

Figure 3-21. Plate with Symmetrical Edge Notches, Tension



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Figure 3-23. Plate with Symmetrical Edge Notches, Bending



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Figure 3-24. Plate with Symmetrical Edge Notches, Bending

Figure 3-25. Plate with Symmetrical Edge Notches, Combined Shear and Bending



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Figure 3-27. Curved Beams, Bending



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Figure 3-28. Circular Shaft with External Shoulder, Tension

Figure 3-29. Circular Shaft with External Shoulder, Torsion



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Figure 3-30. Circular Shaft with External Shoulder, Bending

Figure 3-31. Circular Shaft with narrow Shoulder, Bending



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Figure 3-32. Circular Tube with Internal Groove, Tension

Figure 3-33. Circular Tube with External Groove, Tension



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Figure 3-34. Circular Tube with External Groove, Bending

Figure 3-35. Circular Tube with Internal Groove, Bending



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Figure 3-36. Circular Tube with Internal Groove, Torsion

Figure 3-37. Circular Tube with External Groove, Torsion



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Figure 3-38. Circular Tube with Circular Hole, Tension

Figure 3-39. Circular Tube with Circular Hole, Torsion



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Figure 3-40. Circular Tube with External Shoulder, Tension

Figure 3-41. Circular Tube with Circular Hole, Bending



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