MMU-307 DESIGN OF MACHINE ELEMENTS

FAILURES RESULTING FROM CYCLIC LOADING (FATIGUE)• Fatigue Theories

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Asst.Prof. Özgür ÜNVER December 24th, 2019

Characterizing Fluctuating Stresses• Fluctuating stresses in machinery often take the form of a

sinusoidal pattern because of the nature of some rotating machinery.

• It has been found that in periodic patterns exhibiting a single maximum and a single minimum of force, the shape of the wave is not important, but the peaks on both the high side and the low side are important.

• If the largest force is Fmax and the smallest force is Fmin, then a steady component and an alternating component can be constructed as follows:

•Soderberg,

•Modified Goodman,

•Gerber,

•ASME-elliptic,

•Yield.

Five criteria of failure are diagrammed

Fatigue Failure Criteria for Fluctuating Stress

Fatigue Failure Criteria for Fluctuating Stress

• we want to vary both the midrange stress and the stress amplitude(alternating component).

• The midrange stress line is a 45◦ line from the origin to the tensile strength of the part.

• Yield strength is also plotted on both axes, because yielding would be the criterion of failure if σmax exceedsSy .

Fatigue Failure Criteria for Fluctuating Stress• In general the failure point is little related to the mean stress when it is

compressive, however, it is very much a function of the mean stress when it is tensile.

• Generally, the test data for ductile material fall closer to Gerber parabola but because of scatter in the test points, a straight line relationship is usually preferrred in designing machine parts.

• A Goodman line is used when the design is based on UTS

• A Soderberg line is used when the design is based on yield strength

• If there is a stress concentration, Kf value must be found and used only for variable (alternating) part (if the material is brittle Kt must be used for the mean part)

Fatigue Failure Criteria for Fluctuating Stress

• When the midrange stress is compression, failure occurs whenever σa = Se or whenever σmax = Syc

• Yield strength Sy serves as a reminder that first-cycle yielding rather than fatigue might be the criterion of failure.

Fatigue Failure Criteria for Fluctuating Stress

Soderberg

Modified Goodman

Gerber

Asme Elliptic

Yield

Fatigue Failure Criteria for Fluctuating Stress

Modified Goodman

Yield

Intersection of the static and fatigue

Use line equations instead of these

complex equations, they are the same thing!

ExampleExample-1: A 1.5-in-diameter bar has been machined from an AISI 1050 cold-drawn bar. This part is to withstand a fluctuating tensile load varying from 0 to 16 kpsi. Because of the ends, and the fillet radius, a fatigue stress-concentration factor Kf is 1.85.For 106 or larger life.

Find Sa and Sm and the factor of safety guarding against fatigue and first-cycle yielding, using

(a) the Soderberg and

(b) the Modified Goodman line.

Example-2

Brittle Materials The first quadrant fatigue failure criteria follows a concave upward Smith-Dolan locus represented by

The fatigue diagram for a brittle material differs markedly from that of a ductile material because;

• Yielding is not involved

• UTS_compressive > UTS_tensile

• First-quadrant fatigue failure locus is concave-upward (Smith-Dolan

• More sensitive to midrange stress

• Not enough work has been done on brittle fatigue to discover insightful generalities.

Combinations of Loading Modes

FATIGUE Completely reversing simple loads

(only one type of loading is allowed at a time)

S-N Diagram

Fluctuating simple loads

(only one type of loading is allowed at a time)

Modified Goodman

Soderberg

Combinations of loading modes

(combinations of different types of loading)

Combinations of Loading Modes

kc is used for axial, bending, or torsional loads seperately!

Question: How do we proceed when the loading is a mixture of, say; axial, bending, and torsional loads?

Answer: Distortion energy failure theory proved to be a satisfactory method of combining the multiple stresses on a stress element of a ductile material into a single equivalent vonMises stress. The same approach will be used here.

Combinations of Loading Modes

Generate two stress elements one for the alternating and one for the midrange stresses.

Apply Kf bending, Kfs torsion, Kf axial only for alternating part!

Calculate an equivalent von Mises stress σa and σm.

Select a fatigue failure criterion, modified Goodman or Soderberg, to complete the fatigue analysis.

For the endurance limit, Se, use the endurance limit modifiers, ka, kb, and kc, for bending.

The torsional load factor, kc = 0.59 should not be applied as it is already accounted for in the von Mises stress calculation

The load factor for the axial load can be accounted for by dividing the alternating axial stressby the axial load factor of 0.85.

Combinations of Loading ModesVon Mises stresses for the two stress elements

(you may also use other failure theories as well!)

For the first-cycle localized yielding the maximum von Mises stress is calculated by;• Adding the axial and bending alternating and midrange stresses to obtain σmax• Adding the alternating and midrange shear stresses to obtain τmax. • Substitute σmax and τmax into the equation for the von Mises stress.

Optional

ExampleA rotating shaft

• 42 × 34-mm AISI 1018 cold-drawn steel tubing

• Has a 6-mm-diameter hole drilled transversely through it.

Estimate the factor of safety guarding against fatigue and static failures using the Soderberg and Modified Goodman failure criteria for the following loading conditions:

• The shaft is subjected to a completely reversed torque of 120 Nm in phase with a completely reversed bending moment of 150 Nm

• The shaft is subjected to a pulsating torque fluctuating from 20 to 160 Nm and a steady bending moment of 150 Nm

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