Maximum distortion energy theory - Part

Part – 6: Machine Design

5. Maximum distortion energy theory

Where,

Octahedral shearing stress Failure is said to occur when ^√

This theory is supported quite well by experimental evidences and is identical to Von Mises theory.

6.1.2 Theories of failure for two dimensional stresses:

Taking as zero, the above equations reduce to 1. Maximum principle stress theory

2. Maximum principle strain theory ( )

3. Maximum shear stress theory (a) For like tensile stresses

(b) For unlike stresses ( )

4. Maximum strain energy theory

5. Maximum distortion energy theory

6.1.3 Significance of theories of failure

Mode of failure of a ductile material differs from that of brittle material. It depends on a large number of factors like

 Nature and Properties of the material

 Type of loading

 Shape of member

 Temperature of member, etc.

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If the loading conditions are suitably altered, a brittle material may be made to yield before failure. Hence, design of a member requires the determination of the mode of failure (yielding or fracture), and the factor (such as stress, strain and energy) associated with it. Full scale tests simulating all conditions would be ideal but not practicable.

In practice, in complex loading conditions, the factor associated with failure has to be identified and precautions taken to ensure that this factor does not exceed maximum allowable value determined on the basis of suitable tests (uniform tension or torsion) on the material in the laboratory.

Results of many laboratory tests on ductile material shows shear stress from torsion tests varies between 0.55 and 0.6 of the yield strength determined from tension tests. This result agrees with shear strain energy theory and octahedral shear stress theory. The maximum shear stress theory predicts that the shear yield value is 0.5 times the tensile yield value, which is about 15%

less than the value predicted by the other two theories.

The maximum shear stress theory gives design values on the safe side and is widely used in design with ductile materials.

Part 6.2: Fatigue 6.2.1 Stress concentration

Whenever a machine component changes the shape of its cross-section, the simple stress distribution no longer holds good and the neighbourhood of the discontinuity is different. This irregularity in the stress distribution caused by abrupt changes of form is called stress concentration.

It occurs for all kinds of stresses in the presence of fillets, notches, holes, keyways, splines, surface roughness or scratches etc.

Fig. 6.2.1

In the above member with different cross-section under a tensile load, the nominal stress in the right and left hand sides will be uniform but in the region where the cross-section is changing, a re-distribution of the force within the member must take place. The maximum stress occurs at some point on the fillet and is directed parallel to the boundary at that point.

Theoretical or Form Stress Concentration Factor:

The theoretical or form stress concentration factor is defined as the ratio of the maximum stress to the nominal stress at the same section based upon net area.

Maximum stress Nominal stress

The value of Ktdepends upon the material and geometry of the part.

 In static loading, stress concentration in ductile materials is not so serious as in brittle materials, because in ductile materials local deformation or yielding takes place which reduces the concentration. In brittle materials, cracks may appear at these local concentrations of stress which will increase the stress over the rest of the section.

 In cyclic loading, stress concentration in ductile materials is always serious because the ductility of the material is not effective in relieving the concentration of stress caused by cracks, flaws, surface roughness, or any sharp discontinuity in the geometrical form of the member. If the stress at any point in a member is above the endurance limit of the material, a crack may develop under the action of repeated load and the crack will lead to failure of the member.

Stress Concentration due to Holes and Notches:

Consider a plate with transverse elliptical hole and subjected to a tensile load as shown in the figure.

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Fig. 6.2.2 Stress Concentration due to holes

From the stress-distribution, the stress at the point away from the hole is practically uniform and the maximum stress is induced at the edge of the hole. The maximum stress is given by

( )

and the theoretical stress concentration factor,

( )

When a/b is large (a), the ellipse approaches a crack transverse to the load and the value of Kt

becomes very large. When a/b is small (b), the ellipse approaches a longitudinal slit and the increase in stress is small. When the hole is circular (c), then a/b = 1 and the maximum stress is three times the nominal value.

Fig. 6.2.3 Stress Concentration due to notches

The stress concentration in the notched tension member, is influenced by the depth a of the notch and radius r at the bottom of the notch. The maximum stress, which applies to members having notches that are small in comparison with the width of the plate, may be obtained by the following equation,

( )

Methods of Reducing Stress Concentration:

 Maintain or improve the spacing of the stress flow lines that tend to bunch up and cut very close to the sharp re-entrant corner, by providing

 Fillets and

 Notches ( when not possible to use large radius fillets as in case of ball and roller bearing mountings)

 The stress concentration effects of a press fit may be reduced by making more gradual transition from the rigid to the more flexible shaft

6.2.2 Dynamic loading

 The stresses which vary from a minimum value to a maximum value of the same nature, (i.e. tensile or compressive) are called fluctuating stresses.

Fig. 6.2.4

 The stresses which vary from zero to a certain maximum value are called repeated stresses.

Fig. 6.2.5

 The stresses which vary from a minimum value to a maximum value of the opposite nature (i.e. from a certain minimum compressive to a certain maximum tensile or from a minimum tensile to a maximum compressive) are called alternating stresses.

Fig. 6.2.6

The variable stress, in general, may be considered as a combination of steady (or mean or average) stress and a completely reversed stress component.

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1. Mean or average stress,

2. Reversed stress component or alternating or variable stress,

Note: For repeated loading, the stress varies from maximum to zero (i.e. = 0) in each cycle.

3. Stress ratio,

 For completely reversed stresses, R = – 1

 For repeated stresses, R = 0.

 R cannot be greater than unity.

4. Relation between endurance limit and stress ratio

Where,

Endurance limit for any stress range represented by R.

Endurance limit for completely reversed stresses, and Stress ratio.

6.2.3 Fatigue

When a material is subjected to repeated stresses, it fails at stresses below the yield point stresses. Such type of failure of a material is known as fatigue.

The fatigue of material is effected by the size of the component, relative magnitude of static and fluctuating loads and the number of load reversals.

Fatigue failure is due to crack formation and propagation. A fatigue crack will typically initiate at a discontinuity in the material where the cyclic stress is a maximum.

Discontinuities can arise because of:

 Design of rapid changes in cross section, keyways, holes, etc. where stress concentrations occur

 Elements that roll and/or slide against each other (bearings, gears, cams, etc.) under high contact pressure, developing concentrated subsurface contact stresses that can cause surface pitting or spalling after many cycles of the load.

 Carelessness in locations of stamp marks, tool marks, scratches, and burrs; poor joint design; improper assembly; and other fabrication faults.

 Composition of the material itself as processed by rolling, forging, casting, extrusion, drawing, heat treatment, etc. Microscopic and submicroscopic surface and subsurface discontinuities arise, such as inclusions of foreign material, alloy segregation, voids, hard precipitated particles, and crystal discontinuities.

 Linear-elastic fracture mechanics method

These methods attempt to predict the life in number of cycles to failure, N, for a specific level of loading.

N ≤ 10³ - low-cycle fatigue N > 10³ - high-cycle fatigue Stress-life method:

 Based on stress levels only

 Least accurate approach, especially for low-cycle applications.

 Easiest to implement for a wide range of design applications,

 Has ample supporting data, and

 Represents high-cycle applications adequately.

Strain-life method:

 More detailed analysis of the plastic deformation at localized regions where the stresses and strains are considered for life estimate

 Good Method for low-cycle fatigue applications

 In applying this method, several idealizations must be compounded, and so some uncertainties will exist in the results.

Fracture mechanics method:

 Assumes a crack is already present and detected. It is then employed to predict crack growth with respect to stress intensity.

 Most practical when applied to large structures in conjunction with computer codes and a periodic inspection program.

Fatigue Strength and the Endurance Limit:

The strength-life (S-N) diagram provides the fatigue strength Sfversus cycle life N of a material.

The results are generated from tests using a simple loading (R. R.Moore high-speed

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beam machine) of standard laboratory controlled specimens. The loading often is that of sinusoidally reversing pure bending. The laboratory-controlled specimens are polished without geometric stress concentration at the region of minimum area.

Fig. 6.2.7 Test-specimen geometry for the R.R. Moore rotating beam machine.

Fig. 6.2.8 S-N diagram for steel, normalized; Sut = 116 kpsi; maximum Sut = 125 kpsi.

Fig. 6.2.9 S-N bands for representative aluminum alloys, excluding wrought alloys with Sut < 38 kpsi

For steel and iron, the S-N diagram becomes horizontal at some point. The strength at this point is called the endurance limit [maximum value of the completely reversed bending stress which a polished standard specimen can withstand without failure for infinite number of cycles (usually

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