MMU-307 DESIGN OF MACHINE ELEMENTS

FAILURES RESULTING FROM STATIC LOADING• Maximum-Shear-Stress Theory for Ductile Materials

• Distortion-Energy Theory for Ductile Materials

• Coulomb-Mohr Theory for Ductile Materials

Asst.Prof. Özgür ÜNVER November 12th, 2019

Basics

Strength is a property of a mechanical element.

This property results from;

• Material,

• Heat Treatment,

• Machining,

• Final geometry,

• Loading.

Stochastic vs. Deterministic System

In mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system.

Stochastic refers to systems whose behavior is intrinsically non-deterministic.

The strengths of the mass-produced parts will all be somewhat different from the others in the collection because of (uncertainties);

• variations in dimensions,

• machining,

• forming,

• composition.

Descriptors of strength are necessarily statistical in nature, involving parameters such as mean, standard deviations, and distributional identification.

Static LoadingA static load is a stationary force or couple applied to a member.

To be stationary,

• the force must be unchanging in magnitude, point or points of application and direction.

A static load can produce;

• axial tension or compression,

• a shear load,

• A bending load,

• a torsional load,

• any combination of these.

Failure

Stress Concentration

In some instances it may be due to a surface scratch.

If the material is ductile and the load is static, the design load may cause yielding in the critical location in the notch.

This yielding can involve strain hardening of the material and an increase in yield strength at the small critical notch location.

Since the loads are static and the material is ductile, that part can carry the loads satisfactorily with no general yielding.

In these cases the designer sets the geometric (theoretical) stress-concentration factor Kt=1.

Failure TheoriesUnfortunately, there is no universal theory of failure for the general case of material properties and stress state.

Instead, over the years, several hypotheses have beenformulated and tested.

Ductile materials are normally classified such that εf ≥ 0.05 and have an identifiable yield strength that is often the same in compression as in tension (Syt = Syc = Sy ).

Ductile materials (yield criteria)

• Maximum shear stress (MSS),

• Distortion energy (DE),

• Ductile Coulomb-Mohr (DCM),

Maximum-Shear-Stress Theory for Ductile Materials

When a ductile material is loaded to fracture, fracture lines are seen at angles approximately 45° with the axis of tension.

Since the shear stress is maximum at 45° from the axis of tension, it makes sense to think that this is the mechanism of failure.

MSS theory (Tresca) predicts that yielding begins whenever theMSS in any element equals or exceeds the maximum shear strength.

MSS theory is an acceptable but conservative predictor of failure; and since engineers are conservative by nature, it is quite often used.

• For simple tensile stress, σ = P/A, and the maximum shear stress occurs on a surface 45° from the tensile surface with a magnitude of τmax = σ/2.

• So the maximum shear stress at yield is τmax = Sy/2.

• For a general state of stress, three principal stresses can be determined and ordered such that σ1 ≥ σ2 ≥ σ3.

• The maximum shear stress is then τmax = (σ1 − σ3)/2

Maximum-Shear-Stress Theory for Ductile Materials

Considering factor of safety (n);

Maximum-Shear-Stress Theory for Ductile Materials

• Suppose point a represents the stress state of a critical stress element of a member.

• If the load is increased, it is typical to assume that the principal stresses will increase proportionally along the line from the origin through point a.

• If the stress situation increases along the load line until it crosses the stress failure envelope, such as at point b.

• The factor of safety guarding against yield at point a is given by n = Ob/Oa.

Maximum-Shear-Stress Theory for Ductile Materials

Distortion-Energy Theory (DE) for Ductile Materials

• DE theory predicts that yielding occurs when the distortion strain energy per unit volume reaches or exceeds the distortion strain energy per unit volume for yield in simple tension or compression of the same material.

• Also called Von-Misses and Shear-Energy Theory

• It is the most widely used theory for ductile materials and is recommended for design problems unless otherwise specified.

• DE theory originated from the observation that ductilematerials stressed hydrostatically (equal principal stresses) exhibited yield strengths greatly (rocks below the earth surface)

• Fig.b undergoes pure volume change, that is, no angular distortion.

• Fig.c is subjected to pure angular distortion, that is, no volume change.

Distortion-Energy Theory (DE) for Ductile Materials

For the general state of stress, yield is predicted if;

If we had a simple case of tension σ , then yield would occur when σ ≥ Sy .

Thus, the left of equation can be thought of as an effective stress for the entire general state of stress given by σ1, σ2, and σ3.

Distortion-Energy Theory (DE) for Ductile Materials

Distortion-Energy Theory (DE) for Ductile Materials

Consider a case of pure shear τxy ,where for plane stress σx = σy = 0. For yield;

Distortion-Energy Theory (DE) for Ductile Materials

DE theory gives about 15 percent greater yield margin compared to MSS Theory which was 0.5 Sy

Example: A hot-rolled steel has a yield strength of Syt = Syc = 100 Mpa and a true strain at fracture of εf = 0.55.

Estimate the factor of safety for the following principal stress states:

(a) σx = 70 Mpa, σy = 70 Mpa, τxy = 0 Mpa

(b) σx = 60 Mpa, σy = 40 Mpa, τxy = −15 Mpa

(c) σx = 0 Mpa, σy = 40 Mpa, τxy = 45 Mpa

(d) σx = −40 Mpa, σy = −60 Mpa, τxy = 15 Mpa

(e) σ1 = 30 Mpa, σ2 = 30 Mpa, σ3 = 30 Mpa

Distortion-Energy Theory (DE) for Ductile Materials

Distortion-Energy Theory (DE) for Ductile Materials

• Not all materials have compressive strengths equal to their corresponding tensile values.

• For example, the yield strength of magnesium alloys in compression may be as little as 50 percent of their yield strength in tension.

• The yield strength of gray cast irons in compression varies from 3 to 4 times greater than the tensile strength.

• Therefore, in this theory we will try to predict failure for materials whose strengths in tension and compression are not equal.

Coulomb-Mohr Theory for Ductile Materials

Coulomb-Mohr Theory for Ductile Materials

Example: A 25-mm-diameter shaft is statically torqued to 230 Nm. It is made of cast 195-T6 aluminum, with a yield strength in tension of 160 MPa and a yield strength in compression of 170 MPa.

Estimate the factor of safety of the shaft!

Why do we use yield strength in all our calculations??

Coulomb-Mohr Theory for Ductile Materials

Summary of Failure of Ductile Materials

• For design purposes the maximum-shear-stress theory is easy,quick to use, and conservative (easy to apply).

• If the problem is to learn why a part failed, then the distortion-energy theory may be the best to use. It is generally a better predictor of failure (gives more accurate results).

The cantilevered tube shown is to be made of 2014 aluminum alloy treated to obtain a specified minimum yield strength of 276 MPa. We wish to select a stock-size tube from Table A–8 using a design factor nd = 4. The bending load is F = 1.75 kN, the axial tension is P = 9.0 kN, and the torsion is T = 72 Nm. What is the realized factor of safety?

Example