Course No.: MEBF ZC342 MACHINE DESIGN L1: Stress Analysis Principles, Prof. D. Datta Prof. D. Datta.

Course No.: MEBF ZC342MACHINE DESIGN

L1: Stress Analysis Principles, Prof. D. Datta

Prof. D. Datta

An Overview of the Subject

The Essence of Engineering is the Utilization of resources and the Laws of Nature for the benefit of Mankind

Engineering is an applied science in the sense that it is concerned with understanding scientific principles and applying them to achieve a designated goal.

Mechanical Engineering Design is a major segment of Engineering.

Machine Design is a segment of the Mechanical Engineering Design in which decisions regarding shape and size of Machines or Machine components are taken for their satisfactory intended performance.


Phases in Design

Identification of Need

Definition of Problem

Synthesis

Analysis and Optimization

Evaluation

Presentation

Design is a highly Iterative Process


Steps in DesignIdntify the Need

Collect Data to Describe the System

Estimate Initial Design

Analyze the System

Check Performance Criteria

Is Design Satisfactory? Yes Stop

No

Change the Design based on Experience/Calculation


Design Considerations

1. Functionality2. Strength3. Stiffness / Distortion4. Wear5. Corrosion6. Safety7. Reliability8. Manufacturability9. Utility10. Cost11. Friction12. Weight13. Life

14. Noise15. Styling16. Shape17. Size18. Control19. Thermal Properties20. Surface21. Lubrication22. Marketability23. Maintenance24. Volume25. Liability26. Remanufacturing


Loads and Equilibrium


Restraints


Stresses

Uni-axial Stress

X

AP

x

X

2cos)cos/(cos

A

PA

P cossin)cos/(

sinA

PA

P

A

Normal Stress Shear Stress

Torsional Shear Stress

d

LinearVariation

τmax

J

Tr

J

dT )2/(max

J = Polar Moment of Inertia =32

4d

Angle of Twist

GJ

Tl

Torsion formula with slowly varying area may be used as long as they are circular


Torsional Shear Stress (cont’d)

d

x

y

z

xz

xy

J

dTx

)2/(max

cosxz

sinxy

Here, τxz is negative as it acts opposite to the +z-axisbut τxy is positive as it acts along the +y-axis


Normal Stresses due to Bending

I

Myx

M = Bending Momenty = Distance of the layer from Neutral AxisI = Moment of Inertia of the cross section about the axis of bending

X


P

P/2P/2 L/2L/2

P/2P/2 -+

PL/4

+

b

d

3

12

1bdI

-+

BM Sign

A Problem: Get the Shear Force and BM Distribution


Getting Maximum Normal and Shear Stresses

x

xyxy

X

2sin2cos22 xyxx

2cos2sin2 xyx

Combining the above two equations

22

22

22 xyxx

Equation of a circle

22

minmax, 22 xyxx

Remember this

y

Ductile Materials

• Material exhibits sufficient elongation and necking before fracture

• Yield point is distinct in stress strain curve

• Ultimate tensile and compressive strength are nearly same

• Primarily fails by shear

Engineering Stress Strain Curve

(MPa)

yp

Nyp

0

100

200

300

400

0.1 0.2 0.3 0.4 0.5 0.6 0.7 39.0 39.1 39.2strain /%

+

0

fracture

plastic region,extension uniformalong length

plastic region,necking hasbegun

elasticregion

Engineering Stress Strain Curve (cont’d)

Necking in a Tensile Specimen

Cup and Cone Fracture

Brittle Materials

• Material does not exhibit sufficient elongation and necking before fracture.

• Yield point is not distinct in the stress strain curve, an equivalent Proof

Stress is used in place of the Yield Stress.

• Ultimate tensile and compressive strength are not same, compressive

strength could be as high as three times of the tensile strength.

• Primarily fails by tension.

Proof Stress

Theories of failure for Ductile Materials

• Maximum Principal Stress Theory: Rankine

• Maximum Shear Stress Theory: Tresca

• Maximum Principal Strain Theory: St. Venant

• Maximum Strain Energy Theory: Beltrami and Haigh

• Maximum Distortion Energy Theory: von Mises

Maximum Shear Stress Theory: Tresca’s Theory

Statement

Failure will occur in a material if the maximum shear stress at apoint due to a given set of load exceeds the maximum shearStress induced due to a uniaxial load at the Yield Point.

YP 21For failure not to occur

With factor of safety

NYP 21

For failure not to occur

Maximum Distortion Energy Theory: von Mises Theory

Statement

Failure will occur in a material if the maximum distortion energy at apoint due to a given set of load exceeds the maximum distortionEnergy induced due to a uniaxial load at the Yield Point.


With factor of safety

22

2212

1

NYP


22221

21 YP

Theories of failure for Brittle Materials

• Maximum Principal Stress Theory: Rankine

• Mohr’s Theory

• Coulomb Mohr Theory

• Modified Mohr Theory

Maximum Normal Stress Theory for Brittle Materials

The maximum stress criterion states that failure occurs whenthe maximum principal stress reaches either the uniaxial tensionstrength σt or the uniaxial compression strength σc .

tc },{ 21For failure not to occur

Coulomb Mohr’s Theory of Failure for Brittle Materials

All intermediate stress states fall into one of the four categories in the following table. Each case defines the maximum allowable values for the two principal stresses to avoid failure.

Case Principal Stresses

Criterion Requirements

1. Both in tension

1 > 0, 2 > 0 1 < t, 2 < t

2. Both in Compression

1 < 0, 2 < 0 1 < c, 2 < c

3. One in T and other in C

1 > 0, 2 < 0

4. One in T and other in C

1 < 0, 2 > 0

Variable Loading or Fatigue Loading

S-N Diagram

Failure Criteria for Variable Loading or Fatigue Loading

Gerber (Germany, 1874):

Goodman (England, 1899):

Soderberg (USA, 1930):

Morrow (USA, 1960s):

Failure Lines for Different Fatigue Failure Criteria

Factors Affecting Endurance Limit

· Surface Finish· Temperature· Stress Concentration· Notch Sensitivity· Size· Environment· Reliability

Course No.: MEBF ZC342 MACHINE DESIGN L1: Stress Analysis Principles, Prof. D. Datta Prof. D. Datta.

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