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

L1: Stress Analysis Principles, 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. L1: Stress Analysis Principles, Prof. D. Datta

L1: Stress Analysis Principles, Prof. D. Datta Phases in Design Design is a highly Iterative Process Identification of Need Definition of Problem Synthesis Analysis and Optimization Evaluation Presentation L1: Stress Analysis Principles, Prof. D. Datta

L1: Stress Analysis Principles, Prof. D. Datta Steps in Design Idntify 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

L1: Stress Analysis Principles, Prof. D. Datta Design Considerations Functionality Strength Stiffness / Distortion Wear Corrosion Safety Reliability Manufacturability Utility Cost Friction Weight Life 14. Noise 15. Styling 16. Shape 17. Size 18. Control 19. Thermal Properties 20. Surface 21. Lubrication 22. Marketability 23. Maintenance 24. Volume 25. Liability 26. Remanufacturing L1: Stress Analysis Principles, Prof. D. Datta

L1: Stress Analysis Principles, Prof. D. Datta Loads and Equilibrium Restraints L1: Stress Analysis Principles, Prof. D. Datta

L1: Stress Analysis Principles, Prof. D. Datta Stresses X Uni-axial Stress X A Normal Stress Shear Stress L1: Stress Analysis Principles, Prof. D. Datta

Torsional Shear Stress L1: Stress Analysis Principles, Prof. D. Datta τmax J = Polar Moment of Inertia = Angle of Twist Torsion formula with slowly varying area may be used as long as they are circular L1: Stress Analysis Principles, Prof. D. Datta

L1: Stress Analysis Principles, Prof. D. Datta Torsional Shear Stress (cont’d) Here, τxz is negative as it acts opposite to the +z-axis but τxy is positive as it acts along the +y-axis L1: Stress Analysis Principles, Prof. D. Datta

L1: Stress Analysis Principles, Prof. D. Datta Normal Stresses due to Bending X M = Bending Moment y = Distance of the layer from Neutral Axis I = Moment of Inertia of the cross section about the axis of bending L1: Stress Analysis Principles, Prof. D. Datta

L1: Stress Analysis Principles, Prof. D. Datta A Problem: Get the Shear Force and BM Distribution L1: Stress Analysis Principles, Prof. D. Datta

Getting Maximum Normal and Shear Stresses y X Combining the above two equations Equation of a circle Remember this

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)

Engineering Stress Strain Curve (cont’d)

Cup and Cone Fracture Necking in a Tensile Specimen

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 a point due to a given set of load exceeds the maximum shear Stress induced due to a uniaxial load at the Yield Point. For failure not to occur With factor of safety 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 a point due to a given set of load exceeds the maximum distortion Energy induced due to a uniaxial load at the Yield Point. For failure not to occur With factor of safety For failure not to occur

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 when the maximum principal stress reaches either the uniaxial tension strength σt or the uniaxial compression strength σc . For 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

· Stress Concentration · Notch Sensitivity · Size · Environment Factors Affecting Endurance Limit · Surface Finish · Temperature · Stress Concentration · Notch Sensitivity · Size · Environment · Reliability