Machine Design : Theories of failure and Design for fatigue

Maximum shear stress theory explains failure in ductile materials by peak shear on a plane at 45 degrees to the loading direction, with stress equal to half the tensile strength.

Apply the maximum shear stress criterion to assess failure under 2D and 3D stress by comparing (sigma1 minus sigma3)/2 to the material yield strength.

Explore the maximum shear stress theory under plane and three-dimensional stress states, analyze sigma1, sigma2, sigma3 cases, and define the safe zone boundaries and factor of safety effects.

Examine the maximum distortion energy theory used in design practice, deriving strain energy from the work done under linear elastic loading, with U = 1/2 P X.

Derive strain energy density by dividing energy by volume to obtain a stress–strain form, then show u = sigma1 epsilon1 / 2 in the linear elastic region.

Derive shear strain energy density from the area under the shear stress–strain curve, using elastic limit and rigidity modulus G to show u = 1/2 G gamma^2 and compute U.

Derive the general strain energy density for a 3D state of stress and its reduction to principal normal stresses, linking multi-axial relations to the maximum distortion energy theory.

Explore the maximum distortion energy theory, linking distortion or shape change under hydrostatic loading to ductile material failure, and explain the split of strain energy into distortion and volume components.

The lecture derives the distortion energy component of the strain energy density by separating hydrostatic volume change from distortion, linking distortion energy to the rigidity modulus G.

explain the distortion energy density for the three principal stresses and the maximum distortion energy criterion: a state of stress is safe if distortion energy is below yield energy.

Define von Mises stress and compare it with the maximum shear stress theory. Use an elliptical safe zone for principal stresses to show von Mises is less conservative than Tresca.

Explore Mohr's criterion for ductile materials by constructing tension and compression yield circles and identifying safe states where principal stresses lie within the yield limits.

Derive Mohr's criteria envelope for ductile materials using tension, compression, and torsion data; form a tangent envelope to three Mohr circles to define a safe region for principal stresses.

Explains a modification of the Mohr–Coulomb theory of failure by using tangents to three circles to model compression, tension, and mixed states of stress, and derives the yield criterion.

Analyze cases in the Mohr–Coulomb theory of failure and compare with the maximum shear stress theory. Plot boundary lines for plane-stress principal stresses and assess designs by factor of safety.

Apply the Coulomb-Mohr theory to brittle materials by replacing the stress limits with ultimate tensile strength and ultimate compression strength, keeping the diagram boundary the same.

Maximum principal stress theory for brittle materials predicts failure when any of the three principal stresses reaches ultimate strength, but it is liberal and unreliable in second and fourth quadrants.

Compare brittle and ductile failure theories—maximum principal stress, maximum distortion energy, and maximum shear stress—and apply distortion energy for ductile metals and conservative designs.

Highlight the limitations of traditional static mechanics in design, showing how quasi-static tests and non-reversing loading fail to capture dynamic behavior, material defects, and stress concentrations.

Fatigue failure arises from repeated or fluctuating stresses, with maximum stresses below yield, showing micro-level crack initiation and propagation at stress concentrators.

Identify static loading and dynamic loading, including fluctuating, alternating, and irregular cyclic patterns. Explain stress cycles and frequency in rotating shafts and engine pistons and connecting rods.

Identify stages leading to fatigue failure: initiation at high stress concentrations, crack propagation, and fracture, then apply damage-tolerance design and fracture mechanics to limit crack growth.

Identify the three basic fracture modes in fracture mechanics: mode I opening, mode II in-plane sliding, and mode III out-of-plane shear, with mode I driven by tensile stress.

Surface scratches, internal defects, and discontinuities raise fatigue risk. Temperature and environment influence crack initiation and propagation, while loading conditions and design choices control fatigue life.

Fracture toughness measures a material's resistance to crack propagation from internal flaws under structural, cyclical, and impact loading, influenced by a geometry factor and the plastic zone in ductile materials.

Apply fatigue life (stress-life) method to design and analyze fatigue, using the RR Moore test on a rotating beam under pure bending and plotting an S-N diagram for material-specific design.

Plot fatigue life against stress to reveal low and high cycle regions on the S-N curve. Compare endurance limit versus endurance strength and contrast steel with aluminum.

Learn how endurance strength relates to the endurance limit and how real-world deviations, including surface finish, material variation, manufacturing, heat treatment, environment, size, shape, and loading, guide fatigue life design.

Apply the surface modification factor to adjust endurance strength for different surface finishes. For machine-finished steel with Sut 440 MPa, K ≈ 0.89 reduces endurance strength by about 11 percent.

Explore size and shape modifications to endurance using the K_B factor for bending, torsion, and axial loading, with diameter-based formulas like (D/7.6)^(-0.107) and 1.51 D^(-0.107).

Explain how the loading factor modifies endurance strength for rotating bending, axial loading, and torsion, and why it differs from the size modification factor.

Explore the temperature factor KDE as the modifying factor for endurance strength, showing how operating temperature influences brittleness, creep, and crack propagation compared with room temperature.

The reliability factor accounts for data scatter among specimens by using standard deviation to design for the worst-case endurance limit. A 95 percent reliability standard ensures most parts meet specifications.

Analyze how stress concentrations elevate local stresses near holes and geometric transitions, and define the stress concentration factor K as the ratio of maximum to nominal stress.

Learn how fatigue stress concentration factor multiplies nominal stress near discontinuities such as holes and notches, and how notch sensitivity q guides design under static and dynamic loading.

Compare fluctuating and reverse cyclical loading using sigma max, sigma min, and mean stress to assess fatigue severity; fluctuating loading raises mean stress and is more severe than reverse loading.

Analyze fatigue design using the Goodman diagram to relate alternating and mean stresses to the endurance limit, compare with Soderberg and Goldberg criteria, and apply a factor of safety.