SSRAO

Explanation:
Design philosophy in machine design focuses on functional and engineering aspects: factor of safety (strength vs. load), reliability (probability of satisfactory performance), and manufacturability (ease of production). Pure aesthetic appearance is NOT a primary machine design philosophy consideration. Core design philosophies: (1) Strength-based design, (2) Stiffness-based design, (3) Reliability-based design, (4) Design for manufacturing.

Explanation:
Factor of Safety (FOS) = Failure stress / Allowable (working) stress. For ductile materials: FOS = Yield strength / Working stress. For brittle materials: FOS = Ultimate tensile strength / Working stress. FOS > 1 always; typical values range from 1.5 to 10 depending on application and uncertainty in loading.

Explanation:
Allowable stress = Yield strength / FOS = 250 / 2.5 = 100 MPa. This is the maximum stress the component is designed to withstand under normal operating conditions. FOS of 2.5 means the component can theoretically withstand 2.5 times the design load before yielding.

Explanation:
For components subjected to repeated/cyclic loading, fatigue design is most appropriate. Fatigue failure occurs at stresses well below UTS or yield strength. The endurance limit (Se) is the maximum stress amplitude below which a material can withstand infinite cycles. Key concepts: S-N curve, endurance limit, stress concentration factors, Goodman/Gerber diagrams.

Explanation:
Stress concentration occurs at geometric discontinuities such as holes, notches, keyways, fillets, and abrupt changes in cross-section. Stress concentration factor Kt = sigma_max / sigma_nom. For fatigue: Kf = 1 + q(Kt - 1), where q is notch sensitivity (0 to 1). Stress concentration is critical in fatigue design as cracks initiate at these points.

Explanation:
Kf = 1 + q(Kt - 1) = 1 + 0.8(2.5 - 1) = 1 + 0.8 x 1.5 = 1 + 1.2 = 2.2. Note: q = 0 means material is insensitive to notches (Kf = 1); q = 1 means fully sensitive (Kf = Kt). For the given values: Kf = 2.2.

Explanation:
The endurance limit (Se) is the maximum stress amplitude below which a material can theoretically withstand infinite stress cycles without fatigue failure. For steels: Se approx 0.5 x Sut for Sut <= 1400 MPa. For non-ferrous metals, there is no true endurance limit. Modified endurance limit: Se_prime = ka x kb x kc x kd x ke x Se.

Explanation:
For a plate with a small central circular hole under uniaxial tension, Kt = 3 (Kirsch solution). The maximum stress occurs at the edge of the hole perpendicular to loading: sigma_max = 3 x sigma_nom. For an elliptical hole with semi-axes a and b: Kt = 1 + 2a/b. For circular hole: a = b, Kt = 3.

Explanation:
Surface finish factor (ka) modifies the endurance limit for actual surface roughness vs. polished lab specimens. ka values: Mirror-polished: 1.0, Ground: 0.9, Machined/Cold-drawn: 0.7-0.8, Hot-rolled: 0.6-0.7, As-forged: 0.4-0.6. Modified endurance limit: Se_prime = ka x kb x kc x kd x ke x Se.

Explanation:
Size factor (kb) accounts for the statistical effect that larger components have greater volume of material at high stress, increasing probability of encountering a critical microstructural flaw. kb < 1 for d > 7.62 mm. Shigley formula: kb = 1.24 x d^(-0.107) for 2.79 <= d <= 51 mm. This is the statistical weakest-link theory applied to fatigue.

Explanation:
Rankine Maximum Principal Stress Theory: Failure when sigma_1 = Syt (tensile) or sigma_3 = -Syc (compressive). Suitable for brittle materials (cast iron, ceramics). Does NOT account for effect of other principal stresses. For biaxial stress: failure if sigma_1 >= Syt or sigma_2 <= -Syc.

Explanation:
Tresca Theory: Failure when tau_max = Syt/2. For biaxial stress: (sigma_1 - sigma_2)/2 = Syt/2, so sigma_1 - sigma_2 = Syt. Tresca is conservative (safe side) vs. von Mises. Suitable for ductile materials. Tresca gives hexagonal yield surface; von Mises gives ellipse. Tresca is 15.5% more conservative than von Mises at worst case.

Explanation:
Von Mises stress for biaxial: sigma_vm = sqrt(sigma_1^2 + sigma_2^2 - sigma_1*sigma_2). For triaxial: sigma_vm = (1/sqrt(2)) x sqrt[(sigma_1-sigma_2)^2 + (sigma_2-sigma_3)^2 + (sigma_3-sigma_1)^2]. Failure when sigma_vm = Syt. For 2D: sigma_vm = sqrt(sigma_x^2 + sigma_y^2 - sigma_x*sigma_y + 3*tau_xy^2).

Explanation:
sigma_vm = sqrt(200^2 + 100^2 - 200x100) = sqrt(40000 + 10000 - 20000) = sqrt(30000) = 173.2 MPa. If Syt = 250 MPa, FOS = 250/173.2 = 1.44. Tresca: tau_max = (200-100)/2 = 50 MPa; failure if tau_max = Syt/2 = 125 MPa -> safe.

Explanation:
Rankine Maximum Principal Stress Theory is most suitable for brittle materials (cast iron, glass, ceramics, concrete). Brittle materials fail by fracture along planes of maximum tensile stress. For ductile materials, von Mises and Tresca are preferred. Modified Mohr and Coulomb-Mohr theories are also used for brittle materials when Suc >> Sut.

Explanation:
Coulomb-Mohr (Internal Friction) Theory is used for brittle materials where Suc >> Sut, such as cast iron (Suc approx 3-4 x Sut). Failure criterion: sigma_1/Sut - sigma_3/Suc = 1. For cast iron: Sut approx 200 MPa, Suc approx 600-800 MPa.

Explanation:
Equivalent torque by Tresca: Te = sqrt(M^2 + T^2) = sqrt(1000^2 + 1000^2) = sqrt(2000000) = 1414.2 N.m. Shaft design: tau = 16Te/(pi*d^3) <= Ssy = Syt/2. Equivalent bending moment by Rankine: Me = (M + Te)/2 = (1000 + 1414.2)/2 = 1207.1 N.m.

Explanation:
Modified Goodman line: sigma_a/Se + sigma_m/Sut = 1/FOS. Where Se = endurance limit, Sut = ultimate tensile strength. Other criteria: Gerber parabola: sigma_a/Se + (sigma_m/Sut)^2 = 1. Soderberg: sigma_a/Se + sigma_m/Syt = 1. ASME-Elliptic: (sigma_a/Se)^2 + (sigma_m/Syt)^2 = 1.

Explanation:
Goodman (FOS=1): sigma_a/Se + sigma_m/Sut = 1. sigma_a/200 + 100/400 = 1. sigma_a/200 = 1 - 0.25 = 0.75. sigma_a = 0.75 x 200 = 150 MPa. With no mean stress, sigma_a = Se = 200 MPa. Mean stress reduces allowable alternating stress.

Explanation:
Soderberg: sigma_a/Se + sigma_m/Syt = 1/FOS. Uses Syt (< Sut) for mean stress axis, making it more conservative. Order of conservatism: Soderberg > Goodman > ASME-Elliptic > Gerber. Soderberg is rarely used in practice as it is overly conservative. Goodman is most widely used.

Explanation:
Gerber parabola: sigma_a/Se + (sigma_m/Sut)^2 = 1. Parabolic curve fits experimental fatigue data better than Goodman line. Less conservative than Goodman. Passes through same endpoints (Se on sigma_a-axis, Sut on sigma_m-axis). Goodman preferred for design (conservative, linear). ASME-Elliptic: (sigma_a/Se)^2 + (sigma_m/Syt)^2 = 1.

Explanation:
Von Mises for pure shear (sigma_1 = tau, sigma_2 = -tau): sigma_vm = tau*sqrt(3) = Syt at yield. Ssy = Syt/sqrt(3) = 0.577 x 300 = 173.2 MPa. Tresca: Ssy = Syt/2 = 150 MPa. Von Mises gives higher (less conservative) shear yield strength. Experimental results closer to von Mises for ductile materials.

Explanation:
Distortion Energy Theory (von Mises) gives most accurate predictions for ductile materials. Experimental evidence (Taylor and Quinney, 1931) shows yielding of ductile metals closely follows von Mises criterion. Theory correctly predicts hydrostatic stress does not cause yielding. Von Mises is used in FEA as primary yield criterion for ductile materials.

Explanation:
tau_max = (sigma_1 - sigma_2)/2 = (200 - (-100))/2 = 300/2 = 150 MPa. When sigma_1 and sigma_2 have opposite signs, tau_max = (sigma_1 - sigma_2)/2. For Tresca: failure if tau_max = Syt/2, so Syt = 300 MPa required. Von Mises: sigma_vm = sqrt(200^2 + 100^2 + 200x100) = sqrt(70000) = 264.6 MPa.

Explanation:
Paris Law: da/dN = C*(delta_K)^m, where a = crack length, N = cycles, delta_K = stress intensity factor range = K_max - K_min = delta_sigma*sqrt(pi*a)*F. C and m are material constants (m approx 2-4 for metals). Applies in Region II (stable crack growth). Region I: below threshold delta_K_th. Region III: rapid growth approaching Kc.

Explanation:
Fracture toughness KIc (plane strain) is the critical stress intensity factor for Mode I (crack opening) loading. KIc = sigma_f * sqrt(pi*a) * F. Units: MPa*sqrt(m). Failure when KI >= KIc. High KIc: structural steels (50-200 MPa*sqrt(m)). Low KIc: ceramics (1-5), glass (0.7-1). Used to determine critical crack size: ac = (1/pi)*(KIc/sigma)^2.

Explanation:
Miner Rule: D = Sum(ni/Ni), where ni = actual cycles at stress level i, Ni = cycles to failure at that level. Failure when D >= 1. Assumptions: linear damage accumulation, order-independent. Limitations: actual failure at D = 0.7 to 2.2; does not account for load sequence effects.

Explanation:
D = n1/N1 + n2/N2 = 10^5/(2x10^5) + 5x10^4/10^5 = 0.5 + 0.5 = 1.0. D = 1.0 means component has reached fatigue life. Remaining life = 1 - D = 0.

Explanation:
Octahedral Shear Stress Theory is mathematically equivalent to von Mises. tau_oct = (1/3)*sqrt[(sigma_1-sigma_2)^2 + (sigma_2-sigma_3)^2 + (sigma_3-sigma_1)^2]. At yield in simple tension: tau_oct_yield = (sqrt(2)/3)*Syt. Failure: tau_oct = (sqrt(2)/3)*Syt, identical to von Mises criterion.

Explanation:
Stress ratio R = sigma_min/sigma_max. Fully reversed: R = -1. Repeated (zero to max): R = 0. Static: R = 1. Mean stress: sigma_m = (sigma_max + sigma_min)/2. Alternating stress: sigma_a = (sigma_max - sigma_min)/2. R = -1 is most damaging for given sigma_max.

Explanation:
Von Mises for pure shear: sigma_vm = tau*sqrt(3) = Syt at yield. Ssy = Syt/sqrt(3) = 0.577*Syt. Ssy/Syt = 1/sqrt(3) = 0.577. Tresca: Ssy/Syt = 0.5. Experimental results for ductile metals: Ssy/Syt approx 0.55-0.60, closer to von Mises.

Explanation:
Tresca: tau_max = (sigma_1 - sigma_3)/2 = Syt/2, where sigma_1 > sigma_2 > sigma_3. Therefore sigma_1 - sigma_3 = Syt. For biaxial (sigma_3=0): if sigma_1 > 0 > sigma_2: sigma_1 - sigma_2 = Syt. If sigma_1 > sigma_2 > 0: sigma_1 = Syt. Option D is von Mises criterion.

Explanation:
Modified Goodman with FOS n: sigma_a/Se + sigma_m/Sut = 1/n. Rearranging: n = 1/(sigma_a/Se + sigma_m/Sut). For n=1: failure boundary. For n>1: safe region. Example: sigma_a=100, sigma_m=50, Se=200, Sut=400: n = 1/(0.5+0.125) = 1.6.

Explanation:
For a rotating shaft under constant transverse load, bending stress at any surface point is completely reversed (R = -1). As shaft rotates, point alternates between max tension and max compression. Mean stress sigma_m = 0. Alternating stress sigma_a = sigma_max = 32M/(pi*d^3). Torsional stress for constant torque is static (mean only).

Explanation:
Soderberg: sigma_a/Se + sigma_m/Syt = 1/FOS. Uses Syt for mean stress axis. Most conservative fatigue criterion. Ensures no yielding (sigma_m <= Syt). Rarely used in practice due to excessive conservatism. Goodman (uses Sut) preferred for most engineering applications.

Explanation:
ASME-Elliptic: (sigma_a/Se)^2 + (sigma_m/Syt)^2 = 1. Passes through (0, Se) and (Syt, 0). Lies between Goodman and Soderberg in conservatism. Recommended by ASME Boiler and Pressure Vessel Code for some applications. Better represents actual failure boundary for many ductile materials.

Explanation:
Langer static yield line: sigma_a + sigma_m = Syt. Represents boundary for first-cycle yielding. Points above this line yield on first load application. Goodman diagram has two failure boundaries: (1) Goodman line (fatigue), (2) Langer line (yielding). Safe design is below BOTH lines.

Explanation:
Goodman: sigma_a/Se + sigma_m/Sut = 1/FOS. For sigma_m = 0: sigma_a/Se = 1/FOS. For FOS=1: sigma_a = Se. Fully reversed loading (R = -1) is the most severe fatigue condition for given stress amplitude. The endurance limit Se is the maximum allowable alternating stress for infinite life with zero mean stress.

Explanation:
Haigh diagram (Modified Goodman diagram): x-axis = mean stress (sigma_m), y-axis = stress amplitude (sigma_a). Key points: (0, Se) and (Sut, 0). Goodman line connects these. Langer line: sigma_a + sigma_m = Syt. Safe region is below both lines. FOS determined by ratio of distances from origin to failure line vs. operating point.

Explanation:
Von Mises for 2D (sigma_z=0): sigma_vm = sqrt(sigma_x^2 + sigma_y^2 - sigma_x*sigma_y + 3*tau_xy^2). For pure shear (sigma_x=sigma_y=0): sigma_vm = tau_xy*sqrt(3). Failure when sigma_vm = Syt. Used in FEA post-processing to evaluate yielding.

Explanation:
Notch sensitivity q ranges 0 to 1. q=0: insensitive to notches (Kf=1). q=1: fully sensitive (Kf=Kt). Kf = 1 + q(Kt-1). q increases with notch radius and material strength. High-strength steels: q approx 1. Soft ductile materials: q approx 0. Peterson formula: q = 1/(1 + a/r).

Explanation:
sigma_vm = sqrt(sigma_x^2 + sigma_y^2 - sigma_x*sigma_y + 3*tau_xy^2) = sqrt(150^2 + 0 - 0 + 3*100^2) = sqrt(22500 + 30000) = sqrt(52500) = 229.1 MPa. If Syt = 400 MPa: FOS = 400/229.1 = 1.75. Component is safe.

Explanation:
Design stress = Failure stress / FOS. For ductile: sigma_design = Syt/FOS. For brittle: sigma_design = Sut/FOS. For fatigue: sigma_design = Se/FOS. FOS accounts for uncertainties in material properties, load estimation, stress analysis, manufacturing tolerances, and environmental effects.

Explanation:
KI = sigma*sqrt(pi*a)*F(geometry), F=1 for infinite plate. Units: MPa*sqrt(m). Fracture when KI = KIc. Critical crack size: ac = (1/pi)*(KIc/sigma)^2. Foundation of Linear Elastic Fracture Mechanics (LEFM). For finite plates, F > 1.

Explanation:
Wohler S-N curve: y-axis = stress amplitude (S or sigma_a), x-axis = cycles to failure N (log scale). For ferrous metals: curve becomes horizontal at Se at approx 10^6-10^7 cycles. For non-ferrous: no true endurance limit. Obtained from rotating bending fatigue tests on polished specimens. Represents 50% probability of failure.

Explanation:
For steel with Sut <= 1400 MPa: Se approx 0.5 x Sut = 0.5 x 600 = 300 MPa. For Sut > 1400 MPa: Se approx 700 MPa (constant). This is for standard polished rotating bending specimen (6-8 mm diameter). Actual component endurance limit is lower due to surface finish, size, reliability factors.

Explanation:
Creep: time-dependent permanent deformation under sustained stress at elevated temperatures (T > 0.3-0.4 Tm). Stages: (1) Primary: decreasing creep rate, (2) Secondary: constant minimum creep rate, (3) Tertiary: accelerating rate leading to fracture. Important in: gas turbine blades, boiler tubes, nuclear reactor components.

Explanation:
Toughness = ability to absorb energy before fracture = area under stress-strain curve from zero to fracture strain. Modulus of toughness = integral(sigma*d_epsilon) from 0 to epsilon_f. Approximate: Ut approx (Syt + Sut)/2 x epsilon_f. High toughness: low-carbon steel, copper, aluminum alloys. Low toughness (brittle): cast iron, ceramics.

Explanation:
Pure torsion: sigma_1 = tau, sigma_2 = -tau. Tresca: tau_max = (sigma_1-sigma_2)/2 = tau. Failure when tau_max = Syt/2, so tau = Syt/2. Von Mises: sigma_vm = tau*sqrt(3) = Syt, so tau = 0.577*Syt. Tresca gives Ssy = 0.5*Syt; von Mises gives Ssy = 0.577*Syt. Experimental results closer to von Mises.

Explanation:
Proof resilience = max energy stored per unit volume within elastic limit = Syt^2/(2E). Area under stress-strain curve up to yield point. High resilience materials used for springs (high Syt, low E). Example: Spring steel Syt=1500 MPa, E=200 GPa: Resilience = 1500^2/(2x200000) = 5.625 MJ/m^3.

Explanation:
Thin-walled cylinder (d/t > 20): Hoop stress sigma_h = pd/(2t) = pr/t. Longitudinal stress sigma_L = pd/(4t) = pr/(2t). sigma_h = 2*sigma_L. Critical stress is hoop stress. Design: t = pd/(2*sigma_allow). Spherical vessel: sigma = pd/(4t) in all directions.

Explanation:
Euler: Pcr = pi^2*E*I/Le^2. Effective lengths: Both pinned: Le=L. One fixed, one free: Le=2L (Pcr=pi^2EI/4L^2). Both fixed: Le=L/2 (Pcr=4pi^2EI/L^2). One fixed, one pinned: Le=0.7L. Critical stress: sigma_cr = pi^2*E/(L/k)^2. Valid for long columns (L/k > 120 for steel).

Explanation:
Wahl factor: Kw = (4C-1)/(4C-4) + 0.615/C, where C = D/d = spring index. Corrected shear stress: tau = Kw x 8FD/(pi*d^3). Accounts for: (1) Curvature effect (stress concentration on inner fiber), (2) Direct shear stress. For C=6: Kw approx 1.25. Spring index C should be 4-12 for practical springs.

Explanation:
Hydrostatic: sigma_1=sigma_2=sigma_3=p. Von Mises: sigma_vm = 0 -> no yielding. Tresca: tau_max = 0 -> no yielding. Both correctly predict no yielding under hydrostatic stress. Beltrami (total strain energy) incorrectly predicts yielding. This is consistent with experimental observations for ductile metals.

Explanation:
Rankine-Gordon: Pcr = Syt*A/(1 + a*(L/k)^2), where a = Syt/(pi^2*E). For L/k->0: Pcr->Syt*A (crushing). For L/k->inf: Pcr->pi^2*EI/L^2 (Euler). Smoothly transitions between crushing and buckling. Suitable for intermediate columns (30 < L/k < 120 for steel).

Explanation:
Basquin equation: sigma_a = sigma_f_prime * (2*Nf)^b, where sigma_f_prime = fatigue strength coefficient (approx Sut for steels), 2*Nf = reversals to failure, b = fatigue strength exponent (-0.05 to -0.12). Describes elastic portion of strain-life curve. Basis of stress-life (S-N) approach for HCF.

Explanation:
Coffin-Manson: delta_epsilon_p/2 = epsilon_f_prime * (2*Nf)^c, where epsilon_f_prime = fatigue ductility coefficient, c = fatigue ductility exponent (-0.5 to -0.7). LCF occurs at N < 10^4 cycles with significant plastic deformation. Total strain-life: delta_epsilon/2 = (sigma_f_prime/E)*(2Nf)^b + epsilon_f_prime*(2Nf)^c.

Explanation:
sigma = 32M/(pi*d^3), tau = 16T/(pi*d^3). Von Mises: sigma_vm = sqrt(sigma^2 + 3*tau^2) = sqrt[(32M/pi*d^3)^2 + 3*(16T/pi*d^3)^2] = (32/pi*d^3)*sqrt(M^2 + 0.75*T^2). Failure when sigma_vm = Syt. Tresca gives Te = sqrt(M^2+T^2).

Explanation:
Archard: V = k*W*L/H, where V = wear volume, k = wear coefficient, W = normal load, L = sliding distance, H = hardness of softer material. Wear proportional to load and sliding distance, inversely proportional to hardness. To reduce wear: increase hardness, reduce load, use lubricants, use harder materials.

Explanation:
S-N approach appropriate for HCF (N > 10^4) with predominantly elastic stresses. For LCF (N < 10^3): strain-life (epsilon-N) approach (Coffin-Manson) is more appropriate due to significant plastic deformation. For intermediate cycles: both elastic and plastic strains significant.

Explanation:
sigma = 32M/(pi*d^3), tau = 16T/(pi*d^3). sigma_1 = sigma/2 + sqrt[(sigma/2)^2 + tau^2] = 16M/(pi*d^3) + sqrt[(16M/pi*d^3)^2 + (16T/pi*d^3)^2] = 16/(pi*d^3)*[M + sqrt(M^2+T^2)]. Equivalent bending moment Me = (M + sqrt(M^2+T^2))/2.

Explanation:
Tresca: tau_max = (sigma_1-sigma_2)/2 = (300-(-150))/2 = 225 MPa. Failure when tau_max = Syt/2: 225 = Syt/2 -> Syt = 450 MPa. Yielding if Syt < 450 MPa. Von Mises: sigma_vm = sqrt(300^2+150^2+300x150) = sqrt(157500) = 396.9 MPa. Tresca more conservative.

Explanation:
Temperature factor kd modifies endurance limit for operating temperatures other than room temperature. For steel: kd=1.0 for T<=450 deg C; decreases for T>450 deg C. At 500 deg C: kd approx 0.9; at 600 deg C: kd approx 0.7. Elevated temperatures reduce endurance limit due to reduced strength, increased creep, oxidation.

Explanation:
sigma_vm = sqrt(sigma_1^2 + sigma_2^2 - sigma_1*sigma_2) = sqrt(200^2 + 200^2 - 200x200) = sqrt(40000+40000-40000) = sqrt(40000) = 200 MPa. Equal biaxial tension: sigma_vm = sigma_1 = sigma_2. Both Tresca and von Mises agree for equal biaxial tension.

Explanation:
Mohr circle: graphical method to find (1) Principal stresses sigma_1, sigma_2 from sigma_x, sigma_y, tau_xy, (2) Maximum shear stress tau_max = (sigma_1-sigma_2)/2, (3) Stresses on any inclined plane. Center C = (sigma_x+sigma_y)/2, Radius R = sqrt[(sigma_x-sigma_y)^2/4 + tau_xy^2]. Used to apply failure theories to complex stress states.

Explanation:
Reliability factor kc modifies endurance limit for statistical scatter. Standard Se is for 50% reliability. kc values: 50%: 1.0, 90%: 0.897, 95%: 0.868, 99%: 0.814, 99.9%: 0.753. kc=0.814 for 99% means: if designed to Se_prime = 0.814*Se, there is 99% probability of surviving infinite cycles.

Explanation:
Fretting fatigue: small oscillatory relative motion (1-100 micron amplitude) between contacting surfaces under cyclic loading. Combines fretting wear with fatigue crack initiation. Reduces fatigue life by 50-90%. Common locations: press-fit joints, bolted connections, turbine blade roots. Prevention: surface coatings, shot peening, lubrication.

Explanation:
Palmgren-Miner: D = Sum(ni/Ni) = 1 at failure. Assumptions: (1) Linear damage accumulation, (2) Order-independent, (3) Independent at each stress level. Limitations: actual failure at D=0.7 to 2.2; does not account for load sequence effects. More accurate models: Corten-Dolan, Marco-Starkey (nonlinear).

Explanation:
Neuber rule: Kt^2 = K_sigma * K_epsilon, where Kt = theoretical stress concentration factor, K_sigma = actual stress concentration factor, K_epsilon = actual strain concentration factor. For elastic: K_sigma = K_epsilon = Kt. For elastic-plastic: K_sigma < Kt and K_epsilon > Kt, but product = Kt^2. Used with cyclic stress-strain curve for fatigue life prediction.

Explanation:
For steel Sut <= 1400 MPa: Se approx 0.5 x Sut = 0.5 x 800 = 400 MPa. For Sut > 1400 MPa: Se approx 700 MPa (constant). This is for standard polished rotating bending specimen. Actual component endurance limit lower due to surface finish, size, reliability factors.

Explanation:
Von Mises under hydrostatic (sigma_1=sigma_2=sigma_3=p): sigma_vm = 0 -> no yielding predicted. Physically correct: hydrostatic pressure changes volume but not shape (no distortion), so no yielding. Key advantage over Beltrami theory which incorrectly predicts yielding under hydrostatic stress.

Explanation:
Wohler rotating bending test: standard method for S-N curve. Polished hourglass specimen rotated under constant bending moment. Each rotation = one complete stress cycle (fully reversed, R=-1). Standard specimen: 6-8 mm diameter, Ra < 0.2 micron. Gives endurance limit for 50% reliability.

Explanation:
Rankine-Gordon: Pcr = Syt*A/(1 + a*(L/k)^2), a = Syt/(pi^2*E). For L/k->0: Pcr->Syt*A (crushing). For L/k->inf: Pcr->pi^2*EI/L^2 (Euler). Suitable for intermediate columns. For steel: a approx 1/7500 (both ends pinned).

Explanation:
Euler critical stress: sigma_cr = Pcr/A = pi^2*E/(Le/k)^2, where Le/k = slenderness ratio. Valid for long columns (Le/k > 120 for steel). For steel (E=200 GPa, Syt=250 MPa): critical slenderness = pi*sqrt(E/Syt) = pi*sqrt(800) = 88.9. Euler formula independent of material strength.

Explanation:
FOS by Goodman: n = 1/(sigma_a/Se + sigma_m/Sut). From Goodman: sigma_a/Se + sigma_m/Sut = 1/n. Example: sigma_a=100, sigma_m=50, Se=200, Sut=400: n = 1/(0.5+0.125) = 1.6. Design safe if n >= required FOS (typically 1.5-2.0).

Explanation:
SCC: combined action of sustained tensile stress and specific corrosive environment. Occurs at stresses well below yield strength. Requires specific material-environment combinations (high-strength steel in H2S, stainless steel in chlorides, brass in ammonia). Prevention: reduce tensile stress, use resistant materials, protective coatings, cathodic protection.

Explanation:
Gerber parabola lies above Goodman line (less conservative). For sigma_m = 0.5*Sut: Goodman allows sigma_a = 0.5*Se; Gerber allows sigma_a = 0.75*Se (50% more). Gerber better fits experimental data for ductile materials. Goodman preferred for design (conservative, linear). Gerber used for analysis when accuracy important.

Explanation:
Soderberg: sigma_a/Se + sigma_m/Syt = 1. Goodman: sigma_a/Se + sigma_m/Sut = 1. Since Syt < Sut, Soderberg has smaller intercept on sigma_m-axis. For same mean stress, Soderberg allows less alternating stress. Soderberg line entirely within Goodman line. Soderberg ensures no yielding; Goodman allows mean stresses up to Sut.

Explanation:
Hertz contact theory: stress distribution when two curved elastic bodies are pressed together. Applications: ball/roller bearings, gear tooth contact, cam-follower. For two spheres: contact radius a = (3FR/4E*)^(1/3), max pressure p0 = 3F/(2*pi*a^2). Max shear stress below surface at depth approx 0.48a (spheres) or 0.786b (cylinders), causing pitting.

Explanation:
Beltrami theory: failure when total strain energy per unit volume = strain energy at yield. U = (1/2E)[sigma_1^2 + sigma_2^2 + sigma_3^2 - 2v(sigma_1*sigma_2 + sigma_2*sigma_3 + sigma_3*sigma_1)] = Syt^2/(2E). Not widely used as it incorrectly predicts yielding under hydrostatic stress. Von Mises preferred.

Explanation:
By Maximum Principal Stress Theory: Me = (1/2)*[M + sqrt(M^2 + T^2)]. Shaft design: sigma = 32*Me/(pi*d^3) <= Syt/FOS. By Tresca: Te = sqrt(M^2+T^2) (equivalent torque). tau = 16*Te/(pi*d^3) <= Syt/(2*FOS).

Explanation:
Goodman: sigma_a = Se*(1 - sigma_m/Sut) = 250*(1 - sigma_m/500). Langer: sigma_a = Syt - sigma_m = 350 - sigma_m. At intersection: 250*(1 - sigma_m/500) = 350 - sigma_m. 250 - 0.5*sigma_m = 350 - sigma_m. 0.5*sigma_m = 100. sigma_m = 200 MPa. sigma_a = 150 MPa. For sigma_m > 200: yielding governs; for sigma_m < 200: fatigue governs.

Explanation:
KI = sigma*sqrt(pi*a)*F. Units: MPa * m^(1/2) = MPa*sqrt(m). Fracture toughness KIc has same units. Typical KIc: High-strength steel: 50-100 MPa*sqrt(m); Aluminum: 20-40; Ceramics: 1-5; Glass: 0.7-1. KIc characterizes stress field near crack tip in LEFM.

Explanation:
Rankine = Maximum Principal Stress Theory = Maximum Normal Stress Theory. Tresca = Maximum Shear Stress Theory = Guest Theory. Von Mises = Distortion Energy Theory = Octahedral Shear Stress Theory. Beltrami = Maximum Strain Energy Theory. Saint-Venant = Maximum Normal Strain Theory. Rankine suitable for brittle; von Mises and Tresca for ductile.

Explanation:
Se_prime = 0.8 x 0.9 x 0.868 x 300 = 0.72 x 0.868 x 300 = 0.625 x 300 = 187.5 MPa. Additional factors (kd for temperature, ke for stress concentration) would further reduce Se_prime. The product of all k-factors is typically 0.3-0.5 for real components.

Explanation:
Gerber parabola: sigma_a/Se + (sigma_m/Sut)^2 = 1. Passes through (sigma_m=0, sigma_a=Se) and (sigma_m=Sut, sigma_a=0). Same endpoints as Goodman line but parabolic (curves upward). Goodman is linear between same endpoints. Gerber less conservative than Goodman.

Explanation:
Peterson formula: q = 1/(1 + a/r). Material constant 'a' depends on Sut: For steels: a = 0.0254*(2070/Sut)^1.8 mm (Sut in MPa). Higher Sut -> smaller a -> higher q (more notch sensitive). Soft ductile materials: large a -> low q. High-strength steels: small a -> q approx 1 (fully notch sensitive).

Explanation:
Manson-Coffin: delta_epsilon_p/2 = epsilon_f_prime*(2*Nf)^c. LCF at N < 10^4 cycles with significant plastic deformation per cycle. Applications: pressure vessels under thermal cycling, aircraft structures, nuclear components. Total strain-life combines elastic (Basquin) and plastic (Coffin-Manson) components.

Explanation:
Tresca: Te = sqrt(M^2+T^2). tau = 16*Te/(pi*d^3) = Ssy = Syt/2. d^3 = 16*Te/(pi*Ssy). d = [16*sqrt(M^2+T^2)/(pi*Ssy)]^(1/3). Since Ssy = Syt/2: d = [32*sqrt(M^2+T^2)/(pi*Syt)]^(1/3). Standard formula for shaft design under combined loading.