PE Mechanical Engineering - Machine Design and Materials

Maximum Shear Stress Failure Theory

"Tresca" Ductile Materials If σ1 ≥ σ2 ≥ σ3, then yielding occurs whenever τmax ≥ Sy/2 Sy = Yield strength τmax = (σ1 - σ3)/2

Distortion-Energy Failure Theory

"Von Mises" Ductile Materials Combine all normal stresses and shear stresses into a single value (lookup equation) If σvm ≥ Sy yielding will occur

Tests for out of control (for 3-sigma control limits)

1. A single point falls outside the control limits. 2. Two out of three consecutive points fall on the same side of and more than two sigma units from the centerline. 3. Four out of five consecutive points fall on the same side of and more than one sigma unit from the centerline. 4. Eight consecutive points fall on the same side of the center line. 5. Seven consecutive points trending up or trending down. 6. Eight consecutive points on either side of the centerline and more than one sigma unit from the centerline. 7. Fourteen consecutive points alternating on either side of the centerline. 8. Fifteen consecutive points within one sigma unit of the centerline.

Hierarchy of Controls for Addressing Hazards

1. Eliminate or substitute: design to avoid hazard, or redesign to eliminate hazard 2. Engineering controls: Limit access to the hazard through guards and barriers 3. Administrative controls: Attempt to alert personnel to the presence of hazard through alarms, warnings, labels, instructions, work practices, training 4. Personal protective equipment (PPI): personnel must utilize protective gear because exposure to the hazard cannot be reliably avoided

Elements of Machine Design Methodologies

1. Identifying Requirements 2. Risk assessment 3. Verification and validation

Long Column (euler)

A column is considered long if the slenderness ratio (Sr) is greater than SrD Critical Buckling Load Pcr = pi^2 * EI / (K*L)^2 K = effective-length factor to account for end supports

Intermediate Column

A column is considered to be intermediate if its slenderness ratio (Sr) is less than SrD SrD = sqrt[(2 * pi^2 * E)/(K^2 * Sy)] Sy = yield strength of the material K = effective-length factor to account for end supports Critical Buckling Load Pcr = A[Sy - K^2/E * (Sy*Sr/2/pi)^2]

Force

A force is a vector quantity. It is defined when magnitude, point of application, and direction are known

Friction

A force resisting relative translations between two surfaces F = mu * N F = friction force mu = coefficient of friction (static or dynamic) N = normal force between surfaces in contact

Stress Concentration (Kt)

A local increase of stress due to a geometry change Not usually applicable to static analysis of metals but very important in initiating a fatigue crack

Poisson's Ratio (nu)

A measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading -1 * lateral strain / longitudinal strain

Failure Modes and Effects Analysis (FMEA)

A risk analysis methodology to evaluate product/system components (or functions) for the following: i. Potential failure modes ii. Effects of the failure iii. Severity of the failure effects iv. Cause(s) of the failure v. Likelihood that a specific cause will occur vi. Likelihood that design controls (such as validation/verification) will prevent the occurrence (or reduce the rate of occurrence) of the cause/failure, or detect the cause/failure before the product/system is released to the customer

Moment Couples

A system of two forces that are equal in magnitude, opposite in direction, and parallel to each other is called a couple. M = moment, cross product of radius vector and force r = radius vector F = force M = r x F

Fault Tree Analysis (FTA)

A top-down, logic-based methodology for analyzing a system failure condition and determining the contributing lower-level events. Steps include: i. Identify the system failure condition of interest. ii. Identify faults that could lead to the system failure condition. iii. Identify all known causes that could lead to the identified faults. iv. Connect causes, and link faults to causes, through Boolean logic gates AND and OR. v. Group common fault causes and refine logical connections. vi. Identify countermeasures to address fault causes.

Pitting Corrosion

A very localized form of corrosion similar to crevice corrosion, in which small holes or pits form

Nondestructive Test Methods

Acoustic Emission Acoustic Impact (Tapping) D-Sight (Diffracto) Eddy Current Magento-optic Eddy current Imager Eddy Sonic Electric Current Electrified Particle Filtered particle Infrared (Radiaometry) (Thermography) Leak Testing Magnetic particle Magnetic Field Microwave Liquid Penetrants (Dye or Fluorescent) Fluoroscopy Neutron Radiology Gamma Radiography X-ray Radiology Radiometry X-ray, Gamma Ray, Beta Ray Reverse-Geometry Digital X-ray X-ray computed tomography (CT) Shearography Electronic Thermal Sonic Ultrasonic Thermoelectric Probe

Work

Additional energy added or removed from the system via other bodies acting upon the system Applied forces, friction, etc. over a distance.

Verification

Analysis of whether a product/system complies with codes, regulations, standards, specifications, or imposed conditions: i. Testing to requirements in codes, regulations, and standards ii. Evaluation of whether product/system was installed in accordance with controlling documents pertaining to the design, installation, operation, and performance of the product/system

Validation

Analysis of whether a product/system meets the expectations of customers, end users, or external entities i. Usability studies ii. Prototyping iii. Evaluation of whether specifications are sufficient to create a product/system that meets customer expectations

Maximum Normal Stress Failure Theory

Brittle Materials If σ1 ≥ σ2 ≥ σ3, then failure occurs whenever σ1 ≥ Sut or σ3 ≤ -Suc, where Sut and Suc are tensile and compressive strengths, respectively

Capacitance of a parallel plate capacitor

C = epsilon*A/d epsilon = permittivity of material A = cross-sectional area of the plates d = distance between the plates

Thermal Strain

Change in dimensions due to a change in temperature

Thermal Deformation

Change in size due to a change in temperature Results in strain but no stress deformation = alpha * L * (delta T) alpha = temperature coefficient of expansion

Corrosion

Chemical reaction of a metal with components of its environment. For corrosion to occur, an anode and a cathode must be in electrical contact in the presence of an electrolyte Types: Uniform, galvanic, crevice, pitting, intergranular, selective leaching, erosion-corrosion, stress-corrosion

Coefficient of Restitution

Describes how an impact between two bodies will occur e = (v2' - v1')/(v1 - v2) e = 1, perfectly elastic (energy conserved) e = 0, perfectly plastic (no rebound, both masses act as one)

Uniform Corrosion

Electrochemical corrosion that occurs with equivalent intensity over entire surface

Binary Phase Diagrams

Enable determination of (1) what phases are present at equilibrium at any temperature and average composition (2) the compositions of those phases (3) the fractions of those phases.

Conservation of Energy

Energy cannot be created or destroyed, just change forms KE1 + PE1 = KE2 + PE2 + W KE = Kinetic energy PE = potential energy W = work

Light Drive Fits

FN1, Force or Shrink Fit Light drive fits are those requiring light assembly pressures, and produce more or less permanent assemblies.

Heavy Drive Fits

FN13 Force or Shrink Fit Heavy drive fits are suitable for heavier steel parts, or for shrink fits in medium sections.

Medium Drive Fits

FN2, Force or Shrink Fit Medium drive fits are suitable for ordinary steel parts, or for shrink fits on light sections.

Force Fits

FN4, FN5, Force or Shrink Fit Force fits are suitable for parts that can be highly stressed, or for shrink fits where the heavy pressing forces required are impractical

Fatigue

Failure that occurs after many cycles as stresses below static strength limits

Modified Goodman Failure Theory

Fatigue - infinite life, fully reversed (R=-1) Fatigue failure will* occur whenever: σa/Se + σm/Sut >= 1 or σmax/Sy >= 1 Se = endurance limit Sut = ultimate strength Sy = yield strength σa = alternating stress σm = mean stress σmax = σa + σm *: more accurately "could given enough cycles"

Soderberg Failure Theory

Fatigue - infinite life, fully reversed (R=-1) Fatigue failure will* occur whenever: σa/Se + σm/Sy >= 1 when σm >= 0 Se = endurance limit Sut = ultimate strength Sy = yield strength σa = alternating stress σm = mean stress σmax = σa + σm *: more accurately "could given enough cycles"

Examples of Requirements (for machine design)

Functional performance Limites on size User interfaces External interfaces Production volume Costs Available materials Available porduction technologies Foreseeable usage conditions Foreseeable environmental conditions Foreseeable misuse Required safety factors Life cycle duration product retirement

Risk Assessment Methodologies

Hierarchy of controls for addressing hazards Fault Tree Analysis (FTA) Failure Modes and Effects Analysis (FMEA)

Strain Energy Equations / Castigliano's theorem

If a body of length L is deformed under force F or torque T, the resulting strain energy U is equal to: strain energy: U = 1/2 * F * deformation tension/compression: U = (F^2 * L) / (2 * A * E) torsion: U = (T^2 * L) / (2 * G * J) shear: U = (F^2 * L) / (2 * A * G) bending: U = integral((M^2 * dx)/(2*E*I)) deformation: def_i = partial derivative of U wrt F_i. deformation is at location of F_i

Locational Clearance Fits

LC, Locational Fit Locational clearance fits are intended for parts which are normally stationary, but that can be freely assembled or disassembled.

Locational Interference Fits

LN, Locational Fit Locational interference fits are used where accuracy of location is of prime importance, and for parts requiring rigidity and alignment with no special requirements for bore pressure.

Locational Transition Fits

LT, Locational Fit Locational transition fits are a compromise between clearance and interference fits, for applications where accuracy of location is important, but either a small amount of clearance or interference is permissible.

Uncertainty

Let wR be the uncertainty in the result and w1,w2...wn be the uncertainties in the independent variables. The uncertainty of the equation is the square root of the sum of the squares of each uncertainty multiplied by the derivate of the equation with respect to the variable assoicated with the uncertainty.

Resistivity

Material property that determines the resistance of a resistor rho = R*A/L rho = resistivity of the object R = resistance of the resistor A = cross-sectional area of the resistor L = length of the resistor Conductivity is the reciprocal of the resistivity

Stress-Strain Curve: Ultimate Strength

Maximum stress attained on an ENGINEERING stress/strain curve Point C

Stress in Beams Equations

Normal Stress due to bending: sigma = -M*y / I, max @ y -> c Section modulus: s = I/c Transverse shear stress: tau = V*Q / I*b, maximum tau tabulated for common shapes Transverse shear flow: q = VQ/I M = moment y = distance from neutral axis (NA) I = moment of inertia c = distance from neutral axis to extreme fiber V = shear Q = A'*y_bar. A' is area above where inspecting for shear stress, y_bar is distance from NA to A' centroid b = width of cross section

Erosion-Corrosion

Occurs as a combined consequence of chemical attack and mechanical abrasion due to fluid motion

Stress-Corrosion Cracking (SCC)

Occurs due to the combined influence of an applied tensile stress and a corrosive environment

Intergranular Corrosion

Occurs preferentially along grain boundaries of some alloys in certain environments

Crevice Corrosion

Occurs when concentration differences of ions or dissolved gases exist in an electrolyte system with corrosion occurring preferentially at areas of low concentration

Selective Leaching Corrosion

Occurs when one element is preferentially removed by a corrosion process from a solid solution alloy

Galvanic Corrosion

Occurs when two metals or alloys having different compositions are electrically coupled while exposed to an electrolyte

Cold working

Plastically deforming a metal increases strength and lowers ductility

Close Sliding Fits

RC1, Running Clearance Fit Intended for the accurate location of parts that must assemble without perceptible play

Sliding Fits

RC2, Running Clearance Fit Intended for accurate location, but with greater maximum clearance than RC1. Parts made to this fit move and turn easily but are not intended to run freely, and in the larger sizes may seize with small temperature changes.

Precision Running Fits

RC3, Running Clearance Fit Precision running fits are about the closest fits that can be expected to run freely, and are intended for precision work at slow speeds and light journal pressures, but are not suitable where appreciable temperature differences are likely to be encountered.

Close Running Fits

RC4, Running Clearance Fit Close running fits are intended chiefly for running fits on accurate machinery with moderate surface speeds and journal pressures, where accurate location and minimum play are desired.

Medium Running Fits

RC5, RC6, Running Clearance Fit Medium running fits are intended for higher running speeds, or heavy journal pressures, or both.

Free Running Fits

RC7, Running Clearance Fit Free running fits are intended for use where accuracy is not essential, or where large temperature variations are likely to be encountered, or under both these conditions.

Loose Running fits

RC8, RC9, Running Clearance Fit Loose running fits are intended for use where wide commercial tolerances may be necessary, together with an allowance, on the external member.

Hot working

Raising temperature while deforming the material to cause (1) recovery (stress relief) (2) recrystallization (3) grain growth

Quenching

Rapid cooling from elevated temperature, preventing the formation of equilibrium phases

Endurance Limit Modifying Factors

Real-world parts are not laboratory specimens and specimen fatigue curves can need correction Marin Equation adjusts the rotating beam endurance limit to specific case For finite life analyses different methods are used Factors: Surface finish factor Size Factor Load type factor (axial, bending, torsion) Temperature factor Miscellaneous (corrosion, plating, residual stress, etc)

Hardness

Resistance to penetration Measured by denting a material under known load and measuring the size of the dent

Standard Deviation

Square root of the population variance

Sample Standard Deviation

Square root of the sample variance

Stress-Strain Curve: Breaking Point

Strain at which rupture occurs Point D

Stress-Strain Curve: Yield Point / Yield Strength

Strength at which the stress-strain curve and a line originating at (0.002,0) with slope E intersect (0.2% strain offset definition) Point B

Uniaxial Loading Equations

Stress: stress = load / area Strain: strain = delta length (or deformation) / length Deformation: deformation = (load * length) / (area * modulus of elasticity)

Newton's Second Law

Sum of the applied forces acting on a particle is equal to the derivative with respect to time of mass*velocity F = m*a M = I*alpha

Impact tests

Tests (Charpy, Izod) to determine the amount of energy required to cause failure in standardized test samples Charpy impact test: used to find the energy required to fracture and to identify ductile-to-brittle transition (varying temperature)

Median

The "middle" value of the sample When the discrete data is rearranged in increasing order AND n is odd, the median is the value of the (n+1)/2 items When n is even, the median is the average of the (n/2) and (n/2 + 1) items

Mohr's Circle

The Mohr circle is used to find the stress components and , i.e., coordinates of any point on the circle, acting on any other plane passing through making an angle with the plane

Specific heat (or heat capacity)

The amount of heat required to raise the temperature of a material or an amount of material by 1 degree.

Capacitance

The charge-carrying capacity of an insulating materials Charge held by a capacitor: q = CV q = charge C = capacitance V = voltage

Fracture Toughness

The combination of applied stress and the crack length in a brittle and elastic material. K = Y*sigma*sqrt(pi*a) units: ksi*in^1/2 KIC is the critical stress intensity at which catastrophic crack propagation occurs (under PLANE STRAIN (minimum))

Hardenability

The ease with which hardness can be obtained

Kinetic Energy

The energy contained by a moving object Particle: KE = 1/2 m*v^2 Rigid Body: KE = 1/2 m*vc^2 + 1/2 Ic*omega^2

Potential Energy (in a gravity field)

The energy that can be converted or removed from other sources from a delta change in position in a gravity field PE = m*g*h h = elevation above some specified datum

Weight

The force acting on the object with mass due to an acceleration (ie gravity) W = m*g W = weight (N of lbf) m = mass (kg or lbf-sec^2/ft or lbf-sec^2/in) g = local acceleration of gravity (9.81 m/s^2 or 32.2 ft/sec^2 or 386.4 in/sec^2)

Limiting Friction

The largest frictional force. Any further increase in applied forces will cause motion

Sample range (R)

The largest sample value minus the smallest sample value

Fatigue life

The number of cycles to failure at a specified stress/strain and stress ratio

Stress-Strain Curve: Proportional Limit

The point at which the stress-strain curve is no longer linear Point A

Confidence Level

The probability that the value of the parameter falls within a specified range of values

Endurance Limit

The stress below which fatigue failure in unlikely

Sample Variance

The sum of the squared deviations from the sample mean divided by n-1

Mode

The value that occurs most frequently in a given data set.

Variance

The variance of the population is the arithmetic mean of the squared deviations from the population mean

Creep

Time-dependent deformation under load, usually measured by strain rate. In metals typically due to low energy slip systems becoming active due to raised temperatures (~> 1/2 melting temperature) in addition to a load

Torsional Stress/Strain Equations

Torsion stress (thick walled): tau = T*r/J Torsion stress (thin walled): tau = T / (2*Am*t) Angle of twist: phi = (T*L)/(G*J) Torsional stiffness: k or c = T/phi T = torque L = length of shaft G = torsional modulus J = polar moment of inertia Am = area enclosed by midline of thickness t = wall thickness

Shear Modulus (G)

Torsional stiffness property Can be computed from Modulus of elasticity (E) and Poisson's ratio (nu) G = E / (2*(1 + nu))

Slenderness Ratio

Value to determine if a column is "long", "intermediate", or "short" Sr = L/r L = length of column r = radius of gyration = sqrt(I/A)

Momentum

Vector quantity that describes mass in motion m*v Conserved in a collision m1*v1 + m2*v2 = m1*v1' + m2*v2'

Weighted Arithmetic Mean

Xbar_w = sum(weight_i * Xi) / sum(weight_i)

Miner's Rule

aka Palmgren-Miner Linear Damage Accumulation variable amplitude loading Fatigue failure is assumed to occur when sum(ni / Ni) = C n = cycles at specified maximum stress and stress ratio (min/max) N = cycles that would cause failure at specified stress and stress ratio C = typically equal to 1.0

Engineering Strain

epsilon = delta L / Lo epsilon = engineering strain, in units per unit delta L = change in length of member Lo = ORIGINAL length of member

True Strain

epsilon_t = ln( L / Lo) = ln(1 + epsilon) L = instantaneous length of member

Arithmetic Mean

f X1, X2...Xn represents the values of a random sample of n items, the mean of these items (Xbar) is the sum of the items divided by the number of items

S-N Curve

stress-cycles to fatigue failure curve Handbook gives equation to generate a curve given two data points Sf = aN^b Sf = fatigue stress N = cycles to failure b = log(Sf1/Sf2)/log(N1/N2) a = Sf1/N1^b

Goodman Equivalent Stress

σeq = σa + (Se/Sut)*σm Se = endurance limit Sut = ultimate strength σa = alternating stress σm = mean stress