FE Mechanical formula sheet

Single- and Multivariable Calculus

Limits, derivatives and optimization, the chain rule, integration and the fundamental theorem, partial derivatives and the gradient, and series — the workhorse mathematics of FE Mechanical.

Definition of the derivative

$$f'(x)=\lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$

Instantaneous slope / rate of change of $f$ at $x$; the basis of velocity, acceleration, and gradients.

Chain rule

$$\frac{d}{dx}f\big(g(x)\big)=f'\big(g(x)\big)\,g'(x)$$

Differentiates composed functions; the most-used rule on the exam.

Product and quotient rules

$$(uv)'=u'v+uv',\qquad \left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}$$

Derivatives of products and quotients of functions $u(x)$, $v(x)$.

Extremum test

$$f'(a)=0;\quad f''(a)<0\Rightarrow\text{max},\ f''(a)>0\Rightarrow\text{min}$$

Locates and classifies critical points; check interval endpoints separately.

Fundamental theorem of calculus

$$\int_a^b f(x)\,dx=F(b)-F(a),\quad F'(x)=f(x)$$

Evaluates a definite integral as the change in an antiderivative; converts rates into totals.

Integration by parts

$$\int u\,dv=uv-\int v\,du$$

Integrates products; choose $u$ so that $du$ simplifies (LIATE order).

Gradient

$$\nabla f=\frac{\partial f}{\partial x}\mathbf{i}+\frac{\partial f}{\partial y}\mathbf{j}+\frac{\partial f}{\partial z}\mathbf{k}$$

Vector of partial derivatives; points toward steepest increase, magnitude = max slope.

Taylor series

$$f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n$$

Polynomial expansion about $x=a$; basis of linearization and truncation error.

Geometric series sum

$$\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r},\quad |r|<1$$

Converges for $|r|<1$; used in economics (present worth) and repeated processes.

Where it lives: NCEES FE Reference Handbook — Mathematics (Differential and Integral Calculus, Series) · NCEES FE Reference Handbook — Heat Transfer

Descriptive Statistics, the Normal Distribution and Confidence Intervals

Mean, median, mode, variance and standard deviation, the standard-normal z transform, and the confidence interval that turns a sample into a statement about the true mean.

Arithmetic mean

$$\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i$$

Center of the data; $x_i$ are observations in their physical units, $n$ is the count.

Weighted mean

$$\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$$

Average when observations carry unequal weights $w_i$ (counts, importances).

Sample variance and standard deviation

$$s^2 = \frac{1}{n-1}\sum (x_i-\bar{x})^2, \quad s=\sqrt{s^2}$$

Spread of a sample; divide by $n-1$. $s$ returns to the units of $x$.

Population variance

$$\sigma^2 = \frac{1}{N}\sum (x_i-\mu)^2$$

Use when the full population of size $N$ and its mean $\mu$ are known; divide by $N$.

Coefficient of variation

$$CV = \frac{s}{\bar{x}}$$

Dimensionless relative scatter; lets you compare variability across different units or scales.

Normal density

$$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\,e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$$

Bell curve fixed by mean $\mu$ and standard deviation $\sigma$; area under it is probability.

Standard-normal transform

$$z = \frac{x-\mu}{\sigma}$$

Number of standard deviations $x$ is from the mean; the key into the unit-normal table.

Tail areas

$$R(z) = 1 - F(z), \quad F(-z) = 1 - F(z)$$

$F(z)$ = area left of $z$; $R(z)$ = right tail. Symmetry handles negative $z$.

Standard error of the mean

$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$

Scatter of the sample mean; use $s$ for $\sigma$ when the population value is unknown.

Confidence interval ($\sigma$ known)

$$\bar{x} \pm z_{\alpha/2}\,\frac{\sigma}{\sqrt{n}}$$

Two-sided CI for $\mu$ with known $\sigma$; $z_{0.025}=1.960$ for 95%.

Confidence interval ($\sigma$ unknown)

$$\bar{x} \pm t_{\alpha/2,\,n-1}\,\frac{s}{\sqrt{n}}$$

Small-sample CI; use Student's $t$ with $n-1$ degrees of freedom.

Where it lives: NCEES FE Reference Handbook — Engineering Probability and Statistics · Montgomery & Runger, Applied Statistics and Probability for Engineers

Engineering Ethics, the NCEES Model Rules and Public Welfare

How the NCEES Model Rules structure an engineer's duties, why public health and safety are paramount, and a disciplined method for resolving ethical dilemmas.

Paramount duty

$$\text{Public health, safety, welfare} \succ \text{Employer/Client} \succ \text{Other licensees}$$

Ordering of obligations in Model Rules § 240.15. When two duties conflict, the higher one governs; the public duty is the universal tiebreaker.

Competence test

$$\text{Accept work} \iff \text{qualified by education or experience in that specific field}$$

A licensee undertakes assignments only within demonstrated competence and seals only work prepared under their responsible charge.

Responsible charge

$$\text{Responsible charge} = \text{authority to review/change/approve} + \text{personal awareness} + \text{full responsibility}$$

Definition from the Model Law; required before a licensee may sign and seal a document.

Conflict-of-interest rule

$$\text{Permissible} \iff \text{full disclosure} + \text{written consent of all affected parties}$$

Applies to multi-party compensation and any circumstance that could influence, or appear to influence, professional judgment.

Escalation trigger

$$\text{Judgment overruled} \wedge \text{public endangered} \Rightarrow \text{notify employer/client + appropriate authority}$$

Mandatory escalation under Obligations to the Public when safety is at risk.

Confidentiality

$$\text{Disclose client information} \iff \text{consent} \ \vee\ \text{required by law}$$

Facts and data obtained in a professional capacity stay confidential absent consent or legal compulsion; never used for personal profit.

Dilemma classification

$$\text{Issue} \in \{\text{conceptual (term in dispute)},\ \text{factual (data uncertain)},\ \text{tradeoff (risk vs. benefit)}\}$$

Diagnostic categories from the handbook for analyzing an ethical problem before choosing an action.

Where it lives: NCEES FE Reference Handbook — Ethics and Professional Practice · NCEES Model Rules § 240.15 — Rules of Professional Conduct · NCEES Model Law § 110.20 (Definitions) and § 130.10 (Requirements for Licensure)

Time Value of Money and Annuities

Interest factors (P/F, F/P, P/A, A/P, A/F, F/A), gradients, and nominal-versus-effective rates that turn any cash-flow diagram into a single comparable number.

Single-payment compound amount (F/P)

$$F = P\,(1+i)^{n}$$

Grows a present amount $P$ forward $n$ periods at rate $i$ to a future amount $F$. $i$ is the rate per period; $n$ counts periods, not necessarily years.

Single-payment present worth (P/F)

$$P = F\,(1+i)^{-n}$$

Discounts a future amount $F$ back $n$ periods to its present worth $P$. Reciprocal of the (F/P) factor.

Uniform-series present worth (P/A)

$$P = A\,\dfrac{(1+i)^{n}-1}{i\,(1+i)^{n}}$$

Present worth of an end-of-period annuity $A$ running $n$ periods. First $A$ occurs at $t=1$, one period after $P$.

Capital recovery (A/P)

$$A = P\,\dfrac{i\,(1+i)^{n}}{(1+i)^{n}-1}$$

Level payment $A$ that repays (recovers) a present amount $P$ over $n$ periods at $i$; the standard loan-payment factor. Inverse of (P/A).

Uniform-series compound amount (F/A)

$$F = A\,\dfrac{(1+i)^{n}-1}{i}$$

Future worth at $t=n$ of an end-of-period annuity $A$. Used for savings/replacement funds.

Sinking fund (A/F)

$$A = F\,\dfrac{i}{(1+i)^{n}-1}$$

Level deposit $A$ each period needed to accumulate a future target $F$ in $n$ periods. Inverse of (F/A).

Gradient present worth (P/G)

$$P = G\left[\dfrac{(1+i)^{n}-1}{i^{2}(1+i)^{n}}-\dfrac{n}{i\,(1+i)^{n}}\right]$$

Present worth of an arithmetic gradient that adds $G$ per period; first increment ($G$) lands at $t=2$. Add a separate base annuity for the period-1 amount.

Gradient to uniform series (A/G)

$$A = G\left[\dfrac{1}{i}-\dfrac{n}{(1+i)^{n}-1}\right]$$

Equivalent level annuity of an arithmetic gradient $G$ over $n$ periods. Lets a ramping cash flow be treated as a constant $A$.

Effective annual rate

$$i_e = \left(1+\dfrac{r}{m}\right)^{m}-1$$

Converts nominal annual rate $r$ compounded $m$ times per year to the effective annual rate $i_e$. Continuous limit (not printed in the handbook): $i_e=e^{r}-1$.

Where it lives: NCEES FE Reference Handbook — Engineering Economics · Blank & Tarquin, Engineering Economy · Newnan, Eschenbach & Lavelle, Engineering Economic Analysis

DC Circuit Analysis: Ohm, Kirchhoff and Equivalent Resistance

Ohm's law, KVL and KCL, series-parallel reduction, voltage and current dividers, and power dissipation — the bookkeeping behind every DC problem.

Ohm's law

$$V = IR$$

Voltage across a resistor equals current times resistance. $V$ in volts, $I$ in amperes, $R$ in ohms ($\Omega$).

Kirchhoff's current law (KCL)

$$\sum I_{in} = \sum I_{out}$$

Charge conservation at a node: currents entering equal currents leaving. Use one per independent node.

Kirchhoff's voltage law (KVL)

$$\sum V_{rise} = \sum V_{drop}$$

Energy conservation around a closed loop: rises equal drops. Use one per independent loop.

Resistors in series

$$R_S = R_1 + R_2 + \cdots + R_n$$

Same current through each; resistances add. Equivalent is larger than any single resistor.

Resistors in parallel

$$R_P = \left(\tfrac{1}{R_1} + \cdots + \tfrac{1}{R_n}\right)^{-1}, \quad R_P = \tfrac{R_1 R_2}{R_1+R_2}$$

Same voltage across each; conductances add. Two-resistor form is product over sum; equivalent is smaller than the smallest branch.

Voltage divider

$$V_k = V_s\,\dfrac{R_k}{R_1+R_2+\cdots+R_n}$$

Series string: the target resistor $R_k$ is in the numerator. Voltage splits in direct proportion to resistance.

Current divider (two branches)

$$I_1 = I_s\,\dfrac{R_2}{R_1+R_2}$$

Parallel pair: the OTHER resistor sits in the numerator. Current splits in inverse proportion to resistance.

Power dissipated in a resistor

$$P = VI = I^2 R = \dfrac{V^2}{R}$$

Three equivalent forms; always positive (absorbed). $P$ in watts.

Thévenin equivalent

$$V_{oc} = V_a - V_b, \qquad R_{eq} = R\ \text{seen with sources zeroed}$$

Collapses a two-terminal linear network to one source $V_{oc}$ in series with $R_{eq}$. Zero voltage sources (short) and current sources (open).

Maximum power transfer (DC)

$$P_{L,\max} = \dfrac{V_{oc}^2}{4R_{eq}}\ \ \text{at}\ \ R_L = R_{eq}$$

Matched load receives maximum power; efficiency is only 50% at the match point.

Where it lives: NCEES FE Reference Handbook — Electrical and Computer Engineering · Nilsson & Riedel, Electric Circuits

Free-Body Diagrams, Resultants and Static Equilibrium

Resolve and combine forces, take moments and couples, draw an honest free-body diagram, and solve 2-D and 3-D equilibrium with ΣF=0 and ΣM=0.

Force components

$$F_x=F\cos\theta_x,\quad F_y=F\cos\theta_y,\qquad F_x=\tfrac{x}{L}F,\ \ F_y=\tfrac{y}{L}F$$

Resolve a force by angle or by geometry (run $x$, rise $y$, length $L=\sqrt{x^2+y^2}$). Components in N or lbf.

Resultant of a force system

$$R=\sqrt{\left(\textstyle\sum F_x\right)^2+\left(\textstyle\sum F_y\right)^2},\quad \theta=\arctan\frac{\sum F_y}{\sum F_x}$$

Single equivalent force. Watch the quadrant when taking the arctangent.

Moment of a force

$$\mathbf{M}=\mathbf{r}\times\mathbf{F},\qquad M_z=xF_y-yF_x$$

$\mathbf{r}$ from the moment center to the line of action. In 2-D, magnitude $=F\,d_\perp$ (N·m or ft·lbf).

Couple (pure moment)

$$M=Fd$$

Two equal, opposite, parallel forces a distance $d$ apart; same moment about every point, zero net force.

2-D equilibrium

$$\sum F_x=0,\qquad \sum F_y=0,\qquad \sum M_O=0$$

Three equations for a planar rigid body — solves up to three unknown reactions.

3-D equilibrium

$$\sum F_x=\sum F_y=\sum F_z=0,\qquad \sum M_x=\sum M_y=\sum M_z=0$$

Six equations for a spatial body (ball-and-socket, bearings, space frames).

Resultant of a uniform distributed load

$$W=wL\ \text{at midspan};\quad W=\tfrac{1}{2}w_0 L\ \text{(triangular, at }L/3)$$

Replace a line load by its area resultant acting at the load-diagram centroid. $w$ in N/m or lbf/ft.

Support reactions transmitted

$$\text{roller}:1\ \ \text{pin}:2\ \ \text{fixed}:2\ \text{forces}+1\ \text{couple}$$

Count of unknowns each idealized support adds to the FBD.

Where it lives: NCEES FE Reference Handbook — Statics · Hibbeler, Engineering Mechanics: Statics

Particle Kinematics: Constant Acceleration and Projectiles

Describe how a particle moves — rectilinear and curvilinear paths, the constant-acceleration kinematic equations, projectiles, normal/tangential and polar components, and relative motion.

Kinematic derivatives

$$v = \dfrac{ds}{dt}, \quad a = \dfrac{dv}{dt} = v\,\dfrac{dv}{ds}$$

Defines velocity and acceleration. The third form eliminates time; use it for acceleration given as a function of position $s$ (m), $v$ in m/s, $a$ in m/s².

Constant-acceleration velocity

$$v = v_0 + a t$$

Velocity after time $t$ under constant $a$; $v_0$ = initial velocity (m/s), $a$ in m/s², $t$ in s.

Constant-acceleration position

$$s = s_0 + v_0 t + \tfrac{1}{2} a t^2$$

Displacement under constant acceleration; $s_0$ = initial position (m).

Average-velocity displacement

$$s = s_0 + \tfrac{1}{2}(v_0 + v) t$$

Displacement as average speed times time; valid only for constant $a$. Useful when $a$ is not wanted.

Time-free (velocity-position)

$$v^2 = v_0^2 + 2 a (s - s_0)$$

Relates speed and distance with no time term — use when $t$ is neither given nor sought.

Projectile equations

$$x = v_0\cos\theta\,t, \quad y = v_0\sin\theta\,t - \tfrac{1}{2} g t^2$$

Horizontal $a_x=0$, vertical $a_y=-g$; $\theta$ = launch angle above horizontal, $g = 9.81\ \text{m/s}^2$.

Normal/tangential acceleration

$$a_t = \dfrac{dv}{dt}, \quad a_n = \dfrac{v^2}{\rho}, \quad a = \sqrt{a_t^2 + a_n^2}$$

$a_t$ changes speed, $a_n$ (toward center) changes direction; $\rho$ = radius of curvature (m).

Polar (radial/transverse) acceleration

$$a_r = \ddot r - r\dot\theta^2, \quad a_\theta = r\ddot\theta + 2\dot r\dot\theta$$

$2\dot r\dot\theta$ is the Coriolis term; $r$ in m, $\dot\theta$ in rad/s, $\ddot\theta$ in rad/s².

Plane circular motion

$$v = r\omega, \quad a_n = r\omega^2 = \dfrac{v^2}{r}, \quad a_t = r\alpha$$

Constant-radius case: $\omega$ = angular velocity (rad/s), $\alpha$ = angular acceleration (rad/s²).

Relative velocity

$$\mathbf{v}_A = \mathbf{v}_B + \mathbf{v}_{A/B}$$

Vector addition; $\mathbf{v}_{A/B}$ = velocity of $A$ as seen from a frame translating with $B$.

Where it lives: NCEES FE Reference Handbook — Dynamics · Hibbeler, Engineering Mechanics: Dynamics

Shear/Moment Diagrams, Beam Bending and Transverse Shear

Build shear and moment diagrams from the load-shear-moment relations, then size beams with the flexure formula σ = Mc/I and the transverse shear formula τ = VQ/Ib.

Load-shear relation

$$\dfrac{dV}{dx} = -w(x)$$

Slope of the shear diagram equals minus the distributed load (downward $w$ positive).

Shear-moment relation

$$\dfrac{dM}{dx} = V(x)$$

Slope of the moment diagram equals the shear; $M$ is extremum where $V=0$.

Diagram areas

$$V_2 - V_1 = -\!\int w\,dx, \qquad M_2 - M_1 = \int V\,dx$$

Change in shear = minus area under load; change in moment = area under shear.

Flexure formula

$$\sigma_x = -\dfrac{My}{I}$$

Bending stress at fiber distance $y$ from the neutral axis; $I$ = centroidal moment of inertia.

Maximum bending stress

$$\sigma_{\max} = \dfrac{Mc}{I} = \dfrac{M}{S}$$

$c$ = distance to extreme fiber; $S = I/c$ = elastic section modulus.

Rectangle section properties

$$I = \dfrac{bh^3}{12}, \quad c = \dfrac{h}{2}, \quad S = \dfrac{bh^2}{6}$$

For a solid rectangle of width $b$, depth $h$, about the horizontal centroidal axis.

Transverse shear stress

$$\tau = \dfrac{VQ}{Ib}$$

$Q = A'\bar{y}'$ = first moment of area beyond the level; $b$ = width there. Max at the neutral axis.

Rectangle peak shear

$$\tau_{\max} = \dfrac{3V}{2A}$$

Maximum transverse shear in a rectangular section, $1.5\times$ the average $V/A$.

Simple-span UDL maxima

$$V_{\max} = \dfrac{wL}{2}, \qquad M_{\max} = \dfrac{wL^2}{8}$$

Simply supported beam, full-span uniform load $w$; shear max at supports, moment max at midspan.

Where it lives: NCEES FE Reference Handbook — Mechanics of Materials · Hibbeler, Mechanics of Materials · Gere & Goodno, Mechanics of Materials

The Stress–Strain Diagram and Mechanical Properties

Read the engineering stress–strain curve to extract modulus, yield, ultimate strength, ductility, resilience, and toughness—and know when true stress takes over.

Engineering stress

$$\sigma = \dfrac{F}{A_0}$$

Load $F$ over ORIGINAL area $A_0$ (m² → Pa, in² → psi). The default for design data and offset yield.

Engineering strain

$$\varepsilon = \dfrac{\Delta L}{L_0}$$

Change in gauge length over original gauge length $L_0$. Dimensionless; multiply by 100 for percent.

Hooke’s law (elastic)

$$\sigma = E\,\varepsilon$$

Valid below the proportional limit. $E$ = Young’s modulus (Pa or psi), the slope of the elastic line.

Modulus of resilience

$$U_r = \dfrac{S_y^{\,2}}{2E}$$

Elastic strain energy per unit volume (J/m³). Area of the triangle under the elastic line up to yield.

Percent elongation (ductility)

$$\%EL = \dfrac{L_f - L_0}{L_0}\times 100$$

Permanent strain after fracture, reassembled gauge length $L_f$. A measure of ductility.

True stress

$$\sigma_T = \dfrac{F}{A} = \sigma\,(1+\varepsilon)$$

Load over INSTANTANEOUS area $A$. The $(1+\varepsilon)$ form assumes constant volume, valid before necking.

True strain

$$\varepsilon_T = \ln\!\left(\dfrac{L}{L_0}\right) = \ln(1+\varepsilon)$$

Logarithmic strain; additive over increments. Used in metal-forming and necking analysis.

Hardness–strength (plain-carbon steel)

$$S_u\,[\text{psi}] \approx 500\,\text{BHN}$$

Rough estimate of ultimate strength from Brinell number; $S_u\,[\text{MPa}]\approx 3.5\,\text{BHN}$. Steels only.

Where it lives: NCEES FE Reference Handbook — Materials Science/Structure of Matter (Mechanical Properties) · NCEES FE Reference Handbook — Mechanics of Materials (Uniaxial Loading and Deformation) · Callister & Rethwisch, Materials Science and Engineering: An Introduction

Energy, Continuity and Momentum (Bernoulli)

Conservation of mass, the energy (Bernoulli) equation with pump and head-loss terms, the impulse-momentum principle for forces on bends and vanes, and the EGL/HGL.

Continuity (incompressible)

$$Q = A_1 v_1 = A_2 v_2, \qquad \dot m = \rho A v$$

Volumetric flow constant when $\rho$ is constant; mass flow always constant in steady flow.

Bernoulli equation

$$\dfrac{P_1}{\gamma} + \dfrac{v_1^2}{2g} + z_1 = \dfrac{P_2}{\gamma} + \dfrac{v_2^2}{2g} + z_2$$

Steady, incompressible, frictionless, no machine; each term has units of length (head).

Energy equation with pump and loss

$$\dfrac{P_1}{\gamma} + \dfrac{v_1^2}{2g} + z_1 + h_p = \dfrac{P_2}{\gamma} + \dfrac{v_2^2}{2g} + z_2 + h_f$$

$h_p$ = pump head added (upstream side), $h_f$ = total head loss (downstream side).

Torricelli efflux

$$v = \sqrt{2gh}$$

Free-jet speed from a tank with surface height $h$ above the opening (Bernoulli, surface and jet at atmospheric).

Pressure drop in equal-area pipe

$$P_1 - P_2 = \gamma h_f = \rho g h_f$$

When $z_1=z_2$ and $v_1=v_2$, the full energy equation collapses to friction pressure drop.

Impulse-momentum principle

$$\Sigma F = \rho Q\,(v_2 - v_1)$$

Vector; apply per component. $\Sigma F$ includes pressure forces, weight, and the boundary reaction.

Force on a bend (x, y)

$$\begin{aligned} P_1A_1 - P_2A_2\cos\alpha - F_x &= \rho Q(v_2\cos\alpha - v_1)\\ F_y - W - P_2A_2\sin\alpha &= \rho Q\,v_2\sin\alpha \end{aligned}$$

$\alpha$ = bend angle from inlet axis; $F_x,F_y$ = anchoring force on the fluid (reverse sign for force on the bend).

Force on a fixed deflecting vane

$$F_x = \rho Q\,v(1-\cos\alpha), \qquad F_y = \rho Q\,v\sin\alpha$$

Free jet (gauge $P=0$) of speed $v$ turned by angle $\alpha$; speed unchanged on a frictionless vane.

EGL minus HGL

$$\text{EGL} - \text{HGL} = \dfrac{v^2}{2g}$$

The two grade lines differ by the velocity head; both drop by $h_f$ downstream.

Where it lives: NCEES FE Reference Handbook — Fluid Mechanics · Vennard & Street, Elementary Fluid Mechanics

The First Law, Energy Transfers and Processes

Conservation of energy as a closed-system balance and an open control-volume balance — heat, work, enthalpy, and the isobaric, isothermal, and adiabatic processes that fill the FE.

Closed-system first law

$$Q - W = \Delta U + \Delta KE + \Delta PE$$

Energy balance for a fixed mass. $Q$ positive when added, $W$ positive when done by the system; energies in kJ (or Btu).

Ideal-gas internal energy and enthalpy change

$$\Delta u = c_v\,\Delta T,\qquad \Delta h = c_p\,\Delta T,\qquad c_p - c_v = R$$

Hold for an ideal gas in any process. $c_v,c_p$ in kJ/(kg·K); $\Delta T$ in K (= $\Delta T$ in °C).

Reversible boundary work

$$w_b = \int_1^2 P\,dv$$

Area under the path on a $P$–$v$ diagram; per unit mass in kJ/kg. Positive for expansion.

Constant-pressure work and heat

$$w_b = P\,(v_2 - v_1),\qquad q = \Delta h = c_p\,\Delta T$$

Isobaric closed process. Heat added equals the enthalpy change for an ideal gas.

Isothermal work (ideal gas)

$$w_b = RT\ln\!\dfrac{v_2}{v_1} = RT\ln\!\dfrac{P_1}{P_2}$$

Constant-$T$ ideal gas, $Pv=\text{const}$. Since $\Delta u=0$, $q=w_b$. $T$ absolute (K or °R).

Isentropic process relations (ideal gas)

$$Pv^{k}=\text{const},\quad \dfrac{T_2}{T_1}=\left(\dfrac{P_2}{P_1}\right)^{\frac{k-1}{k}}=\left(\dfrac{v_1}{v_2}\right)^{k-1}$$

Reversible adiabatic ideal gas, $k=c_p/c_v$. $q=0$, so $w=-\Delta u$.

Isentropic / polytropic work (ideal gas)

$$w = \dfrac{P_2 v_2 - P_1 v_1}{1-n} = \dfrac{R(T_2-T_1)}{1-n}\ \ (n\neq 1)$$

Use $n=k$ for isentropic, $n=1$ for isothermal (special form), $n=0$ for isobaric.

Enthalpy definition

$$h = u + Pv$$

Internal energy plus flow work; kJ/kg. Why enthalpy, not internal energy, appears in open systems.

Steady-flow energy equation

$$0 = \dot{Q}_{in} - \dot{W}_{out} + \sum_i \dot{m}_i\!\left(h_i + \tfrac{V_i^2}{2} + gZ_i\right) - \sum_e \dot{m}_e\!\left(h_e + \tfrac{V_e^2}{2} + gZ_e\right)$$

Open control volume, no storage. $\dot{m}$ in kg/s, $h$ in J/kg if SI consistent (or use kJ with care).

Adiabatic turbine / compressor

$$w = h_i - h_e$$

Per unit mass; KE and PE neglected. Turbine $w>0$ (out), compressor $w<0$.

Nozzle / diffuser energy

$$h_i + \dfrac{V_i^2}{2} = h_e + \dfrac{V_e^2}{2}$$

No work, no heat. Enthalpy converts to kinetic energy. In USCS divide $V^2/2$ by $g_c$ and use 778 ft·lbf/Btu.

Throttling process

$$h_i = h_e$$

Valve/porous plug: no work, no heat, $\Delta KE\approx 0$. Isenthalpic — basis of the refrigeration expansion valve.

Where it lives: NCEES FE Reference Handbook — Thermodynamics (First Law of Thermodynamics; Steady-Flow Systems) · Moran, Shapiro, Boettner & Bailey, Fundamentals of Engineering Thermodynamics · Cengel & Boles, Thermodynamics: An Engineering Approach

Conduction and Thermal Resistance Networks

Fourier's law for plane walls and cylinders, the series/parallel thermal-resistance analogy, contact resistance, and the critical insulation radius.

Fourier's law (1-D)

$$\dot{Q} = -kA\,\frac{dT}{dx}$$

Conductive heat rate (W). $k$ = thermal conductivity [W/(m·K)], $A$ = area normal to flow (m²); minus sign gives heat flow down the gradient.

Plane-wall conduction

$$\dot{Q} = \frac{kA\,(T_1 - T_2)}{L}$$

Steady 1-D conduction through a slab of thickness $L$; faces at $T_1>T_2$.

Resistance form

$$\dot{Q} = \frac{\Delta T}{R_{\text{total}}}, \quad R_{\text{total}} = \sum R_i$$

Electrical analogy; series resistances add. $R$ in K/W, $\Delta T$ in K.

Plane conduction resistance

$$R_{\text{cond}} = \frac{L}{kA}$$

Resistance (K/W) of a flat layer; per unit area $R'' = L/k$.

Convection (film) resistance

$$R_{\text{conv}} = \frac{1}{hA}$$

Surface film resistance (K/W); $h$ = convection coefficient [W/(m²·K)].

Cylindrical conduction resistance

$$R_{\text{cyl}} = \frac{\ln(r_2/r_1)}{2\pi k L}$$

Radial shell of length $L$, inner/outer radii $r_1,r_2$; logarithmic, not linear.

Overall coefficient

$$\dot{Q} = UA\,\Delta T, \quad \frac{1}{UA} = R_{\text{total}}$$

$U$ [W/(m²·K)] lumps the whole series chain; $UA$ is the conductance.

Parallel resistances

$$\frac{1}{R_{\parallel}} = \frac{1}{R_1} + \frac{1}{R_2}$$

Side-by-side heat paths (studs, finned regions) combine reciprocally.

Critical insulation radius

$$r_{cr} = \frac{k_{\text{ins}}}{h}$$

Outer radius of maximum heat loss for a cylinder; below it, insulation increases loss.

Contact resistance drop

$$\Delta T_{tc} = \dot{Q}''\,R''_{tc}$$

Interface temperature jump; $R''_{tc}$ = area-specific contact resistance [m²·K/W].

Where it lives: NCEES FE Reference Handbook — Heat Transfer (Conduction, Thermal Resistance) · Incropera & DeWitt, Fundamentals of Heat and Mass Transfer · Çengel, Heat and Mass Transfer: A Practical Approach

Feedback Control Systems and Stability

Block diagrams, open- vs closed-loop transfer functions, the characteristic equation and stability, PID action, and steady-state error by system type.

Transfer function (pole-zero form)

$$G(s)=\dfrac{N(s)}{D(s)}=K\dfrac{\prod_{m=1}^{M}(s-z_m)}{\prod_{n=1}^{N}(s-p_n)}$$

LTI input-output ratio in the Laplace domain. Poles $p_n$ (roots of $D$) set stability and transient modes; zeros $z_m$ (roots of $N$) shape the response.

Closed-loop transfer function (general)

$$\dfrac{Y(s)}{R(s)}=\dfrac{G_1G_2}{1+G_1G_2H}$$

$G_1$ controller, $G_2$ plant, $H$ feedback/sensor. Forward path over one-plus-loop-product.

Unity-feedback closed loop

$$\dfrac{Y(s)}{R(s)}=\dfrac{G(s)}{1+G(s)}$$

Special case $H=1$. The denominator $1+G$ defines the closed-loop poles.

Characteristic equation

$$1+G_1(s)G_2(s)H(s)=0$$

Roots are the closed-loop poles. Stable iff all roots have negative real parts (left half-plane).

Routh condition for a cubic

$$s^3+a_2s^2+a_1s+a_0:\quad a_i>0\ \text{and}\ a_2a_1>a_0$$

Necessary-and-sufficient stability test for third order. At the boundary $a_2a_1=a_0$ the loop oscillates at $\omega=\sqrt{a_1}\,\text{rad/s}$.

Final Value Theorem / DC gain

$$y(\infty)=\lim_{s\to0}sY(s),\qquad \text{dc gain}=\lim_{s\to0}G(s)$$

Valid only when the closed-loop poles are all in the left half-plane. Gives the steady output and the position constant $K_p$.

Static error constants

$$K_p=\lim_{s\to0}G,\quad K_v=\lim_{s\to0}sG,\quad K_a=\lim_{s\to0}s^2G$$

Position, velocity, acceleration constants for a unity-feedback loop; $G$ is the open-loop transfer function.

Steady-state error by input

$$e_{ss}^{\text{step}}=\dfrac{1}{1+K_p},\quad e_{ss}^{\text{ramp}}=\dfrac{1}{K_v},\quad e_{ss}^{\text{accel}}=\dfrac{1}{K_a}$$

Type 0: finite $K_p$, and $K_v = K_a = 0$, so the ramp and acceleration errors are infinite. Each integrator raises the type and zeros the error to one higher input class.

PID controller

$$G_C(s)=K\left(1+\dfrac{1}{T_I s}+T_D s\right)$$

$K$ proportional gain, $T_I$ integral time (s), $T_D$ derivative time (s). Integral action removes offset; derivative action adds damping.

Standard second-order denominator

$$s^2+2\zeta\omega_n s+\omega_n^2=0$$

Closed-loop characteristic form: $\omega_n$ natural frequency (rad/s), $\zeta$ damping ratio. Stable for $\zeta>0$, $\omega_n>0$.

Gain and phase margins

$$\mathrm{PM}=180^\circ+\angle G(j\omega_{0\text{dB}}),\quad \mathrm{GM}=-20\log_{10}|G(j\omega_{180})|$$

Relative-stability measures from the open-loop frequency response. Positive margins indicate a stable closed loop.

Where it lives: NCEES FE Reference Handbook — Instrumentation, Measurement, and Control (Control Systems) · Nise, Control Systems Engineering · Ogata, Modern Control Engineering

Static and Fatigue Failure Theories

When does a part actually break — static yielding by von Mises or Tresca, and fatigue life from the endurance limit, Marin factors, and the Goodman line.

Principal stresses (plane stress)

$$\sigma_{1,2} = \frac{\sigma_x+\sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2}, \quad \sigma_3 = 0$$

Reduces a plane-stress state to principals; the third principal is zero. Inputs in MPa or psi.

Maximum-shear-stress (Tresca)

$$\tau_{max} = \frac{\sigma_1-\sigma_3}{2} \ge \frac{S_y}{2}$$

Ductile yield criterion. Use the 3-D ordered principals; $n = S_y/(\sigma_1-\sigma_3)$. $S_y$ = yield strength.

Distortion-energy (von Mises)

$$\sigma' = \sqrt{\sigma_x^2 - \sigma_x\sigma_y + \sigma_y^2 + 3\tau_{xy}^2} \ge S_y$$

Effective stress for ductile yielding; $n = S_y/\sigma'$. Most accurate of the ductile theories.

Maximum-normal-stress (brittle)

$$\sigma_1 \ge S_{ut} \quad\text{or}\quad \sigma_3 \le -S_{uc}$$

Brittle fracture; uses separate tensile $S_{ut}$ and compressive $S_{uc}$ strengths.

Stress concentration

$$\sigma_{max} = K_t\,\sigma_{nom}, \qquad K_f = 1 + q(K_t-1)$$

$K_t$ geometric factor; $K_f$ fatigue factor with notch sensitivity $q$ ($0$ to $1$).

Stress ratio and amplitudes

$$\sigma_m = \frac{\sigma_{max}+\sigma_{min}}{2}, \quad \sigma_a = \frac{\sigma_{max}-\sigma_{min}}{2}, \quad R = \frac{\sigma_{min}}{\sigma_{max}}$$

Mean and alternating stress for fluctuating loads; $R=-1$ is fully reversed.

Endurance limit estimate (steel)

$$S_e' = 0.5\,S_{ut}\ (S_{ut}\le 1400\ \text{MPa}); \quad S_e' = 700\ \text{MPa}\ \text{otherwise}$$

Rotating-beam endurance limit from ultimate strength, before Marin correction.

Marin corrected endurance limit

$$S_e = k_a k_b k_c k_d k_e\,S_e'$$

$k_a$ surface, $k_b$ size, $k_c$ load, $k_d$ temperature, $k_e$ misc. Each is dimensionless.

Surface and size factors

$$k_a = a\,S_{ut}^{\,b}, \qquad k_b = 1.189\,d_{eff}^{-0.097}\ (8\le d\le 250\ \text{mm})$$

$a,b$ from finish table (MPa or kpsi). $k_b=1$ for axial loading; $k_c=1$ bend, $0.923$ axial ($S_{ut}\le1520$ MPa), $0.577$ torsion.

Modified Goodman criterion

$$\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = \frac{1}{n}$$

Fatigue failure line for tensile mean stress; solve for fatigue factor of safety $n$.

Soderberg criterion

$$\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_y} = \frac{1}{n}$$

More conservative than Goodman; guards against yield as well as fatigue.