FE Mechanical cheat sheet

Derivative rules: d/dx(x^n)=n x^(n-1); d/dx(e^x)=e^x; d/dx(ln x)=1/x; d/dx(sin x)=cos x; d/dx(cos x)=-sin x; product rule (uv)'=u'v+uv'; quotient rule (u/v)'=(u'v-uv')/v^2; chain rule d/dx f(g(x))=f'(g(x))g'(x).

Integrals: ∫x^n dx = x^(n+1)/(n+1)+C for n≠-1; ∫1/x dx=ln|x|+C; ∫e^x dx=e^x+C; ∫sin x dx=-cos x+C; ∫cos x dx=sin x+C; integration by parts ∫u dv=uv-∫v du.

Taylor series about x=a: f(x)=f(a)+f'(a)(x-a)+f''(a)(x-a)^2/2!+…; Maclaurin for e^x, sin x, cos x, ln(1+x), 1/(1-x).

L'Hôpital's rule: if lim f/g is 0/0 or ∞/∞, then lim f/g = lim f'/g' provided the latter exists.

Linear algebra: determinant of 2×2 [[a,b],[c,d]]=ad-bc; inverse exists iff det≠0; eigenvalues solve det(A-λI)=0; for triangular matrices eigenvalues are diagonal entries.

Vector operations: dot product a·b=|a||b|cosθ, scalar; cross product a×b is a vector perpendicular to both with magnitude |a||b|sinθ; a·(b×c) gives the scalar triple product (volume of parallelepiped).

Numerical methods: Newton-Raphson x_{n+1}=x_n - f(x_n)/f'(x_n); trapezoidal rule ∫_a^b f dx ≈ (b-a)/2 [f(a)+f(b)]; Simpson's 1/3 rule with even n: (h/3)[f0+4f1+2f2+4f3+…+fn].

Partial derivatives: for f(x,y), ∂f/∂x treats y as constant; total differential df=(∂f/∂x)dx+(∂f/∂y)dy; gradient ∇f points in direction of greatest increase.

Mean of discrete X: μ=Σ x P(x); variance σ^2=Σ(x-μ)^2 P(x)=E(X^2)-[E(X)]^2; standard deviation σ=√σ^2.

Binomial: P(X=k)=C(n,k) p^k (1-p)^(n-k), mean np, variance np(1-p); Poisson: P(X=k)=e^(-λ) λ^k/k!, mean=variance=λ.

Normal distribution: standardize z=(x-μ)/σ; empirical rule: ≈68% within μ±σ, ≈95% within μ±2σ, ≈99.7% within μ±3σ; central limit theorem: sample mean approx normal for large n.

Confidence interval for mean with known σ: x̄ ± z_(α/2) σ/√n; with unknown σ and small n use t_(α/2, n-1) s/√n.

Hypothesis test: z=(x̄-μ0)/(σ/√n); reject H0 if |z|>z_(α/2) for two-sided test; p-value is probability of observing result as extreme or more extreme given H0 true.

Linear regression: slope b = [nΣxy - ΣxΣy]/[nΣx^2-(Σx)^2]; intercept a = ȳ - b x̄; correlation coefficient r measures linear strength (-1≤r≤1).

Combinatorics: permutations nPr=n!/(n-r)!; combinations nCr=n!/[r!(n-r)!]; number of arrangements of n items with repeats n!/(n1!n2!…).

Probability rules: P(A∪B)=P(A)+P(B)-P(A∩B); P(A|B)=P(A∩B)/P(B); independent events P(A∩B)=P(A)P(B); mutually exclusive P(A∩B)=0.

NCEES Model Rules: engineers shall hold paramount the safety, health, and welfare of the public; perform services only in areas of competence; issue public statements only in an objective and truthful manner.

Conflict of interest: engineers shall disclose all known or potential conflicts of interest to employers or clients and shall not accept compensation from more than one party for the same service without disclosure.

Seal and signature: a licensed PE may seal only work they prepared or directly supervised; sealing work not under their responsible charge is misconduct.

Confidentiality: engineers shall not reveal confidential information without consent except when required by law or when public safety is at risk.

Competitive bidding: engineering services are procured on qualifications, not price alone; QBS (qualifications-based selection) is the preferred method for professional services.

Continuing professional competency: many jurisdictions require PDHs (professional development hours) per renewal cycle; typical requirement is 15–30 PDH per year depending on state.

Whistleblowing: if an employer's actions endanger public safety, the engineer's obligation to the public generally takes precedence; report to appropriate authority after internal channels fail.

Liability: negligence requires duty, breach, causation, and damages; engineers may be liable for errors and omissions; statutes of limitations and repose limit time to file claims.

Time value of money: F=P(1+i)^n; P=F(1+i)^(-n); effective annual rate i_eff=(1+i_nom/m)^m -1 for m compounding periods per year.

Uniform series: (P/A,i,n)=[(1+i)^n-1]/[i(1+i)^n]; (A/P,i,n)=i(1+i)^n/[(1+i)^n-1]; (F/A,i,n)=[(1+i)^n-1]/i; (A/F,i,n)=i/[(1+i)^n-1].

Gradient series: (P/G,i,n) = (1/i)[(P/A,i,n) - n(1+i)^(-n)]; arithmetic gradient converts to equivalent present worth.

Net present worth (NPW): convert all cash flows to present at MARR; accept independent project if NPW≥0; for mutually exclusive choose largest NPW.

Annual worth (AW): AW = NPW(A/P,i,n); equivalent uniform annual cost; choose alternative with lowest equivalent annual cost when comparing costs.

Internal rate of return (IRR): i that makes NPW=0; accept project if IRR≥MARR; multiple sign changes in cash flows can produce multiple IRRs-use NPW instead.

Payback period: simple payback = initial investment / annual net cash flow (uniform); discounted payback uses present worth of cash flows; ignores cash flows after payback.

Depreciation: straight-line D=(B-S)/n; MACRS uses declining balance with half-year convention; book value after k years = B - ΣD; depreciation is not a cash flow but affects taxes.

Breakeven: quantity Q where total revenue = total cost: Q = FC/(price - variable cost per unit); breakeven revenue = price × Q.

Benefit-cost ratio: B/C = PW(benefits)/PW(costs); accept if B/C≥1; incremental B/C analysis for mutually exclusive alternatives.

Equilibrium of a particle: ΣF=0 (ΣFx=0, ΣFy=0, ΣFz=0); for a rigid body in 2D: ΣFx=0, ΣFy=0, ΣM=0 about any point.

Moment of a force about point O: M_O = r × F, magnitude M=Fd where d is perpendicular distance from O to line of action; sign by right-hand rule.

Couple: two equal, opposite, noncollinear forces; moment M=Fd is a free vector, same about any point.

Resultant of distributed load: equivalent force equals area under load diagram; acts through centroid of the area; for uniform load w over length L, R=wL at L/2.

Trusses: method of joints-each joint must satisfy ΣFx=0, ΣFy=0; method of sections-cut through ≤3 members and apply equilibrium to either part; zero-force members at joints with two noncollinear members and no load.

Friction: impending motion F_max=μ_s N; kinetic friction F_k=μ_k N; angle of friction φ_s=tan^(-1) μ_s; belt friction T2=T1 e^(μβ) for flat belt.

Centroid: x̄=∫x dA/∫dA; for composite areas use Σx̄_i A_i / ΣA_i; centroid of semicircle at 4r/(3π) from diameter; triangle centroid at h/3 from base.

Area moment of inertia: I_x=∫y^2 dA; parallel axis theorem I=I_c+Ad^2; rectangle about centroid I=bh^3/12; circle about diameter I=πr^4/4.

Internal forces in beams: shear V and bending moment M; relationship dV/dx = -w(x), dM/dx = V; M maximum where V=0 or changes sign.

Kinematics: v=ds/dt, a=dv/dt=v dv/ds; for constant acceleration: v=v0+at, s=s0+v0t+½at^2, v^2=v0^2+2a(s-s0).

Projectile motion: horizontal x=v0 cosθ t; vertical y=v0 sinθ t - ½gt^2; time of flight for level ground t=2v0 sinθ/g; range R=v0^2 sin2θ/g.

Newton's second law: ΣF=ma; in normal-tangential coordinates: ΣF_t=ma_t, ΣF_n=m v^2/ρ; in polar coordinates include Coriolis-like terms.

Work-energy: U=∫F·dr; work done by weight = -WΔy; work by spring = -½k(x2^2-x1^2); T=½mv^2; U=T2-T1.

Impulse-momentum: ∫F dt = m v2 - m v1; for impact, coefficient of restitution e=(v2'-v1')/(v1-v2); e=1 elastic, e=0 perfectly plastic; momentum conserved in collision if no external impulse.

Rotation: θ, ω=dθ/dt, α=dω/dt; ω=ω0+αt; θ=θ0+ω0t+½αt^2; ω^2=ω0^2+2α(θ-θ0).

Rigid body kinetics: ΣM_G=I_G α; ΣM_O=I_O α for fixed axis O; parallel axis I_O=I_G+md^2; kinetic energy T=½I_G ω^2+½mv_G^2.

Centripetal acceleration a_n=v^2/r=ω^2 r directed toward center; tangential acceleration a_t=dv/dt; total acceleration magnitude √(a_t^2+a_n^2).

Vibrations: undamped natural frequency ω_n=√(k/m) rad/s, f_n=ω_n/(2π); damped frequency ω_d=ω_n√(1-ζ^2); critical damping c_c=2√(km).

Normal stress σ=F/A; normal strain ε=δ/L; Hooke's law σ=Eε in elastic region; Poisson's ratio ν=-ε_lateral/ε_axial.

Shear stress τ=V/A (average); shear strain γ=τ/G; relationship G=E/[2(1+ν)].

Axial deformation δ=FL/(AE); for varying load or area δ=∫F dx/(AE); thermal expansion δ=αLΔT; if constrained, thermal stress σ=EαΔT.

Torsion of circular shaft: τ_max=Tc/J; angle of twist φ=TL/(JG); for solid circle J=πc^4/2; for hollow shaft J=π(c_o^4-c_i^4)/2.

Bending: flexure formula σ=My/I; maximum stress at outer fiber σ_max=Mc/I; beam equation EI d^2y/dx^2=M(x); section modulus S=I/c.

Transverse shear stress in beam: τ=VQ/(It); Q is first moment of area above (or below) the point about neutral axis; maximum at neutral axis for rectangular section τ_max=3V/(2A).

Stress transformation: σ_x'=(σ_x+σ_y)/2 + [(σ_x-σ_y)/2]cos2θ + τ_xy sin2θ; principal stresses σ_1,2=(σ_x+σ_y)/2 ± √[((σ_x-σ_y)/2)^2+τ_xy^2]; maximum shear τ_max=√[((σ_x-σ_y)/2)^2+τ_xy^2].

Mohr's circle: center at (σ_x+σ_y)/2 on σ-axis; radius R=√[((σ_x-σ_y)/2)^2+τ_xy^2]; principal stresses at σ-axis intercepts; max shear = R.

Failure theories: von Mises (distortion energy) σ'=√(σ_1^2-σ_1σ_2+σ_2^2) for plane stress; Tresca (maximum shear) σ_1-σ_3≤σ_Y; use von Mises for ductile metals.

Columns: Euler buckling P_cr=π^2 EI/(KL)^2; effective length factor K depends on end conditions (pinned-pinned K=1, fixed-fixed K=0.5, fixed-pinned K=0.7, fixed-free K=2); slenderness ratio KL/r.