Formula Library

ΣFx = 0, ΣFy = 0, ΣM = 0

F

Force component [kip or kN]

M

Moment about a point [kip·ft or kN·m]

Applications: Reactions, truss analysis, free-body diagrams.

Assumptions: Rigid body; Static loading; Consistent sign convention

Common mistakes: Forgetting a reaction component; Mixing units of force and moment

σ = M·c / I

M

Bending moment [kip·in]

c

Distance to extreme fiber [in]

I

Moment of inertia [in⁴]

Applications: Elastic bending stress checks in beams.

Assumptions: Linear elastic material; Plane sections remain plane; Symmetric bending

Common mistakes: Using gross I on a corroded section; Mixing kip·ft and kip·in

S = I / c

S

Elastic section modulus [in³]

Applications: Preliminary beam sizing.

Assumptions: Elastic behavior

Common mistakes: Confusing elastic S with plastic Z

φb·Mn = 0.90·Fy·Zx (compact, Lb ≤ Lp)

Fy

Yield stress [ksi]

Zx

Plastic section modulus [in³]

Applications: LRFD flexural capacity of compact rolled shapes.

Assumptions: Compact section; Adequate lateral bracing

Common mistakes: Ignoring lateral-torsional buckling when Lb > Lp; Using Sx instead of Zx

φt·Pn = 0.90·Fy·Ag

Ag

Gross area [in²]

Applications: Tension member capacity check.

Assumptions: Concentric load

Common mistakes: Not also checking rupture on the net area with the shear lag factor U

Mn = As·fy·(d − a/2), a = As·fy / (0.85·f'c·b)

As

Tension steel area [in²]

d

Effective depth [in]

Applications: Reinforced concrete beam flexure.

Assumptions: Tension-controlled section; Whitney stress block

Common mistakes: Assuming tension-controlled without checking εt ≥ 0.005

Vc = 2·λ·√f'c·bw·d

f'c

Concrete strength [psi]

Applications: Shear capacity of the concrete contribution.

Assumptions: Normal-weight concrete unless λ adjusted

Common mistakes: Using ksi in a psi equation

Fa = 0.25·f'm·[1 − (h/140r)²]

f'm

Masonry compressive strength [psi]

Applications: Preliminary masonry wall checks.

Assumptions: Slender wall formulation for h/r ≤ 99

Common mistakes: Using the wrong slenderness branch

RF = (C − γDC·DC − γDW·DW) / (γLL·LL(1+IM))

C

Member capacity [kip·ft]

IM

Dynamic allowance [—]

Applications: Bridge load rating for existing members.

Assumptions: Capacity reflects measured section loss

Common mistakes: Using as-built section properties on a corroded member

qult = c·Nc + q·Nq + 0.5·γ·B·Nγ

c

Cohesion [psf]

B

Footing width [ft]

Applications: Shallow foundation sizing.

Assumptions: General shear failure; Homogeneous soil

Common mistakes: Ignoring groundwater correction on γ

Sc = (Cc·H / (1+e0))·log10((σ'0+Δσ)/σ'0)

Cc

Compression index [—]

H

Layer thickness [ft]

Applications: Settlement of normally consolidated clay.

Assumptions: Normally consolidated; One-dimensional compression

Common mistakes: Using this form for overconsolidated soil without Cr

Ka = tan²(45° − φ/2)

φ

Friction angle [degrees]

Applications: Retaining wall lateral pressure.

Assumptions: Smooth vertical wall; Horizontal backfill

Common mistakes: Applying Rankine to a battered wall with wall friction

SSD = 1.47·V·t + V² / (30·(a/32.2 ± G))

V

Design speed [mph]

G

Grade [decimal]

Applications: Geometric design sight distance checks.

Assumptions: 2.5 s reaction time; Deceleration 11.2 ft/s²

Common mistakes: Sign error on grade for downgrades

Rmin = V² / (15·(e + f))

e

Superelevation rate [decimal]

f

Side friction factor [—]

Applications: Horizontal alignment design.

Assumptions: Point-mass model

Common mistakes: Using percent instead of decimal for e

Q = C·i·A

C

Runoff coefficient [—]

i

Rainfall intensity [in/hr]

A

Drainage area [acres]

Applications: Peak flow for small urban watersheds.

Assumptions: Area under ~200 acres; Uniform rainfall

Common mistakes: Using a duration shorter than the time of concentration

Q = (1.49/n)·A·R^(2/3)·S^(1/2)

n

Roughness coefficient [—]

R

Hydraulic radius [ft]

Applications: Open channel and storm sewer capacity.

Assumptions: Steady uniform flow; US customary constant 1.49

Common mistakes: Using 1.49 with SI units instead of 1.0

Re = ρ·V·D / μ

V

Velocity [ft/s]

D

Diameter [ft]

Applications: Flow regime classification.

Assumptions: Circular full pipe

Common mistakes: Mixing dynamic and kinematic viscosity

TF = LS − ES = LF − EF

LS

Late start [days]

Applications: CPM schedule analysis.

Assumptions: Finish-to-start logic

Common mistakes: Confusing total float with free float

P = F / (1+i)^n

i

Interest rate per period [decimal]

n

Number of periods [—]

Applications: Life-cycle cost comparison of alternatives.

Assumptions: Constant discount rate

Common mistakes: Mismatching period length and rate

G = (Δelevation / Δhorizontal) × 100

G

Grade [%]

Applications: Profile and drainage checks.

Assumptions: Straight grade segment

Common mistakes: Using slope distance instead of horizontal distance

x̄ = Σ(Ai·xi) / ΣAi

Ai

Component area [in²]

Applications: Built-up section properties.

Assumptions: Planar area

Common mistakes: Forgetting to subtract holes

I = Ī + A·d²

d

Distance between axes [in]

Applications: Composite moment of inertia.

Assumptions: Parallel axes

Common mistakes: Using the centroid of the wrong component