Mechanical Formula Finder

Static force balance Statics

sum(F) = 0

The starting point for support reactions, free-body diagrams, and equilibrium checks.

• sum(F) = algebraic sum of forces in a chosen direction

Static moment balance Statics

sum(M) = 0

Used constantly for reactions, tipping checks, and bracket equilibrium.

• sum(M) = algebraic sum of moments about a chosen point

Rectangle centroid Statics

xbar = b / 2, ybar = h / 2

A basic geometric relation used when building section properties.

• xbar = centroid x location
• ybar = centroid y location
• b = width
• h = height

Parallel axis theorem Statics

I = Ic + A * d^2

Useful when building custom section properties from simple shapes.

• I = shifted second moment of area
• Ic = centroidal second moment of area
• A = area
• d = offset distance

Factor of safety Statics

n = Strength / Applied load effect

Often the final quick check after a hand calculation is complete.

• n = factor of safety
• Strength = allowable or failure value
• Applied load effect = demand from load, stress, or moment

Pressure force Statics

F = p * A

Useful for cylinder, gasket, cover, and sealing-load checks.

• F = force
• p = pressure
• A = loaded area

Bending stress Strength

sigma = M * c / I

Useful for quick beam and bracket stress checks before moving into deeper FEA.

• sigma = bending stress
• M = bending moment
• c = distance to outer fiber
• I = second moment of area

Cantilever tip deflection Strength

delta = F * L^3 / (3 * E * I)

A common spreadsheet calculation for brackets, arms, and fixture members.

• delta = tip deflection
• F = applied end load
• L = beam length
• E = elastic modulus
• I = second moment of area

Simply supported beam center deflection Strength

delta = F * L^3 / (48 * E * I)

Good for fast checks on plate supports, rails, and cross-members.

• delta = center deflection
• F = center load
• L = span length
• E = elastic modulus
• I = second moment of area

Axial normal stress Strength

sigma = F / A

A first-pass check for rods, ties, columns, and loaded sections.

• sigma = normal stress
• F = axial load
• A = cross-sectional area

Axial strain Strength

epsilon = deltaL / L

Used with Hooke’s law and elongation calculations.

• epsilon = strain
• deltaL = change in length
• L = original length

Hooke’s law Strength

sigma = E * epsilon

A core linear-elastic relation for quick material response checks.

• sigma = stress
• E = elastic modulus
• epsilon = strain

Axial elongation Strength

deltaL = F * L / (A * E)

Common for bolts, rods, tie members, and simple stiffness estimates.

• deltaL = elongation
• F = axial load
• L = member length
• A = area
• E = elastic modulus

Average shear stress Strength

tau = V / A

A useful screening check for pins, lugs, and shear planes.

• tau = average shear stress
• V = shear force
• A = shear area

Beam shear stress Strength

tau = V * Q / (I * t)

Helpful when web shear matters in beams or built-up sections.

• tau = shear stress
• V = shear force
• Q = first moment of area
• I = second moment of area
• t = local thickness

Section modulus relation Strength

S = I / c

Useful when converting between section properties and bending stress checks.

• S = section modulus
• I = second moment of area
• c = distance to outer fiber

Euler buckling load Strength

Pcr = pi^2 * E * I / (K * L)^2

A standard check for slender compression members.

• Pcr = critical buckling load
• E = elastic modulus
• I = second moment of area
• K = effective length factor
• L = unsupported length

Slenderness ratio Strength

lambda = K * L / r

Used to decide whether buckling is likely to govern.

• lambda = slenderness ratio
• K = effective length factor
• L = unsupported length
• r = radius of gyration

Von Mises stress (plane stress) Strength

sigma_vm = sqrt(sigma_x^2 + sigma_y^2 - sigma_x * sigma_y + 3 * tau_xy^2)

Useful when combining hand-calculated normal and shear stresses.

• sigma_vm = equivalent stress
• sigma_x = normal stress in x
• sigma_y = normal stress in y
• tau_xy = in-plane shear stress

Maximum principal stress Strength

sigma_1,2 = (sigma_x + sigma_y)/2 +/- sqrt(((sigma_x - sigma_y)/2)^2 + tau_xy^2)

Helpful for failure assessment and Mohr-circle style checks.

• sigma_1,2 = principal stresses
• sigma_x = normal stress in x
• sigma_y = normal stress in y
• tau_xy = in-plane shear stress

Solid shaft torsional shear stress Machine Design

tau = T * r / J

Useful for coupling, shaft, and drive component screening.

• tau = shear stress
• T = torque
• r = outer radius
• J = polar moment of inertia

Angle of twist Machine Design

theta = T * L / (J * G)

Helpful when stiffness matters more than ultimate strength.

• theta = angle of twist
• T = torque
• L = shaft length
• J = polar moment of inertia
• G = shear modulus

Power, torque, and speed Machine Design

P = T * omega

A common motor, gearbox, and rotating-equipment relationship.

• P = power
• T = torque
• omega = angular speed

Power, torque, and rpm Machine Design

P = 2 * pi * N * T / 60

Useful when catalogs or machine data are expressed in rpm rather than rad/s.

• P = power
• N = rotational speed in rpm
• T = torque

Bearing pressure Machine Design

p = F / A

Useful for pins, bushings, and quick contact-pressure estimates.

• p = bearing pressure
• F = load
• A = projected area

Bolt tensile stress Machine Design

sigma = F / At

A first-pass relation for threaded fastener loading.

• sigma = tensile stress
• F = bolt preload or axial load
• At = tensile stress area

Bolt preload from torque Machine Design

T = K * F * d

A practical shop-floor estimate for preload discussions.

• T = tightening torque
• K = torque coefficient
• F = preload
• d = nominal diameter

Helical compression spring rate Machine Design

k = G * d^4 / (8 * D^3 * n)

Useful for quick spring selection and what-if stiffness checks.

• k = spring rate
• G = shear modulus
• d = wire diameter
• D = mean coil diameter
• n = active coils

Spring deflection Machine Design

delta = F / k

Pairs naturally with spring-rate estimates during concept work.

• delta = spring deflection
• F = spring load
• k = spring rate

Belt or pulley surface speed Machine Design

V = pi * D * N / 60

Useful for drive sizing and quick motion checks.

• V = surface speed
• D = pulley diameter
• N = rotational speed in rpm

Gear ratio Machine Design

i = N_in / N_out = z_out / z_in

A basic but constant relation in transmission layout work.

• i = gear ratio
• N_in = input speed
• N_out = output speed
• z_in = input gear teeth
• z_out = output gear teeth

Gear pitch-line velocity Machine Design

V = pi * D * N / 60

Useful for gear duty and lubrication conversations.

• V = pitch-line velocity
• D = pitch diameter
• N = rotational speed in rpm

Interference per diameter Machine Design

delta = Dhole - Dshaft

The simplest direct relation behind fit-window reasoning.

• delta = fit clearance or interference
• Dhole = actual hole size
• Dshaft = actual shaft size

Mass from density and volume Machine Design

m = rho * V

Common for estimating component mass before full CAD properties are available.

• m = mass
• rho = density
• V = volume

Weight from mass Machine Design

W = m * g

Useful when switching between mass properties and static loading.

• W = weight force
• m = mass
• g = gravitational acceleration

Linear kinetic energy Dynamics

KE = 0.5 * m * v^2

Helpful for impact, motion, and safety-envelope reasoning.

• KE = kinetic energy
• m = mass
• v = velocity

Rotational kinetic energy Dynamics

KE = 0.5 * I * omega^2

Useful for flywheels, spindles, and deceleration checks.

• KE = rotational kinetic energy
• I = mass moment of inertia
• omega = angular speed

Newton’s second law Dynamics

F = m * a

The backbone relation for actuator, acceleration, and inertia sizing.

• F = net force
• m = mass
• a = acceleration

Rotational dynamics Dynamics

T = I * alpha

Useful for drive-train acceleration and motor sizing checks.

• T = net torque
• I = mass moment of inertia
• alpha = angular acceleration

Linear momentum Dynamics

p = m * v

Useful for collision and impulse-style estimates.

• p = momentum
• m = mass
• v = velocity

Impulse Dynamics

J = F * deltaT = delta(m * v)

Useful when peak loads depend strongly on stopping time.

• J = impulse
• F = average force
• deltaT = impact duration
• delta(m * v) = momentum change

Mechanical work Dynamics

W = F * s

A constant relationship in energy-balance and actuator calculations.

• W = work
• F = force in direction of motion
• s = displacement

Gravitational potential energy Dynamics

PE = m * g * h

Useful for lifting, drop, and stored-energy discussions.

• PE = potential energy
• m = mass
• g = gravity
• h = elevation change

Centripetal force Dynamics

F = m * v^2 / r

Helpful for rotating systems, curved paths, and balance reasoning.

• F = centripetal force
• m = mass
• v = tangential velocity
• r = radius

Natural frequency of spring-mass system Dynamics

fn = (1 / (2 * pi)) * sqrt(k / m)

Useful for vibration screening and resonance avoidance.

• fn = natural frequency
• k = stiffness
• m = mass

Damping ratio Dynamics

zeta = c / cc

A useful vocabulary-level formula for quick vibration interpretation.

• zeta = damping ratio
• c = actual damping coefficient
• cc = critical damping coefficient

Linear thermal expansion Heat Transfer

deltaL = alpha * L * deltaT

Great for fit-up, gap, and allowance conversations during layout review.

• deltaL = length change
• alpha = thermal expansion coefficient
• L = original length
• deltaT = temperature change

1D conduction heat rate Heat Transfer

Q = k * A * deltaT / L

Useful for quick insulation and wall-stack comparisons.

• Q = heat transfer rate
• k = thermal conductivity
• A = area
• deltaT = temperature difference
• L = thickness

Convection heat rate Heat Transfer

Q = h * A * (Ts - Tinf)

A first-pass estimate for cooling surfaces and enclosure checks.

• Q = heat transfer rate
• h = convection coefficient
• A = area
• Ts = surface temperature
• Tinf = bulk fluid temperature

Radiation heat transfer Heat Transfer

Q = epsilon * sigma * A * (Ts^4 - Tsur^4)

Useful when hot surfaces or furnaces make radiation non-negligible.

• Q = heat transfer rate
• epsilon = emissivity
• sigma = Stefan-Boltzmann constant
• A = area
• Ts = surface temperature
• Tsur = surroundings temperature

Conduction thermal resistance Heat Transfer

Rth = L / (k * A)

A convenient building block for thermal resistance networks.

• Rth = thermal resistance
• L = thickness
• k = thermal conductivity
• A = area

Convection thermal resistance Heat Transfer

Rth = 1 / (h * A)

Useful for first-pass thermal network models.

• Rth = thermal resistance
• h = convection coefficient
• A = area

Log mean temperature difference Heat Transfer

LMTD = (deltaT1 - deltaT2) / ln(deltaT1 / deltaT2)

A core exchanger relation for rough thermal sizing.

• LMTD = log mean temperature difference
• deltaT1 = end temperature difference 1
• deltaT2 = end temperature difference 2

Sensible heat rate Heat Transfer

Q = m_dot * cp * deltaT

A very common HVAC, cooling, and process estimate.

• Q = heat transfer rate
• m_dot = mass flow rate
• cp = specific heat
• deltaT = temperature rise or drop

Thermal diffusivity Heat Transfer

alpha = k / (rho * cp)

Useful when thinking about response time as well as conductivity.

• alpha = thermal diffusivity
• k = thermal conductivity
• rho = density
• cp = specific heat

Biot number Heat Transfer

Bi = h * Lc / k

Helpful for deciding whether lumped-capacitance assumptions are reasonable.

• Bi = Biot number
• h = convection coefficient
• Lc = characteristic length
• k = thermal conductivity

Fourier number Heat Transfer

Fo = alpha * t / L^2

Useful in transient heat transfer reasoning.

• Fo = Fourier number
• alpha = thermal diffusivity
• t = time
• L = characteristic length

Reynolds number Fluids

Re = rho * V * D / mu

A recurring lookup for deciding laminar versus turbulent assumptions.

• Re = Reynolds number
• rho = fluid density
• V = velocity
• D = hydraulic diameter
• mu = dynamic viscosity

Continuity equation Fluids

Q = A * V

A basic but constant day-to-day relationship in line sizing work.

• Q = volumetric flow rate
• A = flow area
• V = average velocity

Pressure-head conversion Fluids

h = deltaP / (rho * g)

Useful alongside pressure-drop conversations and pump sanity checks.

• h = head
• deltaP = pressure difference
• rho = fluid density
• g = gravity

Bernoulli equation Fluids

p/(rho*g) + V^2/(2*g) + z = constant

The backbone energy relation behind many fluid sanity checks.

• p = pressure
• rho = density
• g = gravity
• V = velocity
• z = elevation

Mass flow rate Fluids

m_dot = rho * Q

Useful whenever thermal and hydraulic calculations need to connect.

• m_dot = mass flow rate
• rho = density
• Q = volumetric flow rate

Dynamic pressure Fluids

q = 0.5 * rho * V^2

Useful in duct, nozzle, and flow-energy estimates.

• q = dynamic pressure
• rho = density
• V = velocity

Darcy-Weisbach pressure loss Fluids

hf = f * (L / D) * V^2 / (2 * g)

A core pipe-flow relation for quick closed-line loss estimates.

• hf = friction head loss
• f = Darcy friction factor
• L = pipe length
• D = pipe diameter
• V = velocity
• g = gravity

Minor loss Fluids

hm = K * V^2 / (2 * g)

Useful when valves, bends, and fittings matter more than straight run.

• hm = minor head loss
• K = fitting loss coefficient
• V = velocity
• g = gravity

Laminar friction factor Fluids

f = 64 / Re

A very common lookup shortcut for laminar internal flow.

• f = Darcy friction factor
• Re = Reynolds number

Hydraulic diameter Fluids

Dh = 4 * A / Pwet

Useful when ducts or non-circular flow paths are involved.

• Dh = hydraulic diameter
• A = flow area
• Pwet = wetted perimeter

Hydraulic power Fluids

P = rho * g * Q * H

Useful for a first-pass pump or system power estimate.

• P = hydraulic power
• rho = density
• g = gravity
• Q = volumetric flow rate
• H = total head

Brake power from efficiency Fluids

Pbrake = Phydraulic / eta

Helps connect hydraulic duty to motor demand.

• Pbrake = shaft or brake power
• Phydraulic = hydraulic power
• eta = pump efficiency

Pump specific speed Fluids

Ns = N * sqrt(Q) / H^(3/4)

Useful as a rough pump-family classification parameter.

• Ns = specific speed
• N = rotational speed
• Q = flow rate
• H = head

Orifice flow rate Fluids

Q = Cd * A * sqrt(2 * deltaP / rho)

Useful for restriction, leak, and nozzle estimates.

• Q = flow rate
• Cd = discharge coefficient
• A = opening area
• deltaP = pressure difference
• rho = density

Mach number Fluids

M = V / a

Useful when compressibility may start to matter.

• M = Mach number
• V = fluid velocity
• a = local speed of sound