Industrial Mechanics
Material strength and applied dynamics tools.
Equations Summary - Strength of Materials
- σ: tension (+), compression (-).
- τ: average shear: τ = V/A
- Hooke: σ = E · ε
- Equilibrium: ΣFx = 0, ΣFy = 0, ΣMz = 0
- Beam: dM/dx = V, dV/dx = -q
- FS: σadm = σlim / FS
- Bearing: σesmag = P/(t · d)
Mechanics Equations
Strength, buckling, and transmission formulas. Leave one field blank.
Normal Stress
Sigma = F / A
Stress from axial force over area.
Yield Stress
Sigma_y = F / A
Yield stress from force and area at the elastic limit.
Shear Stress
Tau = F / A
Average shear stress.
Strain
Epsilon = Delta_L / L0
Strain from length change.
Hooke Law
Sigma = E * Epsilon
Stress-strain relation.
Euler Critical Load
Pcr = (pi^2 * E * I) / (K * L)^2
Buckling critical load.
Euler Critical Stress
Sigma_cr = (pi^2 * E) / ((K * L)/r)^2
Critical buckling stress.
Slenderness Ratio
Lambda = (K * L) / r
Slenderness ratio.
Torque
T = F * d
Torque from force and lever arm.
Rotational Power
P = (T * RPM * 2*pi) / 60
Power from torque and RPM.
Peripheral Speed
v = (pi * D * RPM) / 60000
Peripheral speed from diameter and RPM.
Transmission Ratio
i = n_entrada / n_saida
Speed ratio.
Rectangular Inertia
I = (b*h^3)/12
Second moment of area.
Circular Inertia
I = (pi*d^4)/64
Second moment of area.
Support Reactions (Point Load)
RVB = (P*a)/L; RVA = P - RVB
Reactions for a simply supported beam with eccentric load.
Max Bending Moment (Eccentric Load)
M_max = (P*a*(L-a))/L
Maximum bending moment for eccentric point load.
Bending Stress
Sigma = (M_max*(h/2)) / I
Stress from bending moment and section geometry.
Max Deflection (Eccentric Load)
delta = (P*a*(L-a)^2)/(9*sqrt(3)*E*I*L) * (L + a)
Maximum deflection from the provided formula.
Safety Factor
FS = Limite / Admissivel
Safety factor from limit and allowable.
Allowable Stress
Sigma_adm = Sigma_lim / FS
Allowable stress from limit and FS.
Truss Check
Tipo: M + R vs 2J
Compare M + R vs 2J.
Self-weight Stress
Sigma = rho * g * L
Stress at top of a vertical bar.
Axial Elongation
delta = (F * L) / (A * E)
Elongation from axial load.
Axial Stiffness
k = (A * E) / L
Equivalent axial stiffness.
Thermal Stress
Sigma = E * alpha * DeltaT
Stress from restrained thermal expansion.
Thermal Expansion
delta = alpha * L * DeltaT
Free thermal expansion.
Torsion Shear Stress
Tau_max = (T * r) / J; J = (pi*d^4)/32
Max shear stress in circular shaft.
Angle of Twist
theta = (T * L) / (J * G)
Angle of twist for circular shaft.
Simple Bending
Sigma = (M * y) / I
Bending stress by M, y, I.
Shear Stress (Rectangular)
Tau_med = V/A; Tau_max = 1.5*V/A
Average and max shear for rectangle.
E, G and Poisson
G = E / (2*(1+nu))
Shear modulus from E and nu.
Mohr Circle (Plane Stress)
sigma1/2 and tau_max from sigma_x, sigma_y, tau_xy
Principal stresses and max shear.
von Mises (Plane Stress)
Sigma_vm = sqrt(sigma_x^2 - sigma_x*sigma_y + sigma_y^2 + 3*tau_xy^2)
Equivalent stress.
Section Modulus
W = I / (h/2)
W = I / (h/2).
Simply Supported Beam (UDL)
R = wL/2; Mmax = wL^2/8; Vmax = wL/2; delta = 5wL^4/(384EI)
Reactions, max shear/moment and deflection.
Cantilever (Point Load)
Vmax=P; Mmax=P*L; delta=P*L^3/(3EI); theta=P*L^2/(2EI)
Cantilever with point load at tip.
Cantilever (UDL)
Mmax = wL^2/2; Vmax = wL; delta = wL^4/(8EI)
Cantilever with uniform load.
Simply Supported (Mid Load)
R=P/2; Mmax=P*L/4; delta=P*L^3/(48EI)
Midpoint point load.
Bearing Stress
Sigma = P / (t * d)
Average bearing stress.
Belt Power
P = (T1 - T2) * v
Power from belt tensions.
Force Magnitude
|F| = sqrt(Fx^2 + Fy^2 + Fz^2)
Vector magnitude.
Shear Strain
gamma = tan(phi) ~= phi
Shear strain for small angles.