Equilibrium Equations (2D)
- Sum of Fx = 0 (sum of all forces in the x-direction equals zero)
- Sum of Fy = 0 (sum of all forces in the y-direction equals zero)
- Sum of M_A = 0 (sum of moments about any point A equals zero)
Equilibrium Equations (3D)
- Sum of Fx = 0, Sum of Fy = 0, Sum of Fz = 0
- Sum of Mx = 0, Sum of My = 0, Sum of Mz = 0
- Six independent equations for 3D equilibrium
Moment of a Force
- M = F * d (moment equals force times perpendicular distance)
- M = r x F (cross product for vector moment calculation)
- M = |r| * |F| * sin(theta) (scalar form)
- Varignon's theorem: moment of a force equals the sum of moments of its components
Truss Analysis
- Method of Joints: analyze equilibrium at each joint (2 equations per joint in 2D)
- Method of Sections: cut through truss and solve for member forces (3 equations per section)
- Zero-force members: identified by inspection at joints with only two non-collinear members and no external load
- Determinacy: m + r = 2j (m = members, r = reactions, j = joints)
Friction
- F_friction = mu * N (friction force equals coefficient times normal force)
- Static friction: F_s <= mu_s * N (maximum before motion)
- Kinetic friction: F_k = mu_k * N (during motion, mu_k < mu_s)
- Angle of friction: phi = arctan(mu)
- Belt friction: T2 = T1 * e^(mu * beta) where beta is the contact angle in radians
Centroids
- x_bar = Sum(x_i * A_i) / Sum(A_i) (x-coordinate of centroid)
- y_bar = Sum(y_i * A_i) / Sum(A_i) (y-coordinate of centroid)
- Rectangle centroid: (b/2, h/2) from corner
- Triangle centroid: (b/3, h/3) from right angle vertex
- Circle centroid: at center (r, r)
Moments of Inertia
- Rectangle: I_x = bh^3/12, I_y = hb^3/12 (about centroidal axes)
- Circle: I = pi*r^4/4 (about centroidal axis)
- Parallel axis theorem: I = I_c + A*d^2
- Composite bodies: sum individual I values using parallel axis theorem