Equilibrium of Rigid Bodies
A body is in static equilibrium when the net force and net moment acting on it are both zero. For a two-dimensional system, three independent scalar equations can be written:
- Sum of Fx = 0 (forces in horizontal direction)
- Sum of Fy = 0 (forces in vertical direction)
- Sum of M_O = 0 (moments about any point O)
For three-dimensional problems, six equations are available: three force equations and three moment equations about the coordinate axes.
Free Body Diagrams
A free body diagram (FBD) isolates a body from its surroundings and replaces all contacts and supports with the forces and moments they exert. Key steps include:
- Identify the body of interest and isolate it
- Replace every support with its corresponding reaction forces
- Include all applied loads, weights, and distributed forces
- Pin support: two reaction components (Rx, Ry)
- Roller support: one reaction component perpendicular to the surface
- Fixed support: two force components and one moment (2D)
Truss Analysis: Method of Joints
Trusses consist of two-force members connected at joints. The method of joints analyzes equilibrium at each pin joint:
- Start at a joint with at most two unknowns
- Apply Sum of Fx = 0 and Sum of Fy = 0 at each joint
- Tension (pulling away from joint) and compression (pushing toward joint) are identified by sign
- Proceed joint by joint until all member forces are determined
Truss Analysis: Method of Sections
When only a few member forces are needed, the method of sections is more efficient:
- Cut through the truss, exposing no more than three unknowns
- Draw FBD of one portion of the cut truss
- Apply three equilibrium equations to solve for the cut member forces
- Choose moment points strategically to eliminate unknowns
Friction
Dry (Coulomb) friction opposes the tendency of sliding between surfaces in contact:
- Maximum static friction: F_s = mu_s * N (coefficient of static friction times normal force)
- Kinetic friction: F_k = mu_k * N (applies once motion begins, mu_k < mu_s)
- Friction angle: phi = arctan(mu)
- Wedge problems: apply equilibrium to the wedge and the object separately
- Belt friction: T_tight = T_slack * e^(mu*beta), where beta is the contact angle in radians
Centroids and Moments of Inertia
The centroid locates the geometric center of an area, and the moment of inertia quantifies resistance to bending:
- Centroid: x_bar = Sum(x_i * A_i) / Sum(A_i)
- Rectangle moment of inertia about centroidal axis: I = bh^3/12
- Circle moment of inertia: I = pi*r^4/4
- Triangle moment of inertia about base: I = bh^3/12; about centroidal axis: I = bh^3/36
- Parallel axis theorem: I = I_c + A*d^2 (d = distance from centroidal axis to new axis)
- Composite shapes: add or subtract individual moments of inertia using the parallel axis theorem