Overview
This lecture provides a comprehensive review of key statics concepts for the FE exam, focusing on equilibrium, vector components, distributed loads, moments, trusses, centroids, moments of inertia, and friction, with step-by-step example problems.
Principles of Statics and Equilibrium
- Statics involves ensuring all forces and moments (rotations) acting on a body are balanced so the body remains at rest.
- Equilibrium equations:
- Sum of forces in the x-direction = 0 (ΣFₓ = 0).
- Sum of forces in the y-direction = 0 (ΣFᵧ = 0).
- Sum of moments about a point = 0 (ΣM = 0).
- Use vector components and basic trigonometry (sine, cosine) to resolve and sum forces.
Resultant Forces and Vector Components
- Break forces into x- and y-components using trig functions:
- Fₓ = F × cos(θ); Fᵧ = F × sin(θ).
- Add force components to find the resultant vector's magnitude using the Pythagorean theorem.
Distributed Loads and Moments
- Replace distributed loads with equivalent point loads:
- Rectangular: F = w × L (area under the curve).
- Triangular: F = 0.5 × w × L.
- Locate resultants at centroids:
- Rectangle: L/2 from one end.
- Triangle: 2L/3 from the smaller end.
- Calculate moments as force × perpendicular distance from point.
Equivalent Moments and Force Systems
- Balanced vertical forces do not always mean moments are balanced.
- Find the equivalent moment created by multiple forces by summing moments about a chosen point.
Beam Reactions and Free Body Diagrams
- Use free body diagrams (FBDs) to identify all forces and reactions on a structure.
- Apply equilibrium equations to solve for unknown reactions, ensuring consistent sign conventions.
- Resultant reaction at a support found by combining horizontal and vertical components.
Cables, Trusses, and Frames
- Cables and truss members carry force only along their length (axial tension or compression).
- Use geometry (similar triangles) and trigonometry to relate force components.
- Trusses: For method of joints, apply ΣFₓ = 0 and ΣFᵧ = 0 at the joint to solve for member forces.
Centroids
- The centroid (center of area) of a composite shape can be found using:
- ȳ = (ΣyA) / (ΣA), where y is the distance from reference and A is the area.
Moment of Inertia (Second Moment of Area)
- For standard shapes, use formulas (rectangle: I = bh³/12, circle: I = πr⁴/4).
- For composite shapes, use the parallel axis theorem if centroids do not align:
- I_total = I_centroid + A × d², where d = distance between centroids.
Friction
- Maximum friction force: F_friction = µ × N, where µ is the coefficient of friction and N is the normal force.
- In multi-block or inclined plane problems, resolve forces parallel and perpendicular to the contact surface.
Key Terms & Definitions
- Moment — The tendency of a force to rotate an object about a point (force × distance).
- Centroid — The geometric center or average position of an area or shape.
- FBD (Free Body Diagram) — A diagram showing all external forces acting on a body.
- Resultant Force — A single force representing the vector sum of multiple forces.
- Parallel Axis Theorem — Used to calculate the moment of inertia about any axis, not just the centroid.
Action Items / Next Steps
- Review equilibrium equations and practice drawing FBDs.
- Practice breaking forces into components and using Pythagorean theorem for resultants.
- Work example problems involving distributed loads, moments, and truss analysis.
- Review centroid and moment of inertia formulas; know how to find them in the reference manual.
- Practice friction problems and using calculators for system equations.
- Prepare for next week's topic: Structural Analysis.