Overview
This lecture introduces how to analyze force vectors in structures, including decomposition into components, calculating magnitudes and directions, and applying trigonometry to 2D and 3D vectors.
Purpose of Finding Stresses
- Stresses in structures are calculated to compare with material properties and ensure safety.
- If safe, a factor of safety can be reported (covered in-depth in Mechanics of Materials).
Force Vectors and Their Representation
- External loads cause reaction forces and internal member forces.
- Force vectors have magnitude (in units like newtons or pounds) and direction (usually in degrees).
- Vectors can be written as components along x, y, and z axes (e.g., 3i + 4j or Fx, Fy, Fz).
- Components are found using trigonometric functions (sine, cosine, tangent).
- Vector direction is typically referenced from the positive x-axis, increasing counterclockwise.
Vector Decomposition and Trigonometry
- To find vector components: Fx = F cos(θ), Fy = F sin(θ), but depends on given angle.
- For different reference angles, sine and cosine may swap roles for x and y components.
- Pythagorean theorem finds magnitude: |F| = √(Fx² + Fy²).
- Direction found with tangent: θ = arctan(Fy / Fx), but adjust for quadrant location.
Working with Multiple Vectors
- When adding vectors, add x and y components separately.
- The resultant vector's magnitude and direction are found using the same trigonometric principles.
- If resulting vector is zero, each component sum must also be zero.
- Negative component values indicate opposite direction.
3D Force Vectors
- 3D vectors decompose with two angles and one magnitude.
- Use sine and cosine with corresponding angles to find Fx, Fy, Fz.
- Negative signs denote direction relative to axes.
- Example: F = 24i + 25√3j – 7k.
Key Terms & Definitions
- Force Vector — A quantity with both magnitude and direction, representing a force in space.
- Component — The projection of a vector along an axis (e.g., Fx, Fy, Fz).
- Pythagorean Theorem — Relates the squared length of the hypotenuse to the sum of squares of the other sides in a right triangle.
- Sine/Cosine/Tangent — Trigonometric functions used to relate angles to side lengths in right triangles.
- Arctangent (arctan) — Function used to find an angle from the ratio of two side lengths.
Action Items / Next Steps
- Review trigonometry basics (sine, cosine, tangent, Pythagorean theorem).
- Practice decomposing vectors into components and recombining them.
- Attempt example problems involving vector addition and determining resultant magnitudes/directions.