Overview
This lecture covers the concept of force vectors, how to represent and resolve them into components, and the process of vector addition, which is fundamental for analyzing forces in structures.
Force Vectors and Structural Analysis
- Forces are central to understanding stresses in structural members.
- A force is a vector, meaning it has both magnitude and direction.
- Structural safety is assessed by analyzing the forces and resulting stresses.
Representation and Components of Force Vectors
- A force vector can be drawn with a tail, head, and length proportional to its magnitude.
- The line of action extends infinitely through the head and tail of a vector.
- The point of application indicates where the force acts on a structure.
- Vectors in the XY plane can be represented by X and Y components, forming a right triangle.
- The vector’s magnitude and direction allow calculation of component magnitudes using sine and cosine.
Determining Components Using Trigonometry
- For a force F at angle θ with the x-axis:
FX = F × cos(θ); FY = F × sin(θ).
- Example: A 25 kN force at 36.87° has components FX = 20 kN, FY = 15 kN.
- The Pythagorean theorem relates the magnitude of the vector and its components.
Negative Magnitude and Vector Direction
- A negative magnitude indicates the vector is in the opposite direction.
- Reverse the direction and use the positive magnitude to find components as usual.
Finding Magnitude and Direction from Components
- Given FX and FY, form a right triangle to find the resultant vector.
- The magnitude is F = √(FX² + FY²).
- The direction angle θ = arctan(FY/FX).
Vector Addition
- To add multiple vectors, convert each to X and Y components.
- Add all X components for the resultant X; add all Y components for the resultant Y.
- The resultant vector’s magnitude and orientation are then found using the Pythagorean theorem and trigonometry.
Key Terms & Definitions
- Force Vector — A quantity with both magnitude and direction, representing force.
- Line of Action — The infinite line along which a force acts.
- Component Vectors — The projections of a vector along the coordinate axes.
- Resultant Vector — The single vector equivalent to the sum of two or more vectors.
Action Items / Next Steps
- Practice vector addition and subtraction problems as assigned.
- Review trigonometric relationships for resolving vectors into components.