CIV ENG 2P04: Lecture 3 - Force & Position Vectors Part ii

CIV ENG 2P04: Statics and Strength of Materials - Lecture 3 (CH2: Force & Position Vectors Part ii)

Course Overview and Structure

• Statics (Weeks 1 to 6)

  • CH: 1, 2, 3: Particle Equilibrium, Force Vectors, Position Vectors, 3D Force Vectors

  • CH: 4: Equilibrium of Rigid Bodies, Static (in)determinacy, Stability, Free Body Diagrams, Force System Resultants

  • CH: 5: Centre of Gravity, Centroids & Moment of Inertia

  • CH: 6: Structural Analysis, Internal Forces, Analysis of Trusses

  • Real-world example: Floor beam, stringer, deck in a structure (e.g., bridge or building).

• Mechanics of Materials (Weeks 7 to 13)

  • CH: 7: Stress & Strain

  • CH: 8: Mechanical Properties of Materials

  • CH: 9: Axial Load

  • CH: 10: Torsion

    • Shear stress & strain.

    • Torsion of solid and hollow circular sections.

  • CH: 11: Bending

  • CH: 12: Transverse Shear

  • CH: 13: Combined Loading

  • CH: 14: Stress Transformation

    • Principal stresses.

    • Mohr's circle representation.

Learning Outcomes for This Chapter

By the end of this chapter, students should be able to:

• Add forces and resolve them into components.

• Express force and position as Cartesian vectors.

• Use the dot product to find:

  • The angle between two vectors.

  • The projection of one vector onto another.

Lecture Outline

• Cartesian Vectors

• Addition of Cartesian Vectors

• Position Vectors

• Force Vectors Directed Along a Line

• Dot Product

Cartesian Vectors (Section 2.5)

A) Previous Knowledge: Vector Operations Review

• When asked to determine the magnitude of a resultant force, one cannot always use techniques for coplanar forces if the forces are in three dimensions.

• Difference: Coplanar forces lie in a single plane, while 3D forces operate in three-dimensional space, requiring different vector resolution and addition methods.

B) Rectangular Components of a Vector

• A general vector $extbf{A}$ in 3D space will have three rectangular components along the x, y, and z coordinate axes: $extbf{A}<em>x, extbf{A}</em>y, extbf{A}_z$.

• These components are determined by successive applications of the parallelogram law:

  • First application: $extbf{A} = extbf{A}' + extbf{A}_z$ (where $extbf{A}'$ is the projection of $extbf{A}$ on the xy-plane).

  • Second application: $extbf{A}' = extbf{A}<em>x + extbf{A}</em>y$.

• Combining these, vector $extbf{A}$ is represented by the vector sum of its three rectangular components:

  • $extbf{A} = extbf{A}<em>x + extbf{A}</em>y + extbf{A}_z$

C) Cartesian Vector Representation

• In three dimensions, the Cartesian unit vectors $extbf{i}, extbf{j}, extbf{k}$ are used to designate the directions of the x, y, z axes, respectively.

• The three components of $extbf{A}$ can be written in Cartesian vector form as:

  • $extbf{A} = A<em>x extbf{i} + A</em>y extbf{j} + A_z extbf{k}$

• Advantage: This form simplifies vector addition, subtraction, dot products, and cross products by allowing component-wise operations.

D) Magnitude of a Cartesian Vector

• If $extbf{A}$ is expressed as a Cartesian vector $(A<em>x extbf{i} + A</em>y extbf{j} + A_z extbf{k})$, its magnitude ($A$) can be determined using the Pythagorean theorem in 3D:

  • $A = rac{ rac{ rac{}{ rac{ rac{ rac{}{(A<em>x)^2 + (A</em>y)^2 + (A_z)^2}}}}}}$

E) Coordinate Direction Angles

• The direction or orientation of vector $extbf{A}$ is defined by the angles $extrm{\alpha}$, $extrm{\beta}$, and $extrm{\gamma}$.

• These angles are measured between the vector $extbf{A}$ and the positive X, Y, and Z axes, respectively.

• Their range of values is from $0^ extrm{\circ}$ to $180^ extrm{\circ}$ ($0 extrm{ rad}$ to $extrm{\pi rad}$).

• Using trigonometry,