Engineering Mechanics: Statics, Equilibrium, and Beam Analysis

Fundamentals of Mechanics and Units

• Newton's Laws of Motion:
  • First Law: A particle originally at rest, or moving in a straight line with constant velocity, remains in this state unless acted upon by an unbalanced force.
  • Second Law: A particle acted upon by an unbalanced force $\mathbf{F}$ experiences an acceleration $\mathbf{a}$ in the direction of the force with magnitude proportional to the force:     $\mathbf{F} = m\mathbf{a}$
  • Third Law: Mutual forces of action and reaction between two interacting particles are equal in magnitude, opposite in direction, and collinear.
• International System of Units (SI):
  • Length: Metre ($\text{m}$
  • Time: Second ($\text{s}$
  • Mass: Kilogram ($\text{kg}$
  • Force: Newton ($\text{N} = \frac{\text{kg}\cdot\text{m}}{\text{s}^2}$
• Standard Unit Prefixes:
  • Giga ($\text{G}$): $10^9$
  • Mega ($\text{M}$): $10^6$
  • Kilo ($\text{k}$): $10^3$
  • Milli ($\text{m}$): $10^{-3}$
  • Micro ($\mu$): $10^{-6}$
  • Nano ($\text{n}$): $10^{-9}$

Force Vectors and Component Resolution

• Cartesian Vector Representation:
  • Uses orthogonal dimensionless unit vectors $\mathbf{i}$ and $\mathbf{j}$ along the $x$ and $y$ axes:     $\mathbf{F} = F_x\mathbf{i} + F_y\mathbf{j}$
• Resolution of Coplanar Forces:
  • Calculated via direction angle $\theta$:     $F_x = F\cos(\theta)$$F_y = F\sin(\theta)$
  • Calculated via slope ratios ($a:b:c$):     $F_x = F\left(\frac{a}{c}\right)$$F_y = F\left(\frac{b}{c}\right)$
• Resultant Force of Concurrent Coplanar Forces:
  • Component sums:     $F_{Rx} = \sum F_x$$F_{Ry} = \sum F_y$
  • Resultant magnitude:     $F_R = \sqrt{F_{Rx}^2 + F_{Ry}^2}$
  • Direction angle measured from the reference axis:     $\theta = \tan^{-1}\left(\frac{|F_{Ry}|}{|F_{Rx}|}\right)$

Particle Equilibrium and Free-Body Diagrams (FBD)

• Equilibrium Conditions for a Particle:
  • Net resultant force acting on a stationary or uniformly moving particle is zero:     $\sum F_x = 0$$\sum F_y = 0$
• Free-Body Diagram (FBD) Requirements:
  • Isolate the system or point from its physical surroundings.
  • Identify all external forces: active loads, applied weights ($W = mg$), contact reactions (normal and friction forces), and cable tensions.
  • Set up a reference coordinate frame to resolve oblique vectors.
• Linear Elastic Spring Force:
  • Governed by Hooke's Law:     $F = ks$
  • Parameter $k$ represents spring stiffness ($\text{N/m}$) and $s = l - l_0$ represents elongation or deformation from unstretched length $l_0$.

Moments and Rotational Tendency

• Scalar Moment Calculation:
  • Measures the rotational tendency of a force about a specific point or pivot $O$:     $M_O = Fd$
  • $d$ is the perpendicular moment arm between the point $O$ and the force's line of action.
• Sign Convention and Direction:
  • Direction is perpendicular to the force-arm plane following the right-hand rule.
  • Standard 2D convention treats counterclockwise rotation as positive and clockwise rotation as negative.
• Resultant Moment of a Coplanar Force System:
  • Algebraic sum of individual moments:     $M_{RO} = \sum (Fd)$

Rigid Body Equilibrium and Support Reactions

• Equilibrium Equations in Two Dimensions:
  • Completely prevents translation and rotation:     $\sum F_x = 0$$\sum F_y = 0$$\sum M_O = 0$
• Support Types and Reactive Unknowns:
  • Cable / Weightless Link: 1 unknown; directed in tension along the line of the cable/link.
  • Roller / Rocker / Smooth Surface: 1 unknown; contact force acting perpendicular to the surface.
  • Smooth Pin or Hinge: 2 unknowns; two orthogonal force components ($A_x$, $A_y$).
  • Fixed Connected Collar on Smooth Rod: 2 unknowns; force perpendicular to the rod and a resisting couple moment $M$.
  • Fixed Support: 3 unknowns; two orthogonal force components ($A_x$, $A_y$) and a resisting couple moment ($M_A$).

• Reduction of Distributed Loads:
  • Equivalent Resultant Force ($F_R$): Equal to the total area underneath the distributed loading diagram $w(x)$.
  • Uniform load of intensity $w$ over span $L$:       $F_R = wL$
  • Triangular load from $0$ to peak intensity $w$ over span $L$:       $F_R = \frac{1}{2}wL$
  • Line of Action ($\bar{x}$): Passes through the geometric centroid of the loading curve.
  • Triangle centroid is situated at $\frac{2}{3}L$ from the zero-load vertex ($\frac{1}{3}L$ from the maximum base).

Internal Loadings and Beam Diagrams (SFD & BMD)

• Components of Internal Loading:
  • Normal Force ($N$): Acts perpendicular to the cut cross section.
  • Shear Force ($V$): Acts tangent to the cut cross section.
  • Bending Moment ($M$): Couple moment resisting bending at the cross section.
• Internal Sign Convention:
  • Normal Force: Positive in tension (directed outward from the section).
  • Shear Force: Positive when causing clockwise rotation of the beam element (directed downward on the right-hand cut, upward on the left-hand cut).
  • Bending Moment: Positive when causing sagging / concave-upward curvature (compression at top, tension at bottom fibers).

• Method of Sections for Constructing Diagrams:
  • Find external support reactions using global rigid body equilibrium.
  • Establish position coordinates $x$ from the left end across discrete intervals separated by point loads, moments, or distributed load discontinuities.
  • Section beam at location $x$, apply positive internal forces to the free-body diagram, and solve equilibrium equations to establish equations for $V(x)$ and $M(x)$.
  • Plot Shear Force Diagrams (SFD) and Bending Moment Diagrams (BMD); locations where the shear force passes through zero ($V = 0$) define points of local maximum or minimum bending moment.