Engineering Mechanics: Statics

• Engineering mechanics is the application of mechanics to solve problems involving common engineering elements.
• It is fundamental for the design of many things, including bridges, dams, and vehicles.
• Engineering mechanics generally encompasses:
  • Statics: The study of bodies at rest under the action of forces.
  • Dynamics: The study of the motion of bodies under the action of forces.

Statics

• Statics deals with bodies in equilibrium.
• Equilibrium is the state where the net force and net moment acting on a body are zero.
• Key concepts in statics:
  • Forces: A push or pull acting on a body.
  • Moments: The tendency of a force to cause rotation about a point.
  • Free Body Diagrams (FBDs): A diagram showing all forces and moments acting on a body.
  • Equilibrium Equations: Mathematical expressions enforcing the conditions of equilibrium (sum of forces = 0, sum of moments = 0).
• Common types of forces:
  • Gravitational force (weight).
  • Normal force (perpendicular to a surface).
  • Frictional force (opposing motion along a surface).
  • Tension (force through a cable or rope).
  • Spring force (related to spring stiffness and displacement).
• Static analysis involves:
  • Identifying all forces acting on a body.
  • Drawing a free body diagram.
  • Applying equilibrium equations to solve for unknown forces or moments.

Dynamics

• Dynamics deals with bodies in motion.
• Key concepts in dynamics:
  • Kinematics: The study of motion without considering the forces causing it (displacement, velocity, acceleration).
  • Kinetics: The study of the relationship between forces and motion (Newton's laws of motion).
  • Work and Energy: Concepts related to the energy of a system and its transformation.
  • Impulse and Momentum: Concepts related to changes in motion due to forces acting over time.
• Newton's Laws of Motion:
  • First Law (Law of Inertia): A body at rest stays at rest, and a body in motion stays in motion with the same velocity unless acted upon by a net force.
  • Second Law: The net force acting on a body is equal to the mass of the body times its acceleration (F = ma).
  • Third Law: For every action, there is an equal and opposite reaction.
• Types of motion:
  • Rectilinear motion (motion along a straight line).
  • Curvilinear motion (motion along a curved path).
  • Rotational motion (motion about an axis).
• Dynamic analysis involves:
  • Determining the forces acting on a body.
  • Applying Newton's laws to determine the body's acceleration.
  • Using kinematic equations to determine the body's position and velocity as a function of time.

Strength of Materials

• Strength of materials (also known as mechanics of materials) is the study of the behavior of solid materials under stress and strain.
• It focuses on the internal stresses and strains within a material caused by external loads.
• It is crucial for the design of structural components that can withstand applied loads without failure.

Stress

• Stress is the internal force per unit area within a material.
• It is a measure of the intensity of internal forces acting within a deformable body.
• Types of stress:
  • Normal stress (tensile or compressive): Force acting perpendicular to the area.
  • Shear stress: Force acting parallel to the area.
  • Bending stress: Stress caused by bending moments.
  • Torsional stress: Stress caused by twisting moments.
• Stress is usually expressed in units of Pascals (Pa) or pounds per square inch (psi).

Strain

• Strain is the deformation of a material caused by stress.
• It is a measure of the deformation of a body.
• Types of strain:
  • Normal strain: Change in length per unit length.
  • Shear strain: Change in angle.
• Strain is a dimensionless quantity.

Material Properties

• Key material properties used in strength of materials:
  • Elasticity: The ability of a material to return to its original shape after the removal of stress.
  • Young's modulus (E): A measure of a material's stiffness, relating stress and strain in the elastic region (stress = E * strain).
  • Poisson's ratio (ν): A measure of how much a material deforms in one direction when subjected to stress in a perpendicular direction.
  • Yield strength: The stress at which a material begins to deform permanently.
  • Ultimate tensile strength: The maximum stress a material can withstand before breaking.
  • Ductility: A measure of a material's ability to deform plastically before fracture.
  • Brittleness: The tendency of a material to fracture with little or no plastic deformation.
  • Hardness: Resistance to localized plastic deformation.
• Stress-Strain Diagram: A graphical representation of the relationship between stress and strain for a particular material.

Axial Loading

• Axial loading refers to a force applied along the longitudinal axis of a member.
• It results in either tensile stress (if the force is pulling) or compressive stress (if the force is pushing).
• Axial deformation (elongation or contraction) can be calculated using the formula: ΔL = (PL) / (AE), where:
  • ΔL is the change in length.
  • P is the applied axial force.
  • L is the original length.
  • A is the cross-sectional area.
  • E is Young's modulus.

Torsion

• Torsion refers to the twisting of a member due to an applied torque (twisting moment).
• Torsional shear stress is maximum at the outer surface of a circular shaft.
• Angle of twist (φ) can be calculated using the formula: φ = (TL) / (GJ), where:
  • φ is the angle of twist in radians.
  • T is the applied torque.
  • L is the length of the shaft.
  • G is the shear modulus of the material.
  • J is the polar moment of inertia of the cross-section.

Bending

• Bending refers to the deformation of a member due to an applied bending moment.
• Bending stress varies linearly with distance from the neutral axis.
• The flexure formula relates bending stress (σ) to bending moment (M), distance from the neutral axis (y), and moment of inertia (I): σ = (My) / I.
• Deflection refers to the displacement of a beam under bending. Deflection calculations depend on the loading conditions and beam supports.

Shear Force and Bending Moment Diagrams

• Shear force and bending moment diagrams are graphical representations of the internal shear force and bending moment along the length of a beam.
• These diagrams are essential for determining the maximum shear force and bending moment, which are used to calculate stresses and deflections.
• Key relationships:
  • The slope of the shear force diagram is equal to the negative of the distributed load.
  • The slope of the bending moment diagram is equal to the shear force.
  • The area under the shear force diagram is equal to the change in bending moment.

Combined Stresses

• Combined stresses occur when a member is subjected to multiple types of loading simultaneously (e.g., axial load and bending moment).
• The principal stresses and maximum shear stress can be determined using stress transformation equations or Mohr's circle.
• Failure theories are used to predict when a material will fail under combined stresses.
• Common failure theories include:
  • Maximum principal stress theory.
  • Maximum shear stress theory (Tresca criterion).
  • Distortion energy theory (von Mises criterion).

Columns

• Columns are vertical structural members subjected to axial compressive loads.
• Buckling is a phenomenon where a column suddenly bends or deflects laterally under a critical load.
• Euler's formula is used to calculate the critical buckling load (Pcr) for a slender column: Pcr = (π²EI) / (Le²), where:
  • E is Young's modulus.
  • I is the minimum moment of inertia of the cross-section.
  • Le is the effective length of the column, which depends on the end conditions.
• Effective length (Le) accounts for how the end supports affect the column's buckling behavior.