Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Define and explain stress, strain, and elasticity.
• Calculate stress and strain from applied forces.
• Apply Hooke's Law.
• Calculate and explain the Factor of Safety.
• Identify different types of structural loads.

Lesson summary

Overview: This week, we begin our exploration of applied mechanics and stability in civil structures. This is a fundamental topic in Civil Technology because it explains why buildings, bridges, and other structures stand up and remain safe. Applied mechanics deals with the forces acting on structures and how they behave under those forces. Stability, in turn, describes the ability of a structure to resist overturning, collapsing, or becoming unstable under load. Understanding these principles is crucial for any aspiring civil technician or engineer. Think about the informal settlements in South Africa, where inadequate construction practices can lead to unstable and dangerous structures.

Lesson notes

2.1 Stress, Strain, and Elasticity Stress: Stress is the internal resistance offered by a material to an externally applied force. It's essentially the force acting per unit area within the material. We represent stress as σ (sigma).

Formula: σ = F/A, where: σ = Stress (typically in Pascals (Pa) or N/m², or Megapascals (MPa)) F = Applied Force (in Newtons, N) A = Cross-sectional Area (in square meters, m²)

Types of Stress: Tensile Stress: Occurs when a material is pulled or stretched (tension).

Compressive Stress: Occurs when a material is pushed or compressed.

Shear Stress: Occurs when a force acts parallel to the surface, causing one part of the material to slide relative to another.

Strain: Strain is the deformation of a material caused by stress. It's a measure of how much the material has changed in length or shape relative to its original dimensions. We represent strain as ε (epsilon).

Formula: ε = ΔL/L₀, where: ε = Strain (dimensionless, often expressed as mm/mm or %) ΔL = Change in Length (in meters, m) L₀ = Original Length (in meters, m) Strain is dimensionless because it's a ratio of two lengths. It represents the amount of deformation per unit of original length.

Elasticity: Elasticity is the ability of a material to return to its original shape and size after the applied force is removed. A material is considered elastic as long as the stress applied is within the elastic limit.

Elastic Limit: The maximum stress a material can withstand without permanent deformation. Beyond this limit, the material will exhibit plastic deformation.

Plastic Deformation: Permanent deformation that remains after the applied force is removed. 2.2 Hooke's Law Hooke's Law states that the stress within an elastic material is directly proportional to the strain, provided the elastic limit is not exceeded. This relationship is crucial for understanding how materials respond to forces.

Formula: σ = Eε, where: σ = Stress (in Pa or MPa) E = Young's Modulus (or Modulus of Elasticity) (in Pa or MPa) – A material property representing its stiffness. ε = Strain (dimensionless) 2.3 Factor of Safety (FoS) The Factor of Safety is a design parameter used to ensure that a structure can withstand loads significantly greater than those anticipated in normal use. It's a ratio of the material's ultimate strength (or yield strength) to the actual stress experienced by the structure.

Formula: FoS = Ultimate Strength / Working Stress Ultimate Strength: The maximum stress a material can withstand before failure.

Yield Strength: The stress at which a material begins to deform plastically.

Working Stress: The actual stress experienced by the structure under normal operating conditions. A higher Factor of Safety indicates a more conservative design, providing a larger margin of safety. Factors of Safety vary depending on the application and the potential consequences of failure. For critical structures like bridges, a higher FoS is used than for less critical components. 2.4 Types of Loads Dead Loads: These are permanent, unchanging loads on a structure. They include the weight of the structure itself (walls, floors, roof), as well as any permanent fixtures like air conditioning units or cladding.

Live Loads: These are variable loads that can change over time. They include the weight of people, furniture, equipment, and stored materials. Live loads are typically estimated based on building codes and occupancy types. In South Africa, SANS 10160 provides guidelines for determining live loads.

Wind Loads: These are forces exerted by the wind on a structure. Wind loads are particularly important in tall buildings and structures located in windy areas. The magnitude of wind loads depends on factors such as wind speed, building shape, and terrain.

Other Loads: Structures can also be subjected to other loads such as earthquake loads (seismic loads), snow loads, and hydrostatic pressure. 2.5 Center of Gravity (CG) The Center of Gravity is the point at which the entire weight of an object can be considered to be concentrated. For a uniform object, the CG is located at its geometric center. The location of the CG is crucial for determining the stability of a structure.

Stability and CG: A structure is more stable if its CG is located lower and within its base of support. If the CG falls outside the base of support, the structure will tend to topple over.

Example 1: Calculating Stress and Strain A steel rod with a diameter of 20 mm is subjected to a tensile force of 50 kN. Calculate the stress and strain in the rod, assuming the Young's Modulus for steel is 200 GPa.

Solution: Calculate the cross-sectional area (A): A = πr² = π(10 mm)² = π(0.01 m)² = 3.142 x 10⁻⁴ m² Calculate the stress (σ): σ = F/A = (50 x 10³ N) / (3.142 x 10⁻⁴ m²) = 159.15 x 10⁶ Pa = 159.15 MPa Calculate the strain (ε): σ = Eε => ε = σ/E = (159.15 x 10⁶ Pa) / (200 x 10⁹ Pa) = 0.000796 Example 2: Calculating Factor of Safety A concrete column is designed to support a load of 500 kN.