Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Calculate bending moments and shear forces for simply supported beams.
• Draw and interpret Bending Moment and Shear Force Diagrams (BMD & SFD).
• Explain the relationship between load, shear force, and bending moment.
• Apply these principles to assess simple beam designs.

Lesson summary

This week, we delve deeper into applied mechanics and stability in civil structures, focusing specifically on bending moments, shear forces, and their impact on structural integrity. This is crucial because, as future Civil Technologists, you will be responsible for designing and overseeing the construction of safe and durable infrastructure. From bridges spanning the Orange River to the high-rise buildings shaping our cities, understanding these principles is fundamental to preventing structural failures and ensuring public safety.

Lesson notes

2.1 Bending Moment (M): Bending moment is the internal moment induced in a structural element when an external force is applied that causes the element to bend. It's essentially the sum of the moments about a given point caused by all forces acting on one side of that point. The units are typically Newton-meters (Nm) or kilonewton-meters (kNm). A positive bending moment indicates sagging (tension at the bottom, compression at the top), while a negative bending moment indicates hogging (tension at the top, compression at the bottom). 2.2 Shear Force (V): Shear force is the internal force acting perpendicular to the longitudinal axis of a structural element, causing one part of the element to slide relative to the adjacent part. It's the algebraic sum of all vertical forces acting on one side of a section. The units are typically Newtons (N) or kilonewtons (kN). A positive shear force is generally considered to be upwards to the left of a section or downwards to the right. 2.3 Relationship between Load, Shear Force, and Bending Moment: These three are intrinsically linked. The load is the cause, the shear force is the first effect, and the bending moment is the second effect.

Mathematically: The shear force at any section is the integral of the load distribution up to that section. (V = ∫w dx, where w is the load per unit length) The bending moment at any section is the integral of the shear force up to that section. (M = ∫V dx) Therefore, the derivative of the bending moment is the shear force (V = dM/dx), and the derivative of the shear force is the load (w = dV/dx). This relationship is crucial for sketching BMDs and SFDs. 2.4 Sign Conventions: Consistent sign conventions are vital.

We will use the following: Shear Force: Upward force to the left of the section is positive. Downward force to the right of the section is positive.

Bending Moment: Sagging (tension at the bottom) is positive. Hogging (tension at the top) is negative. 2.5 Simply Supported Beam: A simply supported beam is a beam supported at both ends, allowing it to rotate freely. It's a fundamental structural element in many constructions. We'll focus on this type of beam initially. 2.6 Uniformly Distributed Load (UDL): A UDL is a load that is evenly spread across a length of the beam (e.g., the weight of a concrete slab resting on a beam). It is measured in N/m or kN/m.