TLDR;
This video provides a detailed explanation of cantilever beams, covering theoretical concepts, experimental procedures, and mathematical derivations. It explains how to determine the depression of a cantilever beam under load, including the derivation of formulas for both rectangular and circular cross-sections.
- Introduction to Cantilever Beams
- Experimental Setup and Procedure
- Derivation of Depression Formula
- Special Cases for Rectangular and Circular Cross-Sections
Introduction to Cantilever Beams [0:04]
The video introduces the concept of a cantilever, which is a beam fixed at one end and loaded at the other. It explains that when a load is applied to the free end of the cantilever, it bends, and there's a reaction force at the fixed end acting in the opposite direction. The video also mentions the importance of understanding equilibrium conditions, where the internal bending moment equals the external bending moment.
Experimental Setup and Procedure [3:19]
The video describes a practical experiment to measure the depression of a cantilever beam. A needle is fixed with a clamp, and a microscope is used to observe the needle's movement as weight is added. The process involves adding weight in increments (e.g., 50g, 100g, 150g) and noting the microscope readings at each increment. The difference in readings (depression) is then calculated.
Derivation of Depression Formula [5:21]
The video explains the derivation of the formula for the depression of a cantilever beam. It starts by defining key parameters such as the length of the beam (L) and the distance from the fixed end to a point (P) on the beam (x). It considers a small section of the beam (dx) and uses geometric relationships to find the radius of curvature (R) and the angle dθ. By equating the internal and external bending moments, the video derives the equation for dθ and integrates it to find the total depression (y).
Special Cases for Rectangular and Circular Cross-Sections [22:41]
The video discusses two specific cases for the cross-sectional shape of the cantilever beam: rectangular and circular. For a rectangular cross-section, the formula for the depression (y) is modified by substituting the appropriate value for the area moment of inertia (I). Similarly, for a circular cross-section, the formula is adjusted using the corresponding value for I. The video concludes by noting that the derived formulas can be used to find the modulus of elasticity (y) if the depression is known from experimental measurements.