Cantilever beam - Cosinusoidal distributed load Calculator

Cantilever Beam - Cosinusoidal Distributed Load

A cantilever beam is a structural element fixed at one end and free at the other. When a cosinusoidal distributed load is applied along the length of the beam, the load intensity varies following a cosine function. This type of loading commonly occurs in aerodynamic, wave, and vibration-related applications.

Key Concepts

Cosinusoidal Distributed Load: A varying load described by \( w(x) = w_0 \cos\left(\frac{\pi x}{L}\right) \), where \( w_0 \) is the maximum load intensity at \( x = 0 \), and \( L \) is the beam length.
Fixed End: The end rigidly attached to a support, resisting both rotation and translation.
Free End: The unsupported end where the load intensity is minimum.
Shear Force: Varies non-linearly along the beam due to the cosine distribution of the applied load.
Bending Moment: The moment varies along the beam, with the maximum occurring at the fixed end.
Deflection: The displacement of the beam, maximum at the free end, influenced by the cosinusoidal loading pattern.

Behavior of the Cantilever Beam

Reaction Forces:
- At the fixed end, the beam generates a reaction force \( R_A \) equal to the total equivalent force of the cosinusoidal load.
- A reaction moment \( M_A \) is developed to balance the distributed load.
Shear Force Diagram:
- The shear force is obtained by integrating the cosinusoidal load distribution.
- It varies along the beam, reaching its maximum at the fixed end.
Bending Moment Diagram:
- Calculated by integrating the shear force distribution.
- The maximum bending moment occurs at the fixed end, and it follows a complex non-linear distribution.
Deflection: Maximum at the free end, derived using beam deflection formulas for varying loads.

Applications

Aerodynamics: Modeling aerodynamic loads on wings and blades.
Wave Loading: Used in offshore structures subject to wave-induced forces.
Structural Engineering: Analysis of beams subjected to gradually varying loads.

Formula

Quantity	Formula
Deflection \(y_{AB}\)	\[ y_{AB} = \frac{-w_0 L}{3\pi^4 EI} \left( 48L^3 \cos \frac{\pi x}{2L} - 48L^3 + 3\pi^3 Lx^2 - \pi^3 x^3 \right) \]
Maximum Deflection \(y_{MAX}\)	\[ y_{MAX} = \frac{-2w_0 L^4}{3\pi^4 EI} (\pi^3 - 24) \quad \text{at } x = L \]
Slope \(\theta_{AB}\)	\[ \theta_{AB} = \frac{-w_0 L}{\pi^3 EI} \left( 2\pi^2 Lx - \pi^2 x^2 - 8L^2 \sin \frac{\pi x}{2L} \right) \]
Slope at B \(\theta_B\)	\[ \theta_B = \frac{-w_0 L^3}{\pi^3 EI} (\pi^2 - 8) \]
Moment \(M_{AB}\)	\[ M_{AB} = \frac{-2w_0 L}{\pi^2} \left( \pi L - \pi x - 2L \cos \frac{\pi x}{2L} \right) \]
Shear \(V_{AB}\)	\[ V_{AB} = \frac{2w_0 L}{\pi} \left( 1 - \sin \frac{\pi x}{2L} \right) \]
Reactions \(R_A\)	\[ R_A = \frac{2w_0 L}{\pi} \]

Definitions

Symbol	Physical quantity	Units
E·I	Flexural rigidity	N·m², Pa·m⁴
y	Deflection or deformation	m
θ	Slope, Angle of rotation	-
x	Distance from support (origin)	m
L	Length of beam (without overhang)	m
M	Moment, Bending moment, Couple moment applied	N·m
P	Concentrated load, Point load, Concentrated force	N
w	Distributed load, Load per unit length	N/m
R	Reaction load, reaction force	N
V	Shear force, shear	N