Tapered cantilever beam deflection and stress distribution

In summary, the conversation discusses a beam with a positive moment of 12000Nmm at the tip, applied as two equal and opposite point loads in the x direction at points C and D. The beam is fully fixed along AB and has a Young's modulus of 200*(10^3) MPa and a Poisson's ratio of 0.3. The beam's geometric properties include a width of 10mm, a length of 200mm, and depths of 36mm at AB and 12mm at CD. The goal is to use provided formulas to confirm a theoretical tip displacement of 0.3241mm and calculate the σ_xx stress distribution along the top edge of the beam. The provided document shows
  • #1
Epiktetos
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Homework Statement


A positive moment of 12000Nmm at the tip applied as two equal and opposite point loads of 1000N in the x direction at C and D.
The beam is fully fixed along AB. Young's modulus E = 200*(10^3) MPa and the poisson's ratio = 0.3.
Geometric Properties: Width, b=10mm... Length, L=200mm
Depth, d at AB = 36mm
d at CD = 12mm
Using the formulae provided, confirm that the theoretical tip displacement of the beam is 0.3241mm. In addition, calculate the σ_xx stress distribution along the top edge of the beam.

(see attached image)

Homework Equations


I = [b*{36-(48x/L)}^3]/12 for x≤ L/2 and I'=[b*(12^3)]/12 for x> L/2
v=1/E ( ∫∫M(x)I(x)dx dx ) the integration for both integrals is from 0 to L
σ_xx = (-My)/I



The Attempt at a Solution


I attached my attempted solution as a document.
 

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  • #2
kindly, what is the analysis of the load boundary condition in the problem specification.and the stress distribution along the top edge of the beam
 
  • #3
Necropost. OP is more than a year old.
 

1. What is a tapered cantilever beam?

A tapered cantilever beam is a structural element that is supported at one end and has a gradually decreasing cross-section along its length. This type of beam is commonly used in engineering and construction to support loads and resist bending.

2. How is deflection calculated for a tapered cantilever beam?

The deflection of a tapered cantilever beam can be calculated using the Euler-Bernoulli beam theory, which takes into account the beam's length, modulus of elasticity, moment of inertia, and applied load. There are also various online calculators and software programs that can be used to determine deflection for specific beam designs.

3. What factors affect the stress distribution in a tapered cantilever beam?

The stress distribution in a tapered cantilever beam is affected by the beam's geometry, material properties, and applied load. The taper angle, length, and cross-sectional dimensions of the beam can all impact the stress distribution. Additionally, the type of material used and the magnitude and location of the load can also affect stress distribution.

4. How does the taper angle of a cantilever beam affect its deflection and stress?

The taper angle of a cantilever beam has a significant impact on its deflection and stress. A steeper taper angle can result in a higher deflection and stress at the tip of the beam, while a shallower taper angle can distribute the stress more evenly along the length of the beam. The taper angle should be carefully considered in the design of a tapered cantilever beam to ensure structural stability and safety.

5. What are some common applications of tapered cantilever beams?

Tapered cantilever beams are commonly used in various engineering and construction applications, such as bridges, buildings, and aerospace structures. They are also used in the design of machine components, such as conveyor belts and crane arms, as well as in the manufacturing of sports equipment, such as golf clubs and ski poles.

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