Beam boundary condition problem

In summary, the beam boundary condition problem involves determining how a beam behaves under various constraints at its supports, which can significantly affect its deflection and stress distribution. Different boundary conditions, such as fixed, pinned, or free ends, lead to distinct mathematical formulations and solutions in structural analysis. Accurately modeling these conditions is crucial for predicting the performance of beams in engineering applications.
  • #1
Motorbiker
34
1
Homework Statement
derive the boundary conditions of a beam whichcarries masses and is supported by springs and dampers
Relevant Equations
equations are shown in my working
vibration question.png



In my working, I have drawn a free body diagram of the LHS of the beam, I obtained some equations but after that not sure what I need to do. Usuaully in vibration problems you are given BC's then you plug them into the general solution to obtain the frequency equation.

Please can you help me understand how to do this problem? I'm really struggling to understand how to do it.

vibration question.png


working.jpeg
 
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  • #2
Please use LaTex to render equations. A guide is in the lower left hand corner of your response window.
 
  • #3
Chestermiller said:
Please use LaTex to render equations. A guide is in the lower left hand corner of your response window.
I think I showed the equations in my working?
 
  • #4
Motorbiker said:
I think I showed the equations in my working?
It's unreadable.
 
  • #5
Chestermiller said:
It's unreadable.
I’ve attached another picture of my working, please let me know if you can read this one.

F3C4EF10-986E-4B71-A02C-B6325D4B5520.jpeg
 
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  • #6
It should be $$-ku-C\frac{du}{dt}-mg+Q=\frac{d^2u}{dt^2}$$where Q is the upward shear force exerted by the bar on the mass.
 
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  • #7
Chestermiller said:
It should be $$-ku-C\frac{du}{dt}-mg+Q=\frac{d^2u}{dt^2}$$where Q is the upward shear force exerted by the bar on the mass.
Thanks for this. Could you please explain how you got this equation? I want to understand this better.
 
  • #8
Motorbiker said:
Thanks for this. Could you please explain how you got this equation? I want to understand this better.
it is from the free body diagram on the mass. It is the force balance equation on the mass.
 
  • #9
Chestermiller said:
it is from the free body diagram on the mass. It is the force balance equation on the mass.

I'm guessing this equation applies to both sides?
Also how do we know that there is shear force acting on the beam?
 
  • #10
Please can I get someone help with this? I don't really understand the solution fully, where does the shear force come from?
 
  • #11
Motorbiker said:
Please can I get someone help with this? I don't really understand the solution fully, where does the shear force come from?
Are the masses attached to the end of the bar or not?
 
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  • #12
Chestermiller said:
Are the masses attached to the end of the bar or not?

Yes they are.
 
  • #13
Motorbiker said:
Yes they are.
The shear force makes sure that the mass moves with the bar.
 
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FAQ: Beam boundary condition problem

What is a beam boundary condition problem?

A beam boundary condition problem refers to the analysis of how a beam behaves under various constraints at its ends or along its length. These conditions can include fixed, simply supported, or cantilevered ends, which influence the beam's deflection, moment, and shear force distributions when subjected to loads.

What are the types of boundary conditions for beams?

The common types of boundary conditions for beams include: 1) Fixed (or clamped) ends, where both translation and rotation are restricted; 2) Simply supported ends, where translation is allowed but rotation is restricted; 3) Cantilevered ends, where one end is fixed and the other is free; and 4) Free ends, where no constraints are applied. Each type affects how the beam deforms and reacts to loads.

How do boundary conditions affect beam deflection?

Boundary conditions significantly affect beam deflection by determining how much a beam can bend under a load. Fixed ends will generally result in lower deflections compared to simply supported or cantilevered beams, as the constraints at the ends resist movement. The specific relationship between load, boundary conditions, and deflection can be analyzed using beam theory and differential equations.

What is the significance of boundary conditions in structural analysis?

Boundary conditions are crucial in structural analysis because they define the constraints and support conditions of a structure. They influence the internal forces, moments, and overall stability of the structure. Accurate representation of boundary conditions is essential for predicting how structures will behave under various load scenarios, ensuring safety and performance.

How are boundary conditions applied in beam analysis?

Boundary conditions are applied in beam analysis through mathematical models and equations that describe the beam's behavior. Engineers and scientists use methods such as the Euler-Bernoulli beam theory, finite element analysis, or numerical methods to incorporate boundary conditions into their calculations. These methods allow for the evaluation of deflections, stresses, and other critical factors in beam design and analysis.

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