Mechanics of materials -- deformation problem

In summary, the diagram is very difficult to read and it is not clear what is causing node N to move downwards. Without knowing more about the diagram, it is difficult to say if the method of superposition is correct.
  • #1
Yossi33
22
3
Homework Statement
find the deformation of the structure and find delta(N).
Relevant Equations
deformations equations
Motif-1.jpg

Hi, i'm struggling with that problem , i need to find the distance that point N went down.My way of thinking is that the structure is twice not statically determined because of the beam MN and beacuse of the left support which is also unnecessary in order for equilibrium. My 2 equations of deformation in order to find the variables are d(N)-d(M)=d(Lmn) and d(N) of the left beam equal to the d(N) of the right beam.
i thought to denote the force of the beam as N and then to divide to to each beam (pic 1) then i got the problem that i have a beam with no support and a force that causing it do go down, so i tought (pic 2 ) that the beam NM is equal to support there and got the problem that if there is a support it wont get down. in addition to that i cant figure out if the force that bending one beam is causing the torsion of the other , because the left doesnt resist that its only has a translational resistance. I wonder if my initial analysis is somehow correct and what other perspective there is to solve problems like this. thank you.
 
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  • #2
The posted diagram is very difficult to read.
What is causing node N to move downwards?
 
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  • #3
W the uniform load acts on the beam , that casues the member NM to stretch and as a result , bending down the upper structure.
Note 10 Jan 2023 (1)-1.jpg
Note 10 Jan 2023-1.jpg
 
  • #4
Thank you.
Is node M perfectly articulated in the three directions of the links converging at it?
If so, it seems that link TM can be removed with no consequences for our problem.
 
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  • #5
no , also in the diagram you can see that it is a half node and the upper structure TMS is continous . and i did some extra black there to emphasize the rigidity of the 90 degree.
.- edit - that is why i said that the structure is two times statically indeterminate.
 
  • #6
Have you studied the method of superposition, which states that the deflection at any point on the beam is equal to the resultant of the slopes or deflections at that point caused by each of the load acting separately?
 
  • #7
yes, that is why i stated that my first try was to compare the displacements of point M , and the solution is 2 equations of deformations because the structure is twice statically indeterminate but i try to solve that and i dont know if its right or no . can you tell me if its ok or if im in the right direction? , its in the pdf
 

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  • Note 11 Jan 2023.pdf
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FAQ: Mechanics of materials -- deformation problem

What is deformation in the context of mechanics of materials?

Deformation refers to the change in shape or size of a material body under the action of external forces. It can be elastic (temporary and reversible) or plastic (permanent and irreversible), depending on the material properties and the magnitude of the applied forces.

What are the types of deformation?

There are mainly two types of deformation: elastic deformation and plastic deformation. Elastic deformation is reversible, meaning the material returns to its original shape once the load is removed. Plastic deformation is permanent, meaning the material does not return to its original shape after the load is removed.

What is the difference between stress and strain?

Stress is the internal force per unit area within a material that arises due to externally applied forces. It is measured in units such as Pascals (Pa). Strain, on the other hand, is the measure of deformation representing the displacement between particles in the material body relative to a reference length. It is a dimensionless quantity.

How is the modulus of elasticity related to deformation?

The modulus of elasticity, also known as Young's modulus, is a measure of a material's stiffness. It is defined as the ratio of stress to strain in the elastic region of the material's stress-strain curve. A higher modulus of elasticity indicates that the material is stiffer and deforms less under a given load.

What factors influence the deformation of a material?

Several factors influence the deformation of a material, including the type and magnitude of the applied load, the material's properties (such as yield strength, modulus of elasticity, and ductility), the temperature, the rate of loading, and the presence of any pre-existing flaws or defects in the material.

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