Problem on stiffness matrix of beam and deflection of tapered beam

In summary, the first conversation is about deriving the stiffness matrix for a beam with nodal linkage, where the shear force at the first node is 0 but the usual shear and moment resistance is present for the second node. The person suggests solving the problem by setting the first row of the stiffness matrix to 0 and asks for help in this.The second conversation is about finding the deflection of a tapered beam with an inertia that is a function of x. The person suggests using the relation between EI and moment equation and integrating it, but is unsure of how to proceed. They ask for help in this and mention that it would depend on how the tapered beam is set up.
  • #1
chiraganand
113
1
1.How do i derive the stiffness matrix for a beam with nodal linkage? ie the shear force at the first node is 0 but the usual shear and moment resistance is present for the second node.

I thouth about solving the problem by just putting the first row of the stiffness matrix as 0 . Am i correct? Plee help in this

2. How do i get the deflection of a tapered beam? i have the intertia as a function of x. How do i go about the problem?
I thought of using the the the relation between EI and Moment equation and integrating it. But i a m not sure how to go further. Can someone please help me in this
 
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  • #2
chiraganand said:
2. How do i get the deflection of a tapered beam? i have the intertia as a function of x. How do i go about the problem?
I thought of using the the the relation between EI and Moment equation and integrating it. But i a m not sure how to go further. Can someone please help me in this

Your moment equation would be

EI (d2y/dx2) = -M

where I is a function of x.

So you'd need to just do

d2y/dx2 = -M/EI

then integrate twice inputting your boundary conditions. This would depend on how your tapered beam is set up.
 
  • #3
1. I am having trouble visualizing your beam and the nature of this 'nodal linkage'. Can you provide a picture?
 

FAQ: Problem on stiffness matrix of beam and deflection of tapered beam

What is a stiffness matrix of a beam?

A stiffness matrix of a beam is a mathematical representation of the stiffness of a beam, which is a measure of its resistance to bending. It is a square matrix that relates the applied forces and moments to the resulting displacements and rotations of a beam.

How is the stiffness matrix of a beam calculated?

The stiffness matrix of a beam can be calculated using various methods, such as the finite element method or the direct stiffness method. These methods involve breaking the beam down into smaller elements and using equations to calculate the stiffness of each element. These individual stiffness values are then combined to create the overall stiffness matrix of the beam.

What factors affect the stiffness matrix of a beam?

The stiffness matrix of a beam is affected by various factors, including the material properties of the beam (such as Young's modulus and cross-sectional area), the geometry of the beam (such as length and cross-sectional shape), and the type and magnitude of applied loads.

How is the deflection of a tapered beam calculated?

The deflection of a tapered beam can be calculated using the Euler-Bernoulli beam theory, which takes into account the beam's material properties, geometry, and applied loads. This theory involves solving differential equations to determine the deflection and bending moment at different points along the beam.

What are some common applications of analyzing the stiffness matrix and deflection of tapered beams?

The stiffness matrix and deflection of tapered beams are important considerations in various engineering and construction projects, such as designing bridges, buildings, and other structures. They can also be used in the analysis and optimization of mechanical systems, such as aircraft wings and automobile frames.

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