# Finite Element Methods (global stiffness matrix)

• eWizardII
In summary, the size of the global stiffness matrix K (i.e., Kuu) for the 2-D problem is determined by the total number of nodes multiplied by the number of degrees of freedom for each node. In the solved problem, there are 12 nodes and 2 degrees of freedom per node, resulting in a 24 x 24 matrix. In the unsolved problem, there are 15 nodes and 2 degrees of freedom per node, resulting in a 30 x 30 matrix.
eWizardII

## Homework Statement

I have the following practice problem which is presented as follows:

What is the size of the global stiffness matrix K (i.e., Kuu) for the 2-D problem?

http://imgur.com/KZec3 (Unsolved)

http://imgur.com/piv1J (Solved)

## The Attempt at a Solution

So in the solved problem which I think I did right I counted the total number of nodes which is 12, since each has 2 degrees of freedom the size of the matrix is 24 x 24? Does this sound correct?

If so now I attempt the other problem, where the ? I think should just be theta's but I believe there was a rendering problem. Anyways my attempt would be again count the number of nodes in which case there are 15 nodes. And again the nodes have a total of 2 degrees of freedom so this is a 30 x 30 size matrix?

Does this sound correct? (there are few problems with which to practice this one, so I'm trying to understand the methodology here for an exam, so any insight would be helpful)

the size of Global stiffness matrix = nodesxnodes.

## 1. What is a global stiffness matrix in Finite Element Methods?

A global stiffness matrix is a mathematical representation of the stiffness of a structure or system in Finite Element Methods. It is a square matrix that contains all the stiffness values for each degree of freedom in the system, taking into account the material properties, geometry, and boundary conditions.

## 2. How is a global stiffness matrix created in Finite Element Methods?

A global stiffness matrix is created by assembling individual element stiffness matrices. Each element stiffness matrix is calculated based on the shape, size, and material properties of the element. These individual matrices are then combined to form a larger matrix that represents the entire system.

## 3. What is the significance of the global stiffness matrix in Finite Element Methods?

The global stiffness matrix is a crucial component in Finite Element Methods as it provides a complete representation of the stiffness of a system. It is used to solve for the displacements and stresses in the system under different loading conditions, making it an essential tool for design and analysis in engineering applications.

## 4. How can the global stiffness matrix be solved in Finite Element Methods?

The global stiffness matrix can be solved using numerical methods, such as the Gaussian elimination method or the Cholesky decomposition method. These methods involve calculating the inverse of the matrix, which can then be used to solve for the unknown displacements and stresses in the system.

## 5. Are there any limitations to using the global stiffness matrix in Finite Element Methods?

The global stiffness matrix is an approximation of the true stiffness of a system and can be affected by errors in the modeling and assumptions made in Finite Element Methods. Additionally, the size of the matrix can become very large for complex systems, making it computationally expensive to solve. As such, the accuracy and efficiency of the global stiffness matrix can be limiting factors in the use of Finite Element Methods for certain applications.

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