MAxwell reciprocal theorem and symmetric stiffness matrix

In summary, the conversation discusses the validity of Maxwell's reciprocal theorem for elastic materials and small displacements. It is noted that the stiffness matrix is symmetric according to the theorem. However, the speaker also brings up their own observation of a symmetric stiffness matrix in their written programs for geometric non-linear problems. This leads to a discussion about how the stiffness matrix can be symmetric in cases of non-linear behavior, with the suggestion of providing an example and clarification on the elements in the matrix. The speaker concludes that the stiffness matrix is linear due to the kij constant and the inclusion of beam elements in the matrix.
  • #1
svishal03
129
1
As per Maxwell reciprocal theorem, it is valid only for elastic materials and structures indergoing small displacements.That is k12 = k21, kij = kji hence stiffness matrix is symmetric.

Howbver, I just have been going through MY OWN written programs for geometric non linear problems and I observe that stiffness matrix (assembled stiffness matrix) is well symmetric.

How can stiffness matrix be symmmetric for non linear behaviour of structures?
 
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  • #2
Perhaps if you posted an example?

What elements do you have in your matrix?

kij is just a constant and therefore linear.
 
  • #3
I have beam elements in my matrix. It is a cantilever beema with end moment.
 

Related to MAxwell reciprocal theorem and symmetric stiffness matrix

What is Maxwell reciprocal theorem?

Maxwell reciprocal theorem is a fundamental principle in mechanics that states the forces between two bodies are equal and opposite, regardless of the order in which they act.

How is Maxwell reciprocal theorem applied in engineering?

This theorem is commonly used in structural engineering to analyze the behavior of structures under different loading conditions. It allows engineers to determine the relationship between internal forces and displacements in a structure.

What is a symmetric stiffness matrix?

A symmetric stiffness matrix is a square matrix that represents the stiffness properties of a structure. It is symmetric because the coefficients on either side of the diagonal are equal, reflecting the reciprocal nature of forces in Maxwell's theorem.

How is the symmetric stiffness matrix obtained?

The symmetric stiffness matrix is obtained by solving a set of equations that relate the external forces and displacements of a structure. These equations are based on the equilibrium and compatibility conditions of the structure.

What are the advantages of using Maxwell reciprocal theorem and symmetric stiffness matrix in structural analysis?

One of the main advantages is that it simplifies the analysis of complex structures, making it easier to determine the internal forces and displacements. Additionally, it allows for quick and accurate evaluation of different loading scenarios, making it a valuable tool for engineers in the design process.

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