# Stiffness Influence Coefficient Fea For Plate

• chandran
In summary, the conversation discusses the stiffness matrix generation for a single rectangular element, considering each node having 2 translational dofs. The stiffness influence coefficient for force at ith dof due to a unit displacement at jth dof is calculated for different types of elements, such as rods, beams, solids, and planes. The conversation also delves into the derivation of the stiffness matrix using various methods, such as principle of virtual work and Galerkin method.
chandran
i have put this question in other forum also.

I have understood the stiffness matrix generation for a single rectangular element as below. I consider each node having 2 translational dofs

each in x and y coord system. There are 4 nodes in this rectangular plate element and hence 8 dofs overall.

K11U1+K12U2+K13U3+K14U4+K15U5+K16U6+K17U7+K18U8= F1
K21U1+K22U2+K23U3+K24U4+K25U5+K26U6+K27U7+K28U8= F2
K31U1+K32U2+K33U3+K34U4+K35U5+K36U6+K37U7+K38U8= F3
K41U1+K42U2+K43U3+K44U4+K45U5+K46U6+K47U7+K48U8= F4
K51U1+K52U2+K53U3+K54U4+K55U5+K56U6+K57U7+K58U8= F5
K61U1+K62U2+K63U3+K64U4+K65U5+K66U6+K67U7+K68U8= F6
K71U1+K72U2+K73U3+K74U4+K75U5+K76U6+K77U7+K78U8= F7
K81U1+K82U2+K83U3+K84U4+K85U5+K86U6+K87U7+K88U8= F8

where K(IJ) is the stiffness influence coefficient saying the force at ith dof due to a unit displacement at jth dof.
U1,U2,U3 ETC is the displacement at ith dof.
F1,F2,F3 ETC is the force at the ith dof.

Now i have a question. In nastran or other fea packages how this K(IJ) is calculated for rectangular elements. I know that for rod element it is

AE/L where A is the rod area,E youngs modulus and
L is the length of the rod. But for solid elements and plane elements how this is calculated.

One more question regarding rod or truss elements. There is a line of 10mm length horizontal to x axis. Why shouldn't i divide the
line into 10 rod or truss elements. What will be the error? Can i divide the line with 10 beam elements?

chandran said:
One more question regarding rod or truss elements. There is a line of 10mm length horizontal to x axis. Why shouldn't i divide the
line into 10 rod or truss elements. What will be the error? Can i divide the line with 10 beam elements?

I'm not sure I'm getting this 'easy' part ... can't see why you couldn't, and basic finite elements such as rods, beams etc. typically produce in practise identical results with analytical theories (as long as we're considering typical bending theories and leave nonlinearities etc. out) == extremely accurate, work well with pretty crude meshes.

chandran said:
Now i have a question. In nastran or other fea packages how this K(IJ) is calculated for rectangular elements. I know that for rod element it is

AE/L where A is the rod area,E youngs modulus and
L is the length of the rod. But for solid elements and plane elements how this is calculated.

This goes into the "heart" of FEM ... you can work with simple element formulations using rod, beam and so forth analogies, but in general it's preferable to derive the balance equations using "valid" method of derivation, which in case of FEM means either principle of virtual work, minimum of potential energy, variational derivation or the "best" of all (personally) using weak forms, such as the Galerkin method.

Considering the bilinear 2D solid element, its stiffness matrix is typically derived as follows (the notation I'm using is similar to links I've added to the end):

*Assuming infinitesimal strain - displacement relations and linear - elastic constitutive relations (plane stress or strain), write the Cauchy's equation of motion (equilibrium equation) substituting the kinematic and constitutive relationship to it.
*Using virtual displacements, write for plane elastic body :

$$\int_\textrm{Ve}(\sigma_{ij}\delta\epsilon_{ij}+\rho\ddot{u_{i}}\delta u_{i})dV-\int_\textrm{Ve}f_{i}\delta u_{i}dV-\int_\textrm{Se}t_{i}\delta u_{i}dS=0$$

... so write the virtual displacement energy balance law element - wise

*Introduce the FE approximation functions / interpolations, along the lines
$$u=\sum_{i=1}^n u_{i}^e \psi_{i}^e(x,y)$$

*substitute them to above and you'll end up with a typical FEA matrix equation :

$$[M^e]({\ddot{\Delta^e}})+[K^e](\Delta^e)-(f^e)-(q^e)=0$$

*where the stiffness matrix you're after is:

$$[K^e]=h_{e} \int_\textrm{Ve} [B^e]^T [C^e] [B^e] dxdy$$

... get the specifics of the derivation for example here :

http://www.vector-space.com/TourPE.pdf
http://www.mathsoft.cse.clrc.ac.uk/felib/Docs/html/Intro/intro-node10.html

although looking at common FEA software manuals or any book is probably the best way to go.

Last edited by a moderator:

Thank you for your question. The stiffness influence coefficient (K(IJ)) is calculated using the finite element method (FEM). In FEM, a structure is divided into smaller elements, and the stiffness matrix for each element is derived based on its geometry, material properties, and boundary conditions. These stiffness matrices are then assembled to form the global stiffness matrix for the entire structure. This global stiffness matrix is then used to solve for the displacements and stresses in the structure.

For rectangular plate elements, the stiffness matrix is derived using the equations of elasticity and the shape functions of the element. The exact formulation may vary depending on the type of element (e.g. quadrilateral or triangular) and the type of analysis (e.g. plane stress or plane strain). This calculation is usually done by the FEA software, such as Nastran, and the user does not need to manually calculate the stiffness matrix.

For rod or truss elements, the stiffness influence coefficient is calculated using the formula AE/L, as you mentioned. This is because these elements only have axial stiffness and do not account for bending or shear stiffness. As for dividing a line into multiple elements, it is recommended to use a reasonable number of elements to capture the behavior of the structure accurately. Dividing a line into too many elements may introduce unnecessary complexity and computational cost, while too few elements may result in inaccurate results. The exact number of elements needed depends on the type of analysis and the level of accuracy required.

In summary, the stiffness influence coefficient is calculated using the FEM method, and the exact formulation may vary depending on the type of element and analysis. It is recommended to use a reasonable number of elements to accurately capture the behavior of the structure. I hope this helps clarify your doubts.

## What is a stiffness influence coefficient in FEA for plates?

A stiffness influence coefficient is a numerical value that represents the relationship between a load applied to a plate and the resulting displacement of that plate. It is used in finite element analysis (FEA) to calculate the stiffness of a plate and how it will respond to external forces.

## How is the stiffness influence coefficient determined for a plate in FEA?

The stiffness influence coefficient for a plate is typically determined through a process called meshing, where the plate is divided into smaller elements. The stiffness of each element is then calculated, and the influence coefficients are derived from the stiffness values. This process is repeated for multiple load cases to determine the stiffness influence coefficients for different types of loading.

## What factors can influence the stiffness influence coefficient for a plate?

The stiffness influence coefficient for a plate can be influenced by several factors, including the material properties of the plate, the geometry of the plate, the boundary conditions, and the type of loading applied. Additionally, the mesh size and type of element used in the FEA analysis can also affect the stiffness influence coefficient.

## How is the stiffness influence coefficient used in FEA analysis?

The stiffness influence coefficient is used in FEA analysis to predict the behavior of a plate when subjected to different types of loads. By applying the influence coefficients to the known loads, the displacement and stress distribution of the plate can be calculated, helping engineers and scientists to design more efficient and reliable structures.

## Are there any limitations to using the stiffness influence coefficient in FEA analysis?

While the stiffness influence coefficient is a useful tool in FEA analysis, it is not without its limitations. It assumes linear elastic behavior, which may not accurately reflect the behavior of real-world materials. Additionally, the accuracy of the results can be affected by the mesh size and element type used in the analysis, as well as any simplifications made in the modeling process. As with any analytical tool, it is important to understand the limitations and assumptions of the stiffness influence coefficient when using it in FEA analysis.

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