Stiffness Influence Coefficient Fea For Plate

[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?

 

[PerennialII]

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.

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.

 

[ravenprp]

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.