Research Article
Beams and Plate Bending Macro-Elements
Department of Civil Engineering, Engineering Collage, Al-Isra University, Jordan
The analysis of large structural systems using the conventional finite element method is impractical. This is because of the necessity to use relatively fine mesh to obtain an accurate model. This will lead to a large number of equations to be solved. Therefore, it is advantageous to seek for approaches that reduce the total number of degrees of freedom (dof) needed to successfully model large systems. One of these methods is to use macro-elements.
In this study two types of macro-elements were developed.
The first is for beam element to demonstrate the idea of the macro elements and the second is for plate bending elements.
These macro-elements are based on transformation of many structural finite elements into single equivalent macro-element. This is done by preserving the same potential energies of the structure modeled by finite elements and the same structure modeled by macro-elements (Alani, 1983).
FORMULATIONS OF MACRO-ELEMENTS
In this study formulations of beam macro-elements and plate bending macro elements are developed.
In this modeling, several basic finite elements are combined to form a macro-element. The original structure that consists of many small finite elements will be replaced by an equivalent model containing one or more macro-elements.
The macro-elements are assembled and analysis continued in a manner analogous to that used in the finite element method.
BASIC ASSUMPTIONS FOR MACRO-ELEMENT FORMULATION
The formulation is based on the following assumption:
• | The potential and kinetic energies of the original finite element and the equivalent macro-element models are equal |
• | All the elements that are composing the macro-element must be of the same type such as beam elements, plane stress elements, plate bending elements etc. |
• | The order of the assumed displacement field of the macro-element is at least of the same order as that of the original finite elements |
• | The macro-element behavior follows the theory controls the behavior of the structural elements that compose the macro-element |
• | The compatibility requirements for the macro-elements are the same those of the original finite element |
NECESSARY STEPS NEEDED FOR DEVELOPMENT OF A MACRO-ELEMENT
The necessary steps of the development of macro-elements are as follows:
Step 1: | Divide the original structure that consists of many finite elements into macro-elements |
Step 2: | Select the order of the macro-element displacement function. This step depends on the order and number of the finite-elements composing the macro-element. Accuracy of the results depends greatly on this step |
Step 3: | Set-up the stiffness matrices of the finite-elements forming the macro-element |
Step 4: | Calculate the local coordinates (S, T) for the nodal points of the finite elements with the respect to the macro-element nodes so as to formulate the transformation matrix (T) required in the next step |
Step 5: | Formulate the transformation matrix (T), which relates the nodal degrees of freedom of the macro-element to the nodal degrees of freedom of the original structure modeled by finite elements |
The stiffness matrix of each finite element is multiplied by its corresponding transformation matrix to produce the participation of this element in establishing the macro-element stiffness matrix, as it will be seen later.
The stiffness matrix of the macro-element is formulated by equating the strain energy of the original structure modeled by finite-elements and that of the equivalent model as follows:
(1) |
Where:
Uo | : | The strain energy of the original structure modeled by many finite elements that constitutes one macro-element |
Um | : | The strain energy of the macro-element |
(2) |
Where:
qo | : | Displacement vector of the structure modeled by many finite elements that constitute one macro-element |
q m | : | Displacement vector of one macro-element |
(Sko) | : | The assembled stiffness matrix of all stiffness matrices of the finite elements constituting one macro-element |
(Km) | : | The stiffness matrix of the macro-element |
Let the displacement vector of the original structure, (which constitutes one macro-element) {q o} be related to that of the macro-element, {q m} as:
(3) |
where, T is the transformation matrix for the macro-element.
Substituting Eq. 3 into Eq. 2 gives:
(4) |
In this solution, matrix (SKo) is not needed, only (Ko), the stiffness matrix of a finite element bounded by the macro-element is needed. To explain this let.
n | : | The No. of finite elements comprising the macro-element |
(Te) | : | The finite element transformation matrix |
Every time (Te) carries a partition of the transformation matrix (T) that corresponds to the degrees of freedom of the finite element under consideration. The transformation for each finite element is placed in its proper place in the structural stiffness matrix of the equivalent model, which is the place of (Km) and:
(5) |
The transformation matrix (T) is simply the evaluation of the shape functions of macro-element at the nodes of the finite element. This evaluation is based on local coordinates for the nodal points of the finite elements with respect to the macro-element nodes .
To form a general transformation matrix Ti corresponding to an arbitrary nodal point I of the original structure within a certain macro- element, consider the notation Nki which means the shape function K of node I of this macro-element is evaluated at point I using its local coordinates within the macro-element, then the transformation matrix will depend on the macro-element type as will be seen latter.
Step 6: Construct the Macro-Element Nodal Load Vector
The external loading are applied at nodes of the finite element model. However, these nodes may not necessarily coincide with the macro-elements nodes. It is required to calculate the equivalent nodal load vector of each macro-element.
In general, all forms of loading other than concentrated loads subjected to the original structure nodes must be first reduced to equivalent nodal forces acting on the original structure, as with the conventional finite element method. The nodal load vector of the original structure can then be transformed to equivalent macro-element structural load vector by equating the external work done on the original structure modeled by finite elements and that of the macro-element model as following:
(6) |
Where:
Wo | : | The external work done on the original structure that constitutes one macro-element |
Wm | : | The external work done on the macro-element |
(7) |
Where:
{Fo} | : | The assembled nodal load vector of the finite elements constituting one macro-element |
{Fm} | : | The equivalent nodal load vector of the macro-element |
Substituting Eq. 3 into 7 gives:
(8) |
Where, T is the same transformation matrix used in deriving km.
Step 7: | Assemble all the macro-element stiffness matrices into a structural stiffness matrix and also construct the macro-element structural load vector |
Step 8: | Apply the boundary conditions which will be at the macro-elements nodes. Other boundary conditions corresponding to the eliminated nodes of the finite- elements of the original structure will be ignored |
Step 9: | Solve for the equivalent model nodal displacements in a straight forward manner |
Step 10: | Using results obtained in step 9 the displacements at any point inside the macro-elements may be calculated making use of the macro-elements shape function |
Step 11: | After the structure is analyzed for nodal displacements, the stresses at selected points in each macro-element may be obtained in the usual manner |
FORMULATION OF MACRO-ELEMENTS FOR ONE-DIMENSIONAL BEAM ELEMENTS
This simple element is used to demonstrate clearly the idea behind this new modeling. This element has 2 dof per node of type (W) and (θ) (Cook, 1981).
Using the aforementioned steps, the formulations of the macro-element for one dimensional beam problems, will be considered below. For other types of structures, the formulations are straightforward.
Fig. 1: | Case a: Original beam structure modeled by four structural finite elements |
Fig. 2: | Case b: The beam structure modeled by two macro elements |
Fig. 3: | Beam element in local coordinate |
To clearly demonstrate the formulation of the macro element for one-dimensional beam problems, consider first a simple problem, then generalize the idea to more complicated systems.
Step 1: | Consider a beam divided into four subdivision, each of length (l) and consider only two degrees of freedom per node, Wi θi as shown in Fig. 1 let this system be denoted as case a |
Let this system be modeled by another system, calling it case b, composed of two macro-elements, as shown in Fig. 2.
Step 2: | It is known that the displacement behavior of a beam problem is cubic. If the macro-element is modeled by a cubic displacement function an exact solution is expected. This is effectively achieved, as will be shown later |
Step 3: | Required formulation of the stiffness matrix of the structural element, which is here a beam element as shown in Fig. 3 |
Let the local x-axis be defined as that which passes from node 1 to node 2 let a non-dimensional s-axis be defined whose origin is located at node 1, as shown in Fig. 3.
Now:
(9) |
The beam element is a cubic with a displacement function as:
(10) |
Where:
(11) |
(12) |
But:
(13) |
Also,
To find ∂2w/∂S2, use Eq. 10 and differentiate twice with respect to S.
(14) |
The total potential energy of the element is:
where, P is the distributed load per unit length and Fi is the concentrated load applied at point i in the direction of wi
But:
M = -EIw″ |
Substitute for w and w″ from Eq. 9 and 14, respectively.
Minimize the total potential energy with respect to {q}T:
Where:
To evaluate the stiffness matrix then:
After integration Ke will be:
(15) |
For load vectors:
(16) |
Fig. 4: | Two finite elements from system a, Fig. 1 |
Fig. 5: | One Macro-element from system b, Fig. 2 |
After integration:
(17) |
The shape functions must be evaluated at the point of application of Fi for each load and sum.
Step 4: | Requires establishing the local coordinate of nodes 1, 2, 3 of system a with respect to the nodes 1 and 2 of system b as shown in Fig. 4 and 5 |
In line elements, it is relatively easy to establish the local coordinates. Because we are working with the macro-element its length will be taken as one unit.
The coordinate transformation is simply:
An application for the coordinate transformation is presented here. Let:
The local coordinates of points 1, 2 and 3 of system are calculated as:
Node 1
Node 2
Node 3
Step 5: | Is important because it constructs the transformation matrix (T). To find the relations between {q0} and {qm}, where the subscripts 0 and m refer to original and macro-element, respectively, relations between Wa2 and θa2 and the macro-element displacements are needed. This can be found from the displacement function of the two cases: |
(18) |
And:
(19) |
Substitution of Eq. 11 into Eq. 19 yields:
But:
(20) |
It was shown that the local coordinate of point 2 in system a is equivalent to ½ in system b.
(21) |
But:
Then:
Also,
(22) |
Also,
(23) |
All the ingredients of the transformation matrix (T) are available from Eq. 21-23. Regarding this information in matrix form, one obtains
(24) |
The above example shows clearly that to find the relations between systems (a) and (b) for the displacements Wi and rotations θi, the global coordinates of the points in system (a) must be transformed to local coordinates of system (b) the displacements and rotation relation Wi and θi will be achieved through the evaluation of the shape function N1, N3 and ∂N2/∂x, ∂N4/∂x, respectively, evaluated at the local coordinate values of point 1, 2, 3 system (a).
To form a general transformation matrix (T), let the notation NK|ji mean that the shape-function K is evaluated at node i of system j. Then the transformation matrix will take the form shown in Eq. 25.
(25) |
The structural beam-element stiffness matrix after transformation to global coordinates has size 4x4, but the transformation matrix (T) is of size nx4. Therefore, it is necessary to extract from matrix (T) that the part corresponding to the degrees of freedom of matrix (K0). Let (Te) be required part and its size be 4x4.
Then:
(26) |
The stiffness matrix of the macro-elements is constructed.
Next, the consistent load vector acting on the macro-element and nodes is calculated. This is done using Eq. 16 and 17.
If the distributed load is not constant, then (P) will be P(s) and the integration will be carried out. If the distributed load is discontinuous, then the limits will be taken over the parts where loading are present. For concentrated loading conditions, the consistent load vector is calculated as the distribution of each concentrated load summed over each node using the shape functions of the macro-element. At this stage, the macro-elements and the load vectors are assembled to construct the linear systems over the whole structure.
After assembling and reflecting the boundary conditions, the system equations are ready for solution. Upon solving for the nodal values at the macro-element nodal points, it is easy to find the required values at any point inside the structure using Eq. 18 which is:
PLATE BENDING FINITE ELEMENTS USED IN THE FORMULATIONS OF THE MACRO-ELEMENTS
What follows are brief information about the plate bending finite elements studied and used in the formulations of the macro-elements.
The (Q8) Quadratic Serendipity Finite Element (Fig. 6).
This element has eight nodes with three dof per node (Rock and Hinton, 1976). It is Mindlen type plate bending finite element (Fig. 7).
Fig. 6: | General quadrilateral isoparametric finite element |
Fig. 7: | The quadratic serendipity (Q8) quadrilateral isoparametric finite element |
The displacement vector is
⌊ qi ⌋ = [Wi Wi,y -Wi,x] |
where, i = 1, 2, .8
The (Q9) Quadratic Langrangian Finite Element.
This element is a Mindlen type plate bending element with nine nodes and three dof per node (Pugh et al., 1978) as shown in (Fig. 8).
The displacement vector is:
where, i = 1, 2, .. 9
FORMULATION OF PLATE BENDING MACRO-ELEMENTS:
The stiffness matrix of a macro-element is formulated by equating the strain energy of the original structure modeled by finite-elements and that of the equivalent macro-element model as follows (Alani, 2002):
(27) |
Where:
Uo | : | The strain energy of the original structure modeled by many finite elements that constitute one macro-element (Fig. 9) |
Um | : | The strain energy of the macro-element |
Fig. 8: | The quadratic lagrangian (Q9) quadrilateral isoparametric finite element |
Fig. 9: | General macro-element discretization |
(28) |
Where:
qo | : | Displacement vector of the structure modeled by many finite elements that constitute one macro-element |
qm | : | Displacement vector of one macro-element |
(Sko) | : | The assembled stiffness matrix of all stiffness matrices of the finite elements constituting one macro-element |
(Km) | : | The stiffness matrix of the macro-element |
Let the displacement vector of the original structure, (which constitute one macro-element) {qo} be related to that of the macro-element {qm} as:
(29) |
where, T is the transformation matrix for the macro-element. Substituting Eq. 29 into Eq. 28 gives:
(30) |
In the solution, matrix (SKo) is not needed, only (Ko), the stiffness matrix of a single finite element bounded by the macro-element is needed. To explain this let.
n | : | The No. of finite elements comprising the macro-element |
(Te) | : | The finite-element transformation matrix |
Every time [Te] carries a partition of the transformation matrix [T] that corresponds to the degrees of freedom of the finite-element under consideration. The transformed stiffness matrix for each finite-element is placed in its proper place in the structural stiffness matrix of the equivalent model, which is the place of [Km], as:
(31) |
The transformation matrix [T] is simply the evaluation of the shape functions of the macro-element at the nodes of the finite-element. This evaluation is based on local coordinates for the nodal points of the finite-elements with respect to the macro-element nodes (Fig. 10).
To form a general transformation matrix [Ti] corresponding to an arbitrary nodal point i of a certain finite element within a certain macro-element, consider the notation NkI which means that shape function k of node I of this macro-element is evaluated at point i using its local coordinates within the macro-element. The transformation matrix will depend on the macro-element type as follows:
The (Q8) Quadratic Serendipity Finite-Element
The displacement functions over this finite element are expressed as follow (Armanios and Negm, 1983).
where, the shape functions (Ni) are the same in the above equations.
To construct [Te] of a certain finite element consider (Fig. 11). The transformation matrix [Te] of the finite element [k, L, m, n, o, p, q, r] which is inside the macro-element (1, 2, 3, 4, 5, 6, 7, 8) will be as follows:
Fig. 10: | A general description of the local coordinates (S,T) within AME |
Fig. 11: | The correspondence between the finite element dof and the macro element dof |
Where:
i.e., the participation of node (k) of the finite element that corresponds to node (1) of the macro-element under consideration.
In general:
Where:
i | = | k , L , m , q , r the nodes of the finite element |
j | = | 1, 2 , 3 , 7 , 8 the nodes of the macro element |
Then:
The (Q9) Quadratic Lagragian Finite Element
Here, there are nine nodes with three dof of type w, θx and θy.
Then:
[Te] is 27x27 and
Where:
i | = | k, L, m, n, o, p, q, r, s |
j | = | 1, 2, 3, 4, 5, 6, 7, 8, 9 |
And:
MACRO-ELEMENT LOAD VECTOR
The externals loading are applied at known nodes of the finite-element model. However, these nodes may not necessarily coincide with the macro-elements nodes. It is required to calculate the equivalent consistent nodal load vector of each macro-element.
In general, all forms of loading other than concentrated loads subjected to the original structure nodes must be first reduced to equivalent nodal forces acting on the original structure, as with the conventional finite element method. The nodal load vector of the original structure can then be transformed to equivalent macro-element structural load vector by equating the external work done on the original structure modeled by finite-elements and that of the macro-element model as follows:
(32) |
Where:
Wo | : | The external work done on the original structure that constitute one macro-element |
Wm | : | The external work done on the macro-element |
(33) |
Where:
{Fo} | : | The assembled nodal load vector of the finite-elements constituting one macro-element |
{Fm} | : | The equivalent nodal load vector of the macro-element |
Substituting Eq. 29 into Eq. 33 gives:
(34) |
where, T is the same transformation matrix used in deriving (km).
The assembly of all the macro-element stiffness matrices into a structural stiffness matrix and also the construction of the macro-element structural load vector and solution of the structure equation are the same as that of conventional finite element method.
NUMERICAL APPLICATIONS
One Dimensional Beam Problems
Two problems have been selective and solve using both the finite element method and the equivalent energy method.
• | Problem No. 1: A cantilever beam with two concentrated loads as shown in Fig. 12 was modeled by Fig. 13 and 14. The results are shown in Table 1. |
• | Problem No. 2: A cantilever beam with five concentrated loads and with variable cross-sections as shown in Fig. 15 was modeled by Fig. 16 and 17 the results are tabulated in Table 2. |
Fig. 12: | Cantilever beam with constant cross-section: problem No. 1 |
Fig. 13: | Finite element method modeling of the beam in problem No. 1 |
Table 1: | Results of the beam in problem No. 1 |
Fig. 14: | Macro element modeling of the beam in problem No. 1 |
Fig. 15: | Cantilever beam with many concentrated loads problem No. 2 |
Fig. 16: | Finite element method modeling of beam problem No. 2 |
Fig. 17: | Macro element modeling of beam problem No. 2 |
Table 2: | Results of beam problem No. 2 |
APPLICATIONS FOR PLATE BENDING ELEMENTS
Various problems of plate bending analysis are solved and presented below in order to demonstrate the efficiency of the macro-elements developed.
The accuracy of the equivalent energy macro-elements are checked by using the conventional finite elements method and if available, the exact solution.
The moments and stresses are generally calculated at the Gauss points of the macro-elements in the problems presented below unless it is stated differently.
Problem No. 1
The analysis of thin, square simply supported isotropic plate under a uniformly distributed load, as shown in Fig. 18.
• | L = 10 (in units of length) |
• | T = 0.1 (in units of length) |
• | E = 10.92 x 107 (in units of force/area) |
• | Gxy = Gxz = Gyz = 4.2 x 107 (in units of force/area) |
• | Nu = 0.3 |
• | Qz = 1.0 (In units of force/area) |
The following data are given for this problem:
The results may be expressed in a normalized form as follows:
• | Deflection= C x Qz x L4 x 10-2/D |
• | Rotations (in x or y)= C xQz x L3 x 10-1/ D |
• | Mx , My or Mxy= C xQz xL2x10-4 (for Nu = 0.3) |
where, C x 10-2 represents the value of the function for the data given above.
Due to symmetry only one quarter of plate is analyzed. The analysis is done using the (Q8) elements, as shown in Fig. 18.
The original finite element mesh has (65) nodes and (195) dof. The equivalent energy model has (21) nodes and (63) dof. The total reduction in dof is 67.7%.
Fig. 18: | Quartar of plate for problem No. 1 analyzed with the (Q8) elements |
Fig. 19: | X-axis deflection for problem |
Fig. 20: | Quarter of plate for problem No. 2 analyzed with (Q9) elements |
The results for deflections and rotations are shown in Fig. 19 and 20. The maximum errors are (7.9 and 4.2%), respectively.
Table 3 shows a comparative study for the execution time (CPU), the band width solution operation count (Ne x HBW2), central deflections and their corresponding errors. The analysis is done using (Q8) conventional FE and (Q8) equivalent ME meshes. The bandwidth-solutions operation count (Armanios and Negm, 1983) is a useful measure of the computer time required to solve banded equations.
Table 3: | A comparative study of different (Q8) meshes for problem No. 1 |
Table 4: | Details for problem No. 2 |
The errors in central deflections are measured from those of the original FE meshes.
Problem No. (2)
The analysis of thin and thick, square, simply supported orthotropic plate under a uniformly distributed load. The following data are given for this problem:
L = 7.2 m. |
Thickness t
• | Case A: t= 0.114 m (t/L = 0.02, i.e., thin plate) |
• | Case B: t= 1.080 m (t/L = 0.15, i.e., thick plate) |
• | Ex= 20 x 106 kN m-2 |
• | Ey= 30 x 106 kN m-2 |
• | Gxy= 15x 106 kN m-2 |
• | Gxz= Gyz : Variable and as defined on graphs |
• | Nuxy = 0.15 |
• | Nuyx= Nuxyx Ey / Ex = 0.225 |
Loading Qz
• | Case A:Qz = 2.875 kN m-2 (for thin plate) |
• | Case B:Qz = 1212.807 kN m-2 (for thick plate) |
Due to symmetry, only on quarter of plate is analyzed. The analysis is done using the (Q9) isoparametric elements.
The plate is first considered as a thin plate, i.e., case A, then considered as thick plate, i.e., case B. The same discretizations are used for both cases, which are shown in Table 4. Table 4 shows that the total number of dof is reduced by 69.1% with the equivalent models.
The analysis is done using the technique of reduced integration when the plate is thin and using full numerical integration when the plate is thick.
The results for deflections for both cases along x-axis are shown in Fig. 21 and 22. Also, the results for moments (Mx, My and Mxy) and shears (Vx and Vy) for the thick plate along sec. A-A are shown in Fig. 23-25.
Fig. 21: | X-axis deflection for thin plate of problem No.2 (t/L = 0.02) |
Fig. 22: | X-axis deflection for thick plate of problem No. 2 (t/L = 0.15) |
Fig. 23: | Moments Mx and My along sec. A-A for the thick plate of problem No.2 (t/L = 0.15) |
Fig. 24: | Moments Mx y along sec. A-A for the thick plate of problem No.2 (t/L = 0.15) |
Fig. 25: | Shears Vx and Vy Along Sec. A-A for the thick plate of problem No.2 (t/L = 0.15) |
Table 5: | A comparative study for different (Q9) meshes for the thin plate of problem No. 2 |
Table 5 shows a comparative study for the execution time (CPU), the band width-solution operation count (NexHBW2), central deflections, central moments (Mx and My) and their corresponding errors. The analysis is done on the thin plate using (8 x 8 Q9) original FE mesh and:
Gxz = Gyz = Gxz. The errors in central deflection and moments are measured from the (8x8) original FE mesh analysis.
Table 6: | Comparative study of cpu time between original FE model and ME model of plate bending for problems 1 and 2 |
The central moment values in Table 5 are obtained by extrapolating the moment values at the Gauss points using a technique known as local stress smoothing, which is simply a bilinear extrapolation of the (2x2) Gauss point stress values within an element (Cook, 1981).
The solved problems showed that using the macro-elements in the analysis reduced the number of equations to be solved. When the size of the macro-element used is of moderate, excellent results are achieved with good amount of reduction in dof and computer time.
But when the size of the macro-element is large still acceptable results are achieved with substantial reductions in dof and computer time as shown in Table 3 and 5. Comparative study of cpu time between original FE model and ME model of plate bending for problem 1 and 2 is shown in Table 6.
CONCLUSION AND RECOMMENDATIONS
New modeling of beam and plate bending macro-elements based on beam and plate bending types of finite elements were developed. The solved examples demonstrated that using these macro-elements in the analysis largely reduced the total number of dof required to model a certain structure. This in turn reduced the total number of equations to be solved. Reduction in total number of equations reduced computer time and memory space for storage. This will allow personal computers to analyze relatively large structures. At the same time these ME provided accurate results. In addition, finite elements of different sizes, thicknesses and material properties can easily be used inside the macro-elements if required in the analysis. This developed macro-element theory was applied to different kinds of structural elements like beams, trusses, thin plates and thick plates and good results were achieved in accuracy and time of execution. This theory can be applied to any kind of structures as long as the basic assumptions for macro-element formulations of section-4 are satisfied. It is recommended to apply this theory in the field of shell problems, non-linear problems and cracked structures to reduce the large number of dof required to model crack tips.
The author would like to gratefully acknowledge Al-Isra University for offering me the facilities and time to prepare this study.
NOTATIONS
The following symbols are used in this study
c | : | Clamped edge of plate |
dof | : | Degrees of freedom |
D | : | Flexural rigidity |
Ex, Ey | : | Moduli of elasticity along x and y direction of the plate, respectively |
Fr | : | Free adge of plate |
FE | : | Finite element |
{F} | : | Element nodal load vector |
Gxy, Gxz and Gyz | : | Shear moduli in the Z, Y, and X planes, respectively |
HBW | : | Half band width of the structural stiffness matrix |
(K) | : | The stiffness matrix |
L | : | Side length of a square plate |
m | : | A subscript refers to the macro-element structure |
ME | : | Macro-element |
{M} | : | The vector of generalized stresses at a point |
Ne | : | Total number of equations to be solved in a problem |
Nu, Nuxy, Nuyx | : | Poissons rations |
{N} | : | Vector of shape functions |
o | : | A subscript refers to the original (finite element) structure |
Pz | : | Concentrated force applied on the plate in the z direction |
Pzi | : | Concentrated force at node i of an element, in the z direction |
{q} | : | Element nodal displacement vector |
Qz | : | Uniformly distributed load applied on the plate in the z direction (force per unite area) |
R | : | Radius of annular plate |
S, T | : | Local coordinates of a point in the x and y directions, respectively |
Si , Ti | : | Local coordinates of node i of an element, in the x and y directions, respectively |
SS | : | Simply supported edge of plate |
t | : | Thickness of plate |
(T) | : | The transformation matrix needed in macro-element construction |
wi | : | Vertical displacement at node i of an element, in the z direction |
x, y, z | : | Global coordinates |
x' and y' | : | First derivatives of certain function with respect to x and y, respectively |