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The Linear Triangular Element and
Plane Stress
The Bilinear Quadrilateral Element
Basic Equations
The linear triangular element is atwo‐dimensional finite element with both local and global coordinates.
It is characterized bylinear shape functions.
This element can be used forplane stressorplane strainproblems in elasticity
This element can be used forplane stressorplane strainproblems in elasticity.
It is also called theconstant strain triangle (CST).
The linear triangular element hasmodulus of elasticityE,Poisson’s ratioν, and thickness t.
Each linear triangle hasthree nodeswithtwo inplanedegrees of freedom at each node. The global coordinates of the three nodes are denoted by (xi, yi), (xj, yj), and (xm, ym).
Theorder of thenodes for each element is important.they should be listed in a counterclockwise directionstartingfrom any node.
Thearea of each triangleshould be positive
Thearea of each triangleshould be positive.
In this case theelement stiffness matrixis given by
It is clear that the linear triangular element has six degrees of freedom– two at each node.
Consequently for a structure with n nodes, the global stiffness matrixKwill be of size2n × 2n.
Once the global stiffness matrixK is obtained we have the following structureequation:
The above matrix is solved by partitioning and Gaussian elimination. Finally once the unknown displacements and reactions are found, thep , stress vectoris obtainedfor each elementas follows:
σ is thestress vector in the element(of size 3 × 1) and uis the 6 × 1 element displacement vector.The vectorσis written for each element as {σ} = [σxσyτxy]T.
Th Bili The Bilinear Quadrilateral
Element
Basic Equations
Thebilinear quadrilateral elementis atwo‐dimensionalfinite element with both local and global coordinates.
It is characterized bylinear shape functionsin each of thexandy directions This element can
It is characterized bylinear shape functionsin each of thexandy directions. This element can be used forplane stressorplane strainproblemsin elasticity.
It is a generalization of the4‐node rectangular element.
The bilinear quadrilateral element has modulus of elasticityE, Poisson’s ratioν, and thickness t.
Eachbilinear quadrilateralelementhasfour nodeswithtwo in‐plane degrees of freedomat each node.
The global coordinates of the four nodes are denoted by (x1, y1), (x2, y2),(x3, y3), and (x4, y4).
The order of the nodes for each element is important – they should be listed in a counterclockwise directionstarting from any node.
The area of each element should be positi e
The area of each element should be positive.
Theelement is mapped to a rectanglethrough theuse ofthenatural coordinatesξandη.
In this casethe element stiffness matrix is not written explicitly butcalculated through symbolic integrationwith the aid of the MATLAB Symbolic Math Toolbox.
Thefour shape functionsfor this element are listed explicitly as follows in terms of the natural coordinatesξandη
Stiffness matrix
The integral in the stiffness matrix can also be computed numerically by Gauss quadrature in two dimensions.
where
and his the thickness.
we perform the numerical computation by
HereFi,jis a matrixdependent onthe natural points(ξi,ηj ). Integration points(ξi,ηj) and integration weights depend on the type of integration the user wishes to perform. In the four‐node element we can use a 2 × 2 numerical integrationforexact integration.
we perform the numerical computation by
The Gauss quadraturein two dimensions replacesthe integrationby a summation over Gauss points, by
wherep, qare thenumber of integrating pointsin theξ,ηdirections, respectively,and wi, wjare the correspondingweights.
One Gauss point integration (ξ = 0, η = 0); two Gauss point integration (ξ, η =±1/√3 )
Thestiffness matrixcan be computed using2× 2 Gauss pointsas
It is clear that the bilinear quadrilateral element has eight degrees of freedom– two at each node.
Consequently for a structure with n nodes, the global (
stiffness matrixK will be of size2n × 2n(since we have two degrees of freedom at eachnode).
Once the global stiffness matrixK is obtained we have the following structureequation:
[K]{U} = {F}
Fi ll th k di l t d ti
Finally once the unknown displacements and reactions are found, the stress vector is obtained for each elementas follows
{σ} = [D][B] {u}
The vector σ is a linear function of ξ and η. Usually numerical results are obtained at the centroid of the element whereξξ=ηη= 0.
The function BilinearQuadElementStresses gives two results –the general linear stress functions inξ and η, and thenumerical values of the stresses at the centroid of theelement.
function y = BilinearQuadElementArea(x1,y1,x2,y2,x3,y3,x4,y4)
% BilinearQuadElementArea This function returns the area
% of the bilinear quadrilateral
% l h fi d h
% element whose first node has
% coordinates (x1,y1), second
% node has coordinates (x2,y2),
% third node has coordinates
% (x3,y3), and fourth node has
% coordinates (x4,y4).y
yfirst = (x1*(y2–y3) + x2*(y3–y1) + x3*(y1–y2))/2;
ysecond = (x1*(y3–y4) + x3*(y4–y1) + x4*(y1–y3))/2;
y = yfirst + ysecond;
