Derivation of the Nodal Forces Equivalent to Uniform Pressure for Quadratic Isoparametric Elements RAWB, Last Update: 30 September 2008

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1 Deivation of the odal oces Equivalent to Unifom Pessue fo Quadatic sopaametic Elements RWB, Last Update: 0 Septembe 008 The displacement vecto u at an point within a single element, E, is lineal elated to the displacements of all the nodes of that element via the shape functions,, whee =,,, epesents the nodes of the element, thus, u () E whee is the displacement vecto of node. Suppose nodal foces ae applied to the nodes of element E. The wok done b these foces, on element E alone, if the cause incements of displacement, is, Wok Done b odal oces: () E E j j ltenativel, if a pessue P is applied to a face of element E which is composed of man small elements of aea d, then the wok done would be, Wok Done b pplied Pessue: u Pd Pd () E whee u is the incement of the displacement at an point on the element face in question caused b the applied pessue, and is the coesponding incement in the nodal displacements of the element. Equ.() has been used to deive the RHS of Equ.(). ow if the nodal foces in () ae to be equivalent to the applied pessue in (), then () and () must be equal fo abita nodal displacements. Hence we equie, Pd that is, j Pdj () n Equ.(), is the aea of the face, o faces, of the element to which pessue is applied. Equ.() applies geneall even fo cuved elements with non-plana faces to which non-unifom pessue is applied. ow conside a ectangula element with unifom pessue applied to one face pependicula to the -ais. Equ.() becomes, P dd ; 0 (5) o a D element, one of the integal vaiables is simpl eplaced b the element thickness, t.

2 D Quadatic sopaametic Element The shape functions,, can be witten staight down since the ae the unique functions which ae not moe than quadatic in the coodinates and which ae equal to unit at the node,, in question, but eo at all othe nodes. Thus, in D we have, Cone ode : (6) Midside ode : (7) Hee we ae using the usual isopaametic convention that the cone nodes lie at. ode lies at (-, -), and node at (0, -). Let the element edge which consists of nodes,, have pessue P applied. t s nomal is the -diection, and so Equ.(5) gives, d (8) Consideing fistl the cone node, the nodal foce evaluates to be, d d d (9) ow conside the midside node, d d d (0) Hence, the cone:midside nodal foce atio is :. (PTO

3 D Quadatic sopaametic Element The shape functions,, can be witten staight down since the ae the unique functions which ae not moe than quadatic in the coodinates and which ae equal to unit at the node,, in question, but eo at all othe nodes. Thus, in D we have, fo the face with = -, Cone ode : () 8 Midside ode : () Hee we ae using the usual isopaametic convention that the cone nodes lie at. ode lies at (-, -, -), and node at (0, -, -). Let the element face = - have pessue P applied. t s nomal is the -diection, and so Equ.(5) gives, dd () Consideing fistl the cone node, the nodal foce evaluates to be, d 8 dd dd dd dd () ow conside the midside node, PT d dd dd (5) Hence, the cone:midside nodal foce atio is -:, the same magnitude as in D but the opposite sign.

4 Remembe that the above esults ae fo a single element onl. Whee a node lies on moe than one element, each element will make a contibution to the nodal foice, which will alte the atio between the cone and midside nodes. Conside the edge of a D mesh to which pessue is applied. o nodes not at the ends of such a bounda, each cone node will lie on two elements, but midside nodes on onl one element. Hence the cone:midside nodal foce atio would then be :. Conside the bounda of a D mesh to which pessue is applied. o nodes not at the edges of such a bounda, each cone node will lie on fou elements, but the midside nodes on onl two elements. Hence the cone:midside nodal foce atio would then be -:.

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