5.4 Hybrid elements

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By a special technique triangular elements can be developed which have only

the three degrees of freedom w, w,x , wy at the three nodes, but which yield

better results than simple nonconforming displacement-based elements. The

best known elements of this type are the triangular HSM element (hybridstress-

model), [29], and the DKT element. The latter is a modified Reissner–

Mindlin plate element which will be discussed later. First we discuss the HSM

element.

426 5 Slabs

The starting point is the principle of minimum complementary energy. According

to this principle, the moment tensor M = [mij ] of the exact solution

minimizes the complementary energy of the slab,

Πc(M) = −1

2

_

Ω

M•C

−1[M] dΩ = −1

2

_

Ω

mij κij dΩ (5.26)

on the set Vc, which is the set of all bending moment tensors M that satisfy

the equilibrium condition

− div2M = p or − mij ,ji = p (5.27)

and if necessary, static boundary conditions such as

mij ni nj = ￣mn ,

d

ds

mij ni tj + mij ,i nj = ￣vn on ΓN . (5.28)

Here n = [nx, ny]T and t = [tx, ty]T are the normal- and tangent vectors

at the edge of the slab. The boundary conditions (5.28) mean that along a

portion ΓN of the edge, the moments ￣mn and forces ￣vn are prescribed. Other

combinations of static boundary conditions are possible as well. Geometric

boundary conditions like w = 0 (hinged support) or w = w,n= 0 (clamped

edge) are of no concern for the purpose of this principle.

To construct a subset V c

h of the space V c, a moment tensor Mp is chosen

which satisfies the equilibrium condition (5.27) and the static boundary

conditions (5.28), and this tensor Mp is paired with a string of homogeneous

tensors Mi

V c

h

≡Mp ⊕

_

i

σiMi − div2Mi = 0, (5.29)

where homogeneous also implies that the tensors Mi satisfy the boundary

conditions (5.28) in a homogeneous form, i.e., ￣vn = ￣mn = 0 on ΓN.

This is the same solution technique as in the force method, where

M = M0 + XiMi −M

__

0 = p −M

__

i = 0. (5.30)

The tensor Mp corresponds to the bending moment M0 of the primary state

and the σi are the redundants Xi.

If the bending moments mkj in the tensorsMi are first-order polynomials,

the equilibrium condition −div2Mi = 0 is satisfied in each element, but the

bending moments are discontinuous at the interelement boundaries, which

violates the definition of Vc.

To overcome this obstacle the continuity requirement for the bending moments

is added to the functional Πc, using Lagrange multipliers:

Πc(M, w) = −1

2

_

Ω

M•C

−1[M] dΩ

+

_

i

_

Γi

[(m+n

− m

−

n ) w,n +(v+

n

− v

−

n )w] ds +

_

k

Fk w(xk) , (5.31)

5.4 Hybrid elements 427

where the Γi are the interelement boundaries and the superscripts + and -

denote the force terms on the left- and right-hand side of the interelement

boundary. Each Fk is the sum of the corner forces (discontinuity of mnt) of

the single elements at the nodes xk. The two Lagrange multipliers are the

deflection w and the rotation w,n at the interelement boundary Γi.

The corner forces Fc appear in Green’s first identity if the boundary integral

of mnt ˆ w,s is integrated by parts

_

∂Ωe

mij ni tj∂t ˆ wds = −

_

c

Fc ˆ w(xc) −

_

∂Ωe

d

ds

(mij ni tj) ˆ wds . (5.32)

Here ∂Ωe is the edge of the element and the xc are the corner points of the

element. If this procedure is reversed, the corner forces can be brought under

the integral sign, and because the normal vectors on two adjacent element

edges point in opposite directions, the sum of (v+

n

− v−

n )w, etc., over the interelement

boundaries Γi can be written as a sum over the element boundaries

Γe, so that the following equation is equivalent to Eq. (5.31)

Πc(M, w) = −1

2

_

e

_

Ωe

M•C

−1[M] dΩ

+

_

e

_

∂Ωe

[wqn − w,n mn − w,smnt] ds . (5.33)

The FE program constructs the tensorsMi by linearly interpolating the bending

moments mij . This guarantees that the equilibrium condition

− div2M = div (div M) = mij ,ji

= m11,11 +m12,12 +m21,21 +m22,22 = 0 (5.34)

is satisfied in each element. The deflection w at the edge is interpolated with

cubic polynomials, and therefore the complementary energy of an element

becomes

Πc = −1

2 σT Bσ + σT Cw, (5.35)

with σ and w as the nine nodal variables of the bending moments, and the

nodal deformations wi, w,i

x , w,i

y. The condition that the first variation with

respect to the parameters σi of the bending moments must vanish,

∂Πc

∂σ

= −Bσ +Cw = 0 , (5.36)

implies σ = B

−1Cw, and therefore the stiffness matrix can be expressed in

terms of the nodal values of the deflection w alone:

K(9×9) = CT B

−1 C . (5.37)

428 5 Slabs

Fig. 5.12. Support reactions of a hinged trapezoidal slab subject to a uniform load.

a) At an obtuse corner point the support reactions become singular. b) These effects

vanish if the rotations w,x and w,y are set free at these points

By this detour a stiffness matrix in terms of the natural nodal degrees of

freedom of a triangular plate element is derived. The whole technique is very

similar to the derivation of a DKT element, and as in the case of a DKT

element, a consistent approach for the calculation of the equivalent nodal

forces is not defined. Theoretically we should proceed as in the force method,

where the terms on the right-hand side (called δi0 in the force method and not

fi) are the scalar product between the bending moment M0 and the bending

moments Mi,

δi0 =

_ l

0

M0Mi

EI

dx , (5.38)

that is, the scalar product of the bending moment tensorMp of the particular

solution and the tensors Mi should define the equivalent nodal forces:

fi =

_

Ω

Mp •Mi dΩ . (5.39)

But instead, to achieve the “same effect” a) the deflection w that originally

only lives on the interelement boundaries is continued by a choice of appro-

29.9

30.8

29.0

30.8

516.0

563.1

29.3

563.1

-5.1

-5.1

5.5 Singularities of a Kirchhoff plate 429

5.13. Hinged

singular, mxx →−∞

and myy → +∞

priate global shape functions wi(x) across the whole slab, and b) the scalar

product of the load and these shape functions is calculated

˜ fi =

_

Ω

pwi dΩ . (5.40)

In practice this means that the principle of minimum complementary energy

is only used to derive the stiffness matrix, and that we later switch back to

the principle of minimum potential energy.

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