5.2 The displacement model

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In an ideal slab model the unit deflections ϕi(x, y) of the nodes describe what

happens to the plate if a node deflects, w = 1, or if it rotates by 45◦ about

the y- or x-axis, w,x = 1 and w,y = 1.

These unit deflections form the basis of the trial space Vh, and the FE

solution is an expansion in terms of these 3n unit deflections,

wh(x, y) =

_3n

i=1

ui ϕi(x, y) , (5.12)

where the nodal degrees of freedom ui (in this sequence: u1 = w, u2 =

w,x , u3 = w,y etc.) denote a deflection and two rotations.

422 5 Slabs

A unit load case p i can be associated with each of these unit deflections

and so, as before, the FE load case can be considered a superposition of these

3n unit load cases,

ph(x, y) =

_3n

i=1

ui p i , (5.13)

where the weights ui are chosen in such a way that the FE load case ph is

work-equivalent to the original load case in terms of the 3n unit deflections

δWe(ph, ϕi) = δWe(p, ϕi) i = 1, 2, . . . 3n . (5.14)

Conforming shape functions are C1 but not C2, therefore the curvatures on

opposite sides of the interelement boundaries are different. In a beam, such

abrupt changes in the curvature would be attributed to the action of nodal

moments

− EI κ(x

−

i ) + EI κ(x+

i ) = M(x

−

i ) −M(x+

i ) = M(xi) , (5.15)

while here these discontinuities are attributed to the action of line moments

(kNm/m), and jumps in the Kirchhoff shear to the action of line loads (kN/m),

so that the attributes of a typical FE load case ph (as in Fig. 5.10) are

• line loads—along the interelement edges (kN/m)

• line moments—along the interelement edges (kN m/m)

• surface loads—on each element (kN/m2)

• (eventually) forces Pi—at the nodes (kN)

The forces Pi result if the corner forces Fe of the individual elements are

added at the nodes.

The slab in Fig. 5.10 gives the impression of such a load case ph. The original

loading consists of a uniformly distributed surface load p = 6.5 kN/m2.

The element is a conforming rectangular element with 16 degrees of freedom

which is based on the shape functions of a beam; see Eq. (5.17).

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