5.1 Kirchhoff plates

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The deflection w, the three curvature terms κij and the three bending moments

mij are coupled by a system of seven differential equations, which in

indicial notation reads

κij − w, ij = 0, (3 eqs.),

K{(1 − ν)κij + νκkkδ ij} + mij = 0, (3 eqs.), (5.1)

−mij ,ji = p , (1 eq.) .

The constant

K = Eh3

12(1 − ν2), h= slab thickness (5.2)

is the plate stiffness and ν is Poisson’s ratio.

This system is equivalent to the biharmonic differential equation

K(w,xxxx +2 w,xxyy +w,yyyy ) = KΔΔw = p (5.3)

for the deflection surface w(x, y) of the slab.

The similarity of (5.3) to the beam equation EIwIV = p is obvious. As in

a beam, the bending moments in a slab are proportional to the curvature of

the deflection surface (see Fig. 5.3)

5.1 Kirchhoff plates 417

Fig. 5.4. Bending moments and shear forces

mxx = −K(w,xx +ν w,yy ), myy = −K(w,yy +ν w,xx ) ,

mxy = −(1 − ν)Kw,xy ,

and the shear forces are proportional to the third derivatives:

qx = −K(w,xxx +w,yyx ), qy = −K(w,xxy +w,yyy ) .

The bending moments mxx are accounted for by reinforcement in the xdirection

and the bending moments myy by reinforcement in the y-direction.

The extreme values of the normal curvature κ11 and κ22 at a given point

on a surface are called the principal curvatures, and they occur in the direction

tan 2 ϕ =

2myy

mxx − myy

. (5.4)

The principal curvatures determine the maximum and minimum bending of

a slab at any given point (see Fig. 5.5):

mI,II = mxx + myy

2

±

._

mxx − myy

2

_2

+ m2

xy . (5.5)

In an arbitrary direction the resultant stresses are, in indicial notation,

mnn = mij ni nj, mnt = mij ni tj, qn = qi ni ,

where n = [nx, ny]T is the normal vector and t = [tx, ty]T = [−ny, nx]T is the

tangent vector.

All resultant stresses are resultant stresses per unit length; see Fig. 5.4.

418 5 Slabs

Fig. 5.5. Slab: a) plan view, b) support reactions, c) principal moments

5.1 Kirchhoff plates 419

Fig. 5.6. Poisson’s ratio leads to

transverse compression in the flexural

sion in the flexural tension zone

Fig. 5.7. Influence of Poisson’s ratio ν on the bending moments, the deflection at

midspan and the corner forces F of a hinged slab. Plotted are the ratios with respect

to ν = 0. The bending moments increase if ν increases while the corner force F and

the deflection w decrease

Poisson’s ratio

Poisson’s ratio ν ensures that the concrete widens in the compression zone

and that it narrows in the tension zone; see Fig. 5.6.

The larger ν gets, the more the bending moments increase, while the deflection

and also the corner forces F decrease; see Fig. 5.7. The deflection w

becomes smaller because the slab stiffness K increases; see Eq. (5.2).

In uncracked concrete ν has a value of about 0.2. Many tables are based

on ν = 0.0, and the bending moments for ν       = 0 are obtained with

mxx(ν) = mxx(0) +ν myy(0), myy(ν) = myy(0) +ν mxx(0) . (5.6)

The effects of an incorrect Poisson’s ratio ν may well exceed the approximation

error, as the following example of a quadratic slab with clamped edges

shows. The correct value of Poisson’s ratio is assumed to be ν = 0. Plotted in

Fig. 5.8 is the error in the FE bending moment mxx = myy at the center of

the plate, as a function of the number of elements and ν. Obviously the error

due to deviations in Poisson’s ratio is larger than the approximation error. If

the analysis were based on ν = 0.3, the best FE result would overestimate the

bending stresses for ν = 0 by about 30%. Even at ν = 0.1, which is relatively

compression zone, and transverse ten420

5 Slabs

Fig. 5.8. Error in the

midspan FE bending

moment as compared

elements

Fig. 5.9. Slab with edge anchorage

close to the true value ν = 0 the error on a fine mesh with 144 elements is still

relatively high, 10.96%, while on a mesh with 16 elements but with the correct

value ν = 0 the approximation error is only 9.65%. These results suggest that

a correct assessment of the elastic constants and parameters is very important

for accurate analysis.

Equilibrium

The Kirchhoff shear vn rather than the shear forces qn, maintains the equilibrium

with the applied load. The Kirchhoff shear vn is the shear force qn plus

the derivative of the twisting moment mnt with regard to the arc length s on

the edge,

vn = qn + dmnt

ds

, (5.7)

so that along a vertical (|) or a horizontal (– ) edge, respectively,

vx = qx + dmxy

dy

vy = qy + dmxy

dx

. (5.8)

(ν = 0), as a function

to a series solution

of the number of

5.2 The displacement model 421

The additional force

dmnt

ds

_

kNm/m

m

= kN/m

_

(5.9)

is easily understood if the twisting moment is split into pairs of opposing

forces; see Fig. 5.3.

Because the equivalent nodal forces are work-equivalent to the resultant

stresses along the edge of an element, the nodal forces represent the action of

the Kirchhoff shear, and not of the shear forces qn.

The shear force qn and the twisting moments mnt are not zero along a free

edge, but they are so tuned that the change in mnt is balanced by qn, so that

the total effect vanishes

vn = qn + dmnt

ds

= 0. (5.10)

On a clamped edge the twisting moment is zero, mnt = 0, so the shear force

qn coincides with the Kirchhoff shear vn. For other support conditions the two

are not the same, qn     = vn, but the difference is usually not very large.

Corner force

The jump in the twisting moment at a corner point

F := m+

nt

− m

−

nt (5.11)

can be identified with a corner force F; see Fig. 5.3. If the corner is not held

down, no physical reaction is possible and the slab will tend to move away

from the support. This is the source of the “corner lifting” phenomenon that

can be observed on a laterally loaded square slab with simply supported edges

that do not prevent lifting; see Fig. 5.9. This effect does not appear if the edges

meeting at the corner point are clamped, mnt = 0.

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