1.20 Accuracy

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Each displacement u, v,w, and each stress or stress resultant

σxx, σxy, σyy mxx,mxy,myy, qx, qy (1.255)

the stress σ

1.20 Accuracy 81

Fig. 1.54. Influence functions for a) the deflection w, b) the slope w,x, c) the

bending moment mxx, and d) the shear force qx at the center of the slab

at a node or a Gauss point (see Fig. 1.53) is the scalar product of the associated

Green’s function Gj (the index j corresponds to the index j of the Dirac delta

δj) and the applied load p

mxx(x) =

_

Ω

G2(y, x) p(y) dΩy . (1.256)

The FE program replaces—as shown previously—the exact Green’s function

with an approximation Gh2

, with what it “considers” to be the exact Green’s

function, and therefore the error in the bending moments is proportional to

the distance between G2 and Gh2

:

mxx(x) − mh

xx(x) =

_

Ω

[G2(y, x) − Gh2

(y, x)] p(y) dΩy . (1.257)

This holds true for any other value as well.

Hence the real task of an FE program is not the solution of a single load

case, but an optimal approximation of the Green’s functions, because normally

more than one load case is solved on the same mesh. The truck drives on, but

the Green’s functions, being mesh-dependent, are invariant with respect to

the single load cases that are solved on the mesh, although the accuracy may

depend on where the truck is parked because the error in the Green’s function

varies locally.

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

82 1 What are finite elements?

The shift

In plate bending (Kirchhoff) problems the Green’s functions for

w, w,x , w,y ,mxx,mxy,myy, qx, qy (1.258)

at a specific point x are generated if a point source conjugate to w, w,x , w,y

etc. is applied at x. A source conjugate to w is a single force, a source conjugate

to w,x is a single moment, etc.

As can be seen in Fig. 1.54, the complexity of the deflection surfaces generated

by these sources increases with the order of the derivative. The higher

the derivative the more narrow and focused the influence function and therefore

the more difficult it is to approximate these surfaces with simple shape

functions.

This tendency of the influence functions has to do with the shift of the

kernel functions.

The surface load p that acts on a slab is (in somewhat simplified terms)

the fourth-order derivative of the deflection surface w. The influence function

for the deflection

w(x) =

_

Ω

G0(y, x)

_ _ _

kernel

p(y) dΩy (1.259)

extracts from p—which is the “fourth-order derivative”—the zeroth order derivative.

Thus the kernel G0 integrates four times. Its shift is of order −4.

Table 1.4. The kernels Gi and their shifts

Magnitude Derivative Kernel Action Shift

w(x) 0 G0 force -4

w,i (x) 1 G1 moment -3

mij(x) 2 G2 kink -2

qi(x) 3 G3 dislocation -1

Hence, influence functions are integral operators. They transform the load

p into the deflection w, the normal (or more general directional) derivative

w,n etc. The more they achieve, the more they integrate p, the more negative

the shift, and the more these functions spread in all directions; see Fig. 1.54.

Therefore if the deflection at a point x is to be calculated very precisely the

mesh must have the same quality everywhere, because the integral operator

integrates four times, although because the operator is a very smooth function

a coarse mesh probably is sufficient. But if the focus is on the shear force qx,

the mesh in the neighborhood of the point is critical, because the integral

1.20 Accuracy 83

Fig. 1.55. a) Influence function for the bending moment at x = l/2, and b) the

horizontal displacement at the same point. c) Influence function for the deflection

at the quarter point x = l/4. The dashed curves are the FE approximations

operator in the influence functions for the shear force has the low shift -1.

An operator which does nothing (it has shift zero) is the Dirac delta,

which is usually identified with a point force. Now it must be considered a

displacement, because all influence functions are displacements. In the case

of a Kirchhoff plate it would be the deflection surface w(x) = KΔΔg0(y, x)

with g0 = 1/(8πK)r2 ln r, which in a 3-D picture would be a lone peak

hovering over the mesh. This peak neither integrates nor differentiates what

is placed under the integral sign:

p(x) =

_

Ω

δ0(y − x) p(y) dΩy . (1.260)

It just reproduces the function. Its shift is of order zero. It is truly a local

operator.

There are also integral operators that differentiate, which have a positive

shift. If a bar is stretched by u units of displacement, then the normal force

N(0) is

N(0) = EA

u

l

. (1.261)

The operator u → N = EAu_ is an integral operator which differentiates.

Actually, to see an integral sign would require a 2-D model of the bar, but

evidently u/l is a difference quotient.

It integrates only once. This makes it a nearly local operator.

84 1 What are finite elements?

The bending moment in a beam fixed on the left-hand side and simply

supported on the right-hand side is, if the simple support moves downward

by δ units,

M(0) = −3 EI

l2 δ . (1.262)

This integral operator differentiates twice as can be seen from the l2 in the

denominator.

The maximum error in the Green’s functions

It is not easy to predict where the maximum error in the displacements or

stresses will occur, because the accuracy depends on the accuracy of the approximate

Green’s function and the nature of the load which is applied. It is

only guaranteed that if the exact Green’s function lies in Vh, then the value

will be exact. But the opposite need not be true: recall that if the exact solution

lies in Vh, then the stresses are exact even though the Green’s function

does not lie in Vh.

We thus concentrate on the error in the Green’s function alone. In Figure

1.55 two influence functions are plotted, one for the bending moment of a beam

at x = l/2 and the other for the longitudinal displacement at the same point

if the bar is stretched or compressed. The dashed curves are the approximate

Green’s functions if just one element is used. Obviously the maximum error

occurs at the source point x = l/2 itself. It is easy to see why this happens:

outside of the element Gi = Ghi

, and within the element Ghi

is essentially the

curve obtained if Gi is interpolated at the nodes with a third-degree polynomial

(a homogeneous solution), while the essential part wp, which contains

the peak, EI wIV

p = δi(y − x), is neglected. Obviously the distance between

the smooth interpolating function Ghi

and the peak is at its maximum at the

source point x. This is even more evident in plate bending problems. Recall

the infinite peaks in the influence functions for the bending moments mij in

a slab! Clearly the maximum error will occur at the source point.

Hence the stresses at the foot of a single force are the least reliable, independent

of the problematic nature of single forces. Can we say then that the

more a load is spread, the better the accuracy of the FE results? Do gravity

loads therefore have an advantage over traffic loads?

It is not guaranteed that the maximum error always occurs at the source

point. One counterexample is the influence function for the deflection at the

first quarter point of a beam with fixed ends; see Fig. 1.55 c. The FE approximation

Gh0

is zero, so that the maximum error is identical to the maximum

deflection of G0, which occurs at some distance from the quarter point. But

it can be assumed that this “mismatch” occurs only in influence functions for

displacements, and not resultant stresses, because the peaks in the latter functions

are more pronounced. But note that the influence functions for support

reactions are also of displacement type.

1.20 Accuracy 85

Fig. 1.56. Plate: a) system and loading; b) influence function for the stress σyy near

the corner point; c) influence functions for the internal action Ny in cross-section

A–A

Fig. 1.57. Influence function

for σyy and Ny

86 1 What are finite elements?

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