3.3 Finite elements and the slope deflection method

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If a frame is analyzed with an FE program, is the solution an approximation

or is it exact?

The solution is exact if the frame could also have been analyzed with the

slope deflection method, because in standard frames—no tapered beams, constant

stiffness—the FE result is identical to the results of the slope deflection-

The deflection curve w = w0 + wp of a beam can be split into two parts,

a homogeneous deflection curve w0 and a particular deflection curve wp (see

Fig. 3.15), where

w0 = u1 · ϕ1(x) + u2 · ϕ2(x) + u3 · ϕ3(x) + u4 · ϕ4(x) (3.61)

and where wp is a solution corresponding to fixed ends.

The homogeneous solution solves EIwIV

0 = 0, while the particular deflection

curve wp solves the equation EIwIV

p = p. The deflection w0 “carries” the

end displacements ui while the deflection wp “carries” the load p. The second

function is “mute” at the ends of the beam.

Correspondingly the deformation of a frame (see Fig. 3.16) can be split

into two parts: deformations resulting from the movement of joints (nodal

displacements), and local deformations, i.e., displacements between the nodes

caused by the distributed load p. The local deformations would also occur if

all nodes were fixed. The nodal displacements are the decisive terms, because

they establish the interaction between the individual beams.

Hence, if in the FE method the movements of a frame are expanded with

regard to the unit displacements ϕi of the nodes,

Fig.

method; see Fig. 3.14. The reason is that the reduction of the load into the

nodes does not change the nodal displacements, as is explained below.

290 3 Frames

3.16. The FE deforterms

of the nodal unit disuh(

x) =

_

i

ui · ϕi(x) , (3.62)

the local contributions wp are neglected, i.e., it is assumed that p = 0.

Because such unit displacements can only yield the exact shape if the load

is concentrated at the nodes, the FE method reduces all distributed loads

to the nodes. To this end, it lets the distributed load p act through the unit

displacements ϕi(x), and it places nodal forces ￣ fi at the nodes that contribute

the same amount of work,

δWe(p, ϕi) = ￣ fi · 1 , (3.63)

which means that the nodal displacements u satisfy

Ku = ￣f . (3.64)

But the vector ￣f on the right-hand side

￣f = f + p (3.65)

is identical to the right-hand side in the slope deflection method, because when

the nodes are released and the forces (reactio) which previously prevented any

movement of the nodes are applied in the opposite direction (actio), and when

the equilibrium position u of the frame is determined, then this system (3.64)

is solved.

The first vector f in (3.65) is the vector of the true nodal forces, i.e., the

concentrated loads applied directly at the nodes, while the second vector p

contains the equivalent nodal forces resulting from the distributed load. But

Fig.

mation is an expansion in

placements of the joints. The

expansion is exact if only

nodal forces act on the frame

3.3 Finite elements and the slope deflection method 291

Fig. 3.17. Beam and constant load

these equivalent forces are just the fixed end forces ×(−1) of the distributed

load, because

(−1) × fixed end forces = p i =

_ l

0

pϕi dx

=

_ l

0

pϕi dx = p i = equivalent nodal force .

(3.66)

For this to be true, it is of course necessary that the shape functions be the

exact unit displacements ϕi

Hence, an FE program which employs exact unit displacements is an

implementation of the slope deflection method (in one step). Whether the

distributed load is left out on the beam or reduced to the nodes makes no

difference—the nodal displacements are the same. This remarkable result only

holds for 1-D problems (ordinary differential equations—but see the remark

on the next page).

Of course, between the joints the exact solution (distributed load) and the

FE solution (equivalent nodal forces) differ, but this is of no consequence,

because (3.64) is only used to calculate the nodal displacements. To calculate

the stress resultants and the displacements in the individual beam elements,

for each element (e) we invoke the relation

Keue = f e + pe ⇒ f e = Ke ue − pe . (3.67)

This provides the beam end forces f e that belong to the end displacements

ue and once the fe

i are calculated, the internal actions between the nodes can

be calculated with influence functions or transfer matrices.

A study of the displacements of a simple one span beam with a constant

load will explain this (see Fig. 3.17). The beam has a length of 15 m, the

bending stiffness is EI = 34, 167 kNm2, and the applied load is p = 10 kN/m.

The following table shows results for an analysis with one or two elements.

Exact 1 element 2 elements

Max. moment 281.25 281.25 281.25 kNm

End rotations 41.147 41.147 41.147

Center deflection 19.2876 15.4301 19.2876 mm

of the beam, because these functions are also the influence

functions for the fixed end forces. This duality is the reason why the two

sides in (3.66) are the same, why the equivalent nodal forces are also the fixed

end forces.

292 3 Frames

The maximum bending moments and the rotations are identical, but the

center deflections are not. This is no surprise, given that the shape functions

are cubic polynomials. The deflection curve must be a symmetric function

with respect to the center of the beam, it must be a second-order polynomial!

The end rotations are correct, because a reduction of the distributed load into

the nodes will have no effect on the nodal displacements.

Remark 3.1. The necessary condition that the FE solution interpolates the

exact solution at the nodes is that the Green’s functions of the nodes lie in

Vh. The Green’s functions are piecewise homogeneous solutions. In equations

such as

− EAu

__(x) + cu(x) = p(x) EI wIV (x) + cw(x) = p(x) (3.68)

the homogeneous solutions

u(x) = c1 ex

√

c/EA + c2 e

−x

√

c/EA (3.69)

and

w(x) = eβ x(c1 cosβ x + c2 sinβ x) + e

−β x(c3 cosβ x + c4 sinβ x) (3.70)

β = 4

%

c

EI

(3.71)

are not the typical shape functions of the FE space Vh. Hence, in these cases

the FE solution does not interpolate the exact solution at the nodes. Theoretically

it can also happen that Vh is too “smooth” i.e., for to generate a Green’s

function of the equation −EAu__ = p, we must allow for a discontinuous firstorder

derivative at the nodes. The FE space Vh of an Euler–Bernoulli beam

would not do us the favor.

Hence, in some sense in FE analysis we must strike a balance between

the regularity that is required by the energy and the non-regularity that is

necessary in order to come close to the Green’s functions. And the latter we

achieve by letting h → 0 because then we can model a nearly infinite slope,

1/h→∞, with a nodal unit displacement u = 1.

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