Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

49, 4, pp. 1079-1100, Warsaw 2011

TOPOLOGICAL CLASSES OF STATICALLY DETERMINATE

BEAMS WITH ARBITRARY NUMBER OF SUPPORTS

Agata Kozikowska

Faculty of Architecture, Bialystok University of Technology, Białystok, Poland

e-mail: a.kozikowska@pb.edu.pl

The paper presents all topologies of statically determinate beams with ar-
bitrary number of pin supports. The geometry of each beam with a fixed
topology is optimized by a genetic algorithm,with absolutemaximummo-
ment as the objective function. An equality relation between minimum
values of this function is defined on the set of all topologies as an equiva-
lence relation.This relation partitions the set of topologies into equivalence
classes, called topological classes, for uniform, linear and parabolic gravity
loads. An in-depth description of these classes is provided. Exact formulas
for optimal locations of supports andhinges are found for the uniform load.

Key words: statically determinate beams, topology optimization, geometry
optimization, equivalence classes

Notations

cE,cH,cEH – number of external, internal and all cantilevers
g – number of optimal geometry variants
l, lE, lH,L – lengths of optimal beam segments and length of beam, see

Fig.3
m – number of optimal moment diagrams
Mi,M

n
i – optimal moment value of topology ti and class T

n
i

n,p,r – number of supports, topological classes and no-support
bars

q – intensity of evenly distributed load
{rn} – sequence of class moment ratios
R – equivalence relation of beam topologies
t,ti – beam topology, i=1,2, . . . , |Tn| or i=1,2, . . . , |T2:n|
ti – topological code of support i, i=1,2, . . . ,n
tM – number of topologies with the same optimal moment dia-

gram



1080 A. Kozikowska

Tn,T2:n – set of all topologies with n supports andwith two to
n supports

Tni ,T
2:n
i – topological class with n supports and with two to n

supports
|Tn|, |T2:n|, |Tni | – number of topologies in set Tn,T2:n and class Tni
{Tnk} – sequence of topological classes
x – axial coordinate
yi – dimensionless length of cantilever, i=1,2, . . . ,n
zi – dimensionless length of span, i=1,2, . . . ,n−1
(·)n,(·)2:n,(·)ni – quantities in set Tn,T2:n and class Tni

1. Introduction

Structural topology optimization has been identified as one of the most chal-
lenging and economically the most rewarding tasks in structural design. It is
of substantial practical importance because it can achieve much greater sa-
vings than geometry (shape) or sizing (cross-section) optimization (Kirsch,
1989; Rozvany et al., 1995). Topology optimization may not only considera-
bly enhance the design, but also provide the best configuration for further
comprehensive shape and sizing optimization (Bojczuk and Szteleblak, 2006).

The idea of topology optimization can be extended to support position de-
sign. Design of the optimal support layout is studied in Zhu and Zhang (2006,
2010). Beams are among themost important structuralmembers, particularly
statically determinate cases form the basis of solid mechanics (Pedersen and
Pedersen, 2009). The optimization of support locations of beams can be found
in Mróz and Rozvany (1975), Imam and Al-Shihri (1996), Wang and Chen
(1996), Bojczuk and Mróz (1998), Won and Park (1998), Mróz and Bojczuk
(2003), Wang (2004, 2006), Friswell (2006), Jang et al. (2009). However these
papers concern continuous beams which have a single bar with all supports
attached to it. Therefore, the topology optimization problem which consists
in selecting the pattern of member connections does not concern them. By
contrast, statically determinate beams are chains of bars joined by hinges and
placed on pin supports. There can be numerous associations between the bars
and the supports, but some associations can produce wrong forms. What we
need is a constructive rule for generating only the correct topologies. For the
case ofGerber beams – inwhich all supports aremoved away from the ends of
bars – the topologies were constructed by Golubiewski (1995) in the form of
directed graphs.The problemof construction ofGerber and non-Gerber beam



Topological classes of statically determinate beams... 1081

topologies was solved by Rychter (Rychter and Kozikowska, 2009) in a much
simpler and direct manner.

The optimal design in topological optimization is usually sought in a class
of domains, but not in the full domain. Such an approach does not guarantee
optimality, because terminal topologies depend on a initial layout, which is
adoptedarbitrarily. In this paper, the space of all possible candidate topologies
is known, exhaustive search in this space is carried out and global optima are
found.

Before this exhaustive search is performed, values of the merit function,
which ranks beam topologies, are found by geometry optimization of each
beam with a fixed topology. In most of the practical beam design problems,
reducing the maximal bending moment is of paramount importance (Wang,
2006). Therefore, the objective function in this geometry optimization process
has been defined as the absolute maximum moment for the uniform, linear
and parabolic load. The function is multi-modal, non-smooth, which means
that traditional, gradient-based optimization algorithms fail and much more
robust, randomized search techniques must be employed. Among methods of
probabilistic optimization, genetic algorithms (Goldberg, 1989) have been wi-
dely used because of the simplicity of the search mechanism. Many studies
on optimizations of different structures by genetic algorithms have been re-
ported in the literature, including beam structures (Wang and Chen, 1996;
Lyu and Saitou, 2005). Therefore, a modified version of the specialized gene-
tic algorithm (Rychter and Kozikowska, 2009) has been applied in this paper
to optimize beam geometries for all topologies. For best performance, the al-
gorithm was written by the author in the efficient C programming language
(Kernighan and Ritchie, 1988).

Searching for global optima in the space of topologies can be based on a
more effectivemethod than exhaustive search.However, the aim of this article
is to findnot only the best topologies, but also to discover the structure of this
space.An equality relation betweenminimumvalues of the absolutemaximum
momenthas beendefinedon the set of all topologies as an equivalence relation.
This relation splits that set into a sequence of topological equivalence classes.
Typical features of these classes are extensively discussed.

2. Beam topology

The subject of the paper is the set of all statically determinate beams, resting
on a fixed number of pin supports (Fig.1) or a number of pin supports varying
within a certain interval. The beams only carry loads perpendicular to their



1082 A. Kozikowska

longitudinal axes. Such beams do not undergo horizontal displacements and
forces. Therefore, statical determinacy is secured without introducing roller
supports.

Fig. 1. Beams with topological code: 0 – support at the bar end, 1 – support moved
left of the bar end, 2 – support moved right of the bar end

A statically determinate beam with n pin supports has n− 1 bars. The
bars have n endpoints, two external ends and n− 2 internal, hinged ends.
The topology of a statically determinate beam is represented by a vector of
topological codes of supports

t= [t1, . . . , tn] (2.1)

The topological code ti describes the location of support i relative to the end
of the bar (terminal supports) or two adjacent bars (intermediate supports),
Fig.1. This topology-coding scheme is presented in Rychter and Kozikowska
(2009).

The size |Tn| of the set Tn of all n-support beam topologies

|Tn|=2 ·3 · . . . ·3
︸ ︷︷ ︸

n−2

·2=4 ·3n−2 (2.2)

and the size |T2:n| of the set T2:n of all topologies of beams with two to n
supports

|T2:n|=
n
∑

i=2

|Tn|=
n
∑

i=2

(4 ·3i−2)= 2 ·3n−1−2 (2.3)

grow exponentially with the number of supports.

3. Beam geometry

The geometry of a beam is represented by a set of 2n− 1 parameters di-
vided into two groups. The parameters zi represent the dimensionless leng-
ths of spans between neighbouring supports. The parameters yi describe the



Topological classes of statically determinate beams... 1083

dimensionless external cantilever lengths and the internal cantilever lengths
to span length ratios (Fig.2).

Fig. 2. Beam geometry: span lengths zi, cantilever lengths yi

We assume in this work that all the beams have the same length L, nor-
malized to unity

L= y1+z1+z2+ . . .+zn−1+yn =1 (3.1)

Amoredetailed descriptionof thegeometrical parameters is given inRych-
ter andKozikowska (2009). The number of nonzero terminal cantilevers (non-
zero parameters y1 and yn) is equal to cE. The number of nonzero internal
cantilevers (nonzero parameters yi for i∈{2, . . . ,n−1}) is equal to cH.
Theminimumnumberof geometric variables equals thenumberof supports

and hinges, 2n− 2. We use 2n− 1 variables zi, yi subjected to constraint
(3.1), because this approach is better for geometry optimization by a genetic
algorithm.

4. Equivalence relation of beam topologies

4.1. Geometry optimization of the beam with a fixed topology

In this studywe concentrate on the topological complexity of beams,which
grows exponentially with the number of supports. Therefore, we use simple
gravity load distributions: uniform, linear and parabolic.
Let us consider a beam of unit length, with a fixed topology t, under the

gravity load. The beam is optimized with respect to geometrical variables.
This optimization problemmay be stated as follows

Minimize max
x∈[0,1]

|M(zi,yj,x)|
(4.1)

Subject to











0<zi < 1 i=1,2, . . . ,n−1
0<yj < 1 for tj 6=0 j=1,2, . . . ,n
y1+z1+z2+ . . .+zn−1+yn =1



1084 A. Kozikowska

where maxx∈[0,1] |M(zi,yj,x)| is the objective function representing themaxi-
mum of the absolute bendingmoment, zi are the span lengths, yj denote the
nonzero lengths of cantilevers, which are created bymovements of supports of
nonzero topological codes and x is the axial coordinate. The total number of
designvariables equals the sumof thenumberof spans n−1and thenumberof
nonzero external and internal cantilever lengths cE and cH, respectively. For
a homogeneous beam with a uniform cross-section this optimization process
corresponds to design for minimumweight.
A modified version of the genetic algorithm (Rychter and Kozikowska,

2009) is used for optimization of geometrical parameters for all topologies ti,
where i=1,2, . . . ,4 ·3n−2 or i=1,2, . . . ,2 ·3n−1−2 in accordance with Eq.
(2.2) or Eq. (2.3), respectively. Chromosomes representing n-support beam
withfixed topology are vectors of (n−1)+(cE+cH) real genes zi and yj. Such
chromosomes are compact and suitable for genetic operations, particularly
crossover andmutation. Theminimal value of the absolutemaximumbending
moment Mi is found as a result of geometry optimization of each beam with
topology ti.

4.2. Definition of equivalence relation of beam topologies

T is the set of beam topologies: Tn or T2:n. We define an equivalence
relation R on the set T. Any two topologies ti and tj of the set T are
equivalent with respect to the relation R if the values of the optimalmoments
Mi and Mj of these topologies are equal

ti ≡R tj if Mi =Mj (4.2)

The relation R is an equivalence relation because R satisfies the conditions of
reflexivity, symmetry and transitivity. The relation R partitions the set Tn

into disjoint subsets Tni called equivalence classes of beam topologies or topo-
logical classes. Parameters which concern the class Tni have the superscript n
and subscript i. Similarly, the relation R splits the set T2:n into topological
classes T2:ni .

5. Topological classes for a fixed number of supports under

a uniform load

5.1. Optimal bending moment diagram for a fixed topology

Abeam of length L from the class Tni , with optimal geometry for a fixed
topology, is shown inFig.3.Thebeamis foundwithunique, optimal, uniformly



Topological classes of statically determinate beams... 1085

distributed bending moment diagram. All cnE,i + c
n
H,i +n− 1 local extreme

moment values, at cnE,i+c
n
H,i supports, whichweremoved away from the ends

of bars, and in n−1 spans, are equal to Mni .

Fig. 3. A beamwith optimal geometry for a fixed topology from the class Tn
i
under

a uniform load

Geometry optimization of statically determinate beamswith a fixed topo-
logy, with the absolute maximum bendingmoment as the objective function,
can be found in Imam andAl-Shihri (1996) andWang (2006). Results of such
optimization tasks are given in Siegel (1962), Salvadori andHeller (1975), Ko-
lendowicz (1993) and Allen and Zalewski (2010). All these authors state that
to obtain the optimal geometry, it is desirable to equate themoment absolutes
at the supports and in spans.

Thepaper introduces formulas thatallowonetocalculate theexact optimal
geometry for a fixed topology under a uniform load.We can find values of the
parameters lni , l

n
E,i and l

n
H,i (see Fig.3) solving the system of equations

(n−1)lni + cnE,ilnE,i+(cnE,i+2cnH,i)lnH,i =L
1

2
lni − lnE,i =0 (lni )2−4lni lnH,i−4(lnH,i)2 =0

(5.1)

where lni is the length of each beam segment with the bottom in tension,
lnE,idenotes the lengthof eachnonzero external cantilever and l

n
H,i is the length

of each nonzero internal cantilever or the distance between the zero-moment
point insidea spanandtheclosest support.Thefirst equation in (5.1)describes
the total length of the beam. The second equation represents the comparison
between the lengths of a cantilever anda simply supportedbeamwith the same
values of the absolute maximum moment. The maximum bending moment
value of a simply supported beam of the length lni +2l

n
H,i equals twice this

value of a simply supported beam of the length lni in accordance with the
third equation.



1086 A. Kozikowska

The solution to system (5.1) is given by

lni =
L

dni
lnE,i =

L

2dni
lnH,i =

(
√
2−1)L
2dni

(5.2)

where dni =n+0.5
√
2cnH,i+(

√
2−1)cnH,i−1.

The value of the absolute maximum bending moment Mni can be calcu-
lated as the moment in the middle of a simply supported, uniformly loaded
beam of the length lni

Mni =
1

8
q(lni )

2 (5.3)

The publications about optimization of statically determinate beamswith
a uniform load do not contain exact formulas for optimal geometrical pa-
rameters, only give approximate values. In Fig.4, the exact moment values
calculated from Eq. (5.3) are compared to results of other authors. The ma-
ximum moment values for different optimization tasks are assumed to be all
equal to 100%. The comparison reveals the clear advantage of the accurate
solutions, found by the author.

Fig. 4. Comparison of optimization results: 1 – two-support beam from Salvadori
and Heller (1975), 2 – six-support beam fromKolendowicz (1993), 3 – two-support
beamwith one cantilever fromAllen and Zalewski (2010), 4 – two-support beam

with two cantilevers fromAllen and Zalewski (2010)

Figure 3 presents only one variant of topology. An optimal moment dia-
gram can be equivalent to many topologies with unsupported hinges at va-
rious points of zero moment, left or right of supports (Fig.5). The number of
different topologies with the same optimal moment diagram tnM,i equals the
number of combinations of cnH,i unsupported hinges locations

tnM,i =2
cn
H,i (5.4)



Topological classes of statically determinate beams... 1087

An optimal moment diagram can correspond to evenmore numerous optimal
geometrical parameter sets, related to the locations of single hinges within
spans with two zero-moment points inside (Fig.5).

Fig. 5. All topologies and geometries corresponding to the samemoment diagram:
(a) topology [2,2,2,0], (b) topology [2,2,1,0], (c) topology [2,1,1,0],

(d) topology [2,1,2,0]

5.2. Features of beam topologies and geometries in a topological class

All optimal bending moment diagrams from a topological class, under a
uniform gravity load, are shown in Fig.6.

Fig. 6. All optimal moment diagrams in the class T5
8
under a uniform load

The quality measure of the class Tni is the value of moment M
n
i , which

is dependent on the length lni , in accordance with Eq. (5.3). The length l
n
i

depends on parameters cnE,i and c
n
H,i (see Eq. (5.2)). Thus for two topologies

ti and tj of the set T
n under a uniform load the equivalent condition from

Eq. (4.2) can be expressed as

ti ≡R tj if cE,i = cE,j and cH,i = cH,j (5.5)



1088 A. Kozikowska

where cE,i, cH,i, cE,j, cH,j are the numbers of external and internal cantilevers
for the topology ti and tj, respectively. All topologies of uniformly loaded
beams in the class Tni have the same values of parameters c

n
E,i and c

n
H,i. It

does not make any difference which supports are moved away from the beam
end andhinges. Thus, the total number of different bendingmoment diagrams
in the class Tni , m

n
i equals the product of binomial coefficients

mni =

(

2

cnH,i

)(

n−2
cnH,i

)

(5.6)

The total number of different topologies in the class Tni , |Tni | equals the
product of the number of diverse moment diagrams mni and the number of
moves of cnH,i unsupported hinges 2

cn
H,i

|Tni |=mni ·2
cn
H,i =

(

2

cnH,i

)(

n−2
cnH,i

)

2
cn
H,i (5.7)

The minimal value of the moment Mni , has the first class T
n
1 with all

supports moved away from the ends of bars. An algebraic formula from Eq.
(5.8), found by the author, determines the number of geometry variants in
this class. In each of n− 1 spans of each beam, there are two points of zero
bending moment. Depending on the beam topology, in a span there can be
none or one or two hinges placed at the zero-moment points. To discover this
formula, wemust solve the problem of finding all proper placements of n−2
hinges in 2(n−1) zero crossings of the moment diagram. A beam from this
class consists of three types of bars:with two, one or no supports.Abeamwith
n­ 2 supports has n−1 bars of which r=0,1, . . . ,Floor[n/2]−1 can be no-
support bars, where the function Floor[y] gives the greatest integer less than
or equal to y. Each pair of two-support barsmust be separated by exactly one
unsupported bar, thus giving a (2r+1)-element chain of r+1 two-support
bars and r no-support bars between them. Any number of one-support bars
can be placed anywhere before, inside and after the (2r+1)-element chain.
The chain of 2r+1 elements can be placed arbitrarily in n−1 locations of
bars, preserving their order in the chain, thus giving

(n−1
2r+1

)
combinations of

possible placements (topologies). The remaining (n−1)−(2r+1)=n−2−2r
locations of the total n−1 places are occupied by one-support bars. Each such
bar creates a spanwith one hinge inside and each such hinge has two possible
locations, at two points of zero moment. This yields 2n−2−2r combinations
of single hinge locations (variants of geometry) for each beam topology with



Topological classes of statically determinate beams... 1089

n supportsand rno-supportbars.Multiplying thenumberof topologies
(n−1
2r+1

)

by the number of variants of geometry 2n−2−2r for fixed r, summing over all
possible values of r and using Mathematica software package to simplify the
result, we get the number of optimal geometry variants in the class Tn1

gn1 =

Floor[n/2]−1
∑

r=0

(

n−1
2r+1

)

2n−2−2r =
3n−3
6

(5.8)

5.3. Comparison of topological classes

The whole set of four-support topological classes, under a uniform load,
with all optimal moment diagrams is presented in Fig.7.

Fig. 7. All four-support topological classes with their optimal moment diagrams
under a uniform load: (a) T4

1
, (b) T4

2
, (c) T4

3
, (d) T4

4
, (e) T4

5
, (f) T4

6
, (g) T4

7
,

(h) T4
8
, (i) T4

9

The set of all n-support classes is described by the set of all possible
ordered pairs (cnE,i,c

n
H,i) where c

n
E,i ∈ {0,1,2} and cnH,i ∈ {0,1, . . . ,n− 2}.

Thus the set of all classes is characterized by the Cartesian product of the



1090 A. Kozikowska

three-element and (n−1)-element sets. The total number of classes pn is the
product of the numbers of set members

pn =3(n−1) (5.9)

The set of n-support topological classes can be arranged by their optimal
moment values in the monotonically increasing sequence

{Tnk}
3(n−1)
k=1 = {T

n
1,T

n
2, . . . ,T

n
3(n−1)} (5.10)

The class Tni precedes the class T
n
j in the sequence {Tnk}, if Mni is smaller

than Mnj . For beamswith fewer than five supports, the class T
n
i precedes the

class Tnj if the class T
n
i has more cantilevers

cnE,i+ c
n
H,i >c

n
E,j + c

n
H,j (5.11)

or the class Tni has more external cantilevers with the same total number of
cantilevers

cnE,i >c
n
E,j and c

n
E,i+ c

n
H,i = c

n
E,j + c

n
H,j (5.12)

For beams with five or more supports, the topological class Tni precedes the
class Tnj if condition (5.11) or (5.12) is fulfilled unless the following condition
is satisfied

cnE,i =2 and c
n
E,j =0 and

cnE,i+ c
n
H,i = c

n
E,j + c

n
H,j −1= cnH,j −1

(5.13)

If the compared classes meet condition (5.13), then Tni immediately precedes
Tnj although the total number of cantilevers in T

n
j is greater by one than

in Tni (see Fig.8).

Fig. 8. Two successive topological classes under a uniform load: the class T5
6
(a)

precedes the class T5
7
(b) with the total number of cantilevers greater by one

The number of cantilevers in the two-support class T2i can be computed
from

c2E,i =3− i c2H,i =0 for i=1,2,3 (5.14)



Topological classes of statically determinate beams... 1091

The number of cantilevers in the three-support class T3i is given by the follo-
wing

c3E,i =Floor[(6− i)/2] c3H,i =Mod[i,2] for i=1,2, . . . ,6 (5.15)

where the function Mod[a,b] returns the remainder on division of a by b. The
formulas for the number of cantilevers in classes with at least four supports
are shown in Table 1.

Table 1. Number of cantilevers in n-support topological classes for n ­ 4
under a uniform load

Tni T
n
1 T

n
2 T

n
3 T

n
4 T

n
i , i∈ [5,3n−7] Tn3n−6 Tn3n−5 Tn3n−4 Tn3n−3

cnE,i 2 2 1 2 Mod[i+2,3] 0 1 0 0

cnH,i n−2 n−3 n−2 n−4 A 2 0 1 0
A=(3n−i+1−5Mod[i+2,3])/3

Let us consider the sequence of real numbers, {rn}∞n=2, whose members
are ratios of moment values of extreme classes

rn =
Mn
3(n−1)

Mn1
=
1

2

(2n+
√
2−2

n−1
)2

(5.16)

The sequence {rn} decreasesmonotonically and converges to the limit 2. This
means that the values of class moments become closer to each other with a
growing number of supports, but themoment value of theworst class Tn

3(n−1)
is always more than twice the value of the best class Tn1. A growing rappro-
chement between class moment values is also seen in Fig.9, which compares
the moment values Mni of all topological classes for beams with 2, 3, 4 and 5
supports.

The total number of different optimal moment diagrams in all n-support
topological classes mn is the sum of the number of diverse diagrams in each
class Tni ,m

n
i over all possible values of parameters c

n
E,i and c

n
H,i. Substituting

mni from Eq. (5.6) and simplifying by softwareMathematica, we obtain

mn =
2
∑

cn
E,i
=0

n−2
∑

cn
H,i
=0

mni =
2
∑

cn
E,i
=0

n−2
∑

cn
H,i
=0

(

2

cnE,i

)(

n−2
cnH,i

)

=2n (5.17)

An algorithmwhich calculates the total number of optimal geometry variants
in all n-support topological classes was written by the author. The algorithm



1092 A. Kozikowska

Fig. 9. Optimal moments in topological classes for a fixed number of supports under
a uniform load

generates all topological codes of n-supportbeamsandaddsup thenumbers of
geometry variants associated with them. The sequence of numbers of optimal
geometries was generated by this algorithm. The recurrence formula for the
total number of optimal geometries gn was found using the website search
engine for TheOn-Line Encyclopedia of Integer Sequences (http://oeis.org/).

gn =










4 for n=2

16 fo n=3

4gn−1−gn−2+1 for n> 3
(5.18)

The numbers of topologies and optimal geometry variants are shown in
Fig.10.

Fig. 10. Number of supports versus number of topologies and optimal geometry
variants



Topological classes of statically determinate beams... 1093

6. Topological classes for a fixed number of supports under a

non-symmetric linear or parabolic load

Beams with optimal geometry under a non-symmetric load havemoment dia-
gramswhose local extreme values are the same, like in the case of beamsunder
a uniform load. Optimal beams for a non-symmetric load, with almost equ-
al absolute values of the support and span moments, can be found in Siegel
(1962) andMróz and Rozvany (1975).

Under a non-symmetric load, there is only one moment diagram in each
class. The diverse topologies in the class arise only from different locations
of hinges in zero points of this single diagram. The number of topologies is
equal to 2

cn
H,i, according to Eq. (5.4). The number of classes is equal to the

total number of moment diagrams 2n, according to Eq. (5.17). Figure 11
shows consecutive topological classes under a linear load. The topologies from
Fig.11 formone class under a uniform load (Fig.6). For a non-symmetric load,
each topologywith cantilevers in different places belongs to another class. The
class Tni with the same number of external and internal cantilevers as in the
class Tnj , c

n
E,i = c

n
E,j and c

n
H,i = c

n
H,j, precedes the class T

n
j if the cantilevers

of the class Tni are created under a lower load. If the compared classes T
n
i

and Tnj have a different number of external and/or internal cantilevers, then
Tni precedes T

n
j if condition (5.11) or (5,12) is satisfied for n< 5or if condition

(5.11) or (5.12) is fulfilled unless condition (5.13) is true for n­ 5.

Fig. 11. Successive five-support topological classes with one internal and one
external cantilever under a linear load: (a) T5

18
, (b) T5

19
, (c) T5

20
, (d) T5

21
,

(e) T5
22
, (f) T5

23



1094 A. Kozikowska

7. Topological classes for a fixed number of supports under a

symmetric parabolic load

Figure 12 shows successive topological classes under a symmetric quadratic
load. All the topologies from these classes are members of one class under a
uniform load (see Fig.6) and belong to six classes under a linear load (see
Fig.11). For a symmetric quadratic load, there is a singlemoment diagram or
there are twomomentdiagrams in theclass.Thenumberof different topologies
in the class Tni is equal to 2

cn
H,i or 2

cn
H,i
+1
.

Fig. 12. Successive five-support topological classes with one internal and one
external cantilever under a symmetric parabolic load: (a) T5

12
, (b) T5

13
, (c) T5

14

We need to find the number of topological classes for a symmetric load.
The problem, solved by the author, is easiest to analyse for an odd and even
number of supports separately. For beams with n supports and cEH cantile-
vers (cEH = cE+cH), the number of all distinct optimal moment diagrams is

equal to
( n
cEH

)
. For odd n, among these diagrams

( Floor[n/2]
Floor[cEH/2]

)
are symme-

tric and
( n
cEH

)
−
( Floor[n/2]
Floor[cEH/2]

)
are non-symmetric. Symmetric diagrams form

classes independently, while each class with non-symmetric diagrams has two
such diagrams. We sum over all possible values of cEH and simplify using
Mathematica. Thus, the number of topological classes for an odd number of
supports is equal to

pnodd =
n
∑

cEH=0

1

2

[(

n

cEH

)

−
(

Floor[n/2]

Floor[cEH/2]

)]

︸ ︷︷ ︸

number of classes
with two moment diagrams

+
n
∑

cEH=0

(

Floor[n/2]

Floor[cEH/2]

)

︸ ︷︷ ︸

number of classes
with one moment diagram

=2n−1+2Floor[n/2]
(7.1)



Topological classes of statically determinate beams... 1095

For an even number of supports n and an odd number of cantilevers cEH,
all

( n
cEH

)
optimalmomentdiagramsarenon-symmetric. If nand cEH are even,

there are
( n/2
cEH/2

)

symmetric and
( n
cEH

)

−
( n/2
cEH/2

)

non-symmetric diagrams.

Replacing an odd cEH with 2k+1 and an even cEH with 2k, where k is
an integer, summing over all possible values of k and using Mathematica to
simplify, we finally get the number of classes for an even number of supports

pneven =

n/2−1
∑

k=0

1

2

(

n

2k+1

)

︸ ︷︷ ︸

number of classes
with two moment diagrams
for odd number of cantilevers

+

n/2
∑

k=0

1

2

[(

n

2k

)

−
(

n/2

k

)]

︸ ︷︷ ︸

number of classes
with two moment diagrams

+

n/2
∑

k=0

(

n/2

k

)

︸ ︷︷ ︸

number of classes
with one moment diagram

︸ ︷︷ ︸

even number of cantilevers

=2n−1+2n/2−1
(7.2)

The number of topological classes under a symmetric parabolic load can
be computed from one formula for any number of supports

pn =2n−1+2Floor[(n+1)/2]−1 (7.3)

8. Comparison of topological classes for a fixed number of

supports under different loads

Thenumbersof classes for three loading types are given inFig.13.Thenumber
of classes grows linearly with the number of supports for a uniform load and
exponentially – for the other two types of load.
The optimal moment values in all classes of three-support beams are pre-

sented in Fig.14. The resultants of all the loading types are the same. The
moment values are normalized relative to the largest value in the figure. The
order of topological classes for a uniform load remains the same for a non-
uniform load. If the topology ti belongs to a better class than the topology tj
for a uniform load, then the topology ti also belongs to a better class for
the other two loading types. If the topologies ti and tj are elements of the
same class for a uniform load and their topological differences only concern
movements of the same supports (one or more) in opposite directions, then



1096 A. Kozikowska

Fig. 13. Number of supports versus number of topological classes for different loads

Fig. 14. Comparison of three-support topological classes for different loads

they belong to the same class for the non-uniform load too. If the topologies
ti and tj aremembers of the same class for a uniform load and have different
supportsmoved away from the ends of bars, then the topologies aremembers
of different classes for a non-symmetric load but for a symmetric load they
belong to either the same class or different classes.

9. Topological classes for a different number of supports under

a uniform load

Let us consider the set T2:n consisting of beam topologies with two to n
supports and topological classes T2:ni . The plot in Fig.15 shows the moment
values M2:4i of all classes for beamswith two to four supports under a uniform



Topological classes of statically determinate beams... 1097

load. The classes contain topologies with two successive numbers of supports
(T2:46 , T

2:4
8 and T

2:4
12 ) or topologies with only one number of supports (the

remaining classes). The class T2:46 with its two optimal moment diagrams is
presented in Fig.16.

Fig. 15. Optimal moments in topological classes for a different number of supports
under a uniform load

Fig. 16. Class T2:4
6
with its optimal moment diagrams

The two topologies tki and t
k+1
j with the number of supports k and k+1,

where k∈{2, . . . ,n−1}, are in the same class if theymeet the condition

tki ≡R tk+1j if cE,i =2 ∧ cE,j =0 ∧ cH,i = d ∧ cH,j = d+1 (9.1)

where d∈{0,1, . . . ,k−2} and cE,i, cH,i, cE,j, cH,j are thenumbers of external
and internal cantilevers for the topology tki and t

k+1
j , respectively. Substitu-

ting the numbers of supports and cantilevers into Eq. (5.2), respectively for
both topologies, we get the same length of the beam segmentwith the bottom
in tension and the samemoment value fromEq. (5.3).

10. Conclusions

Topological optimization, which belongs to the complex area of discrete, com-
binatorial optimization, usually refers to finding the optimal layout of the
structure within a specified design domain. The goal of the present paper is



1098 A. Kozikowska

to study the whole space of statically determinate beam topologies and to
present not only the best topologies, but also to give the full description of
the whole space.
Because all topologies of statically determinate beams are known, an

exhaustive search of the space of beam topologies is carried out. The me-
rit function in this search is found as a result of geometry optimization of
each beam with a fixed topology. This optimization process is performed by
a genetic algorithm, with the absolute maximum bendingmoment as the ob-
jective function, for a uniform, linear and parabolic gravity load. The beams
with optimal geometry have uniformly distributedmoment diagrams for each
topology and load. Under a uniform load, the exact formulas for the locations
of supports and hinges of optimal beams have been found for all topologies.
An equality relation between minimum values of the absolute maximum

moment has been defined as the equivalence criterion for the classification of
beam topologies. This criterion partitions the whole space of topologies into
topological classes. Typical features of the classes have been found.
Theresults of thepresentworkcanbeusedasaguide to thebeamstructure

design. Topological classes found here are worthy of further research with
additional design variables, such as cross-sectional and material properties,
with other equivalence relations including constraints on strength, stiffness,
stability, with more complex load distributions, with multiple load cases and
with multiple objective functions.

Acknowledgements

The support fromBialystokUniversity of Technology under grant S/WA/5/07 is

gratefully acknowledged.

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Klasy topologiczne statycznie wyznaczalnych belek o dowolnej liczbie

podpór

Streszczenie

W pracy omówiono wszystkie topologie statycznie wyznaczalnych belek o do-
wolnej liczbie przegubowych podpór. Geometrię każdej belki o ustalonej topologii
zoptymalizowano za pomocą algorytmu genetycznego z bezwzględnie maksymalnym
momentem jako funkcją celu. Relację równości minimalnych wartości tej funkcji zde-
finiowano na zbiorze wszystkich topologii jako relację równoważności. Na podstawie
tej relacji dokonanopodziału zbioru topologii na klasy równoważności, zwane klasami
topologicznymi, pod równomiernym, linowym i kwadratowym grawitacyjnym obcią-
żeniem.Przedstawionoszczegółowącharakterystykętychklas.Znaleziono ścisłewzory
na optymalne położenie podpór i przegubów belek obciążonych równomiernie.

Manuscript received September 27, 2010; accepted for print January 19, 2011