Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

52, 1, pp. 257-269, Warsaw 2014

TOPOLOGICAL CLASSES OF STATICALLY DETERMINATE BEAMS WITH
ARBITRARY NUMBER OF SUPPORTS UNDER THE MOST

UNFAVOURABLY DISTRIBUTED LOAD

Agata Kozikowska

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

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

The paper deals with topological classes of statically determinate beams with an arbitrary
number of pin supports. The beams carry piece-wisely distributed loads which are placed
in such a way that bending moment values are extreme at any section. For such loads, it is
sufficient to consider only two load cases with alternate spans uniformly loaded. Each beam
with a fixed topology is subjected to geometrical optimization with the absolute maximum
moment as the objective function. Exact formulas for optimal values of geometrical para-
meters are found for all topologies. An equality criterion between minimum values of the
objective function is used as an equivalence relation.On the basis of this relation, the set of
all topologies is divided into equivalence topological classes.Typical features of these classes
are found and discussed.

Key words: statically determinate beams, topology and geometry optimization, equivalence
classes; the worst load

Notations

cE,cH,cS,cT – number of external, internal cantilevers, number of segments lS, lT
g – number of optimal geometry variants
h,s – coordinates of hinges and of supports
l, lE, lH, lS, lT ,L – lengths of optimal beam segments and length of beam, see Fig. 2
m – number of optimal envelopes of moment diagrams
Mi,M

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

n
i

n,p – number of supports and of topological classes, respectively
q – maximum intensity of arbitrarily distributed gravitational load
{rn} – sequence of class moment ratios
R – equivalence relation of beam topologies
ti,t
n
i – beam topology, i=1,2, . . . , |Tn| or i=1,2, . . . , |T2:n|

ti – topological code of support i, i=1,2, . . . ,n
Tn,T2:n – set of all topologies with n supports and with two to n supports
Tni ,T

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

|Tn|, |T2:n| – number of topologies in set Tn,T2:n
{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



258 A. Kozikowska

1. Introduction

Beams are encountered all around us in many engineering applications. Statically determinate
beams are also widely used in engineering structures due to their many advantages. Their short
members are well suited for prefabrication, transportation and installation. In these beams, no
stresses are produced by changes of temperature, settlement of supports and imperfections of
assembly. Statically determinate beam cases are the basis of solid mechanics (Pedersen and
Pedersen, 2009). They have not been fully explored yet and still attract attention of researchers
(Golubiewski, 1995; Choi et al., 2004; Pennock and Alwerdt, 2007; Liu et al., 2009).

Topology optimization is a rapidly expanding research area of structural mechanics (Kirsch,
1989; Rozvany et al., 1995; Bojczuk and Szteleblak, 2006). Topological optimization of beams is
an important part of this area and can be found, among others, in articles ofMróz andRozvany
(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). These papers, however, do not concern multispan hinged beams and the assignment of
supports to bars. Topological optimization of statically determinate multispan hinged beams
with arbitrary many supports was the subject of the author’s earlier articles (Rychter and
Kozikowska, 2009; Kozikowska, 2011). The topology in these papers is understood as the way
how supports are connected to bars. The first paper introduces the space of these beams and a
genetic algorithm for their topology and geometry optimization. The secondpresents topological
classes of these beams. In both articles, the beams carry stationary loads which remain in a
fixed position. In many practical situations, however, beams are subjected to a load whose
position may vary. Such optimization tasks usually come down to problems with multiple load
cases. There is a limited number of papers about beam optimization involving multiple loading
conditions (Mayeda and Prager, 1967; Karihaloo and Kanagasundaram, 1988; Rozvany et al.,
1988; Bryant and Heinlein, 1994). Therefore, this article deals with topology and geometry
optimization of statically determinate beams under theworst piece-wisely distributed load. It is
assumed that the beamsunder consideration are loaded so slowly that the loadmay be regarded
as quasi-static. In order to determine themost unfavourable arrangements of this load, influence
lines are used. Since the beams primarily must resist bending due to action of transverse loads,
the absolute maximum bending moment was chosen as the objective function to rank beam
topologies, like in Wang (2006) and Xing and Wang (2012). According to influence lines, the
maximum possible value of this function in each cross-section of the beam corresponds to only
two quasi-static load cases with a distributed load, which covers all odd or all even spans.

Given the complexity of topology design spaces, topological optimization is usually not car-
ried out in the full domain. Since the whole space of statically determinate beam topologies
is known (Rychter and Kozikowska, 2009), exhaustive examination of all possibilities can be
performed and a division of this space into topological equivalence classes can be found. This
partition is based on the equivalence relation defined as an equality criterion between values
of the absolute maximum moment of beams with optimal geometry. Geometry optimization of
each beamwith a fixed topology was carried out by amodified version of the genetic algorithm,
which was presented in Rychter andKozikowska (2009). Typical features of optimal geometries
and exact formulas for optimal locations of supports and hinges are shown. A comparison of
topological classes for a uniform load and the most unfavourably distributed load is reported.

2. Beam topology and geometry

In this paper,we studyall statically determinatebeams restingonafixednumberof pin supports
or on a number of pin supports varying within a certain interval. Such beams were analysed
in Rychter and Kozikowska (2009) and Kozikowska (2011). To find the topology of a beam, we



Topological classes of statically determinate beams... 259

start with the simply supported beamwith all supports at the ends of the bars. Thenwe regard
each support as topologically moveable: it can bemoved inside its associated bar (first and last
support) or inside any of its two associated bars (intermediate supports). The topology of a
n-support statically determinate beam is a vector of these support shifts relative to the ends of
bars: move left (code 1), move right (code 2), and nomove (code 0)

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

The geometry of the beam is represented by a set of n−1 dimensionless lengths of spans be-
tween neighbouring supports (parameters zi), two dimensionless lengths of external cantilevers
(parameters y1 and yn) and n−2 ratios of internal cantilever lengths to span lengths (para-
meters yi for i∈{2, . . . ,n−1}). The size |Tn| of the set Tn of all n-support beam topologies
and the size |T2:n| of the set T2:n of all topologies of beams with two to n supports are given
in Kozikowska (2011).

3. Equivalence relation of beam topologies

3.1. Geometry optimization of the beam with a fixed topology

The beams considered in Kozikowska (2011) were subjected to stationary loading. In this
article, as in many practical situations, beams carry piece-wisely distributed loads that can
occupy different positions. The maximum intensity of the arbitrarily distributed gravitational
load is q. It is assumed that the rate of load changes is slow enough so that the load can be
considered as a quasi-static. The effects of such loads are studied bymeans of influence lines in
Fig. 1.

Fig. 1. A beamwith the twomost unfavourable cases of piece-wisely distributed load:
(a) and (b) influence lines for the bending moment at the section A and B, (c) the load that causes the
maximum possible value of bending moment with the top in tension in all sections of odd bars and with
the bottom in tension in all sections of even bars, (d) the load that causes the maximum possible value
of bending moment with the bottom in tension in all sections of odd bars and with the top in tension in

all sections of even bars

Weobserve that thebendingmoment reaches themaximumvaluewith the top or thebottom
in tension when piece-wisely distributed loads occupy all the spans of the beam, over which the
influence line does not change sign. Furthermore, themost unfavourable load out of loads of any
distribution is uniform of maximum intensity q. Themost dangerous loads can be extended to
some spans of the beam where the ordinates of the influence line are equal to zero. This gives
the two most unfavourable load cases with the load on all odd or all even spans, regardless of
the topology of the beam and the place where the bending moment is calculated, as shown in
Fig. 1. The first load case from Fig. 1c creates the largest possible moment values with the top
in tension in all sections of odd bars andwith the bottom in tension in all sections of even bars.
The second load case fromFig. 1d produces the largest possiblemoment values on the opposite



260 A. Kozikowska

side of he beam: for odd bars tension appears at the bottom fibres and for even bars at the top
fibres.
We consider a beamof unit length,with afixed topology.Thebeam is optimizedwith respect

to geometrical variables for both the most unfavourable load cases. This optimization problem
may be formulated as follows

Minimize max
x∈[0,1]

|M(zi,yj,x)|

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

(3.1)

where maxx∈[0,1] |M(zi,yj,x)| is theobjective function representingthemaximumof theabsolute
bendingmoment for both load cases, zi denote lengths of spans, yj represent nonzero lengths of
internal and external cantilevers with nonzero topological codes and x is the axial coordinate.
The total number of design variables is equal to the sumof the number of spans and the number
of nonzero external and internal cantilever lengths.
Optimization of geometrical parameters for all topologies ti for both load cases was made

usingamodifiedversion of the genetic algorithm(Rychter andKozikowska, 2009). Chromosomes
representing a beamwith afixed topology are vectors of real genes zi and yi. Such chromosomes
are compact and suitable for genetic transformations, particularly crossover, which operation is
hard to design for statically determinate beams. The minimal value of the absolute maximum
bending moment Mi is a result of geometry optimization of each beam with topology ti for
both load cases.

3.2. 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 optimal moments Mi and Mj of these topologies are equal

ti ≡R tj if Mi =Mj (3.2)

The relation R creates a decomposition of the set Tn into disjoint equivalence classes of beam
topologies or topological classes Tni and the set T

2:n into topological classes T2:ni , respectively.
Parameters from the class Tni have the superscript n and subscript i.

4. Topological classes for a fixed number of supports

4.1. Optimal envelope of bending moment diagrams for a fixed topology

Abeamof length L from the class Tni , with optimal geometry for a fixed topology, is shown
in Fig. 2. The beam is found with two unique bending moment diagrams, drawn with a solid
line or dashed line, for both themost unfavourable load cases. The optimal envelope of the two
moment diagrams has the same local extrememoment values equal to Mni . These values occur
at the bottom of the beam at mid spans or close to them, over all supports which were moved
away from the ends of bars and over whole spans which were created by moving away both
supports from the ends of bars towards themiddle of the bars. Bothmoment diagrams are never
on the same side of the beam. The envelope may have portions with non-zero values only on
one side of the beam. Such portions are cantilevers at the end of the beam and spans with two
zero moment points inside or at the end (such as hinges and/or ends of the beam). Zero values
of the envelope occur only in hinges and at both ends of the beam. Unsupported hinges can



Topological classes of statically determinate beams... 261

not change their locations without changing themoment diagrams like in the case of stationary
loading (Kozikowska, 2011) and the optimal envelope of moment diagrams is equivalent to only
one topology.

Fig. 2. A beamwith optimal geometry for a fixed topology from the class Tni

The paper provides exact formulas that allow one to calculate the optimal geometry for any
topology, under themost unfavourably distributed load. To get the values of the parameters lni ,
lnE,i, l

n
H,i, l

n
S,i and l

n
T,i (see Fig. 2) we need to solve the system of equations given below

(n− cnS,i−cnT,i−1)lni +cnE,ilnE,i+cnH,ilnH,i+ cnS,ilnS,i+ cnT,ilnT,i =L
1

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

lnS,i =
1

2
lni +
1

2

√

(lni )
2− lni lnH,i l

n
T,i =

√

(lni )
2− lni lnH,i

(4.1)

where lnE,i and l
n
H,i denote the lengths of nonzero external and internal cantilevers, respectively,

lni , l
n
S,i and l

n
T,i are the lengths of optimal beam segments between two closest supports and/or

hinges with a parabolic moment diagram at the bottom of the beam for one of the two load
cases. Zero moment values for this load case occur at both ends of the segment lni , at only one
end of the segment lnS,i and do not occur at any point of the segment l

n
T,i. The first equation

in (4.1) describes the total length of the beam. The parameters cnE,i, c
n
H,i, c

n
S,i, c

n
T,i are the

numbers of the segments lnE,i, l
n
H,i, l

n
S,i and l

n
T,i, respectively. The second equation represents

the comparison between the lengths of a cantilever and a simply supported beamwith the same
values of the absolute maximum moment. The maximum bending moment value of the simply
supported beam of the length lni +2l

n
H,i equals twice this value of the simply supported beamof

the length lni in accordancewith the third equation.The forth and the last equations in (4.1) are
computed from the system of equations (4.2) and (4.3) and are explained graphically in Fig. 3
and Fig. 4, respectively

lnS,i = l
n
i − lX

lnH,i
MX
=
lni
Mni

MX =
1

2
qlni lX −

1

2
q(lX)

2 Mni =
1

8
q(lni )

2 (4.2)

Fig. 3. Graphic explanation of equations (4.2)

The first equation in (4.2) specifies the difference of the segment lengths lni and l
n
S,i (see

Fig. 3). The second equation is obtained from the similarity of the right angle triangles with



262 A. Kozikowska

the bases lnH,i and l
n
i . The bendingmoment values MX and M

n
i are calculated for the simply

supported, uniformly loaded beam of the length lni : MX at the distance lX from the end of this
beam and Mni in the middle of this beam (the third and the last equation)

lnT,i = l
n
i −2lX

lnH,i
MX
=
lni
Mni

MX =
1

2
qlni lX −

1

2
q(lX)

2 Mni =
1

8
q(lni )

2

(4.3)

Fig. 4. Graphic explanation of equations (4.3)

The first equation in (4.3) determines the length of the segment lnT,i (see Fig. 4). The other
equations in (4.3) are the same as in (4.2).
The solution to system (4.1) is

lni =
L

dni
lnE,i =

L

2dni
lnH,i =(

√
2−1) L

2dni

lnS,i =

(

√

3
2
−
√
2
2
+1

)

L
2dn
i

lnT,i =
√

3
2
−
√
2
2
L
dn
i

(4.4)

where

dni =n+
1

2

(

cnE,i+(
√
2−1)cnH,i

)

+

(

√

3

2
−
√
2

2
−1
)

(1

2
cnS,i+ c

n
T,i

)

−1

The lengths of two-hinged spans of the optimal beam are equal to lni +2l
n
H,i. One-hinged

spans have the lengths of lni + l
n
H,i or l

n
S,i+ l

n
H,i. The lengths of spans without hinges are equal

to lni or l
n
S,i or l

n
T,i. The values of the parameters c

n
E,i, c

n
H,i, c

n
S,i, c

n
T,i can be calculated from the

beam topology tni (see Eq. (2.1)). The value of the parameter c
n
E,i is equal to the number of

non-zero elements in the first and last position of the code tni . The value of the parameter c
n
H,i

is the number of non-zero elements in positions 2 through n−1 of the code tni . The value of the
parameter cnS,i equals thenumberof pairs of neighbouring code elements equal to 11 or 22 (where
for example 111 denotes two pairs) excluding pairs in the sequence 2211. The parameter cnT,i
can be calculated as the number of the sequences 2211 in the topological code tni . Algorithms
to calculate the coordinates of supports and hinges (on the basis of the beam topology and the
lengths lni , l

n
E,i, l

n
H,i, l

n
S,i, l

n
T,i and L) are given by a pseudo code in Appendix.

4.2. Features of beam topologies in a topological class

All optimal bendingmoment diagram pairs from a topological class, under the most dange-
rous piece-wisely distributed load, are shown in Fig. 5.
All topologies in the topological class Tni have the same value of moment M

n
i , which is

dependent on the length lni , in accordance with Eqs. (4.2) and (4.3). According to Eq. (4.4),
the length lni depends on the number of supports and the values of the parameters c

n
E,i, c

n
H,i,



Topological classes of statically determinate beams... 263

Fig. 5. All optimal envelopes of moment diagrams in the class T5
10
: (a)-(d) c5E,10 =1, c

5
H,10 =3,

c5S,10 =2, c
5
T,10 =0, (e)-(f) c

5
E,10 =1, c

5
H,10 =3, c

5
S,10 =0, c

5
T,10 =1

cnS,i and c
n
T,i. In addition, the parameter c

n
S,i = 2 is equivalent to the parameter c

n
T,i = 1 (see

Fig. 5). Thus for two topologies ti and tj of the set T
n under themost dangerous piece-wisely

distributed load, the equivalent condition from Eq. (3.2) can be expressed as

ti ≡R tj if cE,i = cE,j ∧ cH,i = cH,j ∧ cS,i+2cT,i = cS,j +2cT,j (4.5)

where cE,i, cH,i, cS,i, cT,i, cE,j, cH,j, cS,j, cT,j are the numbers of appropriate segments for the
topology ti and tj, respectively.

4.3. Comparison of topological classes

Thewhole set of three-support topological classes, under themost unfavourably distributed
load,withall optimal envelopes ofmomentdiagrams ispresented inFig. 6.Thebetter topological
classes are, themore cantilevers and the less one-hinged spans their beams have. In other words,
better classes have larger values of the parameters cnE,i and c

n
H,i and smaller values of the

parameters cnS,i and c
n
T,i. The values of the parameters c

n
E,i, c

n
H,i, c

n
S,i and c

n
T,i for the classes

fromFig. 6 are given inTable 1. The best topologies in the first class Tn1 with an oddnumber of
supports have only one one-hinged span (see Fig. 6a and Fig. 7a). The number of topologies in
the best classes with the odd number of supports is equal to n−1. The best topological classes
with an even number of supports have only one topology, which does not have any one-hinged
spans (see Fig. 7b).

Table 1.Values of the parameters cnE,i, c
n
H,i, c

n
S,i and c

n
T,i in tree-support topological classes

T3i T
3
1 T

3
2 T

3
3 T

3
4 T

3
5 T

3
6 T

3
7

cnE,i 2 2 1 1 1 0 0

cnH,i 1 0 1 1 0 1 0

cnS,i 1 0 0 1 0 0 0

cnT,i 0 0 0 0 0 0 0

Information about the ways of hinge placement in statically determinate multispan beams
can be found in many books on structural mechanics (Darkov and Kuznetsov, 1970; Nowacki,
1976;Kolendowicz, 1993; Hartsuijker andWelleman, 2006; Gambhir, 2009, 2011; Karnovsky and
Lebed, 2010). The authors distribute hinges at spans and/or on supports of continuous beams
and then they study properties of generated statically determinate beams. They state that
by choosing an adequate place for the hinges, it is possible to influence the absolute maximum
bendingmomentpositively. Theynotice that thismoment in amultispanbeamwithhinges away
from supports is smaller than in a series of simple beamswith the same spans (when hinges are
on supports). Moreover, some authors recommend the hinge layout with the smallest possible



264 A. Kozikowska

Fig. 6. All three-support topological classes with their optimal envelopes of moment diagrams: (a) T31,
(b) T3

2
, (c) T3

3
, (d) T3

4
, (e) T3

5
, (f) T3

6
, (g) T3

7

Fig. 7. The first five- and six-support topological classes: (a) T51, (b) T
6
1

number of one-supportbars (Nowacki, 1976;Gambhir, 2009, 2011). This layout is optimal for the
absolutemaximumbendingmoment undermost unfavourably distributed load, but the authors
do not justify their recommendation in this way. They advise tomake the number of one-hinged
spans as small as possible because the failure of any part of such a beam does not cause the
entire beam to collapse.
An algorithm which calculates the number of topological classes for any number of support

waswritten by the author. The sequence of numbers of classes for a growing number of supports
wasgeneratedby this algorithm.Theclosed formof the formula for the total numberof classes pn

was found on the basis of this sequence usingMathematica software package

pn =
1

4
[6n2−12n+11+(−1)n] (4.6)

Let us consider the sequence of real numbers {rn}∞n=2 whosemembers are ratios of moment
values of extreme classes. Calculating for both classes lni fromEq. (4.4) and M

n
i fromEq. (4.3)

and dividing onemoment value by the other, we obtain

rn =
Mn
the last class
Mn
the first class

=
1

(n−1)2
[

n+
1

2
(
√
2−1)(n−2)+1

2

(

√

3

2
− 1
2

√
2−1
)

(nmod 2)
]2
(4.7)

The sequence {rn} converges to the limit (3+2
√
2)/4, which is smaller than the limit for statio-

nary uniform loading equal to 2 (Kozikowska, 2011). The moment value of the last topological



Topological classes of statically determinate beams... 265

class without any cantilevers is the same for both the most unfavourably distributed load and
uniform load. Themoment value of the first class is obviously greater for themost unfavourably
distributed load, which was selected from a variety of loads, including uniform over the entire
beam. Therefore, elements of the sequence corresponding to the same value of n and the limit
of the sentence are smaller for the most unfavourable load. The rapprochement between class
moment values with the growing number of supports is also seen in Fig. 8, which compares
moment values Mni of all topological classes for beams with 2, 3, 4 and 5 supports.

Fig. 8. Optimal moments in topological classes for a fixed number of supports

The number of different optimal envelopes of moment diagrams mn and the number of
optimal geometry variants gn in all n-support topological classes are equal to the number of
topologies |Tn|, because an optimal envelope corresponds to only one topology and only one
optimal geometry

mn = gn = |Tn|=4 ·3n−2 (4.8)

The formula for |Tn| is presented in Rychter and Kozikowska (2009).

5. Comparison of topological classes for a fixed number of supports under the
most unfavourably distributed load und stationary uniform load

Statically determinate beams which are optimal for the absolute maximum moment have uni-
formly distributed bendingmoment diagrams for the stationary load (Kozikowska, 2011), whe-
reas these beams have evenly distributed moment diagram envelopes for the worst piece-wisely
distributed load. Topological classes are usually different for the stationary load than for the
worst distributed load. Two topologies are members of the same class Tni under the stationary
load if they have equal values of the parameters cnE,i and c

n
H,i. For the most unfavourable load

in addition to these parameters, the values of the parameters cnS,i and c
n
T,i also decide on the

membership of topologies in the class Tni . The first two- and three-support topological classes
are the same in terms of topology for both types of loading. But optimal locations of supports
and hinges are different. If n > 3 then the first class for the most unfavourable load with all
supports shifted away from the ends of bars and with theminimal number of one-hinged spans
(one for odd n and zero for even n) is a part of the first class for the stationary loading. But
optimal beams from the first class under the stationary load have all spans of equal lengths,
while span lengths under themost unfavourable load are diverse. Two-hinged spans are the lon-
gest, one-hinged spans are shorter and the lengths of spanswithout hinges are the smallest. The
optimal lengths of external and internal cantilevers are shorter under the stationary loading.
The last topological classes with all supports under the ends of bars and with equal lengths of
spans for these two loads are identical both in terms of topology and optimal geometry.



266 A. Kozikowska

6. Topological classes for a different number of supports

Let us consider the set T2:n of beam topologies with two to n supports and topological classes
T2:ni . The plot in Fig. 9 shows the class moment values M

2:4
i for beams with two up to four

supports under the most unfavourably distributed load. The classes T2:416 and T
2:4
21 contain

topologies with two successive numbers of supports. The remaining classes consist of topologies
with only one number of support.

Fig. 9. Optimal moments in topological classes for a different number of supports

The beams from Fig. 10 belong to the same topological class T2:ni with two up to n ­ 7
supports. For a fixed number of supports, the six-support beams from Fig. 10a are elements of
the topological class T62, the seven-support beams fromFig. 10b aremembers of the topological
class T744.

Fig. 10. Optimal beams with a different number of supports and equal values of the absolute maximum
moment: (a) two topologies from the class T62, (b) two topologies from the class T

7
44

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

k∈{2, . . . ,n−1}, are members of the same class T2:ni if they satisfy the condition

tki ≡R tk+1j if cE,i =2 ∧ cE,j =0 ∧ cH,i = cH,j ∧ cS,i+2cT,i = cS,j+2cT,j (6.1)

where cE,i, cH,i, cS,i, cT,i, cE,j, cH,j, cS,j, cT,j are the numbers of the segments lE, lH, lS, lT for
the topology tki and t

k+1
j , respectively.

7. Conclusions

The paper presents topology and geometry optimization of statically determinate beams. The
beams can have any fixed number of supports or a number of supports from a certain interval.
The objective of the optimization is to minimize the absolute maximum bending moment due
to the most unfavourably distributed load. The maximum possible moment with the top and



Topological classes of statically determinate beams... 267

the bottom in tension occurs at any cross-section of a beamwhen alternate spans are uniformly
loaded. A genetic algorithm is used to optimize the geometry of any beamwith a fixed topology
for two load cases. An optimal envelope of twomoment diagrams has equal local extreme values
for each beam. Algebraic formulas that determine values of optimal geometrical parameters are
obtained by analyzing properties of the envelope. Beam topologies are sorted into topological
classes according tominimal values of the objective function.The characteristic features of these
classes are described and compared with those for stationary loading.

The results obtained here provide valuable guidance for the design of beam structures under
the worst distributed load. Topological classes for the worst combination of distributed loads
with fixed and themost unfavourable positions may be an important area of further research.

Acknowledgements

This workwas supported byBialystokUniversity of Technology grant S/WA/1/2011.The reviewer’s

suggestions are greatly appreciated.

Appendix

Pseudocode for the algorithms to calculate the coordinates of supports andhinges of the optimal
beam in a one-dimensional coordinate system with the origin at the left end of the beam

FUNCTION calculating support coordinates()
INPUT: the topological code of beam: n-element sequence t; the lengths l, lE, lH, lS, lT ,L
OUTPUT: n coordinates of supports: n-element vector s
IF t1 is equal to 0 THENASSIGN s1 the value zero
ELSEASSIGN s1 the value lE
END IF
FOR i starts at 2, i <n, increment i DO

IF ti−1 is equal to 0 THEN
IF ti is equal to 0 THENASSIGN si the value si−1+ l
ELSE IF ti is equal to 1 THEN
IF ti+1 is equal to 1 THENASSIGN si the value si−1+ lS
ELSEASSIGN si the value si−1+ l
END IF

ELSEASSIGN si the value si−1+ l+ lH
END IF

ELSE IF ti−1 is equal to 1 THEN
IF ti is equal to 0 THENASSIGN si the value si−1+ l+ lH
ELSE IF ti is equal to 1 THEN
IF ti+1 is not equal to 1 THENASSIGN si the value si−1+ l+ lH
ELSEASSIGN si the value si−1+ lS + lH
END IF

ELSEASSIGN si the value si−1+ l+2lH
END IF

ELSE
IF ti is equal to 0 THEN
IF (i is greater than 2) and (ti−2 is equal to 2) THENASSIGN si the value si−1+ lS
ELSEASSIGN si the value si−1+ l
END IF

ELSE IF ti is equal to 1 THEN
IF (i is equal to 2) or ((i is greater than 2) and (ti−2 is not equal to 2)) THEN

IF ti+1 is not equal to 1 THENASSIGN si the value di−1+ l
ELSEASSIGN si the value si−1+ lS
END IF

ELSE



268 A. Kozikowska

IF ti+1 is equal to 1 THENASSIGN si the value si−1+ lT
ELSEASSIGN si the value si−1+ lS
END IF

END IF
ELSE
IF (i is equal to 2) or ((i is greater than 2) and (ti−2 is not equal to 2)) THEN

ASSIGN si the value si−1+ l+ lH
ELSEASSIGN si the value si−1+ lS + lH
END IF

END IF
END IF

ENDFOR
IF tn is equal to 0 THENASSIGN sn the valueL
ELSEASSIGN sn the valueL− lE
END IF
ENDFUNCTION

FUNCTION calculating hinge coordinates()
INPUT: the topological code of beam: n-element sequence t; n coordinates of supports: n-element

vector s; the length lH
OUTPUT: n−2 coordinates of hinges: (n−2)-element vector h
FOR i starts at 1, i <n−1, increment iDO

IF tn+1 is equal to 0 THENASSIGN hn the value sn+1
ELSE IF tn+1 is equal to 2 THENASSIGN hn the value sn+1− lH
ELSEASSIGN hn the value sn+1+ lH
END IF

ENDFOR
ENDFUNCTION

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Manuscript received October 20, 2012; accepted for print October 8, 2013