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Latin American Journal of Solids and Structures

Print version ISSN 1679-7817On-line version ISSN 1679-7825

Lat. Am. j. solids struct. vol.13 no.4 Rio de Janeiro Apr. 2016

http://dx.doi.org/10.1590/1679-78252306 

Articles

Geometry and Topology Optimization of Statically Determinate Beams under Fixed and Most Unfavorably Distributed Load

Agata Kozikowskaa  * 

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

Abstract

The paper concerns topology and geometry optimization of statically determinate beams with an arbitrary number of pin supports. The beams are simultaneously exposed to uniform dead load and arbitrarily distributed live load and optimized for the absolute maximum bending moment. First, all the beams with fixed topology are subjected to geometrical optimization by genetic algorithm. Strict mathematical formulas for calculation of optimal geometrical parameters are found for all topologies and any ratio of dead to live load. Then beams with the same minimal values of the objective function and different topologies are classified into groups called topological classes. The detailed characteristics of these classes are described.

Keywords: Statically determinate beams; geometry and topology optimization; genetic algorithm; stationary load; most unfavorable load

NOMENCLATURE

, bi

assignment of lengths lBj to left or right side of supports, i = 1,2,...,n

CE, CH

number of external and internal cantilevers

CBj

number of j -th cantilevers from the top of the beam interaction scheme, j = 1,2,...n-1

g

beam chromosome

h

coordinates of hinges

l,lE ,lH ,L

lengths of optimal beam segments and length of beam, see Fig. 1

lBj

distance from intersection of optimal moment diagram (with maximum value at the bottom) with beam axis to the nearest support, j = 1,2,...n-1, see Fig. 1 and Fig. 2

Mi ,

optimal moment value of topology ti and class

n

pn , p 2: n

number of topological classes in set Tn and T 2: n

q

dimensionless intensity of uniformly distributed gravity dead load for constant sum of gravity dead load and maximum gravity live load intensities, equal to one, 0 ≤ q ≤ 1

1 - q

maximum dimensionless intensity of arbitrarily distributed live load for constant sum of both load intensities, equal to one

R

equivalence relation of beam topologies

s

coordinates of supports

ti

beam topology

ti

topological code of support i, i = 1, 2,...n

Tn ,T 2: n

set of all topologies with n supports and with two to n supports

topological class with n supports and with two to n supports

x

axial coordinate

yi

dimensionless length of cantilever, i = 1, 2,...n

zi

dimensionless length of span, i = 1, 2,...n

(·) n ,

quantities in set Tn and class

1 INTRODUCTION

Structural optimization, which includes sizing, geometry, and topology optimization, has been a very common topic of research. Topology optimization is a relatively new but fast growing field of this research (Kirsch, 1989; Rozvany et al., 1995; Eschenauer and Olhoff, 2001; Fancello and Pereira, 2003; Marczak, 2008; Rozvany, 2009; Lopes et al., 2015). In recent years, structural topology optimization has received a boost due to the recognition that topological parameters can lead to a significant improvement in the quality of structures. Thanks to the widespread availability of high-speed computers and the development of powerful computational methods for the structural analysis, scientists can return to problems considered to be investigated and can make new interesting discoveries. The topological optimization of statically determinate beams is such an insufficiently explored problem.

Beams have been widely used in civil and mechanical engineering and their optimization has been extensively studied in the literature. Studies in beam optimization originated many years ago and are attributed to Galileo Galilei who dealt with an improvement in the shape of statically determinate beams (Timoshenko, 1953). Since then, many researchers were engaged in the optimization of statically determinate and indeterminate beams for different objective functions and loading conditions (Mróz and Rozvany, 1975; Imam and Al-Shihri, 1996; Wang and Chen, 1996; Dems and Turant, 1997; Bojczuk and Mróz, 1998; Mróz and Bojczuk, 2003; Wang, 2004, 2006; Jang et al., 2009). But beam topology optimization problems concerning locations of supports relative to the ends of bars (hinges) have not yet been resolved completely. Previous articles of the author concerned only a part of the issue - topology and geometry optimization of statically determinate beams under fixed load of different distributions (Rychter and Kozikowska, 2009; Kozikowska, 2011) and under the most unfavorably distributed load (Kozikowska, 2014).

Beam loads are generally a combination of dead and live loads. Dead load is essentially constant and can be treated as uniformly distributed, especially for beams with constant cross-sections. Live load can vary during the life of the structure. In the paper both loads occur simultaneously. It is assumed that live load is characterized by a relatively slow increase of the magnitude and it is regarded as static (without dynamic effects).

Beams are usually subjected to transverse loadings, which result in internal shear forces and bending moments. In the article we consider beams which are relatively long in comparison with their thickness and depth. Bending stresses have the greatest effect on the behavior of such beams and they can be designed mainly against bending moment resistance. Therefore the structural measure of beams is defined in the paper as the absolute maximum bending moment, like in (Wang, 2006). For beams with uniform cross-sections this measure corresponds to the design for minimum weight. The most adverse distributions of the live load for all cross-sections of a beam can be obtained with the help of influence lines for bending moment.

Due to the complexity of the geometric search space of statically determinate beam with any number of supports, geometry optimization of beams is performed using a genetic algorithm. This method of probabilistic optimization have been applied to a great variety of structural optimization problems (Wang and Chen, 1996; Castro and Partridge, 2006; Rychter and Kozikowska, 2009).

Results of topology optimization, occurring in the literature, usually depend on an initial layout, which is adopted arbitrarily. The final solutions are then obtained by exploring only some parts of the full search space and they are not necessarily the best topological layouts. In the paper the space of all possible beam topologies is known, exhaustive search in this space is carried out and global optima are determined. Moreover, the paper presents not only globally optimal beam topologies, but classifies all topologies into equivalence classes with equal minimum values of the absolute maximum moment. Typical features of these classes are discussed.

2 BEAM TOPOLOGY AND GEOMETRY

The subject of the paper is the space of all statically determinate beams with different topologies, with two or more pin supports. The construction of all possible topologies of beams with n pin supports, solved by Rychter (Rychter and Kozikowska, 2009), starts with the topology, where all n ends of all bars are supported. Then each support can be shifted from the end of a bar (first and last support), or the common hinged end of two adjacent bars (intermediate supports), into the interior of a bar, but not to a distant bar. The topology ti of an n -support statically determinate beam is represented by n topological codes of supports ti :

(1)

where ti is equal to 0 for no shift of support i, 1 for shift of support i to the left, and 2 for shift to the right. For example, the beam in Fig. 1 has the topology [2,2,1,1,1,0,...,0,2,1].

The geometry of a beam is described by two sets of geometric parameters: zi and yi . The parameters zi are dimensionless lengths of spans between neighbour supports:

(2)

The parameters yi represent dimensionless lengths of external and internal cantilevers:

(3)

When support i is at the end of the beam or at the hinge, no cantilever is created, and the parameter yi equals zero - the first row in Eq. (3). Otherwise, the parameter yi takes real value from the interval (0,1) - the second row in Eq. (3). The third row in Eq. (3) prevents the cantilevers from overlapping. For the external cantilevers the parameters yi, yn are dimensionless lengths. For the internal cantilevers the parameters y 2, ..., y n-1 are ratios of lengths of cantilevers to the lengths of spans in which the cantilevers reside.

The total length of a beam is the sum of the lengths of all spans and the lengths of the external cantilevers. All beams have the same length, normalized to unity:

(4)

A more detailed description of the topological and geometrical parameters is given in (Rychter and Kozikowska, 2009).

3 GEOMETRY OPTIMIZATION OF A BEAM WITH A FIXED TOPOLOGY

3.1 Problem Formulation

Beams are mainly used in flooring systems of buildings and bridges. In most of these applications beams are prismatic (straight with uniform cross-section) and loaded perpendicularly to the longitudinal direction. Loads of the beams can be categorized into two groups: dead (fixed) loads and live (temporary) loads. Dead loads are gravity loads due to the self-weight of the beams and all other material and equipment permanently attached to them. The magnitude and spatial distribution of the dead loading are constant over time. Dead load is sometimes the most important part of the beam loading, particularly for beams with long spans and built of heavy materials. In some cases the importance of this load can be reduced in relation to other loads, but this load should not be ignored. For prismatic beams this load is mainly uniformly distributed and this case is considered in the article. Live loads are usually gravity-type (possibly piecewise) loads of regularly or irregularly varying magnitudes and/or varying positions caused by the use of the structure. Examples of temporary loads are stored items, furniture, occupants in buildings and pedestrians on footbridges. Although the loads are movable it is assumed in the paper that they are applied slowly and there is no dynamic amplification. In such a case, these moveable loads are considered as quasi-static arbitrarily distributed loads. Because of the variability of the load, we have to consider all possible live load combinations and find the ones that result in the maximum values of bending moment. The issues discussed in the article do not depend on the absolute values of the dead and live load intensities, but only on their ratio. Therefore we normalize both intensities so that their sum is constant, equal to one. The intensity of dead load is equal to q and the maximum intensity of live load is equal to 1 - q. Each intensity can take values from 0 to 1.

Beams under arbitrarily distributed transverse live loads, considered in the author's article (Kozikowska, 2014), had two most unfavorable load cases for the maximum bending moment. Each case included uniformly distributed load of maximum intensity on alternate spans. If we take dead and live load into account, we also have two load cases. One of the cases comprises uniform load of intensity 1 (sum of both loads) on odd spans and uniform dead load of intensity q on even spans. The second case also includes uniform loads: q on odd spans and sum of loads equal to 1 on even spans.

We assume that each beam is of unit length with a fixed topology ti . The geometry optimization problem is defined as follows:

(5)

(6)

where denotes the maximum of the absolute bending moment (objective function) for both load cases, zi are span lengths given by Eq. (2), yj are nonzero lengths of cantilevers, which are created by shifts of supports of nonzero topological codes tj from ends of bars, given by Eq. (3), and x is the axial coordinate.

This geometry optimization has been carried out by a modified version of the genetic algorithm (Rychter and Kozikowska, 2009), written by the author in C/C++ programming language.

3.2 Genetic Algorithm

The genetic algorithm for the optimization of the geometry of statically determinate beams follows the general scheme of genetic algorithms (Goldberg, 1989). Populations of chromosomes are evolved over several generations, subject to random mutation, random crossover (recombination), and selection pressure.

The chromosome g representing a geometry of a n-support statically determinate beam with a fixed topology ti is a string of n - 1 real genes z i and (CE + CH ) real nonzero genes yj :

(7)

where z i, yj are given by Eq. (2), (3), and (4).

The creation of an initial population involves the assignment of random real values from the interval (0,1) to all genes, and then the adjustment of the genes to the conditions contained in the third row of Eq. (3) and in Eq. (4).

The designed chromosomes allow for easy random mutation and crossover, without producing incorrect beams. After these operations, the chromosomes only have to be adjusted to the conditions given in the third row of Eq. (3) and in Eq. (4). The most efficient versions of mutation, crossover and selection have been determined through extensive simulations.

Gaussian mutation, in which a Gaussian distributed random value is added to the value of the chosen gene, has turned out to be the best mutation method.

The performance of algorithm with single-point crossover was the same as with multi-point crossover. Yet the former is simpler and faster than the latter. Therefore, single-point recombination has been used where two parent chromosomes are cut in one random point, and both chromosome parts are swapped to produce two children.

Three selection strategies have been studied: proportional roulette-wheel, proportional deterministic, and ranking tournament. The tournament selection involves running several tournaments (groups of chromosomes) chosen at random from population members. The winners of all tournaments go into the next population. Moreover, in the applied tournament strategy, the best beam must fall into at least one tournament group and so will always survive selection. Numerical simulations have shown the superiority of this tournament selection with binary tournaments over proportional selection methods.

The minimal value of the absolute maximum bending moment Mi has been found for each topology ti as a result of the optimization by this genetic algorithm.

3.3 Optimization Results

A beam with optimal geometry for the fixed topology is presented in Fig. 1. The beam is shown with two unique bending moment diagrams, drawn with a solid line or a dashed line, for both the most unfavorable load cases. The optimal envelope of the two moment diagrams has the same local extreme moment values equal to Mi . These values are present over the supports which were moved away from ends of bars and at the bottom at mid spans or close to them. The envelope has zero values exclusively in hinges and at both ends of the beam and is equivalent to only one topology, unlike in the case of dead load alone (Kozikowska, 2011).

Figure 1 A beam with optimal geometry for the fixed topology [2,2,1,1,1,0,...,0,2,1]. 

We are interested in analytical expressions for optimal geometrical parameters for any topology, under stationary load and the most unfavorably distributed load. In order to determine the values of the parameters l, lE, lH , and lBj for j = 1,2,...,n-1 (see Fig. 1) we solve the system of equations:

(8)

(9)

(10)

(11)

(12)

where lE and lH denote the lengths of nonzero external and internal cantilevers, respectively, l is the length of each segment with at least one of the two optimal moment diagrams at the bottom, with the maximum value of this moment equal to Mi and zero values of this moment at both ends of the segment, lBj for j = 1,2,...,n-1 is the distance from intersection of optimal moment diagram (with maximum value Mi at the bottom) with beam axis to the nearest support, moreover there is no hinge in zero moment point. The lengths lBj neighbour on the external and internal cantilevers. The indices j in lBj are consecutive numbers of these neighbouring cantilevers, counted from the top of the interaction scheme of the beam (see Fig. 2). The neighbouring cantilevers, which form consecutive levels (steps) in the interaction diagram, create sequences. The number of terms (cantilevers) in such a sequence is the length of the cantilever sequence. The pseudo code for an algorithm to assign the lengths lBj to supports on the basis of the beam topology is given in appendix A. The algorithm returns a vector b i of n integer elements bk . The element bk is equal to j if the length lBj is on the right side of the support k, is equal to -j if the length lBj is on the left side of the support k, and is equal to zero if there is no length lBj next to support k (the support k is at the end of the beam or under a hinge). For example, b i = [1, 2, ,-3, -2, -10,...,0,1,-1] for the beam from Fig. 2.

Figure 2 Locations of lengths lBj depending on consecutive numbers of cantilevers in the interaction diagram of the beam. 

Equation (8) describes the length of the beam as the sum of individual segment lengths. The parameters CE, CH, CBj for j = 1,2,...n - 1 are the numbers of the segments lE, lH, lBj for j = 1,2,...n - 1, respectively. The lengths of a cantilever and a simply supported beam with the same values of the absolute maximum moment under uniformly distributed dead and live load are compared in Eq. (9). The maximum bending moment value of a simply supported beam of the length l + 2lH equals twice this moment value of a simply supported beam of the length l in accordance with Eq. (10). Eq. (11) is explained graphically in Fig. 3. The equation was established by comparing the moment value at the support calculated on the basis of the moment diagram on the left of the support with this moment value calculated according to the moment diagram on the right. Eq. (12) was found from the moment diagram, drawn with a solid line in Fig. 4.

Figure 3 Graphic explanation of Eq. (11). 

Figure 4 Graphic explanation of Eq. (12). 

The solution to the system of equations (8)-(12) is given by:

(13)

(14)

(15)

(16)

where

The values of the parameters CE and CH can be determined from the beam topology ti . The value of the parameter CE is equal to the number of nonzero elements in the first and last position of the code ti . The value of the parameter CH is the number of nonzero elements in positions 2 through n - 1 of the code ti . An algorithm to calculate the values of parameters CBj for j = 1,2,...n - 1 on the basis of the vector b i (assigning the lengths lBj to supports) is given by a pseudo code in appendix B. The value of the absolute maximum bending moment Mi can be calculated as the moment in the centre of a simply supported beam of the length l under uniform load equal to the sum of both loads (intensity equal to one):

(17)

Algorithms to calculate the coordinates of supports and hinges (on the basis of the beam topology, the vector b i , and the lengths l, lE, lH , and lBj for j = 1,2,...n - 1) are given by a pseudo code in appendix C.

3.4 Dependence of Optimal Geometrical Parameters on Dimensionless Dead Load Intensity q

The formulas (13)-(16) enable us to calculate the optimal lengths of the segments l, lE, lH ,, lBj for j = 1,2,...n - 1 for any number of supports, for any topology, and for any value of the dimensionless dead load intensity 0 < q < 1. For the extreme values of q, equal to 0 or 1, we receive special cases with specific values of lBj .

For q = 0 (only most unfavorably distributed load) regardless of the beam topology, the value of the parameter lB1 is equal to 0, and the values of the parameters lBj for j = 2,...n - 1, calculated from the formula (16), are less than 0. The negative value of the parameter lBj means that the segment lBj is on the same side of the support as the segment lH (see Fig. 5a for the beam with the topology [2,2,1,1]).

For q = 1 (only fixed load) regardless of the beam topology, both load cases come down to one case with dead load on the entire beam, and all segments lBj for j = 1,...n - 1 have the same length equal to the length lH (see Fig. 5c for the beam with the topology [2,2,1,1]).

For 0 < q <1 (fixed load and most unfavorably distributed load acting simultaneously) regardless of the beam topology, all optimal segment lengths have different values. The segments lBj are shorter than lH , while l, lE are longer than lH . The value of the parameter lB1 is more than zero, but for small values of q (less than about 0.2) the lengths lBj for j = 2,...n - 1 are less than zero.

The lengths of the optimal beam segments can also be calculated in case of live load alone using the formulas presented in (Kozikowska, 2014) and in case of dead load alone using the formulas given in (Kozikowska 2011). The sets of optimal geometrical parameters which have been used in the previous author's articles consist of a smaller number of parameters than the set used in this paper. Therefore, the sets used in (Kozikowska 2014, 2011) cannot be applied to describe the optimal geometry of the statically determinate beams for 0 < q <1.

Figure 5 The beam with optimal geometry for the topology [2,2,1,1] and for different values of the dimensionless dead load intensity q: (a) q = 0, (b) q = 1/2, (c) q = 1. 

The dependence of optimal segment lengths on values of q is illustrated in Fig. 6 with regard to the beam from Fig. 5 (with the topology [2,2,1,1]). The values of the parameters l, lE, lH are more than zero for all values of 0 < q <1, and the values l, lE, lH are the biggest for q = 0. For q = 0, the value of the parameter lB1 is equal to zero, but lB2 is less than zero. Next, the values of the parameters l, lE, lH decrease with increasing q, lB1 and lB2 increase with increasing q, and lB1 and lB2 reach the value lH for q = 1. For q greater than zero but less than a value of about 0.2, the value of the parameter lB2 is less than 0. The dependence of the parameters l, lE, lH, lBj for j = 1, 2,...n - 1on q is the same for beams with any number of supports and any topology.

Figure 6 Lengths l, lE, lH, lB1 , and lB2 of the beam from Fig. 5 (with the topology [2,2,1,1]) for different values of the dimensionless dead load intensity q

4 TOPOLOGY OPTIMIZATION OF BEAMS WITH A FIXED NUMBER OF SUPPORTS FOR 0 < Q < 1

Topology optimization of beams with a fixed number of supports for q = 0 (live load alone) is presented in (Kozikowska, 2014) and for q = 1 (dead load alone) in (Kozikowska 2011).

4.1 Equivalence Relation of Beam Topologies

T is the set of n-support beam topologies: T n or T2 n . Any two topologies ti and tj of the set T are equivalent with respect to the relation R if the minimal values of the absolute maximum moments Mi and Mj of these topologies are equal:

(18)

Based on this relation R, the set T n can be divided into disjoint equivalence classes of beam topologies called topological classes , and the set T2:n into topological classes .

4.2 Features of Beam Topologies in a Topological Class

All optimal bending moment diagram pairs from the topological class (the eighteenth class of all six-support classes ordered by increasing values of moments ), under a fixed uniformly distributed load and the most unfavorably piece-wisely distributed load, are shown in Fig. 7.

Figure 7 All optimal envelopes of moment diagrams in the class. : = 1, = 4, = 3, = 1, = 1, = 0, = 0 

All topologies in the topological class have the same values of moment and lengths , , , for k = 1,2, ...n-1. The lengths , , , for k = 1,2, ...n-1, given by Eq. (13)-(16), depend on the number of supports, the values of the param1eters , , and for k = 1,2, ...n-1, and the value of q. Thus for two topologies ti and tj of the set T n under a fixed and the most unfavorably distributed load the equivalent condition from Eq. (18) can be expressed as:

(19)

where CE,i, CH,i, CBk,i for k = 1,2, ...n-1 , CE,j, CH,j, CBk,j for k = 1,2, ...n-1 are the numbers of the appropriate segments for the topology t i and t j , respectively.

4.3 Comparison of Topological Classes

The whole sets of topological classes under a fixed uniform and the most unfavorably distributed load, with all optimal envelopes of moment diagrams are presented in Fig. 8 (for three support and equal intensities of dead and live loads, q = 1/2), Fig. 9 (for three support and live load intensity eight times greater than dead load intensity, q = 1/9), and Fig. 10 (for four support and equal intensities of dead and live loads, q = 1/2).

Figure 8 All three-support topological classes with their optimal envelopes of moment diagrams: (a) , (b) , (c) , (d) , (e) , (f) , (g) (q = 1/2). 

Figure 9 All three-support topological classes with their optimal envelopes of moment diagrams: (a) , (b) , (c) , (d) , (e) , (f) , (g) (q = 1/9). 

Figure 10 All four-support topological classes with their optimal envelopes of moment diagrams: (a) , (b) , (c) , ..., (p) (q = 1/2). 

The division of beam topologies into topological classes does not depend on the value of the dead to live load ratio. This ratio only affects the optimal values of the geometrical parameters, which can be calculated from the formulas (13)-(16). The lengths , , , and the moment value are greater for smaller values of the ratio (for smaller values of dimensionless dead load intensity q) for all classes except for the last class whose optimal moment is independent of q. The dependence of values on q in three-support classes is shown in Fig. 11. It is observed that the growth of q (smaller share of live load) makes moment values decrease, except for the last class .

Figure 11 Optimal moments in three-support topological classes for different values of the dimensionless dead load intensity q. 

The division of all topologies into topological classes depends on the number of cantilevers and their locations in the interaction diagram of the beam (see Fig.2). The topological classes are the better, the more cantilevers their beams have (the more external cantilevers for the same total number of cantilevers) and the shorter the lengths of the cantilever sequences are in the interaction schemes. In other words, better classes have larger values of the parameters and (have larger values of the parameters than the parameters for the same sum of and ), and have more zero parameters . The values of the parameters , , and for j = 1,...n - 1 for the three-support classes (from Fig. 8 and Fig. 9) and for the four-support classes (from Fig. 10) are given in Table 1 and Table 2, respectively. The best topological class with an odd number of supports have n - 1 topologies, each with a single one-hinged span (see Fig. 8a and Fig. 9a). The best single topology in the first class with an even number of supports does not have any one-hinged spans (see Fig. 10a).

Table 1 Values of the parameters , , for j = 1, 2 in the three-support topological classes. 

Table 2 Values of the parameters , and for j = 1, 2, 3 in four-support topological classes. 

The set of all n-support topological classes is described by the set of all possible (n + 1)-element sequences (, , for j = 1, 2, ..., n - 1) where ∈ {0, 1, 2}, ∈ {0, 1, ... n - 2}, and values of the parameters for j = 1, 2, ..., n - 1 meet the following conditions:

(20)

(21)

(22)

(23)

(24)

The numbers of j-th cantilevers from the top of the interaction scheme are nonnegative (see Eq. (20)). External cantilevers are always at the top of the interaction diagram (they are always the first from the top of the interaction scheme) in accordance with Eq. (21). The number of j-th cantilevers from the top of the interaction diagram must be equal to or larger than the number of (j + 1)-th cantilevers because (j + 1)-th cantilevers are below j-th cantilevers according to Eq. (22). A bar with two supports is always at the bottom of the interaction diagram. For beams with the maximum number of external and internal cantilevers equal to n, the first cantilever is on both sides of each two-support bar, at the top of a cantilever sequence. Thus, if the number of external and internal cantilevers is maximal, then the number of the first cantilevers is equal to double the number of two-support bars which means that is even (see Eq. (23)). Eq. (24) compares the number of cantilevers in the interaction scheme and in the topology. The total number of classes pn can be calculated by an algorithm that counts the number of the sequences (, , for j = 1, 2, ..., n - 1) and is given by a pseudo code in appendix D. The numbers of n-support topological classes for n ∈ {2,3,...,16} are shown in Table 3.

5 TOPOLOGY OPTIMIZATION OF BEAMS WITH A DIFFERENT NUMBER OF SUPPORTS

Assume T2 :n is the set of beam topologies with two to n supports and is the topological class from this set.

For q = 0 (only live load) or q = 1 (only dead load), some classes contain topologies with two successive numbers of supports. Such a class is then the sum of k-support class and (k + 1)-support class for 2< k < n-1. It happens because the length of parameter lB1 is equal to zero for q = 0, and the lengths of all parameters lBj for j = 1,...n - 1 are equal to lH for q = 1. Therefore, the total number of topological classes p 2: n in the set T2: n is for these loads less than the sum of numbers of classes in all sets from T2 to Tn:

(25)

For 0 < q < 1, all classes consist of topologies with only one number of support. Therefore, the total number of topological classes p 2: n in the set T2: n is for these loads equal to the sum of numbers of classes in all sets from T2 to Tn:

(26)

Table 3 Number of supports vs number of topological classes. 

supports 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
topological classes 3 7 16 28 49 78 123 183 272 390 556 774 1072 1459 1977

6 CONCLUSIONS

The paper presents results of geometry and topology optimization of statically determinate beams with an arbitrary number of supports. The beams are exposed to uniform dead load and live load of the most unfavorable distribution. The fixed topology problem involving the geometry optimization for a given topology is solved for each beam. The absolute maximum bending moment is the objective function in this optimization. For this function, it suffices to consider only two load cases, each with dead load on all spans and uniform live load of the maximum possible intensity on alternate spans. Exact formulas for optimal geometrical parameters have been obtained for all topologies and for any dead to live load ratio on the basis of properties of the optimal moment diagram envelopes. Beams of different topologies and equal minimum values of the absolute maximum moment have been assigned to the same topological classes. It has been found that the division of the beam topologies into the topological classes depends on the number of beam cantilevers and their locations in beam interaction diagrams. It has also been found that this division does not depend on the dead to live load ratio for 0 < q < 1. Topologies with the maximum number of external and internal cantilevers and with minimal lengths of cantilever sequences in interaction schemes have been found to be the best options.

The article provides some practical guidelines on how to design statically determinate beam structures with the minimum weight. The examination of all topologically different beams provides tremendous design opportunities because it offers a variety of satisfactory solutions, not only the best ones.

Acknowledgements

This work was supported by Bialystok University of Technology grant S/WA/2/2016.

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Appendix A - Pseudo Code for the Algorithm to Assign the Lengths lBj to Supports for Beam Topology

FUNCTION assigning_lengths_lB_to_supports()

INPUT: the topological code of beam: n-element vector t

OUTPUT: n parameters assigning lengths lBj to supports: n-element vector b

FOR i starts at 1, in, increment i DO ASSIGN to bi the value of 0

END FOR

(* assigning lengths lB to supports for topological codes 2 *)

ASSIGN to i the value of 1

WHILE i is less than or equal to n DO

IF ti is equal to 2 THEN

ASSIGN to bi the value of 1

WHILE (i is less than n) and (ti+1 is equal to 2) DO

ADD 1 to i

ASSIGN to bi the value of bi-1 +1

END WHILE

ADD 1 to i

ELSE ADD 1 to i

END IF

END WHILE

(* assigning lengths lB to supports for topological codes 1 *)

ASSIGN to i the value of n

WHILE i is greater than or equal to 1 DO

IF ti is equal to 1 THEN

ASSIGN to bi the value of -1

WHILE (i is greater than 1) and (ti-1 is equal to 1) DO

SUBTRACT 1 from i

ASSIGN to bi the value of bi+1 -1

END WHILE

SUBTRACT 1 from i

ELSE SUBTRACT 1 from i

END IF

END WHILE

END FUNCTION

Appendix B - Pseudo Code for the Algorithm to Calculate the Parameters cB j for j = 1,2,... n -1 for Beam Topology

FUNCTION calculating parameters cB()

INPUT: n parameters assigning lengths lBj to supports: n-element vector b

OUTPUT: (n - 1)-element vector cB

FOR i starts at 1, i < n, increment i DO ASSIGN to cBi the value zero

END FOR

FOR i starts at 1, in, increment i DO

IF bi is not equal to 0 THEN ADD 1 to cB|bi|

END IF

END FOR

END FUNCTION

Appendix C - Pseudo Code for the Algorithms to Calculate the Coordinates of Supports and Hinges of Optimal Beam in a One-Dimensional Coordinate System with the Origin at the Left End of the Beam

FUNCTION calculating_support_coordinates()

INPUT: n parameters assigning lengths lBj to supports: n-element vector b; the lengths l, lE, lH, lBj for j = 1,2,..., n - 1

OUTPUT: n coordinates of supports: n-element vector s

IF b 1 is equal to 0 THEN ASSIGN to s 1 the value zero

ELSE ASSIGN to s 1 the value lE

END IF

FOR i starts at 2, in, increment i DO

IF bi-1 is equal to 0 THEN

IF bi is equal to 0 THEN ASSIGN to si the value si-1 + l

ELSE IF bi is greater than 0 THEN ASSIGN to the value si-1 + l + lH

ELSE ASSIGN si the value si-1 + l + lB|bi|

ELSE IF bi-1 is greater than 0 THEN

IF bi is equal to 0 THEN ASSIGN to si the value si-1 + lBbi-1 + l

ELSE IF bi is greater than 0 THEN ASSIGN to si the value si-1 + lBbi-1 + lH

ELSE ASSIGN si the value si-1 + lBbi-1 + l + lB|bi|

ELSE

IF bi is equal to 0 THEN ASSIGN to si the value si-1 + lH + l

ELSE IF bi is greater than 0 THEN ASSIGN to si the value si-1 + l + 2lH

ELSE ASSIGN si the value si-1 + lH + l + lB|bi|

END IF

END FOR

END FUNCTION

FUNCTION calculating_hinge_coordinates()

INPUT: the topological code of beam: n-element vector 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 i DO

IF tn+1 is equal to 0 THEN ASSIGN to hn the value sn+1

ELSE IF tn+1 is equal to 2 THEN ASSIGN to hn the value sn+1 - lH

ELSE ASSIGN to hn the value sn+1 + lH

END IF

END FOR

END FUNCTION

Appendix D - Pseudo Code for the Algorithm to Count the Number of Topological Classes

FUNCTION counting_number_of_classes()

INPUT: the number of supports n

OUTPUT: the number of classes pn

ASSIGN to pn the value 0

CREATE an empty vector c B

FOR cE starts at , cE ≤ 2, increment cE DO

FOR cH starts at , cH n - 2, increment cH DO

FOR cB1 starts at cH, cB1 cE + cH , increment cB1 DO

IF (cE is equal to ) and (cH is equal to n - 2) and (cB1 is odd) THEN

SKIP to the next iteration of the loop

END IF

CALL generating_parameters_cB(cE + cH, cB1 )

END FOR

END FOR

END FOR

END FUNCTION

FUNCTION generating_parameters_cB(expected_sum,current_sum)

(* Function generates parameters cBj for j = 2,... n - 1 *)

IF vector c B has n - 1 elements THEN

IF expected_sum is equal to current_sum THEN ADD 1 to pn

END IF

RETURN

END IF

IF expected_sum is less than current_sum THEN RETURN

END IF

FOR i starts at last element of vector cB, i ≥ 0, decrease i DO

ADD i at the end of vector cB

ADD i to current_sum

CALL generating_parameters cB(expected_sum,current_sum)

DELETE last element of vector cB

SUBTRACT i from current_sum

END FOR

END FUNCTION

Received: July 24, 2015; Revised: December 04, 2015; Accepted: January 11, 2016

*Corresponding author email: a.kozikowska@pb.edu.pl

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