NOMENCLATURE
assignment of lengths l_{Bj} to left or right side of supports, i = 1,2,...,n 

C_{E}, C_{H} 
number of external and internal cantilevers 
C_{Bj} 
number of j th cantilevers from the top of the beam interaction scheme, j = 1,2,...n1 
g 
beam chromosome 
h 
coordinates of hinges 
l,l_{E} ,l_{H} ,L 
lengths of optimal beam segments and length of beam, see Fig. 1 
l_{Bj} 
distance from intersection of optimal moment diagram (with maximum value at the bottom) with beam axis to the nearest support, j = 1,2,...n1, see Fig. 1 and Fig. 2 
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 
t_{i} 
beam topology 
t_{i} 
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 
y_{i} 
dimensionless length of cantilever, i = 1, 2,...n 
z_{i} 
dimensionless length of span, i = 1, 2,...n 
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 highspeed 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 AlShihri, 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 crosssections. 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 crosssections this measure corresponds to the design for minimum weight. The most adverse distributions of the live load for all crosssections 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 t_{i} of an n support statically determinate beam is represented by n topological codes of supports t_{i} :
where t_{i} 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: z_{i} and y_{i} . The parameters z_{i} are dimensionless lengths of spans between neighbour supports:
The parameters y_{i} represent dimensionless lengths of external and internal cantilevers:
When support i is at the end of the beam or at the hinge, no cantilever is created, and the parameter y_{i} equals zero  the first row in Eq. (^{3}). Otherwise, the parameter y_{i} 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 y_{i,} y_{n} are dimensionless lengths. For the internal cantilevers the parameters y _{2, ...,} y _{n1} 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:
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 crosssection) 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 selfweight 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 gravitytype (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 quasistatic 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 t_{i} . The geometry optimization problem is defined as follows:
where denotes the maximum of the absolute bending moment (objective function) for both load cases, z_{i} are span lengths given by Eq. (^{2}), y_{j} are nonzero lengths of cantilevers, which are created by shifts of supports of nonzero topological codes t_{j} 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 nsupport statically determinate beam with a fixed topology t_{i} is a string of n  1 real genes z _{i} and (C_{E} + C_{H} ) real nonzero genes y_{j} :
where z _{i,} y_{j} 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 singlepoint crossover was the same as with multipoint crossover. Yet the former is simpler and faster than the latter. Therefore, singlepoint 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 roulettewheel, 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 M_{i} has been found for each topology t_{i} 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 M_{i} . 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}).
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, l_{E}, l_{H} , and l_{Bj} for j = 1,2,...,n1 (see Fig. 1) we solve the system of equations:
where l_{E} and l_{H} 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 M_{i} and zero values of this moment at both ends of the segment, l_{Bj} for j = 1,2,...,n1 is the distance from intersection of optimal moment diagram (with maximum value M_{i} at the bottom) with beam axis to the nearest support, moreover there is no hinge in zero moment point. The lengths l_{Bj} neighbour on the external and internal cantilevers. The indices j in l_{Bj} 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 l_{Bj} 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 b_{k} . The element b_{k} is equal to j if the length l_{Bj} is on the right side of the support k, is equal to j if the length l_{Bj} is on the left side of the support k, and is equal to zero if there is no length l_{Bj} 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.
Equation (^{8}) describes the length of the beam as the sum of individual segment lengths. The parameters C_{E}, C_{H}, C_{Bj} for j = 1,2,...n  1 are the numbers of the segments l_{E}, l_{H}, l_{Bj} 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 + 2l_{H} 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.
The solution to the system of equations (^{8})(^{12}) is given by:
where
The values of the parameters C_{E} and C_{H} can be determined from the beam topology t_{i} . The value of the parameter C_{E} is equal to the number of nonzero elements in the first and last position of the code t_{i} . The value of the parameter C_{H} is the number of nonzero elements in positions 2 through n  1 of the code t_{i} . An algorithm to calculate the values of parameters C_{Bj} for j = 1,2,...n  1 on the basis of the vector b _{i} (assigning the lengths l_{Bj} to supports) is given by a pseudo code in appendix B. The value of the absolute maximum bending moment M_{i} 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):
Algorithms to calculate the coordinates of supports and hinges (on the basis of the beam topology, the vector b _{i} , and the lengths l, l_{E}, l_{H} , and l_{Bj} 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, l_{E}, l_{H} ,, l_{Bj} 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 l_{Bj} .
For q = 0 (only most unfavorably distributed load) regardless of the beam topology, the value of the parameter l_{B1} is equal to 0, and the values of the parameters l_{Bj} for j = 2,...n  1, calculated from the formula (^{16}), are less than 0. The negative value of the parameter l_{Bj} means that the segment l_{Bj} is on the same side of the support as the segment l_{H} (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 l_{Bj} for j = 1,...n  1 have the same length equal to the length l_{H} (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 l_{Bj} are shorter than l_{H} , while l, l_{E} are longer than l_{H} . The value of the parameter l_{B1} is more than zero, but for small values of q (less than about 0.2) the lengths l_{Bj} 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.
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, l_{E}, l_{H} are more than zero for all values of 0 < q <1, and the values l, l_{E}, l_{H} are the biggest for q = 0. For q = 0, the value of the parameter l_{B1} is equal to zero, but l_{B2} is less than zero. Next, the values of the parameters l, l_{E}, l_{H} decrease with increasing q, l_{B1} and l_{B2} increase with increasing q, and l_{B1} and l_{B2} reach the value l_{H} for q = 1. For q greater than zero but less than a value of about 0.2, the value of the parameter l_{B2} is less than 0. The dependence of the parameters l, l_{E}, l_{H}, l_{Bj} for j = 1, 2,...n  1on q is the same for beams with any number of supports and any topology.
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 nsupport beam topologies: T ^{n} or T^{2} ^{n} . Any two topologies t_{i} and t_{j} of the set T are equivalent with respect to the relation R if the minimal values of the absolute maximum moments M_{i} and M_{j} of these topologies are equal:
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 sixsupport classes ordered by increasing values of moments ), under a fixed uniformly distributed load and the most unfavorably piecewisely distributed load, are shown in Fig. 7.
All topologies in the topological class have the same values of moment and lengths , , , for k = 1,2, ...n1. The lengths , , , for k = 1,2, ...n1, given by Eq. (^{13})(^{16}), depend on the number of supports, the values of the param1eters , , and for k = 1,2, ...n1, and the value of q. Thus for two topologies t_{i} and t_{j} of the set T ^{n} under a fixed and the most unfavorably distributed load the equivalent condition from Eq. (^{18}) can be expressed as:
where C_{E,i}, C_{H,i}, C_{Bk,i} for k = 1,2, ...n1 , C_{E,j}, C_{H,j}, C_{Bk,j} for k = 1,2, ...n1 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).
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 threesupport 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 .
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 threesupport classes (from Fig. 8 and Fig. 9) and for the foursupport 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 onehinged 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 onehinged spans (see Fig. 10a).
The set of all nsupport 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:
The numbers of jth 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 jth 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 jth 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 twosupport 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 twosupport 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 nsupport topological classes for n ∈ {2,3,...,16} are shown in Table 3.
5 TOPOLOGY OPTIMIZATION OF BEAMS WITH A DIFFERENT NUMBER OF SUPPORTS
Assume T^{2} ^{: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 ksupport class and (k + 1)support class for 2< k < n1. It happens because the length of parameter l_{B1} is equal to zero for q = 0, and the lengths of all parameters l_{Bj} for j = 1,...n  1 are equal to l_{H} for q = 1. Therefore, the total number of topological classes p ^{2:} ^{n} in the set T^{2:} ^{n} is for these loads less than the sum of numbers of classes in all sets from T^{2} to T^{n}:
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 T^{2:} ^{n} is for these loads equal to the sum of numbers of classes in all sets from T^{2} to T^{n}:
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.