On the insurmountable size of truss-like structures

On the Insurmountable Size of Truss-like Structures Olmedo Rojas, Carlos 1 ; Vázquez Espí, Mariano 2 ; Cervera Bravo, Jaime 3 ABSTRACT Galileo postulated the existence of an insurmountable size for stone columns bearing a useful load as the size for which the structure is only able to resist its self-weight. Herein a method for the determination of the unsurmountable size for truss-like structures is shown, given the form of these structures and the ratio between the allowable stress and the specific weight of the material (the material structural scope). Three types of bars are considered: straight bars, with solid and hollow rectangular cross-section, and catenary bars with circular cross-section —a limit and theoretical case for estimating a meaningful upper bound of the structural scope—. An approximate rule to estimate the structural efficiency —here named GA rule— is shown, and is compared with numerical solutions using the proposed method. Keywords: structural design, insurmountable size, structural scope, trusses, self-weight. 1. THE GALILEO PROBLEM AND THE AROCA RULE In a first approximation, we can represent the physical cost of a structure by its self-weight, as many cost during the manufacturing, but not all, are approximately proportional to the self-weight of the structure: CO2 emissions, mineral resources consumption, etc. For a given structural problem, we define the structural efficiency as the ratio between the useful load and the whole load (i.e., the useful load plus the self-weight) required to solve that problem in a particular structure. Galileo [1] postulated the existence of insurmountable sizes for structures, as well as the relationship between the size of a structure and its ability to resist a useful load: let us imagine a cylindrical stone column, at the limit of its resistance only as a result of its self-weight; we name structural scope of the column to the height (ℒ) of this column, which cannot resist any additional load, hence being null its efficiency. A useful column with the same base must have therefore a height 𝐿 smaller than ℒ. This new column can resist an additional useful load (𝑄), the value of which is at most the weight difference between the two columns. According to the previous definition of efficiency: 𝑟 ≤ ℒ−𝐿 ℒ=1−𝐿 ℒ=1−𝑡 (1) 1 Departamento de Estructuras y Física de Edificación. Universidad Politécnica de Madrid (Spain). [email protected] (Corresponding author) 2 Departamento de Estructuras y Física de Edificación. Universidad Politécnica de Madrid (Spain). mariano.vazq[email protected]s 3 Departamento de Estructuras y Física de Edificación. Universidad Politécnica de Madrid (Spain). jaime.ce[email protected] 477 On the Insurmountable Size of Truss-like Structures Third International Conference on Mechanical Models in Structural Engineering University of Seville. 24-26 june 2015. We define structural (or relative) size (𝑡) as the ratio between the height 𝐿 of a column and its scope ℒ. This structural size 𝑡 can have values between 0 and 1. Note that Eq.(1), that we name Galileo’s rule, is exact in the case of linear pieces with no tangential stress, but it is not proved that it would be a general rule. We define cost (𝑘) as the inverse of efficiency, hence always higher than unity. Then, the self-weight of the column is: 𝑃=(𝑘−1)𝑄 with 𝑘=1/𝑟 (2) Let us define the material structural scope (𝒜), a characteristic length, as the ratio between the admissible stress and its specific weight [2]. This amount is the only information that we will need to obtain the structural scope of the structure itself, besides its geometry and the useful load distribution to be supported. In the case of cylindrical columns, the scope of the columns is the scope of the material: let 𝑓 be the admissible stress of the material, and let 𝐴 be the area of the cross-section, equating the strength capacity of the column base to the weight of the column of maximum height, we obtain 𝑓 𝐴=ℒ 𝐴 𝜌 (where 𝜌 is the specific weight of the material). Then: ℒ=𝑓 𝜌= 𝒜 (3) Maxwell [3] found the way to compare structural costs for a given structural problem. We name [4] Maxwell problem to the problem that consist of defining a structure capable of supporting a equilibrated set of external forces defined both in position and magnitude, and we name Maxwell structure to the structure that resolves a Maxwell problem and is built out of elements that work uniaxially, in tension or in compression. Maxwell showed that for two of those structures solving the same problem, the difference in cost is proportional to the difference in the stress volume 𝒱 of each, defined as: 𝒱= ∑|𝑒|ℓ (4) where 𝑒 is the value of the internal force in each element of the structure and ℓ its length. Later Michell [5] showed that the self-weight of a structure is minimal if its stress volume 𝒱 is also minimal. Also he found a necessary criterion so that a Maxwell structure would be an absolute minimum, from which are derived some optimal layouts for some specific problems. We have defined [6] Michell’s number (𝜈) to the dimensionless ratio between the stress volume of a structure and the product of the total useful load times the size of the problem (the height in the case of columns, the span for beams, etc.): 𝜈= 𝒱 𝑄𝐿 (5) 478 Carlos Olmedo Rojas1, Mariano Vázquez Espí2 and Jaime Cervera Bravo3 Then a structural form is an absolute minimum when its Michell number is lesser or equal to any other structure solving the same Maxwell problem. The contributions of Maxwell and Michell on the measure of structural efficiency do not take into account the self-weight, as it can be seen in the definition of Maxwell problem. Ricardo Aroca [2], [4], [6], [7], [8] joined the theories of Galileo and the theories of Maxwell and Michell: given a Maxwell problem and a structure with stress volume 𝒱 that resolves it, the volume and the self-weight of such structure are: 𝑉= 𝒱 𝜎 𝒱=𝜈𝑄𝐿 𝑃=𝑉𝜌= 𝒱 𝒜= 𝜈𝑄𝐿 𝒜 (6) But previous expressions are only accurate for the null size, since they do not take into account that the self-weight must also be equilibrated. So for structures with 𝑡>0: 𝒱≠𝜈𝑄𝐿 𝑃≠ 𝜈𝑄𝐿 𝒜 (7) Usually the distribution of self-weight will be different to the useful load, but in many cases of interest (for example, in bending structures) both distributions can be represented by distributions of similar forces. We can then estimate the scope of the structural form as the size 𝐿=ℒ for which the structure just resists its self-weight, without possibility of adding any additional load. It results then the following approximate expression, substituting the useful load 𝑄 by the self-weight 𝑃 in the last side of last expression into Eq. (6): 𝑃≈ 𝜈𝑃ℒ 𝒜⇒ ℒ ≈ 𝒜 𝜈 (8) These expressions are Aroca’s rule. Substituting the last expression in Eq. (1), we have the following rule for the efficiency (which we name GA rule, honoring both Galileo and Aroca): 𝑟≈ 1−𝜈𝐿 𝒜 (9) This estimate of the efficiency will be exact for linear structures whose distribution of self-weight is isomorphic to useful load distribution and whose equilibrium requires no tangential stress. In any other case it will be only an approximation whose usefulness must be proved. 2. THE GALILEO PROBLEM FOR TRUSS-LIKE STRUCTURES Here we propose a similar approach to that described above, now for the case of trusses with useful load arranged as forces applied at the nodes, to obtain the structural scope of these structures, subjected to the following limitations: (i) bars are of constant cross-section; (ii) identical tensioncompression patterns on bars due to both useful load and self-weight (i.e., equal sign of the internal force in each bar in these two independent load conditions); (iii) the material has the same absolute value of allowable stress in tension than in compression; and (iv) buckling of compressed bars is not 479 On the Insurmountable Size of Truss-like Structures Third International Conference on Mechanical Models in Structural Engineering University of Seville. 24-26 june 2015. taken into consideration. The aim is to obtain a first estimate of the structural scope for a given structural form, and the structural efficiency. The method proposed here aim to include the cost of transmitting the distributed self-weight along the length of each bar to its extremes, because in large structure this cost will generally be important in relative terms. Rozvany [9] proposed an optimization method that include self-weight but that require bars with variable cross-section —excluded in our approach, limitation (i)—, in fact adopting an exponential function. Such form for bars (close to the so named ’constant maximum stress design’) require a stress tensor that does not fulfil the differential equations of equilibrium [17], so the solutions obtained cannot be considered feasible solutions for the problem. Other authors [10], [11], [12], circumvent the problem of having to consider any bending effects including only half of the bar weight in each of its nodes, but then smart algorithms, as Simulated Annealing, will choose solutions with very large length, as the bending is free of cost [10]. The difficulties of tackling with self-weight in truss-like structures disappear in continuous structures using for example FEM [13], [14]. The selected limitations are justified for several reasons. (i) has a practical meaning. With (ii), we avoid special cases whereby the self-weight of a structure can be equilibrated by the external loading, as pointed out by Bendsoe [15]. With (iii) and (iv), we keep the model simple, but (iii) it is not difficult to overcome and (iv) maybe be suppressed in future research applying new results on this subject [16] . 2.1. Equilibrium equations with useful loads and self-weight Let be 𝐍𝐐 the internal forces in a Maxwell structure under the action of the useful load 𝐐 (hereafter bold capital denote vector or arrays). Suppose that we have solved the problem of designing with bars that include their self-weight. Such bars represent an additional load due to its self-weight that we can introduce using statically equivalent forces at their ends, P. The local equilibrium of the selfweight at interior points of the bar depends on the type of bar: for straight beams, on its bending; for cables or bars without bending stiffness, on the curvature of the bar itself. The Maxwell structure will have to develop additional internal forces for these new loads, NP . We can consider that the design problem is solved if it results that for each bar an axil internal force in the 𝜒 direction defined by its ends is developed, 𝑁𝜒= 𝑁𝑄+𝑁𝑃, and the bar is dimensioned to strictly resist the resulting stresses. The above (ii) limitation can now be expressed saying that sgn(𝑁𝑃)=sgn(𝑁𝑄), and hence sgn(𝑁𝜒)=sgn(𝑁𝑄). The equilibrium equations are the same for both set of loads: Q= HNQ P=HNP Q + P= HN𝜒 ((10) Let 𝜔𝑖 be the ratio between the equivalent weight due to self-weight in the vertex 𝑖 of the bar and the internal force, i.e., 𝜔𝑖= 𝑃𝑖𝑁𝜒 ⁄. Then: P= 𝛀𝐿N𝜒 (11) 480 Carlos Olmedo Rojas1, Mariano Vázquez Espí2 and Jaime Cervera Bravo3 In this expression 𝛀𝐿 is the matrix of coefficients 𝜔𝑖𝑗 of each bar 𝑗 for each component 𝑖 of P, depending on L , the size of the structure. Of course, each column of 𝛀𝐿 has only two non-null components. Therefore, Q+ 𝛀𝐿N𝜒 = HN𝜒 Q = (H − 𝛀𝐿)N𝜒 (12) Note that these equations are nonlinear, as 𝛀𝐿 depends on the size 𝐿 of the structure and on the sign pattern of N𝐐, which is given. When Q →0, i.e., when the structure cannot resist more than its self-weight (and its size 𝐿 is then equal to the scope ℒ of its form), results: (H − 𝛀ℒ)N𝜒 =𝟎 (13) And the value of ℒ is determined as the lowest value of 𝐿 for which (H− 𝛀𝐿) is singular, excluding 𝐿=0. 2.2. Beams (straight bars) Let ℓ be the length of the bar, and 𝛽 the angle formed by the bar and the horizontal, see Fig. 1. The expressions for the internal forces along the bar (tension is positive, compression is negative) are: 𝑁(𝑠)=𝑁𝜒+ 𝜌𝐴sin𝛽(𝑠−1 2ℓ); 𝑀(𝑠)=1 2𝜌𝐴cos𝛽·𝑠(ℓ− 𝑠 ) (14) Figure 1. Beam. We will use the following design conditions: the bars have defined the depth ℎ and the radius of gyration 𝑖 as fractions of the length ℓ of the bar: ℎ=𝑘1·ℓ 𝑖=𝑘2·ℎ=𝑘3·ℓ The maximum normal stresses depending on 𝑠 (position on the 𝜒 axis of the bar) are: 481 On the Insurmountable Size of Truss-like Structures Third International Conference on Mechanical Models in Structural Engineering University of Seville. 24-26 june 2015. 𝜎(𝑠)=𝑁(𝑠) 𝐴±𝑀(𝑠) 𝑊 (15) 𝜎(𝑠)=𝑁(𝑠) 𝐴±ℎ𝑀(𝑠) 2𝐴𝑖2=𝑁(𝑠) 𝐴±𝑘1𝑀(𝑠) 2𝐴𝑘3 2ℓ (16) 𝐴𝜎(𝑠)=𝑁(𝑠)±2𝑘4𝑀(𝑠) ℓ; 𝑤𝑖𝑡ℎ 𝑘4=1 4𝑘1 𝑘3 2=1 4𝑘1𝑘2 2 (17) 𝐴𝜎(𝑠)=𝑁𝜒+𝜌𝐴𝑠((1−ℓ 2𝑠)sin𝛽±𝑘4(1−𝑠ℓ)cos 𝛽) (18) The stress is maximum for 𝑠=𝑘5𝑖ℓ, with 𝑘5𝑖= 𝑘4𝑐𝑜𝑠 𝛽±sin𝛽 2𝑘4cos𝛽 (19) 𝑘5𝑇=1 2 (1+1 𝑘4tan𝛽 ) , in the situation with lower compression or with higher tension (20) 𝑘5𝐶= 1 2 (1−1 𝑘4tan𝛽 ) , in the situation with lower tension or with higher compression (21) which means that for 𝛽 such that tan𝛽≥𝑘4, the stress is maximum at the ends of the bar. Taking 𝑘6𝑇=min{1,𝑘5𝑇}: 𝐴𝜎𝑇=𝑁𝜒+𝜌𝐴 𝑘6𝑇(sin𝛽 (1− 1 2𝑘6𝑇)+𝑘4cos𝛽(1−𝑘6𝑇)) ℓ (22) By grouping the parameters corresponding to the bar, with the following definition of a new constant 𝑘𝑖: 𝑘𝑖= 𝑘6𝑇(sin𝛽 (1− 1 2𝑘6𝑇)+𝑘4cos𝛽(1−𝑘6𝑇)) (23) we can write: 𝐴𝜎𝑇=𝑁𝜒+𝜌𝐴𝑘𝑖ℓ (24) and similarly: 𝐴𝜎𝐶=𝑁𝜒−𝜌𝐴𝑘𝑖ℓ (25) 482 Carlos Olmedo Rojas1, Mariano Vázquez Espí2 and Jaime Cervera Bravo3 Being 𝑓 the allowable stress of the material; making 𝜎𝑇= −𝜎𝐶= 𝑓 to select the appropriate constant cross-sectional area 𝐴, it results, depending on the sign of 𝑁𝜒: 𝐴 𝑁𝜒= ± 1 𝑓−𝜌𝑘𝑖ℓ (26) The coefficients 𝜔𝑖 of the matrix 𝛀 are: 𝜔𝑖= ±𝜌ℓ 2𝐴 𝑁𝜒=sgn(𝑁𝑄)1 2(𝒜 ℓ−𝑘𝑖) (27) As the Eq. (18) only considers the component 𝜎𝜒 of the stress tensor with the model based on the hypothesis of Navier only appropriate for very slender beams, for which the effects of Saint Venant’s principle can be neglected, the expression in Eq. (27) is a good approximation for the self-weight of very slender beams. We keep us on this simple model in this initial work for the sake of simplicity. More accurate models will be used in future research. 2.2.1. Bars with rectangular section With rectangular section, 𝑘2=1/√12, being the width 𝑏 the free design parameter. The crosssectional area is =ℎ·𝑏=𝑘1𝑏·ℓ . Using this value of 𝑘2 to calculate 𝑘𝑖, from Eq.(27) we can calculate the values of 𝜔𝑖 for given values of 𝑘1. 2.2.2. Bars with hollow rectangular section The radius of gyration of a hollow rectangular section with depth ℎ, width 𝑏 and thickness 𝑡 is: 𝑖= √12ℎ𝑡3−6ℎ2𝑡2−6𝑏ℎ𝑡2+3𝑏ℎ2𝑡−8𝑡4+4𝑏𝑡3+ℎ3𝑡 −24𝑡2+12ℎ𝑡+6𝑏𝑡 (28) Eliminating terms with powers of 𝑡, we obtain the value of 𝑖 when 𝑡→0: 𝑖= √3𝑏ℎ2+ℎ3 12(ℎ+𝑏) (29) Taking for this case as an additional design decision that the width 𝑏 is proportional to the depth (𝑏=𝑘𝑏ℎ), resulting the value of 𝑘2: 𝑘2=1 √12√3𝑘𝑏+1 𝑘𝑏+1 (30) being the thickness 𝑡 the free design parameter. Now for given values of 𝑘1 and 𝑘𝑏 , we can take the area of the section as 𝐴=2(ℎ+𝑏)𝑡=2(1+𝑘𝑏)𝑘1𝑡·ℓ, and using this value of 𝑘2 we can calculate the values of 𝜔𝑖. 483 On the Insurmountable Size of Truss-like Structures Third International Conference on Mechanical Models in Structural Engineering University of Seville. 24-26 june 2015. 2.3. Catenary bars of constant cross-section Among all the alternatives for design with constant cross-section and self-weight, the catenary arc, see Figure 2(a), is the better known in respect to efficiency, because there is no tangential stress involved [17]. Figure 2. Catenary arc. Of course this is a theoretical solution, very difficult to build in practice. But as the optimal solutions from Michell’s theory –in fact, funicular structures and hence intrinsic instable ones– their study lead to theoretical limits that no other solution with constant cross-section can exceed. Let 𝑝,𝑛 be axes such that axis 𝑝 follows direction 𝜒, see Figure 2(b). The gravity axis is 𝑔, so 𝛽 is the angle formed by horizontal axis and 𝑝. The chord of the catenary arc is 𝑐, its height is 𝑣, and its base is ℎ (ℎ=𝑐cos𝛽; 𝑣=𝑐sin𝛽). The basic equation result from the equilibrium of a differential arc d𝑠: the variation of the internal force plus the weight must be null: d 𝑁 󰇍 󰇍 +𝜌𝐴(−sin𝛽,−cos𝛽)d𝑠=0 󰇍 (31) and it can be integrated as: 𝑁 󰇍 󰇍 =𝜌𝐴(𝑠sin𝛽+ 𝐾1,𝑠cos𝛽+𝐾2) (32) Let 𝑠0 be the point of the curve with parallel tangent to 𝑝 axis: 𝑁 󰇍 󰇍 (𝑠0)=𝑁(𝑠0)·(1,0)=𝜌𝐴(𝑠0sin𝛽+𝐾1,𝑠0cos𝛽+𝐾2) (33) 484 Carlos Olmedo Rojas1, Mariano Vázquez Espí2 and Jaime Cervera Bravo3 Furthermore, 𝑁(𝑠0) is the oblique component of the internal force, 𝑂 hereafter, i.e., the component in the chord direction for the oblique 𝑝,𝑔 axes, i.e. 𝑁𝜒. Hence: 𝑁 󰇍 󰇍 (𝑠)=(𝑁𝑝,𝑁𝑛)=𝜌𝐴((𝑠−𝑠0)sin𝛽+𝑘,(𝑠−𝑠0)cos𝛽), with 𝑘= 𝑂 𝜌𝐴 (34) Now, changing to the 𝑙,𝑡 axes of the figure, 𝑠0=0, 𝑙(0)=0, y 𝑡(0)=0: 𝑁 󰇍 󰇍 (𝑠)=(𝑁𝑙,𝑁𝑡)=𝜌𝐴(𝑠sin𝛽+𝑘,𝑠cos𝛽), with 𝑘= 𝑂 𝜌𝐴 (35) Note the definition of the 𝑠 coordinate: 𝑠 grows with the concavity on the left, i.e., in the growing direction of d𝑡/d𝑙. In the figure, 𝑠 grows to the right and the concavity points up. As a result, 𝑐 will be negative for a compression arch, accordingly with the definition of ℎ and 𝑣 as function of 𝑐 and 𝛽 above. Defining tan𝛼 as the slope of the tangent to the curve, tan𝛼=𝑁𝑡 𝑁𝑙 cos𝛼= 𝑁𝑙 √𝑁𝑙2+𝑁𝑡2 sin𝛼= 𝑁𝑡 √𝑁𝑙2+𝑁𝑡2 (36) and the parametric equations for this curve can be obtained by integration : 𝑙(𝑠)=∫cos𝛼 d𝑠 𝑡(𝑠)= ∫sin𝛼 d𝑠 (37) Defining 𝜙1(𝑠)=𝑘sinh−1 (𝑠+𝑘sin𝛽 𝑘cos𝛽 ) 𝜙2(𝑠)=√𝑘2+𝑠2+2𝑘𝑠sin𝛽) (38) The parametric equations are 𝑙(𝑠) =(𝜙1(𝑠)− 𝜙1(0))cos2𝛽+(𝜙2(𝑠)− 𝜙2(0))sin𝛽 (39) 𝑡(𝑠) =−(𝜙1(𝑠)− 𝜙1(0))sin𝛽cos𝛽+ (𝜙2(𝑠)− 𝜙2(0))cos𝛽 (40) Note that the limit of 𝑙(𝑠) for ±𝜋/2 is ±(𝑠−|𝑘|)+𝑘, and the limit of 𝑡(𝑠) is null (a vertical line). The modulus of the axial force is: |𝑁 󰇍 󰇍 (𝑠)|=𝜌𝐴 √𝑘2+𝑠2+2𝑘𝑠sin𝛽 =|𝑂 𝑘|𝜙2(𝑠) (41) The coordinates 𝑠1 and 𝑠2 are determined by the fact that the ends of the arc must be the points (𝑙1,𝑡1) and (𝑙2,𝑡2) with the conditions 𝑙2−𝑙1=𝑐 and 𝑡1=𝑡2, i.e.: 𝑙(𝑠1)=𝑙1, 𝑡(𝑠1)=𝑡1; 𝑙(𝑠2)=𝑙1+𝑐, 𝑡(𝑠2)=𝑡2; (42) 485 On the Insurmountable Size of Truss-like Structures Third International Conference on Mechanical Models in Structural Engineering University of Seville. 24-26 june 2015. Let us outline that the above procedure consist in a few and very simple calculations at the first step of the design procedure, providing we have at our disposal the kind of data of the Table 1 for the Maxwell problem under consideration. The teaching of this practice and the associated design theory are included into the curriculum for graduate students of the School of Architecture of Madrid since three decades ago. 5. CONCLUSIONS In this paper we have presented a numerical method to deal with the self-weight load before the cross-sections of its bars are known. When applied to well definite Maxwell problems the method can be used to resolve concrete structural form determining the thickness of its bars, its total self-weight and its structural efficiency. Besides, the method can be used to explore the space of solutions associated to the Maxwell problem and to a definite structural schema, determining the relationships between size, slenderness, insurmountable size, and efficiency of each solution in the search space. With this method we have checked the GA rule (i.e., Galileo-Aroca rule), an approximated but simple formulation to determine the insurmountable size and efficiency of structural forms or schemata, outlining its advantages and drawbacks, and suggesting further research to resolve the latter ones. The proposed method could be improved in several ways: (i) incorporating a more realistic model of the stress distribution for beams (straight bars with bending); (ii) managing different values for tension and compression allowable stress; (iii) considering the local buckling of beams and compressed catenary arcs. In this way, this work can be considered as a reconsideration of the seminal problem of a theory of structural design, formulated by Galileo in 1638, and the approximate solution envisaged by Ricardo Aroca for first time in the eighties of the last century. We hope that this work can contribute to new and promising researches in the future. REFERENCES [1] Galilei, Galileo (1638). Discorsi e Dimostriazioni Matematiche. Leiden: Elsevierii. [2] Cervera Bravo, J. (1990). Las estructuras y el peso propio. Informes de la Construcción, 42.407, 73- 85. [3] Maxwell, J. C. (1870). On reciprocal figures, frames and diagrams of forces. In Scientific Papers II (pp. 160-202). Cambridge Univ. Press. [4] Cervera Bravo, J. (1989). Tres teoremas fundamentales de la teoría del diseño de estructuras. Informes de la Construcción, 40.339, 57-66. [5] Michell, A.G.M. (1904). The limits of economy of material in frame-structures. Philosophical Magazine Series 6, 8.47, 589-597. 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