Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 649341, 10 pages

http://dx.doi.org/10.1155/2015/649341

## On Galileo’s Tallest Column

^{1}Grupo de Investigación en Arquitectura, Urbanismo y Sostenibilidad (GIAU+S UPM), Avenida Juan de Herrera 4, 28040 Madrid, Spain^{2}ETS de Arquitectura de Madrid (UPM), Avenida Juan de Herrera 4, 28040 Madrid, Spain

Received 2 April 2015; Revised 19 June 2015; Accepted 28 June 2015

Academic Editor: Reza Jazar

Copyright © 2015 Mariano Vázquez Espí et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The height at which an unloaded column will fail under its own weight was calculated for first time by Galileo for cylindrical columns. Galileo questioned himself if there exists a shape function for the cross section of the column with which it can attain a greater height than the cylindrical column. The problem is not solved since then, although the definition of the so named “constant maximum strength” solids seems to give an affirmative answer to Galileo’s question, in the form of shapes which seem to attain infinite height, even when loaded with a useful load at the top. The main contribution of this work is to show that Galileo’s problem is (i) an important problem for structural design theory of buildings and other structures, (ii) not solved by the time being in any sense, and (iii) an interesting problem for mathematicians involved in related but very different problems (as Euler’s tallest column). A contemporary formulation of the problem is included as a result of a research on the subject.

#### 1. Introduction

In 1638, in his* Discorsi e Dimostrazioni Matematiche* [1], Galileo postulated the existence of the tallest column, that is, a cylindrical column, such that it attains the maximum height once the area of its cross section and the strength of its material are prescribed. Therefore, Galileo’s tallest column is in the limit of resistance only bearing its own weight. The rationale of his proof gave rise to the square-cube law, a mathematical principle that considers the relationship between the flow through the surface of a volume and the stock into the latter, in the mechanical case, for example, the stress with the weight. This principle has been very useful and applied in a variety of scientific fields, mainly in biology [2–4].

In the following century, Euler [5] pointed out a very different problem, that is, to find the shape of a stable column, axially symmetric with respect to the vertical axis, such that it attains the maximum height once its volume, specific weight, and Young’s module are given, buckling due to the action of a load bear at its top.

In our view, both problems are not yet completely solved nowadays, although Galileo’s problem has received much less attention than Euler’s. Furthermore, we think that Galileo’s problem is more meaningful for a theory of design of structures subjected to small limit on strains and displacements, as buildings and other structures in civil engineering [6, 7].

Hereafter, we consider the solid continuum with the following standard assumptions:(1)The scope of the analysis is the classical theory of Elasticity.(2)The process of deformation is isothermal and quasi-static; heat or kinetic energy is not taken in consideration into energy balancing.(3)We are only interested in solutions whose strains and displacements will be very small; hence equilibrium and compatibility equations approximately hold in the geometry of the undeformed body.

Section 2 outlines important aspects of Galileo’s problem, comparing it to Euler’s and enlightening its importance and profound meaning for a theory of structural design. Our main working hypothesis is formulated there. Section 3 deals with some clues that support our working hypothesis; that is, an insurmountable size exists for a fairly large set of structural problems, as it is the case of cylindrical columns of Galileo’s first insight into the tallest column. This section covers the main aim of this paper which is to attract mathematicians to work out Galileo’s problem, because we are architects and our mathematical knowledge is, to say the least, limited, but to solve the problem is a key point to continue the development of structural design theory. Finally, Section 4 is devoted to formulate the problem formally in contemporary terms.

#### 2. Galileo’s Problem on the Tallest Column

Proposition VII. Among heavy prisms and cylinders of similar figure, there is one and only one which under the stress of its own weight lies just on the limit between breaking and not breaking so that every larger one is unable to carry the load of its own weight and breaks while every smaller one is able to withstand some additional force tending to break it. (GALILEO, 1638)

Consider a cylindrical column of height and diameter , of a lineal elastic material defined by Young’s Modulus , allowable compressive stress **,** and specific weight and subjected to the sole action of its own weight.

Such a column will be unsafe in simple compression if the applied load exceeds the column strength , where is the area of the cross section—if the base is a circle, —or, which is the same, if the applied stress exceeds . This fact means that the height of such a column may not be greater than a characteristic length of the material, . We name this length “structural scope” of the material, . And we name “structural scope” of cylindrical columns to the maximum height of safe columns. In this simple case, is numerically equal to the material scope , but generally is related to but is not equal to [6]. Therefore the first conclusion of Galileo can be expressed asLater, Galileo considers in which way this insurmountable limit can be increased. He envisaged two main ways: (a) to increase the material scope , or (b) to change the shape of the column. In the latter case, he reasons—in a funny paragraph—that if the giants exist, they would have a very different aspect and proportions compared to human beings; specifically the bones of their legs would have a greater diameter/length ratio, because otherwise their weight that increases in proportion to would be greater in proportion to their strength that increases as , and as a result the giants would suffer stresses—which increase as —greater than human beings, and considering the bone material very similar in all the mammals, the giants would be unable to perform in their life as well as human beings. A few centuries later, this result could be confirmed comparing dinosaurs of different sizes but of same suborder or family (i.e.,* Theropoda* or* Tyrannosauridae*) [8].

As a material with infinite strength or null specific weight does not exist, it is clear that following first Galileo’s way we can only increase the insurmountable height but remaining finite. If we adopt the latter way, the main question arises: does there exist an optimal shape which has infinite height? We are looking for an answer to this question because it is a key into the theory of structural design. If the answer is “Yes,” then Galileo’s problem has a solution for any size considered. But if the answer is “No,” there exist instances of structural problems which have no solution; that is, there are unsolvable problems in structural design. Furthermore, as we will show below, near the unsurmountable size, any solution for the problem will have an unaffordable physical cost, so it would be infeasible from a practical view.

Our working hypothesis is that a finite insurmountable size exists for a fairly large set of structural problems (not only for Galileo’s problem). Moreover, the optimal shape for each problem—which maximises the finite insurmountable size—is a sound reference to measure the efficiency of all other shapes with size lesser than the one of the optimal shape [7, 9, 10].

In a first approximation, we can represent the physical cost of a structure by its self-weight, as many costs during the manufacturing, but not all, are approximately proportional to the self-weight of the structure: CO_{2} emissions, mineral resources consumption, and so forth. For a given structural problem, we define the structural efficiency as the ratio between the useful load and the total load (i.e., the useful load plus the self-weight) required to solve that problem in a particular structure. Galileo postulated also the relationship between the size of a structure and its ability to resist a useful load. Let us consider a cylindrical column of size . It can resist an additional useful load , the value of which is at most the weight difference between this column and the column of insurmountable height . Hence, according to the previous definition, the efficiency of such a column will beNote that (2), which we name Galileo’s rule, is exact in the case of cylindrical columns, but it is not proved that it would be a general rule. The best result we get up to date is that Galileo’s rule is a very good estimate in canonical problems like bending of beams and bridges [10]. We define the load cost as the inverse of efficiency, hence always higher than unity, . Then, the self-weight of the column isAs a reward, Galileo’s rule, apart from the cost, gives us a sound estimate for the self-weight that it is a required datum for the final project but unknown in the preliminary phases of design of large structures.

##### 2.1. Comparison between Euler’s and Galileo’s Problems

Remember the cylindrical column of height and diameter . As we saw, such a column will be unsafe if its height is equal to or greater than material scope . But the column may also fail by elastic buckling. According to Landau and Lifshitz’ Course [11], the critical height for buckling is related to the diameter byThe ratio is another characteristic length of the structural material. Whereas the scope is its specific strength, is its specific stiffness. Let us define the geometrical slenderness of the column as the ratio . ThenTherefore, the safety of a given column bearing only its own weight requires the fact that two conditions hold: (i) and (ii) . It is worth noting that it is always possible to satisfy the second condition, as for each height we can choose such that . However the first condition is an absolute one, as it only depends on the properties of material. Hence, the height of a safe, cylindrical column would be lesser than or equal to .

This limit, as noted above, only could be modified in two ways: changing material’s properties or changing the shape of the column (or both). The interesting point here is that to answer Galileo’s question we must elucidate if a finite structural scope (related to the material scope ) exists for any shape of the column. The advances in the analysis of the solution for Euler’s problem [12–14] are useless to this aim, mainly because of ignoring the limit that strength condition imposes on the shape (condition (i) for cylindrical columns).

Furthermore, as it is well-known, the classical solutions of Euler’s problems on buckling are contradicting the experimental data. “This contradiction between theory and experiments is not surprising. The ideal appearance of a phenomenon is always more or less influenced in practice by multiple causes that can deform it to the point of leaving none but a caricature. In the problems of instability, the theory considers only perfect elements, both form and structure indefinitely elastic and resistant. The test pieces, or the members actually built, are very far from perfection; the materials are inhomogeneous, and they are approximately elastic and within certain limits.” [15] As a consequence, in the engineering practice, there is no bifurcation between two different equilibria. On the contrary, as the slenderness approaches to its critical value , the failure changes continuously from simple flattening to bending with net compression. In fact, the so named critical load (or Euler’s load) is not a “load” at all but a stiffness of the column against lateral displacement, and the failure occurs for real loads numerically lesser than this stiffness. (Unfortunately, this stiffness can be expressed in load units (), but it is better understood with stiffness unit (), showing that it is a ratio between the bending moment and the lateral displacement.) Moreover, although the buckling of real structural members (with negligible self-weight) is a nonlineal problem in a first, mathematical view, therefore candidate to a numerical solution, it is possible to overcome the difficulties and to solve the problem by a direct albeit nonexact formulation when structural strength is taken into account [16].

#### 3. On the Existence of a Finite Height for Galileo’s Tallest Column

Our epistemological situation confronting Galileo’s problem is analogous to the situation that algorithm designers are confronting when using the well-known Theory of NP-Completeness [17]. We, the structural designers, do not know if a finite insurmountable size exists for the problem at hand; hence we cannot know in advance if our problem is solvable or not. But if we believe that this limit exists, we can manage at least a rough estimate of its value and, armed with this knowledge, take a decision about the solvability of the problem. Indeed, if we know the size limit that different types of structures can reach for our problem, we can evaluate approximately the relative merit of each type and select the most promising one for the actual size of our problem. So the existence (or not) of a finite height for Galileo’s tallest column is a key point for our everyday work.

Let us consider the two main approaches to the problem: first that such a limit does not exist because it is easy to find the corresponding shape and second that such a limit probably exists because it is very hard to find out any shape that can overcome a given finite limit on its height.

##### 3.1. The Known Solutions with Infinite Height Are Unfeasible

Although Euler’s tallest column and related problems have received very much attention up to date, some researchers while studying that problem spent a few minutes to study, too, problems related with that of Galileo. This is the case of Karihaloo and Hemp [18] that study the “maximum strength design” of structural members. In their approach, all cross sections of a structural member attain the maximum allowable stress for the given material; therefore the solution is also referred to as “constant maximum strength design.”

Let us examine with some detail the constant maximum strength design of Karihaloo and Hemp for tension members; see [18] (Section 2.1) and Figure 1. Consider a cable of length and cross-sectional area , hung on its top edge () and when its bottom edge () is a free boundary. If the gravity and the external load are acting in the negative direction of -axis, the condition of the constant maximum strength isThe solution isas it can be checked obtaining its derivative and comparing the result with (6). If the cross section is circular, the radius gives us the contour of the member as follows:This solution can have an infinite height with constant stress and bear a useful load at the bottom edge (, ). Anyhow, its volume grows exponentially with its size (), so in practice it is an “intractable” solution—in the same meaning that the term is used in algorithm complexity theory [17]—with a load cost as follows: