## Methods of Structural Analysis

*Editor’s Note: The following is an excerpt from an original text by Charles Sanders Peirce. *

May 2017

## Transcriber’s Comments

This text comes from Peirce’s manuscript 1357 as maintained by the Houghton Library at Harvard University, cataloged in 1967 by Richard S. Robin, and made available online by the Scalable Peirce Interpretation Network (SPIN) at http://fromthepage.com/collection/show?collection_id=16. It constitutes a partial draft of the “*Report on Live Loads*” that he prepared in the mid-1890s for George S. Morison’s proposed span across the Hudson River, and immediately follows the excerpt that appeared in the Editorial (*The Esthetics of Structures*) in the February 2017 issue of STRUCTURE.

Peirce suggested that “the method of moments” used by most engineers in his day was not well-suited to complicated structures, such as suspension bridges. He advocated the now-familiar theory of virtual work instead, although he did not use that term, instead referring to it as “the method of Lagrange.” The equation that Peirce provided for elastic strain energy, the area under the linear force-deformation curve, uses (*q*–**q**) for the virtual displacement and “cessiciosity” – from the Latin *cessicius*, which means ceding, conceding, or surrendering – for the inverse of stiffness, or what we today call flexibility; i.e., σ =*L/EA* for an axially loaded member.

In describing the first step of any engineering analysis, Peirce characterized it as diagrammatic reasoning, just as he did in the 1898 article that I quoted throughout Part 2 of my recent “Outside the Box” series on *The Logic of Ingenuity* (STRUCTURE, October 2016). He acknowledged that by employing “a skeleton diagram,” the process “does not solve the real problem, but only a fictitious one.” His subsequent digression into “logical reflections” is worth pondering for its insights into how engineers think and why it (usually) works.

I first made approximate calculations following the method developed by Prof. Melan in the second edition of the *Handbuch der Ingenieurwissenschaften* [*Handbook of Engineering Sciences*] (Vol. II. chap. xii.). But a practical application of this method led me to consider it inconvenient both in its analysis and its modes of numerical computation.

Melan endeavors to adhere, as far as possible, to the method of moments. I believe it ought to be entirely abandoned at the outset. A problem involving *n* elements of any kind and capable of some sort of solution almost always becomes essentially simplified when *n* = 2. By saying it becomes *essentially* simplified, I mean that some special theorem then becomes applicable which practically revolutionizes the solution. For example, an algebraic equation of the nth degree becomes conveniently soluble for *n* = 2. Usually, in such cases, even though *n* exceeds 2, as long as it remains very small, it is still possible to employ a generalization of the theorem for *n* = 2. But as n increases, the advantage of any such generalization soon becomes converted into an enormous disadvantage. Thus, the solution of an algebraic equation of the nth degree, which is so exceedingly handy for *n* = 2, is also possible for *n* = 3 and even for *n* = 4, though it is seldom practically wise to employ it in the last case.

The method of moments is somewhat analogous. For a beam of infinitesimal flexibility resting on two supports, nothing could be happier. Even when there are more than two supports, Clapeyron’s theorem of three moments enables us to use this method. But as the structure becomes more complicated, and especially when the effect of flexures upon the moments have to be taken into account, it is far better, in my opinion, to leave moments out of account until the deformations have been ascertained. If the deformations are insensible, each piece can be considered separately, and the usual graphical methods are in most cases perfectly satisfactory.

But in cases in which the deformations are sufficient sensibly to displace the lines of action of the forces, although they do not take place so swiftly that the momentum of the masses has to be taken into account, I would recommend equations so formed that the only unknown quantities are displacements and deformations. As this method, which is substantially that of Lagrange, is not used by engineers, I will endeavor to describe it that it may be readily understood by those who are rusty in their mathematics.

The first step, whenever a practical problem is set before a mathematician, is to form the mathematical hypothesis. It is neither needful nor practical that we should take account of the details of the structure as it will exist. We have to reason about a skeleton diagram in which bearings are reduced to points, pieces to lines, etc. and [in] which it is supposed that certain relations between motions are absolutely constrained, irrespective of forces. Some writers call such a hypothesis a fiction, and say that the mathematician does not solve the real problem, but only a fictitious one. That is one way of looking at the matter, to which I have no objection to make: only, I notice, that in precisely the same sense in which the mathematical hypothesis is “false,” so also is this statement “false,” that it is false. Namely, both representations are false in the sense that they omit subsidiary elements of the fact, provided that element of the case can be said to be subsidiary which those writers overlook, namely, that the skeleton diagram is true in the only sense in which from the nature of things any mental representation, or understanding, of the brute existent can be true. For every possible conception, by the very nature of thought, involves generalization; now generalization omits, means to omit, and professes to omit, the differences between the facts generalized.

In following out these logical reflections, I am not wandering from the matter in hand so far as might be supposed. For there are many men who will fancy that the question whether a balanced force can be said to exist or not to exist depends upon whether force is a “reality,” or “fact,” or “entity,” or whether it is a “mere mathematical expression.” Such persons can certainly not be accused of any unusual confusion of ideas when we find such a mind as Prof. Tait arguing the similar question of whether energy has “objective existence,” or not. But all such persons confound the quite anti-rational experience of an outward reality, or force, with any possible rational conception of a reality.

That is just the confusion into which Bayle and a hundred others have fallen in accusing the great Cynic [Diogenes] of committing an *ignoratio elenchi* [fallacy of ignorant conclusion] when in answer to the Velean’s [Zeno’s] argument against motion he just got up and walked. He could not have committed a fallacy of any kind since he did not reason at all. He simply repeated, for his part, the experience which blindly and brutally forced the individual fact upon him. Zeno having argued that motion was unreal because unintelligible, Diogenes recognized that, whether intelligible or not, as a fact it compels recognition. But he said nothing because he saw that that sort of reality has to be experienced here and now. If a man carrying a pole on his shoulder knocks one in the eye, there is a reality in that force which I am brutally compelled to see. But when it is a question, not of an experience here and now, but of a general description of things, we can attach no other meaning to the reality of that, than that the statement that it exists is true; and to say that it is true means that there is no understanding the facts without acknowledging this statement.

If a ball rests upon a table, it is perfectly true to say that ball receives continually a component acceleration downward from gravity* plus* an equal component acceleration upward from the elasticity of the table. For we cannot state the behaviour [sic] of the ball under gravity without enunciating a general proposition from which this is a corollary. Since, then, it is true to say there is a component acceleration downward, that component acceleration is a real fact. Its reality consists in its being a case, though only a limiting case, of a general law involving continuity. Generality and continuity are almost the same thing. We might even allow ourselves to say that the ball has a component acceleration east *plus* an equal component acceleration west, provided there was any general truth from which this statement could receive any meaning. On the other hand, there is never any falsity in keeping silence about, or otherwise not referring to, wholly irrelevant circumstances.

Now if we are exclusively concerned with the statical problem of the ball on the table, and if the table neither suffers any sensible compression nor is [in] danger of breaking under the weight, then the fact that the ball would fall under other circumstances is utterly irrelevant, and it is perfectly true to say that it receives no acceleration whatever. Here, then, is [an] acceleration of 32 feet a second [per second] which really exists or does not exist, just as we choose to think it. Only if it exists, another equal upward acceleration exists; while if the former does not exist, neither does the latter. We can save the appearance of verbal absurdity either by saying that the component accelerations exist in any case, but “may be neglected” in the statical problem, or by saying that the component accelerations, or other vectors, which balance one another, are mere mathematical expressions without real existence. Both of these are mere *façons de parler* [manners of speaking], which do not affect the meaning of what is said at all.

In the method, I propose every force is looked upon as the acceleration of a mass coupled with that mass. In the statical problem, every force is balanced. If any independent element of the deformation of the structure is known, we can if we please take cognizance of it and call it a *constraint* against which there are no forces. If, however, we wish to determine one of the forces which are balanced in that element of the deformation, we annul the constraint and use our knowledge of the amount of that deformation to eliminate the sum of the forces balanced against the one we wish to determine. The maxim will be: *where there is constraint there is no force*, meaning by “where” at the same points and in the same directions. A corollary from that maxim is this: *whatever relative displacement does not take place may be treated as impossible*.

Whenever there is a numerous series of parts near to one another and in all important respects regularly related to one another in their serial order, there will be an essential simplification in generalizing the description so far as to represent the whole as a continuously deformable body. Thus, a chain will be treated as a cord and a Warren girder as an elastic line. It is true that the resulting differential equation may make a puzzle for the computer [problem solver], from which a temporary resort to the discrete hypothesis may help to extricate him. Nevertheless, there are conclusions which can be drawn from a differential equation which would not follow from the corresponding equation of finite differences, and such conclusions almost always hold good for the structure.

The second step would consist in forming an expression for the work required to produce any possible given deformation of the structure. By a possible deformation, I mean any deformation which under the conditions we can suppose to be the state of equilibrium that we wish to study. It is not necessary that the expression should hold good for any other deformation. Moreover, the expression may have a constant term which we need not trouble ourselves to determine.

The work required to overcome a force, F, acting along a curve so as to tend to increase, *s*, the arc of that curve, is -∫Fd*s*. Hence, if P is a weight at a depth *y* below a fixed level, the part of the work due to this is -P*y*.

If one of the deformations of the structure consists in changing the value of a quantity *q* and such deformation is in the line of an elatistic [*sic*] force tending to give *q* the special value, **q**, and if what I call the elastic *cessiciosity*, or yieldingness, of *q* is σ, then the corresponding term of the work is (*q*–**q**)_{2}/2σ.

The third step is as simple as the second. In a statical problem treated in the manner proposed, the work is always a continuous function of the elements of the deformation and has a differential coefficient. Where there is equilibrium, every differential coefficient relative to a possible deformation vanishes. Applying this principle, we obtain an equation for each degree of freedom of the rigid parts and a sufficient number of differential equations to determine the deformations of the continuously deformable parts.

The latter have to be treated with circumspection, or the wrong signs will be obtained. Attention must be paid also to the terminal conditions.

The fourth step consists in solving the equations so obtained, which will be of a complicated nature. The best way will be to substitute for each unknown the sum of an assumed approximation *plus* an infinitesimal unknown, and so reduce the equations to linear forms, which will give a new approximation.

This method may fail, though it can hardly do so if the first assumptions are sufficiently near the truth. For the exceptional cases in which it cannot be made to work, I have no general recommendation to make.

The fifth step will consist in determining from the displacements the values of such moments and forces as may be wanted.

I repeat that there is nothing novel about this method, except that it is not in practical use…

*Charles Sanders Peirce was an internationally acclaimed scientist, engineer, and philosopher who lived from 1839 to 1914. He is widely recognized as the founder of pragmatism, the only major school of philosophy native to the United States, as well as semiotics, the study of signs and sign-action.▪*