Historical development of the beam bending equation M equals fS
If a beam cantilevers outward from a wall, then how large does it have to be in order to support a weight at its far end? This question was answered through experience and private design rules for most of our history, without calculating stresses and strains. Alternately, there were some more or less scientific solutions to the problem of bending, but these were either inaccurate or had unnecessarily complex derivations which left the building industry unsatisfied.
The importance of having an accurate solution was recognized at an early date. Galileo observed in the 1600s that beam sizes could not simply be increased by proportion to resist increased live load, for example. His solutions to the problem of bending were improved upon by mathematicians and scientists for a period of about 200 years before an accurate and workable approach was devised. The methods and assumptions that we take for granted were simply not available to researchers in those earlier days.
The most common building materials in the 1700s and the 1800s were timber and masonry. Iron was available, but was not a common building material in America until the early to mid 1800s. Concrete was known to the Romans, but was likewise rare as a building material in America until fairly modern times. Innovations with new materials occurred a little earlier in Europe, but not by much. Therefore, to most of our ancestors, a beam would invariably have had a rectangular cross section – as do most timber beams to this day. This uniformity of cross section makes the calculation of stresses and stains less complex, and will be assumed throughout much of this discussion.
If a stone cantilever is loaded too heavily then it will snap off next to the wall into which it is embedded and the phenomena that caused this might not be obvious. The behaviour of a timber cantilever is more easily visualized, however, because timber is not brittle and will flex as internal stresses increase. Many people would have observed this, but the popular understanding of flexure and the stresses that it generates was nonetheless influenced by how different people visualized the failure mechanism.
Leonardo da Vinci (1452 to 1519) was possibly the first person to illustrate flexure in some sketches. He is not credited with developing any mathematics, however.
Galileo (1564 to 1642) was the first person to apply mathematics to flexure, by considering a cantilever beam. He then extended his thinking to the problem of simply supported beams.
Galileo’s cantilever was timber since his sketch indicated wood grain. On the other hand, the failure mode seemed to be brittle because the beam was imagined to break starting at point ‘A’ and to rotate downward about a hinge point ‘B’. Flexural deflection was not illustrated in his diagram. Galileo therefore brought a special insight to the problem, that flexure was responsible for failure of a beam regardless of whether the material was brittle or ductile.
Galileo’s paper was written as a conversation between a teacher, Sagredo, and two students Salviati and Simplicio. Poor Simplicio required more detailed explanations than did Salviati. The conversation was in paragraph form with only very limited resort to mathematical formulas.
Sagredo first defined a beam’s “absolute resistance to fracture” as its tensile strength. He then surmised that the applied moment about the hinge at point B was resisted by a uniform tensile force on surface AB. The cantilever would fail when this tensile force exceeded the beam’s absolute resistance to fracture.
Two New Sciences; Fig. 17, page 116.
Sagredo (Galileo) recognized that the absolute resistance to fracture was proportional to the cross sectional area of the member. However, he chose to express his results in terms of absolute resistance to fracture and not in terms of stresses. Nor did he try to calculate the value of the absolute resistance to fracture, preferring instead to determine how strong one cross section was as compared to another.
If we were to state Sagredo’s opinions in more modern terms, then we would say that cross section AB was subjected to a moment of M=WL and that this was balanced by a moment of fbdd/2; or M=fS where S=bd2/2 for a rectangular section. This was incorrect by a factor of three since there was no concept of a neutral axis with opposing tensile and compressive stresses. It has also been pointed out that, while Galileo’s model provided moment equilibrium, it did not satisfy equilibrium of forces in either the vertical or the horizontal directions. Many errors that would have arisen from this omission were eliminated by his only dealing in the relative strength of different beams. Therefore, his correct formulation of d2 overcame the incorrect denominator of ‘2’ to some extent and allowed many of his conclusions to remain correct.
It would be unwise to criticize Galileo’s work too severely since his logic led him to several groundbreaking conclusions:
- The weight that can be supported at the end of a cantilever beam is directly proportional to its depth ‘d’ if the absolute resistance to fracture is taken as a given and if the beam’s self-weight is ignored. This is correct within the limits of Galileo’s understanding, the concept of an absolute resistance to fracture being flawed.
- By extension, a beam that is placed on edge will have a greater bending strength than one placed on its flat. Galileo was correct that this improvement in strength was equal to d/b.
- The dead load moment at the support of a cantilever beam is proportional to L2. This was correct.
- The resisting moment of a beam with a circular cross section is directly proportional to ‘r3’. This was correct. Galileo concluded that the strength of a cylinder of constant length was proportional to its volume, which we would interpret as an unnecessary conclusion reflecting his search for fundamental meaning.
- The weight that can be supported at the end of a cantilever is inversely proportional to the length of the cantilever, all other dimensions being held constant. It is at least correct that M=WL.
- A beam which is twice as large as another beam in all respects cannot likewise carry twice the load. In fact, as beams are scaled larger and larger they become proportionally less strong and will eventually fail under their own weight. He was correct that dead load moments increase with the cube of the scale factor.
- If a cantilever beam of length ‘L’ could just sustain its own weight, then the same beam cross section would also be suitable as a simply supported beam with a length of ‘2L’.
- A simple span beam which is loaded at its centerline is more likely to fail than one that is loaded off-center.
- Ignoring the self weight of a cantilever beam, it should be possible to reduce the depth of the section in a parabolic fashion from the embedded end toward the free end without altering its bending strength.
- Hollow beams are relatively stronger than solid sections containing the same amount of material.
“What shall I say, Simplicio? Must we not confess that geometry is the most powerful of all instruments for sharpening the wit and training the mind to think correctly?”
Robert Hooke (1635 to 1703) was an experimental scientist who made many important discoveries. His studies included elasticity and the linear relationship between stresses and strains. He applied these theories to demonstrate that a beam in flexure was subject to tension on one face and to compression on the other and that flexural strains were the direct result of these stresses. Tacit in this understanding was that a rectangular section had a neutral axis located at its mid-depth.
Edmé Mariotte (1620 to 1684) was a contemporary of Hooke and also studied elasticity. To this day, Boyle’s law is known as Mariotte’s law in France. Mariotte applied Galileo’s model of a failure hinge to the problem of a cantilever beam but modified the mathematics to place the neutral axis at mid-depth and to include triangular stress distributions. This resulted in what we would call a section modulus of bd2/3 for a rectangular section.
Mariotte began with Galileo’s formulation as illustrated in Sketch ‘a’. According to this, M=Rd/2. He recognized, however, that a triangular stress distribution would result and that the failure stress ‘f’ was more relevant than Galileo’s resistance to fracture, ‘R’. Therefore he adjusted the mathematics to conform to Sketch ‘b’ and obtained M=Rd/3. One final adjustment was to model the behaviour according to Sketch ‘c’ by substituting d/2 for ‘d’ and doubling the result to represent the sum of the moments provided by the two triangles. This also yielded M=Rd/3. Now, by definition, R=fbd; so that we could represent Mariotte’s final result as M=fS where S=bd2/3 for a rectangular section. Had Mariotte substituted R/4 for R/2 in sketch ‘c’, in addition to substituting d/2 for d, then he would have correctly formulated M=fS where S=bd2/6 for a rectangular section. In finding that M=Rd/3 whether or not his final adjustment was made, Mariotte had shown to his satisfaction that the position of the neutral axis had no influence on the result and that, for simplicity of calculation, a hinge at the bottom fiber could continue to be assumed.
Mariotte’s mathematics was more cumbersome than it needed to be because it modified Galileo’s vision of the behavior rather than replacing it outright. He had all of the background necessary to arrive at a correct solution and it is unfortunate that one error kept him from doing so. In the end, his solution was only closer to the truth than Galileo’s.
Mariotte’s work generated some interest and, for a few years, both his and Galileo’s methods were in use. Galileo’s method remained in vogue for masonry beams while Mariotte’s was used for timber beams. This was likely due to how the failure mechanism was envisaged for brittle materials versus ductile materials, and was possibly also intended to include an additional safety factor for the former case.
Gottfried Wilhelm Freiherr von Leibniz (1646 to 1716) used the new mathematics of calculus to determine section properties. He repeated Galileo’s assumption of a hinge along the bottom edge of the beam, however, and arrived at wrong conclusions. His mathematics would yield a section modulus of bd2/3. It was necessary to forgive the few of the shortcomings of Galileo’s solution – if only because he was Galileo. The same will be necessary for von Leibniz who was a Renaissance man of the age of the Enlightenment. “He was a lawyer, scientist, inventor, diplomat, poet, philologist, logician, moralist, theologian, historian, and a philosopher …” and may be forgiven a slight error in composing ‘bd2/6’.
Jacob Bernoulli (1654 to 1705) made much more significant use of calculus in studying beams than had Leibniz, but applied his studies to deflections rather than to strength. The problem was therefore still not solved.
Antoine Parent (1666 to 1716) also considered the problem of bending. His studies began with the basic assumption made by Galileo that a failure hinge existed at the bottom fiber of the cantilever beam. He knew, however, that Mariotte had postulated the existence of tensile and compressive stresses across the cross section, albeit Mariotte did not think that the location of zero flexural stress was important. Parent performed some calculations based on static equilibrium in order to explain Mariotte’s beliefs, but these calculations were not very helpful except to satisfy Parent that there were, in fact, tensile and compressive stresses. Parent then replicated Galileo’s calculation based on an absolute resistance to fracture but corrected it for a triangular tensile reaction in the same manner as had Mariotte. His next step, however, was to correctly formulate the solution for triangular tensile and compressive stress blocks – without the Mariotte error of not having substituted R/4 for R/2. He had therefore correctly arrived at M=Rd/6 where R is the absolute resistance to fracture. Substituting R=bdf would result in the equation M=fbd2/6; or M=fS where S= bd2/6 for a rectangular section.
Parent had some difficulty in rationalizing his calculations with experiments that took materials beyond their elastic limits, and formulated a failure model that shifted the location of the neutral axis at the point of failure. This was the first time that the problem had been correctly formulated (aside from the point concerning a shift in the location of the neutral axis), very early in the 1700’s. Unfortunately, Parent’s significant insights were not carried into practice and Galileo’s and Mariotte’s methods continued in general use.
Charles Augustin Coulomb (1736 to 1806) correctly formulated the entire problem of bending of a cantilever beam in a paper published in 1773. This was done by methods which we would find familiar, including equilibrium of a cut section which remains plain after rotation under bending stresses. With this, he arrived at M=fS where S= bd2/6. This was without reference to Parent’s earlier discovery of the same result. Coulomb did not promote his findings among building professions, who found the calculations difficult, tedious and of little benefit. His work, like Parent’s, was therefore soon forgotten and many researchers reverted to the earlier theories of Galileo and Mariotte.
P.S. Girard published a work in 1798 which included a calculation of bending. He recognized that both tension and compression acted on the cross section of a beam and that Mariotte’s mathematics which discounted the location of the neutral axis was flawed. He agreed with Mariotte, however, that the final result of the calculation would be the same whether the more correct approach was taken or not. Girard therefore missed the opportunity to arrive anew at Parent’s and Coulomb’s conclusions and to bring the correct solution into common use.
Claude Louis Marie Henri Navier (1785 to 1836) correctly solved the problem again, and without reference to either Parent or Coulomb.
Earlier in his career, Navier thought that Mariotte was correct in assuming that the position of the neutral axis was not important and that the solution to the bending problem could be derived by considering rotation about the bottom fiber of the cantilever beam, as had Galileo. He corrected this opinion later on, however. The familiar principles that sections that are plane before bending remain plane after bending; that the neutral axis of a rectangular section is at its mid-depth; and that the analysis applies only up to the elastic limit are all from Navier. The limitation that the solution only applied up to the elastic limit was not elucidated until 1826.
Navier’s work was distinguished by the practical applications to which he could apply it. Thus, unlike Parent and Coulomb, his solution became well recognized and was actually used in design. The relationship M=fS was finally established.
Approximate Methods for Calculation of Bending:
It may be that Sagredo was correct when he said that “this demonstration, Salviati, is rather long and difficult to keep in mind from a single hearing”, and that this would explain why the formula M=fS was not immediately embraced. It was probably not that simple, however, since Navier was not working in a vacuum and other design methods were being developed at the same time. There was also a growing preference for design methods based on test results which seemed a more reliable approach. The aversion to direct calculation of stresses was encouraged by a lack of understanding of exactly what allowable stresses should be used.
In England, before the middle of the 19th century, a design method was developed whereby the strength of a rectangular beam was given by the expression W=AdC/L: where W was the supported load in tons or pounds depending upon the author; A was the cross sectional area of the beam in square inches; d was the beam depth in inches; and L was the span length in inches. C was a constant which depended upon the loading condition and material type. C also included a conversion to tons in some design tables. William Fairbairn used a very similar expression as late as 1864. Rearranging this equation, and substituting S for bd2/6 and substituting the constant recommended in W. Davis Haskoll’s Railway Guide of 1848 for Memel fir would yield an ultimate bending strength of 10,380 psi for a cantilever of rectangular cross section. He recommended that the calculated load capacity be divided by three for design, for an equivalent of 3,460 psi on fir. In that context, approximations in the analysis seem less critical than prevailing beliefs as to material strength. Haskoll also thought that variations in fabrication were significant enough to make detailed analysis useless in some cases:
“By calculation, a scientifically framed truss may be made; the almost impossibility in practice of making such a perfect assemblage … renders such complicated combinations unwise; … [and so he recommended] a more simple mode of treatment.”
The first iron bridge in America was Dunlop’s bridge, built in 1839. The Europeans were a little further along, however, and W. Davis Haskoll’s Railway Guide of 1848 included tables of cast iron beams available in England together with a recommended design method. Haskoll’s design procedure gave credit to work by Eaton Hodgkinson and Thomas Tredgold. Tredgold and Peter Barlow were the original authors of the method which appeared in Barlow’s handbook of 1817 and Tredgold’s works of 1820 – 22.
The cast iron beams ranged in depth from 10-1/2 inches to 36 inches, the smaller ones being wide flange or HP shapes and the larger ones being inverted Tees with large bottom flanges and small top flanges. They were usually symmetric about their ‘y’ axis but not always so. It seems that they were intended to be placed side-by-side with the outstanding bottom flanges touching in order to frame a floor.
A Typical Cast Iron Beam
The Assistant Engineer’s Railway Guide, page 129.
The design procedure for cast iron beams used a similar formulation to what we have seen for timber; W=AdC/L. In the case of iron beams, however, A was the area of the tension flange only, and W was uniformly reported in tons. The value of C was determined by tests and published in design tables.
Haskoll’s design example for a railroad application noted several principles:
- Beams for railroads should be designed for a concentrated load at mid-span equal to two tons for each foot of span length. This was without regard to how many beams were used or how they were arranged and was not a design load as we would know it. It was more of a design allowance.
- The required depth of the beam could be determined from the depth of another beam which passed a loading test, simply by adjusting that depth in proportion to the span length. Haskoll’s example determined the beam depth for a 30 foot span based on a test for a 7 foot span, for example.
- The required area of the bottom flange, A, could be determined from the equation W=AdC/L, where d and L were known and C was taken equal to 25.
- The top flange area should be about equal to one third of the bottom flange; and the total area should be about equal to twice the area of the bottom flange.
- The minimum web thickness should be one inch, and need not be greater than that. If the other design criteria resulted in an overly thick web then the design depth could be adjusted downward and the procedure repeated.
Much of this was unscientific. If the beam, as designed, was subjected to the specified load at mid span, then a tensile stress of 14,934 psi would result. If the moment due to the self weight of the beam and some light decking and ballast was accounted for separately, then the total tensile stress would be around 16,000 psi on cast iron. Again, the deficiencies of the analysis were outweighed by the stresses that were being accepted. Haskoll added a note, however:
“In designing for cast iron we must never for a moment lose sight of its being the most treacherous material we have to deal with; cast of excessive thicknesses, it has been known to break under, comparatively speaking, a most trifling pressure, and yet when the fractured pieces were tested, they were found to be considerably above the average strength. Again, a very slight blow has been known to break a girder which has for a long time borne the rapid passage of heavy trains; hence, ample dimensions are absolutely necessary, as well as experienced selection of the kind of iron used, and careful superintendence during the casting and testing.”
By 1872, many designers still viewed mathematical models for bending to be unreliable and the use of experimental data continued in design. John Anderson expressed one point of view when he said that “scarcely any of these theories fully explains the law which gives the results found by experiment.” He then added dismissively that results based upon experiments were “quite sufficient for the greater number of constructions in ordinary practice, … owing to the defective state or our knowledge on this part of the subject.”
To his credit, Anderson thought that the rote application of formulas was dangerous:
“This reliance upon mere formulae is a great mistake, and often leads to expensive errors. In some cases, formulae, originally intended for one kind of structure, have been applied in designing a different kind of structure, and the incorrect result arrived at has only been discovered, when it was too late to avoid the consequences of the mistake.”
John Anderson’s design methods were therefore very similar to W. Davis Haskoll’s; and he cited the same authorities including Hodgkinson, Tredgold, Barlow and others. He recognized that sections in bending were subjected to both tension and compression and that they remained plane after bending. There was therefore a neutral axis. He also noted the possibility of buckling if a top flange was too small. He tried to bring practical considerations to bear, including the effects of unintended rotations of fixed joints.
Analyses of stresses and deflections at this time were often inaccurate due to the assumptions that were used. Some designers lacked certainty of how to locate a neutral axis for a complex cross section for example, since “until the position of the neutral axis of such beams as are employed in practice is accurately determined, it [experiment] is the only course open.” Stresses and deflections were generally approximated using the depth between flanges or the total beam depth in place of the depth between flange centroids; webs were usually ignored in determining bending strengths; and the moments of inertia of flanges about their own centroids were also ignored.
John Anderson’s approach probably represented the state of practice when he was writing in 1872. For him, the labor involved in calculating the section modulus of a cast iron beam with a curvilinear outline was not worth the trouble. A reading of Squire Whipple’s 1873 update of his book on bridge building supports this view. For a rectangular pine beam, Whipple recommended that a resisting force be taken about a fulcrum (a hinge) at the bottom fibre, and that this force be equal to the cross sectional area of the beam times 250 psi with a lever arm equal to the depth of the beam. Converting this to the form M=fS would yield an allowable bending stress of 1,500 psi, which was appropriate depending upon the grade of pine. There was still a preference to work with the analogy of an absolute resistance to fracture – even given the correct understanding of the more scientific form, M=fS.
These practices did not reflect the state of knowledge, however, and John Anderson’s very strong defense of the methods ring hollow in retrospect. Professor Allan acknowledged in 1875 that “some question still exists as to the exact location of the neutral axis”. He was confident, however, that the neutral axis passed through the centroid on the section – at least for beams with small deflections. Professor Allan also had no difficulty in finding the centroid of a complex cross section, or in calculating its section modulus. He even offered a method for finding the centroid of a section without tedious calculation, by tracing the shape onto a piece of tin; cutting it out; and suspending it from different points with a plumb bob to locate its center of weight. William Gillespie could not only determine the location of a centroid and the value of a section modulus in 1870, but could also apply calculus to determine curvatures and deflections.
In general, John Anderson’s book reveals the methods that were in use in the early 1870s but does not demonstrate any appreciation for the changes that were already taking place. The latitude granted to Galileo and to Leibniz can therefore not also be extended to John Anderson.