## Columns: The long and the short of it from 1729 to about 1900

*Beginnings*

If a column with a cross sectional area of ‘A’ is acted upon by a compressive load ‘P’ then the stress in the column is given by f_{c} = P / A. Therefore, if the allowable compressive stress is known for the column material, then we also know how large the column has to be to support the load. This approach almost never works for columns of usual proportions, however, since columns buckle before they crush. The column problem is therefore one of those situations where P / A is not enough.

It has been known for millennia that if a column is too slender then it will collapse. The relationship between slenderness and strength was not documented for a long time, however, and builders relied on rules such as not to make a column more than 15 times as high as it was across. Rules like this would have woven themselves into architectural design and the proportioning of columns would have become a matter of ‘making them look right’.

*A long process*

Reliable column design equations did not come quickly or easily. Euler is given major credit for his equation for long columns, but a different approach was needed for short and intermediate columns. Other theoretical equations were proposed to fill the gap but the theories were flawed to an extent of which practitioners were not even aware. Extensive tests were carried out with imprecise modeling of column end conditions and design methods were proliferating at a time when there were no accepted standards.

*1729, Musschenbroek: Strength proportional to b d ^{2 }/ L^{2}*

The Dutch physicist Pieter van Musschenbroek was accomplished in many areas of physical science and was the first to show by experiment that the strength of a thin column was inversely proportional to the square of its length; and was directly proportional to its greater thickness; and also directly proportional to the square of its lesser thickness.

i.e., Strength ~ b d^{2 }/ L^{2}

These findings were reported in his *Physicae Experimentales et Geometricae*, published in 1729, which also dealt with many other topics. This had little or no effect on the building industry, however.

*1744, Euler*

Leonard Euler also demonstrated that the strength of a column was inversely proportional to the square of its length, but he did this through calculation rather than by experiment. His finding, like that of van Musschenbroek was also just a small part of a larger work entitled *Methodus Inveniendi Curvas Lineas* and was published in 1744. The derivation of Euler’s equation was along the following lines:

If a column is compressed by an axial load and the column is not perfectly straight, then a PΔ moment will be generated. Now, the curvature (d^{2}y / dx^{2}) of the bent column at any point is equal to M / EI, so that a second order differential equation results. Solving the equation and substituting for boundary conditions gives Euler’s equation

f_{c} = E π^{2} / L^{2}

Euler could not relate the solution to a section property such as I or A, so he included a quantity k^{2}. His mathematical model was also for a column that was fixed at its base, so that a factor ‘4’ appeared. His actual formulation was therefore

f_{c} = E k^{2} π^{2} / (4 L^{2})

*Problems with Musschenbroek and Euler*

The difficulty with van Musschenbroek’s relationship and with Euler’s equation was that they were not accurate for a full range of column lengths. Very short columns would crush before they failed by elastic instability; and columns of intermediate length would buckle at loads below what Euler’s equation would have predicted. This was pointed out by A. Duleau who showed that Euler’s equation was accurate only for the longest of columns. Something more was needed since most columns were neither very short nor very long.

A modern statement of Euler’s equation for the case of a column with both ends pinned is

σ_{allow} = π^{2 }E / ^{ }(L / r)^{2}

and E. Lamarle proposed in 1846 that this should apply only up to the elastic limit so that

Min. (L / r)^{2} = π^{2 }E / σ_{yp}

At this point, the problem of elastic buckling of long columns had been considered by van Musschenbroek and Euler. Duleau and Lamarle had added their thoughts, and the problem was also considered by Lagrange. There came a time, however, when this work was set aside in favor of simpler approaches. Many observers praised Euler’s brilliant solution, and these testimonials seem today as veiled admissions that others did not want to deal with second order differential equations. In any event, most columns were not so long as to be even approximated by Euler’s solution. Euler would eventually have his due and be recognized as the father of column theory.

*1822, Tredgold’s equations*

Thomas Tredgold developed a series of equations which appeared in his 1822 book *Practical Essay on the Strength of Cast Iron and Other Metals*. These were based on calculations of equilibrium for eccentrically loaded and bent columns. They included factors to account for unintended eccentricities and other imperfections. The equations for column strength all took the form

* *X b d^{3}

4 d^{2} + Y L^{2}

in pounds, where X and Y were constants, L was the column length in feet, and b and d were the cross sectional dimensions in inches with d ≤ b. The constants varied depending upon the material being used (cast iron, wrought iron or oak) and the cross sectional shape. The factor ‘4’ had to do with his recommendation that a design eccentricity of d / 4 be assumed. For cast iron he used a yield strength was 15,300 psi and a yield strain of 1/1,204 and these numbers also contributed to the constants. Overall, the constants resulted from calculation and did not rely on test results.

These equations gained favor in the building industry and survived in one form or another for nearly a hundred years. Tredgold recommended the capacities as ‘safe’, i.e., conservative, but he still intended them as failure loads and they did not include conventional safety factors. The selection of safe working stresses, as opposed to failure loads was up to the builder. Tredgold’s estimates of column failure strengths were very conservative, however.

*1840, Hodgkinson’s tests*

As the 19^{th} century progressed, there was an increasing reliance upon test results in design. Hodgkinson performed a large number of tests on columns and these became available in 1840 with his *Experimental Researches on the Strength of Pillars of Cast Iron and other Materials*.* *These tests and Hodgkinson’s equations were referenced extensively and took over, for a while, from Tredgold’s equations.

Hodgkinson’s tests were intended to measure the ultimate capacities of columns and, for cast iron columns, this would be governed either by elastic buckling or by breaking of the brittle iron before it reached its buckling strength. Hodgkinson’s final load reading for each column might therefore have been the load at which it buckled, whereupon it broke; or the load at which it broke before reaching elastic instability. A fuller discussion of these tests is included in the Appendix.

Hodgkinson’s formula for the breaking load of a column with a length of at least 30 times its diameter was

* *A h^{3.6} / L^{1.7}

where A was a constant depending upon the column end conditions and whether the column was solid or hollow; h was the least cross sectional dimension in inches; and L was the column length in feet.

John Anderson’s 1887 commentary on columns shows how difficult the design process had become. Designers were individually reinterpreting the Hodgkinson test results and reinventing the formula. Tests were referenced to relate capacities to fractional powers of column lengths and section widths, depending upon the shape of the section, its dimensions and end conditions. Anderson noted that local plate buckling of thin-walled columns was another complicating factor, independent of column length, and that there was conflicting advice as to how thick such plates should be. He concluded that tube columns were impractical. Realizing that hollow columns were common, he added that the walls needed to be stiffened against crippling.

Squire Whipple’s 1847 and 1883 works also relied heavily on Hodgkinson’s test data for design of columns. For columns where 15 ≤ L / D ≤ 40 he recommended that test data be used with results adjusted to the average of D^{3} / L^{2} and D^{3} / L; and for 40 ≤ L / D he recommended that D^{3} / L^{2} be used alone. Whipple presented a table of column capacities, but these capacities could not be replicated using his guidelines. Whipple also recognized Gordon’s formula but by 1883 he still had no firm recommendation as to which approach was best.

* **1840s, Gordon’s formula*

Thedgold had expressed his equation for column capacity in a different form as follows

f b d^{2}

d + 6 a

where ‘f’ was the basic allowable material stress in psi and ‘b’ and ‘d’ were the dimensions of the rectangular cross section in inches (d < b). The quantity ‘a’ was the assumed eccentricity of the applied load. The eccentricity measured at mid-height of the column would increase as the load was applied, to a new value given by

a + L^{2} ε / (4 d)

Substituting this value for ‘a’ in the equation for column capacity, and assuming an initial load eccentricity of zero at the column ends (which he did not recommend for design), Tredgold obtained a new equation for column capacity as follows

F b d^{2}

d + a L^{2}

where a = 6 ε / (4 d)

and where ‘a’ was measured in inch^{-1} units while L was measured in feet. The units required a factor of 144 in addition to substituting 15,300 psi for ‘f’ and 1 / 1,204 for ε. The equation for column capacity then reduced to

15,300 b d

1 + 0.18 L^{2} / d^{2}

The form of this equation is known as Gordon’s formula although Tredgold published it in the 1831 edition of his work and possibly in the 1822 first edition. Gordon re-defined the constants, however, based on Hodgkinson’s test results. The constants in Gordon’s formula were then entirely empirical and could only be determined by tests on different materials and for different column end conditions.

*1850s, Gordon-Rankine formula*

William Rankine inserted the least radius of gyration in Gordon’s formula as follows, with the variable constants generalized

f K

1 + a L^{2} / k^{2}

This is the Gordon-Rankine formula, where ‘f’ and ‘a’ were the constants; ‘K’ was the cross sectional area; and ‘k’ was the least radius of gyration.

Tredgold’s equation was thought to have some theoretical basis, but Hodgkinson’s test results were the best authority to date as to the actual strength of columns. It is therefore significant that Gordon and then Rankine reverted to the Tredgold form but substituted new values for the constants in order to reflect the Hodgkinson test results.

Some effort was made to define the constants on a more theoretical basis. For example, Reuleaux proposed a formula for column capacity in 1862 as follows

C K

1 + C L^{2} / (8 E k^{2})

where C is a safe level of stress less than F_{y} and E is Young’s modulus.

Another variation was by J.D. Crehole who proposed in 1870 to substitute π^{2} for the factor ‘8’ in Rouleau’s formula. This was for a column with round ends. For a column with square ends he recommended to substitute 4π^{2} instead of π^{2}. For one round end and one square end he would substitute 16/9 π^{2}.

Some confidence must have been lacking in the work of Tredgold, Hodgkinson, and Gordon-Rankine, since some designers were using a single allowable stress for all short columns, and another stress for longer columns regardless of the value of L / r. The Mohawk River bridge at Schenectady was an example of this. In other cases, allowable compressive stress depended upon what sort of member was being considered. It seems that practical ranges of L / r were being assumed.

*1860, A sad state of affairs*

Daniel Treadwell read a report on the state of column design to the American Academy of Arts and Sciences in 1860 which summarized developments to that time. He noted “glaring discrepancies and contradictions contained in the practical rules and tables in common use by builders”. People such as Hodgkinson had determined failure loads for columns but the designer still had to “determine for himself how far he will keep within this limit”. Cast iron was of a treacherous character, and through caution many design practices had proven safe but only “with a prodigal use of iron”.

*1870s, Launhardt’s and Weyrauch’s formulas*

Wöhler began tests on the effects of repeated loads in 1859, and Spangenberg continued this work into the 1870s. In 1873, this lead to Launhardt’s formula which determined a working stress for fluctuating tension based in part upon a stress ratio. Weyrauch sought to extend Launhardt’s formula for the case of stress reversals. One of the variables in Weyrauch’s equation was known only for the case of tension, but it was assumed that this could also be applied to compression.

*Practice toward the end of the century*

By the 1880s John Anderson and Squire Whipple and others were still using methods based on Hodgkinson’s tests. The Gordon-Rankine formula was the more commonly used equation in bridge design, however. Launhardt’s equation was used, but not often.

Seventeen of twenty two specifications itemized by James Clark for the period from 1871 to 1904 used adaptations of the Gordon-Rankine formula with the constants being either specified or to be determined by tests. Theodore Cooper used a straight line relationship for allowable load versus L / r in two of his specifications (1890 and 1896). Launhardt’s formula was used by the Pennsylvania Railroad Company only in three editions of their specifications before reverting to the use of the Gordon-Rankine formula. In 1885, the Pennsylvania Railroad Company required that Weyrauch’s equation be used in the case of stress reversals.

It would be 1887 before Bauschinger performed what Timoshenko called “the first reliable tests” on columns. Bauschinger had too few test results to devise a formula to replace Gordon-Rankine, however.

The 1905 AREMA specification for iron and steel structures used a straight line relationship similar to Cooper’s of 1890 and 1896.

*Heroes of this story*

If we were awarding a prize for contributions to column design in the earlier days then it would have to go to Leonard Euler. His equation was applicable only for very slender columns, but we would be severely criticized for any other choice.

However, the unofficial hero of the story has to be Thomas Tredwell who devised equations in 1822. His equations were flawed, but they survived in one form or another until the 20^{th} century.

Way t’ go Tom!

**Appendix**

**Observations Concerning Hodgkinson’s Tests**

E. Lamarle (1846) showed that Euler’s equation should only be applied to columns where (L / r)^{2} ≥ π^{2} E / σ_{yp}. Taking σ_{yp} in compression as 14,000 psi for cast iron and E as 13,000,000 psi yields a minimum slenderness ratio for a long cast iron column of 95.7. Therefore, for the case of long columns, this discussion of Hodgkinson’s column tests is limited to solid square sections with an effective L/d of 27.6 or more; and to solid round sections with an effective L/d of 23.9 or more.

Hodgkinson’s tests were intended to measure the ultimate capacities of columns and, for cast iron columns, this would be governed either by elastic buckling or by breaking of the brittle iron before it buckled. Hodgkinson therefore reported the deflection at the middle of the test column as it was loaded which would describe the column’s behaviour as it approached buckling; together with how it broke and at what load level.

The final load reading for each column might have the load at which it buckled, whereupon it broke; or the load at which it broke before reaching elastic instability. The former has been assumed in this discussion. Notes such as “broke immediately after the deflection was taken” and “broke in about five minutes” illustrate the ambiguity of the test results.

A surer way of measuring buckling strength today would be to increase compressive strain and to simultaneously measure the corresponding stresses. Buckling could then be defined as the point at which the stress began to decrease even as the strain was further increased. Hodgkinson’s tests were stress-controlled so that the point of buckling could only be surmised, however.

End conditions were also not well defined. Hodgkinson and all of his contemporaries defined column ends as being either round or flat rather than pinned or fixed. The tests were therefore also carried out with round or flat ends, so that the test specimens with rounded ends were more or less pinned, while the specimens with flat ends were less than fully fixed. Hodgkinson included a third set of tests with ends that were flat and upset like a bolt head. End fixity was increased in this case, but was still not ideally modeled. Values of K were assumed in this discussion to be 1.0 for pinned ends; 0.5 for fixed ends; and 0.66 for one end fixed and the other end pinned.

Many of the test slenderness ratios were very high, more than found in actual structures.

In general, there was a large amount of scatter in the test results. The experimental failure load divided by the Euler load (where 1.0 could have been expected) ranged from 0.83 to 1.39. The majority of the test values for round columns with round ends, for which the sample size was largest, were greater than 1.0. Average test results for flat ends were closer to the Euler prediction than for round ends but there were fewer tests and the range of results was still high. It is speculated that the round ended samples had some unintended end-fixity; while the better results for the flat ends would seem to justify the assumed value of Young’s modulus. There was some tendency for failure loads to be higher relative to Euler for higher values of K L / r, and this is not explained.

While Hodgkinson’s test results for long columns did not agree well with Euler, the tests as a whole varied with K L / r in a convincing manner.

Hodgkinson therefore devised formulas based on his tests, and these were envelopes intended as ‘safe’ estimates. They were still failure loads, however, not design loads. Comparing Hodgkinson’s equations with 55 of his test results for solid round columns with round ends, only eight tests failed at loads less than would have been predicted. Three of these results were for k L / r ratios in excess of 244, which would be unlikely in an actual structure. The average under-estimate of failure strength for the other 47 tests was 12.8% with a maximum of about 50% which demonstrates that the formulas were intended as ‘safe’.

Gordon’s formula as originally proposed matched Hodgkinson’s equation very well since Gordon based his constants on these test results.

**References:**

American Railway Engineering and Maintenance of Way Association, *Proceedings of the Sixth Convention*, Volume 6,Chicago, 1905.

Anderson, John, *The Strength of Materials and Structures*, Longman, Green and Company, London, 9^{th} edition, 1887.

Clark, James G., untitled manuscript dated March, 1956, from his Masters thesis from the University of Illinois entitled* A Study of the History of Specifications for the Design of Iron and Steel Bridges Prior to 1905*, 1939.

Dubois, Augustus Jay, *The Strains in Framed Structures*, John Wiley and Sons,New York, 1883.

Fraser, Craig G., *Mathematical Technique and Physical Conception in Euler’s Investigation of Elastica*, Centauris journal of the Institute for the History and Philosophy of Science and Technology, Vol. 34, Issue 3, p. 211-246, 1991.

Gillespie, W.M., *Notes on the Strength of Materials and the Stability of Structures*, Wiseman, Russell and Company, Schenectady, New York, 1870.

Hodgkinson, Eaton, *Experimental Researches on the Strength of Pillars of Cast Iron and Other Materials*, Philosophical Transactions of the Royal Society of London, 130 (0), 1840, p. 385-456.

Rankine, William, *A Manual of Civil Engineering*, Charles Griffin and Company, London, 1867.

Robinson, S.W., *Strength of Wrought Iron Bridge Members*, D. Van Nostrand, New York, 1882.

Timoshenko, Stephen P., *History of Strength of Materials*, McGraw Hill, New York, 1953.

Tredgold, Thomas, *Practical Essay on the Strength of Cast Iron and Other Metals*, London, 1831 third edition of the 1822 original.

Todhunter, Isaac C., *A History of the Theory of Elasticity and the Strength of Materials*, Cambridge University Press, 1886.

Tredwell, Daniel, *On the Strength of Cast Iron Pillars*, Proceedings of the American Academy of Arts and Sciences, Vol. IV from May, 1857 to May, 1860, Welch, Bigelow and Company, 1860.

Whipple, Squire, *A Work on Bridge Building*, Utica, New York, 1847.

Whipple, Squire, *Elementary and Practical Treatise on Bridge Building*, enlarged and improved edition of the 1873 original, D. Van Nostrand, New York, 1883.

So, I take it that 20th Century finite element analysis was the grandson of these?

W. WoodJuly 15, 2011 at 12:23 PM

Well, I guess not. Finite elements rely upon analysis of equilibrium and compatibility; which is a different thing.

johnwood1946July 17, 2011 at 10:16 AM