Review: On the horizontal thrust of a mass of sand (G. Darwin, 1883)

By Michael Bennett, P.E., M.ASCE (A.G.E.S., Inc., King of Prussia, PA)

 

In 1883, the British Empire stood unrivaled as the most powerful nation on Earth.  The British Army and Royal Navy ensured that Queen Victoria’s government dictated the terms of the world order, and the era known as the Pax Britannica was in full swing.  The Union Jack flew atop flagpoles from Canada to Australia to India to Egypt, and it was little exaggeration to say that the British Empire even reigned supreme over time and space.  It was during the 1880s that world leaders first defined universal systems of coordinates and time zones, and, not coincidentally, the UK wound up at the center of both systems via the Prime Meridian and Greenwich Mean Time.  The Empire’s power in 1883 was particularly visible in the era’s scientific and technological gains, many of which came through London-based forums such as the Royal Society and the Institution of Civil Engineers, or ICE.  The fruits of these advances were readily visible in the British capital.  Since the mid-19th century, civil engineers had transformed London from the grim city of Dickens’s novels into a modern metropolis with brick tunnels to carry sewage out to sea, embankments to control flooding on the River Thames, and a subway system to relieve urban congestion and speed travel (LTM 2024, Mann 2016).

 

One glum development in British science during the 1880s occurred in 1882 when one of the Isles’ greatest-ever scientists, Charles Darwin, died.  His impact on modern biology cannot be overstated, and his theory of evolution by natural selection – hugely controversial when he first introduced it – is now recognized as one of history’s most important scientific discoveries.  Somehow, Darwin found time to be the father both of natural selection and of 10 children, 7 of whom lived to adulthood.  Victorian constraints on women’s roles circumscribed what his surviving daughters, Elizabeth and Henrietta, were allowed to achieve, and little is known about Elizabeth’s life, but Henrietta served ably as her father’s editor and her mother’s biographer.  Her five surviving brothers also enjoyed successful careers.  William became a respected banker, and Leonard spent years first in the British Army and then in the House of Commons.  Horace founded the Cambridge Scientific Instrument Company, a prominent manufacturer of scientific equipment for generations, while Francis worked as his father’s laboratory assistant and eventually became a leading plant physiologist (Berra 2013, Cambridge 2024).

George, the second-oldest Darwin son, may have had the most unique career of the many distinguished and interesting paths taken by Charles and Emma Darwin’s children.  The polymathic George became renowned the world over as an expert on mathematics, astronomy, and tidal oceanography.  Ironically, though, one of his most important contributions to scientific and engineering advancement may have come from a paper he presented to ICE in 1883 which languished for decades after its publication.  The paper, entitled, “On the horizontal thrust of a mass of sand,” only achieved widespread recognition when Karl Terzaghi encountered it during his pioneering research on soil mechanics in the 1910s.  Terzaghi was so impressed with Darwin’s work that, when he published his first major piece on the geotechnical behavior of sands in 1920, he cited his predecessor’s 1883 ICE paper as a lonely bright spot in a largely unilluminated field of knowledge.  Another geotechnical giant, Sir Alec Skempton, agreed with Terzaghi’s high opinion of Darwin’s 1883 write-up.  In the 1960s, Skempton and two colleagues included Darwin’s paper when they assembled a commemorative volume of prominent ICE papers on soil mechanics from the 1840s to the mid-1940s.  Decades later, Sir Alec presented an overview of 35 of geotechnical engineering’s most important early works at the 11th International Conference on Soil Mechanics and Foundation Engineering, held in 1985.  Once again, Darwin’s paper made Skempton’s list (Darwin 1883, Skempton 1985, Terzaghi 1920).

 

Within his 1883 paper, George Darwin clearly followed the six steps of the research process which are colloquially known as the scientific method.  Elementary and middle school students often learn these steps using the mnemonic “Pretty red heads eat animal crackers,” signifying Problem, Research, Hypothesis, Experiment, Analysis, and Conclusion.  In the late 1870s, Darwin identified his research problem when he noted that extant theories of lateral earth pressure failed to adequately explain the mechanics of sand behavior.  It turned out that scant research had been done on the issue, leaving him little to review and next to nothing on which to base a hypothesis.  Darwin therefore went directly from formulating his problem to beginning his experiments.  He selected a sand with an angle of repose of roughly 35⁰ – the internal friction angle of soil, the analogous property in modern geotechnical practice, had yet to be defined – and assembling an apparatus to measure the sand’s “thrust,” now known as its active lateral earth pressure (Darwin 1883).

Darwin’s apparatus has long since been superseded but remains strikingly simple, ingenious, and technically sound.  It consisted of a cubic wooden box measuring 14 inches high, 12 inches long, and 8.5 inches wide and having an open top.  Three sides of the box were secured directly to the bottom, but the fourth was a door which could fall outward around a bottom hinge.  As Darwin filled the box with sand before each of his experiments, he kept the door in place using an external bolt and a wire connecting the top of the door to a spring scale.  Once Darwin had filled the box, he pulled the wire taut and carefully removed the bolt.  He then gradually reduced the tension on the wire until the sand underwent active earth failure by pushing the door outward.  At the instant (or, for less distinct cases, instants) the sand failed, Darwin noted the reading on the spring scale.  Later, he used this reading to compute the force on the door at failure.  Darwin minimized the influence on his results of friction between the sand and the wooden box by painstakingly gluing sand to the door.  In modern geotechnical parlance, this meant that he set the interface friction angle, δ, between the sand and the box equal to the sand’s internal friction angle, φ’, making his results simpler to analyze (Darwin 1883).

 

 

 

Next, Darwin spent months experimenting with the effect of the depth and configuration of the sand on its active lateral earth pressure at failure.  During his experiments, Darwin assessed the behavior of sand placed in layers in the box in six distinct configurations.  For the first two configurations, Darwin placed horizontal sand layers in the box.  Darwin either placed the layers loosely without compaction (Series 1) or densified them by rodding after he placed them (Series 2).  For the next two configurations, Darwin placed sand layers in the box at the sand’s angle of repose and sloping either downward (Series 3) or upward (Series 4) from the box’s door.  For these states, he leveled the top of the sand after placement in the box.  For the final two states, Darwin again placed the sand layers in the box at the sand’s angle of repose and sloping either downward (Series 5) or upward (Series 6) from the box’s door.  For these states, however, Darwin did not level the top of the sand after placement in the box.  Darwin performed several dozen tests using each configuration (Darwin 1883).

 After finishing his tests, Darwin analyzed his experimental data by calculating regression functions for each of the six series.  Francis Galton, the father of modern statistics, was only beginning to develop such regression techniques in the late 1870s as George Darwin toiled on his experiments.  Francis, ironically, was George’s half-cousin, and the family connection could have aided Darwin in his data analysis.  He eventually determined that he could fit a regression function to the data from each configuration to express the tension in the cable at failure, T, in terms of the cube of the depth of sand in the box, l.  21st-century readers might find Darwin’s derivation of the function somewhat tough to follow, which is understandable in hindsight.  Both statistical regression and Mohr’s circle were still being developed in the late 1870s, and modern geotechnical analyses lay decades away.  However, Sir Alec Skempton noted in his presentation on landmark papers in early geotechnical engineering that Darwin’s intricate cubic regression functions can be streamlined using modern soil mechanics to T = Ka × cos (δ) × l 3, where Ka is the coefficient of active lateral earth pressure and T, δ, and l remain as defined previously (Gillham 2009, McCarty 1947, Skempton 1985).

 

 

Finally, Darwin analyzed the values of the Ka × cos (δ) coefficients from his six sand configurations.  He first compared his coefficients from the four configurations involving level backfill against the door – 0.180 for Series 1, 0.132 for Series 2, 0.165 for Series 3, and 0.189 for Series 4 – to the active failure coefficients predicted for sand with an angle of repose of 35⁰ by the Rankine (0.271) and Boussinesq (0.199) theories.  Darwin quickly concluded that Rankine’s coefficient was too much higher than his own results to be reliable and that, for his analyses, the theory could be “safely neglected.”  The disparity arises from the assumption in Rankine theory of zero friction between a retaining wall and the soil behind it – a faulty premise for assessing Darwin’s experiments (Darwin 1883).

Darwin also noted that the difference between his and Boussinesq’s coefficient values, while smaller, remained too large to “be fairly put down to errors of observation.”  He chalked this discrepancy up to the assumptions inherent in Boussinesq’s theory and derived a new equation for the coefficient.  Darwin assumed within his derivation that the horizontal pressure of a mass of sand could be estimated using a hydrostatic approximation and that the wedge of sand immediately behind the wall (or door) was supported entirely by sand-wall interface friction and the remainder of the mass of sand.  From these premises, Darwin derived a new expression for the Ka × cos(δ) coefficient.  He found that his equation gave values comparable to his experimental results not only for Series 1 through 4 but also for Series 5 and 6 – neither of which could be accurately evaluated using Rankine or Boussinesq theory (Darwin 1883).

Darwin then closed his analysis by examining the considerable difference in coefficient values between Series 1 and 2.  He posited that the density of the sand was the primary driver of this difference, since moving tightly packed sand grains would require more energy than moving loose sand grains.  “The coefficient of internal friction of sand is a function … not merely of the pressure then existing,” Darwin concluded, “but also of the pressure and shaking to which at some previous period that portion of the mass of sand has been subjected.”  He thus simply yet elegantly recognized the influence of relative density and stress history on sand behavior, and both concepts remain fundamental to understanding the soil mechanics of sands nearly 150 years later.  Darwin also concluded that, during his testing of dense sand layers in Series 2, the sand must have had to – in his words – “unsettle” during failure by rotating and briefly occupying a larger volume.  Later experiments involving triaxial testing on dense sands would bear him out, and modern geo-professionals refer to this phenomenon as dilatancy (Darwin 1883).

George Darwin noted that he had been “interrupted by other occupations” after finishing his regimen of lateral earth pressure experiments with sands and had taken his research no further; unfortunately for posterity, he never did.  However, the published discussion of his write-up among members of ICE in 1883 reflects that Joseph Boussinesq himself extended Darwin’s work just a bit further.  Boussinesq back-calculated angles of repose for Darwin’s loose Series 1 sand (37⁰) and dense Series 2 sand (43⁰) and thereby made explicit Darwin’s implied conclusion that the sand’s angle of repose was directly related to its relative density.  This principle, modified to consider internal friction angle instead of angle of repose, remains a linchpin of geotechnical practice.  Darwin’s work would pay dividends again decades later when Karl Terzaghi began taking the next steps toward understanding the geotechnical behavior of sands and used Darwin’s data to supplement his own experimental results.  Clearly, and rather appropriately, George Darwin – son of the first champion of the theory of evolution – wound up playing a key role in the evolution of geotechnical engineering (Darwin 1883, Skempton 1985, Terzaghi 1920).

 

Acknowledgments

Sebastian Lobo-Guerrero, Ph.D., P.E., BC.GE., M.ASCE (A.G.E.S., Inc.: Canonsburg, PA), the author’s colleague, reviewed the entry’s technical content.  Thomas Kennedy (Geopier: Davidson, NC), the author’s Virginia Tech classmate, co-authored a previous version of the entry posted in 2021 on an independent webpage.

 

References

Amazon. 2024. “George Howard Darwin (1845-1912).” Amazon. Accessed Feb. 18, 2024. https://www.amazon.com/1845-1912-Astronomer-Geophysicist-Photograph-Magazine/dp/B07C4H6B6N

Berra, T.M. 2013. “Ten facts about Charles Darwin’s ten children.” Oxford University Press Blog, Nov. 2. Accessed Feb. 18, 2024. https://blog.oup.com/2013/11/ten-facts-about-charles-darwins-ten-children/

Britannica Kids. 2024. “1871 cartoon of Charles Darwin.” Britannica Kids. Accessed Feb. 18, 2024. https://kids.britannica.com/students/assembly/view/210233

Cambridge (University of Cambridge). 2024. “Darwin, William Erasmus, 1839-1914.” University of Cambridge ArchiveSearch. Accessed Feb. 18, 2024. https://archivesearch.lib.cam.ac.uk/agents/people/8367

Darwin, G. H. 1883. “On the horizontal thrust of a mass of sand.”  Min. Proc. Inst. Civ. Eng., 71, 350-378.  Reprinted in A century of soil mechanics: classic papers on soil mechanics published by the Institution of Civil Engineers, 1844-1946, L.F. Cooling, A.W. Skempton, and A.L. Little (eds.), 1969.  London, UK: William Clowes and Sons, Ltd.

Gillham, N.W. 2009. “Cousins: Charles Darwin, Sir Francis Galton and the birth of eugenics.” Significance, 6 (3), 132-135. https://doi.org/10.1111/j.1740-9713.2009.00379.x

LTM (London Transport Museum). 2024. “The Metropolitan line.” London Transport Museum. Accessed Feb. 18, 2024. https://www.ltmuseum.co.uk/collections/stories/transport/metropolitan-line

Mann, E. 2016. “Story of cities #14: London’s Great Stink heralds a wonder of the industrial world.” The Guardian. Accessed Feb. 18, 2024. https://www.theguardian.com/cities/2016/apr/04/story-cities-14-london-great-stink-river-thames-joseph-bazalgette-sewage-system

McCarty, L.E. 1947. “Application of the Mohr circle and stress triangle diagrams to test taken taken with the Hveem stabilometer.” In Proc., 26th Annual Meeting. Washington, DC, USA: Highway Research Board.

NPR (National Public Radio). 2015. “‘Dirty old London’: a history of the Victorians’ infamous filth.” National Public Radio, Mar. 12. Accessed Feb. 18, 2024. https://www.npr.org/2015/03/12/392332431/dirty-old-london-a-history-of-the-victorians-infamous-filth

Skempton, A. W. 1985. “A History of Soil Properties, 1717-1927.”  In Proc., 11th Int. Conf. on Soil Mech. Found. Eng., San Francisco, CA, USA: ISSMFE, 95-121.

Terzaghi, C. 1920. “Old earth-pressure theories and new test results.” Eng. News-Rec., 85 (14), 632-637. Reprinted in From theory to practice in soil mechanics: selections from the writings of Karl Terzaghi, L. Bjerrum, A. Casagrande, R. B. Peck, and A. W. Skempton (eds.), 1960. New York, NY, USA: John Wiley and Sons.