Breakthroughs in Structural Engineering#

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“Strength is essential but otherwise unimportant.” – attributed to Hardy Cross

Builders raised cathedrals and bridges for centuries by rule and precedent. Structural engineering became a science in three steps. Seventeenth- and eighteenth-century natural philosophers asked how strong a beam is and when a column buckles. Engineers of the nineteenth century, faced with iron railway bridges, worked out the theory of beams, trusses and statically indeterminate structures, and the energy theorems that tie loads to displacements. The twentieth century added plastic collapse, relaxation methods for frames and, finally, matrix methods run on a computer. The entries trace the ideas behind engineeringkit.structural_engineering. Each links to the implementing function and to a gallery example devoted to it.

1638 – Galileo’s Cantilever#

In the second day of the Two New Sciences, Galileo Galilei asked what load breaks a beam built into a wall. He treated the beam as a lever turning about the bottom edge of the wall section, with the whole section pulling with the same stress. The breaking moment then grows with the width and the square of the depth, so a plank set on edge is stronger than the same plank laid flat, and large animals and buildings need proportionally thicker members. The scaling was right. The constant was not: it took Navier’s neutral axis at the centroid to divide it by three.

Implementation: engineeringkit.structural_engineering.rectangular_section_modulus() and engineeringkit.structural_engineering.rectangular_second_moment(). The example compares Galileo’s breaking moment with the elastic one and checks the \(bh^2\) scaling.

References: G. Galilei, Discorsi e dimostrazioni matematiche intorno a due nuove scienze (Elsevier, Leiden, 1638), second day.

Galileo’s cantilever (1638)

Galileo's cantilever (1638)

1675 – Hooke’s Inverted Chain#

At the end of A Description of Helioscopes Robert Hooke printed a Latin anagram for “the true mathematical and mechanical form of all manner of arches”. Decoded after his death, it read ut pendet continuum flexile, sic stabit contiguum rigidum inversum: as hangs the flexible line, so but inverted will stand the rigid arch. A hanging chain can only pull, so it takes the shape in which its loads cause no bending. Turned upside down, the same shape is an arch that only pushes. A three-hinged arch is statically determinate, and its moment is the simple-beam moment less the thrust times the rise, \(M = M_0 - Hy\), which vanishes on the funicular shape.

Implementation: engineeringkit.structural_engineering.three_hinged_arch() and engineeringkit.structural_engineering.ArchResult. The example checks that a parabolic arch under a load uniform along the span carries no moment and the same force as the matching cable, and shows the bending a point load causes.

References: R. Hooke, A Description of Helioscopes, and Some Other Instruments (John Martyn, London, 1676 [1675]), postscript; the solution was published in R. Waller (ed.), The Posthumous Works of Robert Hooke (1705). Dates checked against secondary accounts only.

Hooke’s inverted chain (1675)

Hooke's inverted chain (1675)

1678 – Hooke’s Ut tensio, sic vis#

Robert Hooke published his law as an anagram in 1676 and its solution two years later: “as the extension, so the force”. Springs, wires and wooden rods all stretched in proportion to the load. Written for a bar of length \(L\), area \(A\) and modulus \(E\) (Young’s, 1807), the extension is \(\delta = PL/(AE)\), and the energy stored is the area under the load-extension line, \(U = P\delta/2\). The linear law is the constitutive relation that turns every later structural theory from statics into a calculation of displacements.

Implementation: engineeringkit.structural_engineering.axial_deformation() and engineeringkit.structural_engineering.strain_energy_axial(). The example repeats Hooke’s experiment on a steel wire and checks the extension and the stored energy.

References: R. Hooke, Lectures de Potentia Restitutiva, or of Spring (John Martyn, London, 1678).

Hooke’s law: ut tensio, sic vis (1678)

Hooke's law: ut tensio, sic vis (1678)

1691 – The Catenary#

Galileo had guessed that a hanging chain is a parabola. In 1690 Jacob Bernoulli asked the readers of Acta Eruditorum for its true shape, and in 1691 three answers appeared there, by his brother Johann, by Leibniz and by Huygens: the catenary \(y = a(\cosh(x/a) - 1)\), where \(a = H/w\) is the horizontal tension divided by the weight per unit length. The tension at a support is \(T = H + wf\). A cable that carries a deck uniform along the span, rather than its own weight, hangs in a parabola with \(H = wL^2/(8f)\), and for shallow cables the two curves are nearly the same.

Implementation: engineeringkit.structural_engineering.catenary_cable() and engineeringkit.structural_engineering.parabolic_cable(). The example sizes a power-line conductor both ways and shows where the parabola stops being a good approximation.

References: Joh. Bernoulli, G. W. Leibniz and C. Huygens, solutions of the catenary problem, Acta Eruditorum (1691). Checked against secondary accounts only.

The catenary (1691)

The catenary (1691)

1744 – Euler’s Column Buckling#

In an appendix on elastic curves to his book on the calculus of variations, Leonhard Euler found the load at which a straight elastic column can stay in equilibrium bent as well as straight. For a pin-ended column it is \(P_{cr} = \pi^2EI/L^2\). Other end conditions change the length of the buckled half-wave, written today as an effective length \(KL\). Euler’s load does not depend on the strength of the material at all, which is why slender columns fail by stability and stocky ones by crushing. Engineers distrusted the formula for a century, because the columns they tested were not slender enough for it to apply.

Implementation: engineeringkit.structural_engineering.euler_buckling_load(), engineeringkit.structural_engineering.radius_of_gyration() and engineeringkit.structural_engineering.slenderness_ratio(). The example finds Euler’s load as the lowest eigenvalue of the finite-difference column equation and plots the Euler hyperbola against yield.

References: L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes (Bousquet, Lausanne, 1744), Additamentum I, “De curvis elasticis”.

Euler’s column buckling (1744)

Euler's column buckling (1744)

1784 – Coulomb’s Torsion#

Charles-Augustin Coulomb hung a weight from a thin wire, twisted it and timed the oscillations. The restoring torque was proportional to the angle of twist, to the fourth power of the diameter and inversely to the length, the law \(\phi = TL/(GJ)\) with \(J = \pi d^4/32\). The torsion balance built on it measured the electric force the next year and, in Cavendish’s hands, the density of the Earth. For engineers it gave the shear stress \(\tau = Tr/J\) and stiffness of shafts.

Implementation: engineeringkit.structural_engineering.torsion_angle() and engineeringkit.structural_engineering.torsion_shear_stress(). The example simulates Coulomb’s torsion pendulum, recovers the shear modulus from its period and checks the fourth-power law.

References: C.-A. Coulomb, “Recherches théoriques et expérimentales sur la force de torsion et sur l’élasticité des fils de métal,” Hist. Acad. Roy. Sci. (1784, published 1787) 229-269.

Coulomb’s torsion (1784)

Coulomb's torsion (1784)

1807 – Young’s Eccentric Column#

In his Lectures on Natural Philosophy Thomas Young analysed a column loaded off its axis. The load bends it, the bending increases the lever arm of the load, and the peak stress follows the secant formula, \(\sigma_\max = (P/A)[1 + (ec/r^2)\sec((KL/2r)\sqrt{P/EA})]\). Stress grows faster than load and becomes unbounded at the Euler load, so a column’s factor of safety must be applied to the load. Solving the formula for the load at first yield gives a family of column curves, one for each eccentricity ratio \(ec/r^2\).

Implementation: engineeringkit.structural_engineering.secant_formula_stress() and engineeringkit.structural_engineering.secant_formula_load(). The example shows the nonlinear rise of stress with load and draws the secant-formula column curves.

References: T. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts (Joseph Johnson, London, 1807), vol. 2. The attribution follows the secondary literature; the lectures were not consulted.

Young’s eccentric column (1807)

Young's eccentric column (1807)

1829 – Poisson’s Ratio#

A bar stretched along its axis gets thinner. Siméon Denis Poisson’s molecular theory of elasticity predicted that the lateral strain is always one quarter of the axial strain. Experiments by Wertheim and others soon showed that the ratio \(\nu\) depends on the material, and the argument over it led to the modern two-constant theory of isotropic elasticity. \(\nu\) lies between -1 and 1/2 for a stable material, and \(\nu = 1/2\) means no change of volume.

Implementation: engineeringkit.structural_engineering.poisson_lateral_strain(). The example compares the contraction of steel, rubber and cork and checks the first-order volume change \((1 - 2\nu)\varepsilon\).

References: S.-D. Poisson, “Mémoire sur l’équilibre et le mouvement des corps élastiques,” Mém. Acad. Roy. Sci. Inst. France 8 (1829) 357-570.

Poisson’s ratio (1829)

Poisson's ratio (1829)

1833 – Lamé’s Thick-Walled Cylinder#

Gabriel Lamé and Émile Clapeyron, engineers in the Russian service, sent the Paris Academy a memoir on the internal equilibrium of elastic solids, published in 1833. Among its applications is the hollow cylinder under internal and external pressure: \(\sigma_r = A - B/r^2\) and \(\sigma_\theta = A + B/r^2\). The hoop stress is greatest at the bore. Adding thickness helps less and less, which drove the shrink-fitted and wire-wound gun barrels of the nineteenth century. For a thin wall the solution reduces to the boiler formula \(pr/t\).

Implementation: engineeringkit.structural_engineering.thick_wall_cylinder_stresses() and engineeringkit.structural_engineering.thin_wall_vessel_stresses(). The example plots the stresses through a thick wall and finds the radius ratio, about 10, at which the thin-wall formula is 5% low.

References: G. Lamé and É. Clapeyron, “Mémoire sur l’équilibre intérieur des corps solides homogènes,” Mémoires présentés par divers savans à l’Académie royale des sciences de l’Institut de France 4 (1833) 465-562. The volume and page are from secondary bibliographies.

Lamé’s thick-walled cylinder (1833)

Lamé's thick-walled cylinder (1833)

1847 – Whipple’s Method of Joints#

American railway bridges of the 1840s were built from rules of thumb, and some of them fell down. Squire Whipple, a canal engineer from Utica, showed that the force in every member of a pin-jointed truss follows from statics alone. Each joint is in equilibrium, so its two equations give the forces in two members. Work from a support across the truss, two unknowns at a time, and every force is found. His book was the first to analyse trusses rather than test them, and Herman Haupt reached the same results independently in 1851.

Implementation: engineeringkit.structural_engineering.method_of_joints(). The example solves a six-panel Pratt truss and checks every chord force against the method of sections, \(M/h\).

References: S. Whipple, A Work on Bridge Building (H. H. Curtiss, Utica, 1847).

Whipple’s method of joints (1847)

Whipple's method of joints (1847)

1856 – Jourawski’s Shear Stress#

Building timber bridges for the St Petersburg-Moscow railway, Dmitrii Ivanovich Jourawski saw large beams split along their length at mid-depth, where the bending stress is zero. He balanced the change in bending force on the part of the section beyond a cut against the shear on the cut. That gave the shear flow \(q = VQ/I\), which the keys of a built-up beam must carry, and the shear stress \(\tau = VQ/(It)\). The analysis was done in 1855 and published in Paris the next year. Saint-Venant later found the exact solution, but Jourawski’s formula is the one engineers use.

Implementation: engineeringkit.structural_engineering.transverse_shear_stress(), engineeringkit.structural_engineering.shear_flow(), engineeringkit.structural_engineering.composite_section() and engineeringkit.structural_engineering.SectionProperties.first_moment(). The example checks that the shear stress in a rectangle and an I-beam adds up to the shear force, and spaces the screws of a flitch beam from its transformed section.

References: D. J. Jourawski, “Sur la résistance d’un corps prismatique et d’une pièce composée en bois ou en tôle de fer à une force perpendiculaire à leur longueur,” Annales des ponts et chaussées, 3e série, 12 (1856) 328-351.

Jourawski’s shear stress (1856)

Jourawski's shear stress (1856)

1857 – Clapeyron’s Three-Moment Equation#

A beam continuous over several supports is statically indeterminate: the support moments cannot be found from equilibrium alone. Émile Clapeyron met the problem while rebuilding the Asnières railway bridge over the Seine. He required the slope of the beam to be continuous over each interior support, which gives one linear equation per support linking its moment to those of its two neighbours. The system is tridiagonal and small, and it made continuous girders, lighter than a row of simple spans, practical to design.

Implementation: engineeringkit.structural_engineering.three_moment_equation(). The example checks the textbook coefficients for equal spans, then integrates the curvature of an unequal three-span beam to show it does not move at the interior supports.

References: É. Clapeyron, “Calcul d’une poutre élastique reposant librement sur des appuis inégalement espacés,” C. R. Acad. Sci. Paris 45 (1857) 1076-1080.

Clapeyron’s three-moment equation (1857)

Clapeyron's three-moment equation (1857)

1864 – Maxwell’s Rigidity Count and Reciprocal Theorem#

James Clerk Maxwell’s short paper on frames made two contributions. A plane frame of \(j\) joints needs \(2j - 3\) bars to be stiff, or, counting the \(r\) support reactions, \(m + r = 2j\) for a statically determinate truss. The count is necessary but not sufficient, since bars can be put where they are not needed. The paper also stated the reciprocal theorem: in a linear elastic structure, the deflection at \(i\) caused by a unit load at \(j\) equals the deflection at \(j\) caused by a unit load at \(i\). Betti generalised it in 1872, and Mohr built the force method for indeterminate trusses on it.

Implementation: engineeringkit.structural_engineering.truss_determinacy() and engineeringkit.structural_engineering.maxwell_betti_reciprocity(). The example counts four trusses, one of which passes the count and is still a mechanism, and checks that the flexibility matrix of a redundant truss is symmetric.

References: J. C. Maxwell, “On the calculation of the equilibrium and stiffness of frames,” Phil. Mag. 27 (1864) 294-299; E. Betti, “Teoria dell’elasticità,” Il Nuovo Cimento 7-8 (1872) 5-21.

Maxwell’s rigidity count and reciprocal theorem (1864)

Maxwell's rigidity count and reciprocal theorem (1864)

1867 – Winkler’s Influence Lines#

A railway bridge must be checked for a train standing in every position. Emil Winkler, and Otto Mohr independently the next year, plotted the value of one response, such as the moment at a chosen section, against the position of a single unit load. With this influence line the response to any train is the sum of the axle loads times the ordinates under them, and the worst position of the train can be read off the plot. Influence lines remain the standard tool for moving loads on bridges and crane girders.

Implementation: engineeringkit.structural_engineering.influence_line(). The example finds the worst position of a four-axle locomotive on a 20 m span and checks it against superposed point-load solutions.

References: E. Winkler, Die Lehre von der Elasticität und Festigkeit (Dominicus, Prague, 1867); O. Mohr, “Beitrag zur Theorie der Holz- und Eisen-Constructionen,” Z. Architekten- und Ingenieur-Vereins zu Hannover 14 (1868) 19-51.

Winkler’s influence lines (1867)

Winkler's influence lines (1867)

1874 – Mohr’s Unit-Load Method#

Otto Mohr applied the principle of virtual work to trusses. To find how far one joint moves, put an imaginary unit load there. Its member forces \(n\), working through the real extensions \(NL/(AE)\), equal the unit load times the displacement: \(\delta = \sum nNL/(AE)\). Both force systems come from statics, so a determinate truss needs no stiffness matrix. The same argument gives \(\int mM/(EI)\,dx\) for beams, and, with the redundant forces as unknowns, the force method for indeterminate structures that Maxwell had begun ten years earlier.

Implementation: engineeringkit.structural_engineering.unit_load_truss_deflection() and engineeringkit.structural_engineering.unit_load_beam_deflection(). The example finds the midspan deflection of a Warren truss member by member and checks it against the direct stiffness method.

References: O. Mohr, “Beitrag zur Theorie des Fachwerks,” Zeitschrift des Architekten- und Ingenieur-Vereins zu Hannover 20 (1874) 509-526.

Mohr’s unit-load method (1874)

Mohr's unit-load method (1874)

1879 – Castigliano’s Theorem#

In his 1873 engineering thesis at Turin, extended into a book in 1879, Alberto Castigliano showed that the partial derivative of the strain energy with respect to a load is the displacement at that load, in its direction. For a beam this is the integral of \((M/EI)\,\partial M/\partial P\) along the span. A dummy load, set to zero after differentiating, gives the displacement where no load acts. The companion theorem of least work found the redundant forces of an indeterminate structure by minimising the energy.

Implementation: engineeringkit.structural_engineering.castigliano_deflection() (integrating \(M\,\partial M/\partial P/(EI)\) with Simpson’s rule) and engineeringkit.structural_engineering.strain_energy_bending_constant_moment(). The example finds the tip deflection of a tapered cantilever, checks it by double integration and by the energy balance, and uses a dummy load at midspan.

References: A. Castigliano, Théorie de l’équilibre des systèmes élastiques et ses applications (A. F. Negro, Turin, 1879).

Castigliano’s theorem (1879)

Castigliano's theorem (1879)

1882 – Mohr’s Circle#

Otto Mohr showed that the normal and shear stresses on every plane through a point, plotted against each other, fall on one circle. Its centre is the mean normal stress, its radius the largest in-plane shear stress, and its intersections with the axis are the principal stresses. Turning the element by \(\theta\) moves the point by \(2\theta\) around the circle. Culmann had drawn the stresses in a beam graphically in 1866; Mohr’s construction made the transformation one picture, and he used it to state his theory of failure. For a three-dimensional state he drew three circles, one through each pair of principal stresses; the largest gives the absolute maximum shear \((\sigma_1 - \sigma_3)/2\).

Implementation: engineeringkit.structural_engineering.mohr_circle(), engineeringkit.structural_engineering.stress_transformation() and engineeringkit.structural_engineering.principal_stresses_3d(). The example traces the circle by rotating an element, checks the principal stresses against the eigenvalues of the stress tensor, finds the plane of maximum shear, and draws the three circles for a tank wall, where the largest shear acts out of the plane.

References: O. Mohr, “Über die Darstellung des Spannungszustandes und des Deformationszustandes eines Körperelementes,” Zivilingenieur 28 (1882) 113-156.

Mohr’s circle (1882)

Mohr's circle (1882)

1889 – Engesser’s Tangent-Modulus Load#

Columns of intermediate length failed well below Euler’s load. Friedrich Engesser argued that once the stress passes the proportional limit, a column bends against the material’s tangent modulus \(E_t = d\sigma/d\varepsilon\), so the critical stress solves \(\sigma = \pi^2E_t(\sigma)/(KL/r)^2\). Engesser later replaced it by a “double-modulus” load, which was thought correct for half a century, until F. R. Shanley showed in 1947 that real columns start to bend at the tangent-modulus load. Column curves in design codes descend from it.

Implementation: engineeringkit.structural_engineering.tangent_modulus_load() (solving for the buckling stress with Brent’s method). The example builds the column curve of a Ramberg-Osgood aluminium alloy and checks each stress against Engesser’s equation.

References: F. Engesser, “Ueber die Knickfestigkeit gerader Stäbe,” Z. Architekten- und Ingenieur-Vereins zu Hannover 35 (1889) 455-462; F. R. Shanley, “Inelastic column theory,” J. Aeronaut. Sci. 14 (1947) 261-268.

Engesser’s tangent-modulus load (1889)

Engesser's tangent-modulus load (1889)

1893 – Johnson’s Parabola#

Tests on real columns fell well below Euler’s hyperbola at low slenderness, where columns yield or buckle inelastically. J. B. Johnson, C. W. Bryan and F. E. Turneaure fitted a parabola that starts at the yield stress and meets Euler’s curve tangentially at half of it, at the transition slenderness \(C_c = \sqrt{2\pi^2E/\sigma_y}\). End conditions enter through the effective length \(KL\). With a factor of safety, the same curve was the American steel column formula for most of the twentieth century.

Implementation: engineeringkit.structural_engineering.johnson_critical_stress() and engineeringkit.structural_engineering.effective_length_factor(). The example draws the design curve, checks that the two branches meet with the same slope, and compares four end conditions.

References: J. B. Johnson, C. W. Bryan and F. E. Turneaure, The Theory and Practice of Modern Framed Structures (Wiley, New York, 1893). The date follows the secondary literature; the first edition was not consulted.

Johnson’s parabola (1893)

Johnson's parabola (1893)

1896 – Bredt’s Thin-Walled Torsion#

Rudolf Bredt, an engineer with a crane-building firm, showed that in a closed thin-walled tube of any shape the shear flow \(q = \tau t\) is the same all round the wall, \(q = T/(2A_m)\), where \(A_m\) is the area enclosed by the wall’s centreline. The twist follows as \(\phi = (TL/4A_m^2G)\oint ds/t\). A closed section is far stiffer in torsion than the same wall slit open: the ratio is \(3(r/t)^2\) for a circular tube. Box girders, wing boxes and monocoque fuselages are designed with Bredt’s two formulas.

Implementation: engineeringkit.structural_engineering.thin_walled_torsion_shear_stress() and engineeringkit.structural_engineering.thin_walled_torsion_angle(). The example checks Bredt against Coulomb for a circular tube and compares it with the same tube slit along its length.

References: R. Bredt, “Kritische Bemerkungen zur Drehungselastizität,” Zeitschrift des Vereines deutscher Ingenieure 40 (1896) 785ff. The page is from secondary bibliographies.

Bredt’s thin-walled torsion (1896)

Bredt's thin-walled torsion (1896)

1914 – Kazinczy’s Plastic Hinge#

Gábor Kazinczy loaded steel beams built into concrete walls in Budapest and found that they carried far more than elastic theory allowed. When the most stressed section yielded through its whole depth, it went on turning at a constant plastic moment \(M_p\), a plastic hinge, and the load moved to other sections until enough hinges had formed to make a mechanism. Plastic design of steel frames grew from his tests, through Maier-Leibnitz in Germany and Baker’s group at Cambridge, into the limit-state codes.

Implementation: engineeringkit.structural_engineering.plastic_section_modulus() and engineeringkit.structural_engineering.plastic_collapse_load(). The example follows a built-in beam under a uniform load from first yield through hinges at the walls to collapse at \(16M_p/L^2\).

References: G. Kazinczy, “Kísérletek befalazott tartókkal” (Experiments with clamped girders), Betonszemle 2 (1914); J. F. Baker, M. R. Horne and J. Heyman, The Steel Skeleton, vol. 2 (Cambridge University Press, 1956).

Kazinczy’s plastic hinge (1914)

Kazinczy's plastic hinge (1914)

1915 – Maney’s Slope-Deflection Method#

George A. Maney at the University of Minnesota wrote the moment at each end of a member in terms of the rotations of its ends and of its chord: \(M_{ij} = (2EI/L)(2\theta_i + \theta_j - 3\psi) + M^F_{ij}\). Moment equilibrium at each free joint then gives one equation per unknown rotation. Support settlements enter through the chord rotation \(\psi\). Hardy Cross’s moment distribution (1930) solves the same equations by relaxation, and the stiffness method is their matrix form. Johnson’s Modern Framed Structures credits Otto Mohr with the idea before Maney found it independently.

Implementation: engineeringkit.structural_engineering.slope_deflection() and engineeringkit.structural_engineering.SlopeDeflectionResult. The example analyses a three-span girder, checks it against moment distribution and lets a pier settle.

References: G. A. Maney, “Secondary stresses in rigid frames,” Studies in Engineering (University of Minnesota, Minneapolis, 1915). Checked against secondary accounts only.

Maney’s slope-deflection method (1915)

Maney's slope-deflection method (1915)

1919 – Macaulay’s Singularity Functions#

Integrating \(EIy'' = M\) piece by piece between loads leaves two constants of integration per piece, tied together by continuity conditions. W. H. Macaulay, a Cambridge mathematician, wrote the moment once for the whole beam with brackets \(\langle x - a\rangle^n\) that vanish to the left of \(a\). Integrated as ordinary powers, they leave only two constants. Clebsch had used the idea in 1862, but Macaulay’s two-page note brought it into English textbooks. With the reactions as unknowns too, statically indeterminate beams are solved the same way as determinate ones. Superposition of standard cases, the older method, gives the same answers.

Implementation: engineeringkit.structural_engineering.Beam and engineeringkit.structural_engineering.superpose_beam_results(). The example solves a propped cantilever under a point load, a couple and a triangular load, and repeats it by superposition.

References: W. H. Macaulay, “A note on the deflection of beams,” Messenger of Mathematics 48 (1919) 129-130; A. Clebsch, Theorie der Elasticität fester Körper (Teubner, Leipzig, 1862).

Macaulay’s singularity functions (1919)

Macaulay's singularity functions (1919)

1921 – Maillart’s Shear Centre#

Carl von Bach’s tests at Stuttgart had shown channel beams twisting under loads through the web. The Swiss engineer Robert Maillart explained why. The shear flows in the flanges form a couple, so a load bends the beam without twisting it only if it passes through a point behind the web, the shear centre, at \(e = 3b^2t_f/(6bt_f + ht_w)\). In a section such as an angle the principal axes are inclined, so a vertical load also bends the beam sideways. The neutral axis then follows the generalized flexure formula, not the plane of the load.

Implementation: engineeringkit.structural_engineering.channel_shear_center(), engineeringkit.structural_engineering.unsymmetrical_bending_stress(), engineeringkit.structural_engineering.neutral_axis_angle(), engineeringkit.structural_engineering.channel_section() and engineeringkit.structural_engineering.angle_section(). The example locates the shear centre of a channel from the flange couple and the neutral axis of an angle under a vertical load.

References: R. Maillart, “Zur Frage der Biegung,” Schweizerische Bauzeitung 77 (1921) 195-197.

Maillart’s shear centre (1921)

Maillart's shear centre (1921)

1930 – Hardy Cross’s Moment Distribution#

Continuous beams and building frames led to large sets of simultaneous equations that no engineer wanted to solve by hand. Hardy Cross proposed relaxation instead. Lock every joint and apply the fixed-end moments, then release one joint at a time: its unbalanced moment is shared among the members in proportion to their stiffness, and half of each share is carried over to the far end. Repeat until the joints balance. The method fitted on a sheet of paper, converged in a few cycles and was the main tool of frame analysis until computers replaced it.

Implementation: engineeringkit.structural_engineering.moment_distribution(). The example balances a three-span beam built in at one end, checks the end moments against the stiffness method and plots the decay of the unbalance.

References: H. Cross, “Analysis of continuous frames by distributing fixed-end moments,” Proc. ASCE 56 (1930) 919-928.

Hardy Cross’s moment distribution (1930)

Hardy Cross's moment distribution (1930)

1938 – Simmons and Ruge’s Bonded Strain Gauge#

Stress cannot be measured, only strain. Edward E. Simmons at Caltech (from 1936) and Arthur C. Ruge at MIT (April 1938) independently cemented fine resistance wire to a surface, so that the wire’s resistance changed with the strain beneath it. The device was sold as the SR-4 gauge. One gauge gives the strain in one direction. A rosette of three at known angles gives the full plane strain state, and Hooke’s law turns it into stress. Strain gauges made experimental stress analysis a routine laboratory and field method.

Implementation: engineeringkit.structural_engineering.strain_rosette() and engineeringkit.structural_engineering.plane_stress_from_strains(). The example simulates a rosette on a shaft under bending and torsion and recovers the torque from three noisy readings.

References: E. E. Simmons, US Patent 2,292,549 (11 August 1942). The dates of Simmons’s and Ruge’s work are from secondary accounts.

Simmons and Ruge’s bonded strain gauge (1938)

Simmons and Ruge's bonded strain gauge (1938)

1956 – Turner, Clough, Martin and Topp: the Direct Stiffness Method#

At Boeing, M. J. Turner, R. W. Clough, H. C. Martin and L. J. Topp needed the stiffness of swept jet wings, which hand methods could not give. They wrote a stiffness matrix for each element, added the matrices into one global matrix according to the joints each element connects, and solved \(K_{ff}u_f = F_f\) on a computer. Indeterminate structures cost no more than determinate ones. The paper is the origin of the finite element method; Clough gave it that name in 1960. A near-mechanism gives a matrix that is invertible in floating point but meaningless, so the condition number of \(K_{ff}\) is part of the answer.

Implementation: engineeringkit.structural_engineering.PlanarTruss (with \(\operatorname{cond}(K_{ff})\) and a warning near a mechanism). The example checks a redundant truss against Clapeyron’s energy theorem, then flattens a joint until the warning fires.

References: M. J. Turner, R. W. Clough, H. C. Martin and L. J. Topp, “Stiffness and deflection analysis of complex structures,” J. Aeronaut. Sci. 23 (1956) 805-823.

Turner, Clough, Martin and Topp: the direct stiffness method (1956)

Turner, Clough, Martin and Topp: the direct stiffness method (1956)

1956 – Livesley’s Plane Frames by Computer#

Moment distribution assumes that the joints do not move sideways. A building frame under wind does sway, and hand methods needed an extra correction for every storey. R. K. Livesley programmed the stiffness method for rigid-jointed frames on the Manchester computer. Each member got a matrix with axial and bending terms, three degrees of freedom per joint, and the sway came out of the same solve as the rotations. Matrix frame analysis became the everyday tool of structural design offices.

Implementation: engineeringkit.structural_engineering.PlanarFrame. The example sways a fixed-base portal frame, checks the result against the rigid-beam and pinned-beam limits and checks the equilibrium of the reactions under wind and gravity.

References: R. K. Livesley, “The application of an electronic digital computer to some problems of structural analysis,” Struct. Eng. 34 (1956) 1-12.

Livesley’s plane frames by direct stiffness (1956)

Livesley's plane frames by direct stiffness (1956)

See Also#