References#

The source texts and papers underpinning the formulations used in PyCBA are collected below: the matrix (direct) stiffness method and Euler–Bernoulli beam element; the consistent nodal (fixed-end) load formulae; the non-prismatic (variable-\(EI\)) element and imposed-curvature load; the Timoshenko (shear-deformable) element; and the nonlinear (Generalized Clough) plastic-hinge analysis. Each entry is hyperlinked to a DOI or a stable URL, and is cross-referenced from the Theoretical Basis page by the labelled anchors shown.

Matrix (direct) stiffness method and beam elements#

McGuire, W., Gallagher, R.H. & Ziemian, R.D. (2000). Matrix Structural Analysis, 2nd edn. John Wiley & Sons, New York. The companion MASTAN2 software and the full text are freely available at https://www.mastan2.com/textbook.html. The standard graduate reference for the direct stiffness method, element stiffness matrices, and global assembly.

Przemieniecki, J.S. (1968). Theory of Matrix Structural Analysis. McGraw-Hill, New York. (Dover reprint, 1985, ISBN 978-0486649481.) Catalogue record: https://search.worldcat.org/title/911747. The classic derivation of element stiffness and consistent (work-equivalent) nodal load matrices from energy principles.

Weaver, W. & Gere, J.M. (1990). Matrix Analysis of Framed Structures, 3rd edn. Springer, New York. DOI: 10.1007/978-1-4684-7491-2. Detailed treatment of beam and frame elements, fixed-end actions, support displacements and the handling of member releases.

Ghali, A., Neville, A.M. & Brown, T.G. (2017). Structural Analysis: A Unified Classical and Matrix Approach, 7th edn. CRC Press, Boca Raton. DOI: 10.1201/9781315275215. Unified classical (flexibility/slope-deflection) and matrix treatment, including support settlement and temperature/curvature effects in indeterminate beams.

Hibbeler, R.C. (2017). Structural Analysis, 10th edn. Pearson, Hoboken. Catalogue record: https://search.worldcat.org/title/953777155. An accessible undergraduate development of the stiffness method for beams, with extensive worked examples of fixed-end moments and member releases.

Timoshenko, S.P. & Young, D.H. (1965). Theory of Structures, 2nd edn. McGraw-Hill, New York. Catalogue record: https://search.worldcat.org/title/1473595. Classical reference for the slope-deflection and force methods that underlie the released-element formulation.

Cook, R.D., Malkus, D.S., Plesha, M.E. & Witt, R.J. (2001). Concepts and Applications of Finite Element Analysis, 4th edn. John Wiley & Sons, New York. Catalogue record: https://search.worldcat.org/title/45286597. Source for consistent nodal loads, static condensation, and the imposition of boundary conditions by direct elimination.

Felippa, C.A. (2004). Introduction to Finite Element Methods (lecture notes), University of Colorado Boulder. Freely available at https://quickfield.com/advanced/felippa_introduction_to_FEM.pdf. A clear, openly-available reference for element stiffness, assembly, consistent nodal loads and constraint handling.

Caprani, C.C. Structural Analysis IV — The Matrix Stiffness Method (course notes). Available at http://www.colincaprani.com/structural-engineering/courses/structural-analysis-iv. The lecture notes from which the original CBA program was developed, covering the beam element, assembly, and member-release condensation used in PyCBA.

Consistent nodal loads (fixed-end forces)#

Budynas, R.G. & Sadegh, A.M. (2020). Roark’s Formulas for Stress and Strain, 9th edn. McGraw-Hill, New York. Catalogue record: https://search.worldcat.org/title/1145991188. Closed-form reactions, fixed-end moments and deflections for point, uniform, partial and trapezoidal loadings.

American Institute of Steel Construction (2017). Steel Construction Manual, 15th edn, “Beam Diagrams and Formulas”. AISC, Chicago. https://www.aisc.org/publications/steel-construction-manual-resources/. Tabulated fixed-end moments and shears for the standard load types implemented in PyCBA’s load module.

Non-prismatic (variable-\(EI\)) elements and imposed curvature#

Ghali, A., Favre, R. & Elbadry, M. (2002). Concrete Structures: Stresses and Deformations, 3rd edn. Spon Press, London. DOI: 10.1201/9781482271782. Chapter 13 develops the flexibility-integrated stiffness of a variable-rigidity member, \([S^{*}] = [f^{*}]^{-1}\); Section 13.7 gives the corresponding imposed-curvature (creep, shrinkage and thermal) fixed-end forces in continuous members.

Hulse, R. & Mosley, W.H. (1986). Reinforced Concrete Design by Computer. Macmillan Education, London. DOI: 10.1007/978-1-349-18496-4. Section 2.6 builds a pure-flexural haunched-beam element by Simpson-rule flexibility integration, giving the stiffness and carry-over factors directly.

Portland Cement Association (1958). Handbook of Frame Constants: Beam Factors and Moment Coefficients for Members of Variable Section. PCA, Skokie, IL. Catalogue record: https://search.worldcat.org/title/3992779. Tabulated stiffness factors, carry-over factors and fixed-end-moment coefficients for haunched and tapered (non-prismatic) members.

Elbadry, M.M. & Ghali, A. (1989). “Serviceability design of continuous prestressed concrete structures.” PCI Journal 34(1), 54–91. DOI: 10.15554/pcij.01011989.54.91. Restraint of creep, shrinkage and thermal (imposed) curvatures in continuous members — the engineering context for PyCBA’s imposed-curvature load.

Ghali, A. & Favre, R. (1994). Concrete Structures: Stresses and Deformations, 2nd edn. E & FN Spon, London. Catalogue record: https://search.worldcat.org/title/30894947. Earlier edition of the imposed-curvature / variable-rigidity formulation refined in the 3rd edition.

Timoshenko (shear-deformable) beam elements#

Timoshenko, S.P. (1921). “On the correction for shear of the differential equation for transverse vibrations of prismatic bars.” Philosophical Magazine, Series 6 41(245), 744–746. DOI: 10.1080/14786442108636264. The original beam theory that augments Euler–Bernoulli bending with a transverse shear deformation, the basis of PyCBA’s shear-flexible element.

Cowper, G.R. (1966). “The shear coefficient in Timoshenko’s beam theory.” Journal of Applied Mechanics 33(2), 335–340. DOI: 10.1115/1.3625046. Derives the cross-section shear coefficient \(k\) (and hence the shear area \(A_v = kA\)) used to form the shear rigidity \(GA_v\) supplied to a Timoshenko member.

Friedman, Z. & Kosmatka, J.B. (1993). “An improved two-node Timoshenko beam finite element.” Computers & Structures 47(3), 473–481. DOI: 10.1016/0045-7949(93)90243-7. The interdependent-interpolation two-node element that is free of shear locking and reduces exactly to the Euler–Bernoulli element as \(GA_v \to \infty\) — the \(\Phi\)-parameterised \(4\times4\) stiffness matrix implemented in PyCBA (see also Przemieniecki, 1968).

Numerical integration#

Stroud, A.H. & Secrest, D. (1966). Gaussian Quadrature Formulas. Prentice-Hall, Englewood Cliffs, NJ. Catalogue record: https://search.worldcat.org/title/1527634. Reference for the Gauss–Legendre rules used to evaluate the non-prismatic element flexibility integrals exactly (for polynomial \(EI\)) or to machine precision.

Virtanen, P. et al. (2020). “SciPy 1.0: fundamental algorithms for scientific computing in Python.” Nature Methods 17, 261–272. DOI: 10.1038/s41592-019-0686-2. Provides the cumulative-trapezoidal and Simpson integration (scipy.integrate.cumulative_trapezoid, scipy.integrate.simpson) used for member deflection recovery and the flexibility/curvature integrals.

Harris, C.R. et al. (2020). “Array programming with NumPy.” Nature 585, 357–362. DOI: 10.1038/s41586-020-2649-2. The linear-algebra back end (numpy.linalg.solve, numpy.linalg.eigvalsh, numpy.polynomial.legendre.leggauss) used to solve the assembled system, run the stability eigenvalue test, and generate the Gauss nodes.

Stability / mechanism detection#

Golub, G.H. & Van Loan, C.F. (2013). Matrix Computations, 4th edn. Johns Hopkins University Press, Baltimore. Catalogue record: https://search.worldcat.org/title/824733531. Source for the symmetric eigenvalue problem, condition number, and the singularity criterion used in the free-DOF mechanism check.

Bathe, K.-J. (2014). Finite Element Procedures, 2nd edn. Prentice Hall / K.-J. Bathe, Watertown, MA. Freely available at https://web.mit.edu/kjb/www/Books/FEP_2nd_Edition_4th_Printing.pdf. Establishes that the free (unconstrained) stiffness partition of a stable structure is symmetric positive-definite, and that a mechanism renders it singular.

Nonlinear analysis and plastic-hinge models#

Clough, R.W. & Johnston, S.B. (1966). “Effect of stiffness degradation on earthquake ductility requirements.” Proceedings of the Japan Earthquake Engineering Symposium, Tokyo, pp. 227–232. Report record: https://search.worldcat.org/title/30950305. The stiffness-degradation (“Clough”) model underlying PyCBA’s concentrated-plasticity element.

Li, G.-Q. & Li, J.-J. (2007). Advanced Analysis and Design of Steel Frames. John Wiley & Sons, Chichester. DOI: 10.1002/9780470059715. Chapter 4 gives the interpolated element-stiffness degradation between fixed-fixed, pinned-fixed and fixed-pinned matrices used in the nonlinear engine.

Neal, B.G. (1977). The Plastic Methods of Structural Analysis, 3rd edn. Chapman & Hall, London. DOI: 10.1007/978-94-009-5764-6. Foundations of plastic-hinge theory, collapse mechanisms and the upper/lower bound theorems.

Plastic analysis (virtual work / collapse load factors)#

Baker, J.F., Horne, M.R. & Heyman, J. (1956). The Steel Skeleton, Volume 2: Plastic Behaviour and Design. Cambridge University Press, Cambridge. Catalogue record: https://search.worldcat.org/title/575393. Foundational treatment of plastic collapse of steel frames.

Heyman, J. (1971). Plastic Design of Frames, Volume 1: Fundamentals. Cambridge University Press, Cambridge. DOI: 10.1017/CBO9781139106740. Fundamentals of the plastic theorems used to interpret the collapse load factors.

Horne, M.R. (1979). Plastic Theory of Structures, 2nd edn. Pergamon Press, Oxford. Catalogue record: https://search.worldcat.org/title/4136860. Concise development of plastic-hinge collapse analysis.

Bridge loading and moving-load analysis#

McCarthy, L.A. (2012). Probabilistic Analysis of Indeterminate Highway Bridges Considering Material Nonlinearity. MPhil Thesis, Dublin Institute of Technology. DOI: 10.21427/D7C30J. Application of the nonlinear continuous-beam analysis to indeterminate highway bridges under moving loads.

Caprani, C.C. (2006). Probabilistic Analysis of Highway Bridge Traffic Loading. PhD Thesis, University College Dublin. Available at http://www.colincaprani.com/files/Caprani PhD Thesis.pdf. Background to the influence-line and moving-load machinery in PyCBA.

Prestressed concrete and equivalent loads#

Gilbert, R.I., Mickleborough, N.C. & Ranzi, G. (2017). Design of Prestressed Concrete to AS3600-2009, 2nd edn. CRC Press, Boca Raton. ISBN 978-1466572690. Publisher record. The standard Australian text on prestressed concrete; Example 11.1 (continuous beam) derives the equivalent loads and the total, primary and secondary moments induced by prestress — the worked example reproduced by the pycba.prestress preprocessor.

Structural Data Inc. (2000). PT Designer — Post-Tensioning Design and Analysis Programs: Theory Manual. PDF. Chapters 5–6 define the 12-profile tendon library (shared with RAPT) and the equivalent-load (balanced-load) formulae implemented in pycba.prestress.

Differential temperature (thermal effects)#

Standards Australia (2017). AS 5100.2:2017 — Bridge design, Part 2: Design loads. Standards Australia, Sydney. Clause 18.3 specifies the design vertical (differential) temperature gradient for bridge superstructures — for concrete, the Priestley fifth-order positive gradient over the top 1200 mm — which the Creep, Shrinkage and Thermal tutorial models as an imposed curvature.

Priestley, M.J.N. (1976). Design of concrete bridges for temperature gradients. Origin of the fifth-order design temperature gradient through the top 1200 mm of the superstructure that AS 5100.2 (and US bridge practice) adopt. A contemporaneous treatment is Thermal Stress Analysis of Concrete Bridge Superstructures, Transportation Research Record 607 (1976), PDF.

Beams on elastic foundations#

Hetényi, M. (1946). Beams on Elastic Foundation: Theory with Applications in the Fields of Civil and Mechanical Engineering. University of Michigan Press, Ann Arbor. Publisher record. The classic treatment of the beam on a Winkler foundation; its infinite-beam point-load deflection and moment are reproduced by the pycba.foundation super-element and used to validate it (see the foundation tutorial).

O’Brien, E.J., Keogh, D.L. & O’Connor, A.J. (2014). Bridge Deck Analysis, 2nd edn. CRC Press, Boca Raton. ISBN 978-1482227239. Publisher record. Chapter 4 treats integral bridges and soil–structure interaction; §4.5 (Run-on slab) notes that a run-on slab spans a settlement trough behind the abutment rather than resting on a continuous bed — the practical caveat to the linear Winkler foundation in the foundation tutorial.