Course 2024/2025
dr.ir. P.C.J. Hoogenboom (email p.c.j.hoogenboom@tudelft.nl)
5 Credits (EC)
Many structures can be modelled as thin elastic shells. Examples are pressure vessels, ancient domes, LNG storage tanks, airplane fuselages, industrial chimneys, oil pipes and free form architecture.
The objective of this course is to predict the behaviour of shell structures. After completing the course you can answer the following questions about your shell designs. Will it deflect too much? Will it yield? Will it crack or break? Will it vibrate annoyingly? Will it buckle? Will it be unsafe? What causes this and how can I improve it?
Reader: P.C.J. Hoogenboom, Notes on Shell Structures, 2025
| ..4 Sept. | ..8:45-10:30 | 3.02 | Scope of the course, Sagitta, a/t Ratio, Corbel dome, Arches, Catenary, Pressure lines, Middle third rule, Optimal arch shape, Barlow's formula, CAD essentials, Thin or thick shells, Section forces and moments, Thickness, Force flow in a dome, Pantheon | page 1 - 14 |
| ..5 Sept. | 10:45-12:30 | F | Edge disturbance, Comparison of moments in arches and domes, Fully stressed dome, Optimal dome shape, Curvature, Savill building, Gaussian curvature and mean curvature, Orthogonal parameterisation | page 15 - 30 |
| 11 Sept. | ..8.45-10:30 | 3.02 | Curvilinear coordinate system, Torus, Lame parameters, Equation of Gauss, Queens palace, In plane curvature, Shell membrane equations, Membrane forces in a spherical dome, Duomo di Firenze, Saint Paul's Cathedral, Derivation of the equations | page 31 - 43 |
| 12 Sept. | ..8:45-10:30 | F | Stresses in a simply supported tube, Comparison to a beam model, Bernoulli's theorem, Deformation of a simply supported tube, Misfit at the boundaries | page 44 - 55 |
| 12 Sept. | 10:45-12:30 | F | Monocoque, Sanders-Koiter equations, Compatibility equation, Shell differential equations, Plate boundary conditions | page 56 - 66 |
| 18 Sept. | ..8:45-10:30 | 3.02 | Concentrated shear force, Edge shear stresses, Palazzetto dello sport, In plane shear forces, Derivation of Sander-Koiter equation 18, Shell boundary conditions | page 67 - 77 |
| 19 Sept. | 10:45-12:30 | F | Dependent quantities, Generalised edge disturbance, Beam supported by springs, Influence length, Boiler drums, Saturn V, Finite difference method (Python), Shell finite element types, Nodes and integration points | page 78 - 82 |
| 25 Sept. | ..8:45-10:30 | 3.02 | Element aspect ratio, Mesh refinement, Model accuracy, Selecting the element type, Shear locking, Hourglass mode, Finite element boundary conditons, Canopy finite element results, Singularities, Maximum number of nodes, Moore's law, Arithmetic accuracy, Benchmarks, Averaging at nodes | page 83 - 95 |
| 26 Sept. | ..8:45-10:30 | F | Practical excercise: Install SCIA Engineer, Finite element analysis of a 4-hypar shell, (energy check) | handout |
| 26 Sept. | 10:45-12:30 | F | Interpreting shell finite element results, Hypar curvature, Zeckendorf plaza, Reinforcement, Particle-spring method, Timber grid shells | page 96 - 107 |
| ..2 Oct. | ..8:45-10:30 | 3.02 | In-extensional deformation, Viking ships, Storage tanks, Theorema egregium, Kresge Auditorium, Deitingen petrol station, Gauß-Bonnet theorem, Formulas for point loads, Trajectories, Umbilical patterns | page 109 - 129 |
| ..3 Oct. | 10:45-12:30 | F | no lecture | |
| ..9 Oct. | ..8:45-10:30 | 3.02 | Practical excercise: Use SCIA Engineer, Finite element analysis of a conoid, Natural frequencies, Linear buckling analysis, Non-linear analysis | handout |
| 10 Oct. | ..8:45-10:30 | F | Tucker High School collapse, Cylinder buckling shapes, Ring buckling of an axially compressed cylinder, Differential equation for shell buckling, Buckling load factor, Design check of buckling, Imperfection sensitivity, Experiment, Puzzle, Exceptions to imperfection sensitivity | page 135 - 143 |
| 10 Oct. | 10:45-12:30 | F | Koiter's law, Knock down factor, Linear buckling analysis, Nonlinear finite element analysis, Model for imperfection sensitivity, Buckling, yielding or crushing?, Buckling curves for computational analysis, Measuring shape imperfections, Stiffeners, CNIT | page 144 - 155 |
| 16 Oct. | ..8:45-10:30 | 3.02 | Experiment on your shell in the Stevin Laboratory (previous results) | |
| 17 Oct. | 10:45-12:30 | F | no lecture | |
| 23 Oct. | ..8:45-10:30 | 3.02 | Hyperboloid, Ferrybridge collapse, Modal analysis, Rigid body modes, Equation of motion, Wave numbers, Vibration experiments, Vibration of hemispheres, spheres and cylinders, Natural frequency of a square shell, Enneper's surface, Measuring vibrations, Transient analysis, Damping ratio | page 155 - 170 |
| 24 Oct. | ..8:45-10:30 | F | Limit state function, Approximation of the limit state function, Convexity of the limit state, Monte Carlo analysis (Excel program), Joint probability distribution, Drawing a number, Software, Human error, Annual failure probability, Personal safety, Henk, Economic safety | page 171 - 179 |
| 24 Oct. | 10:45-12:30 | F | no lecture | |
| ..7 Nov. | ..9:00-12:00 | AE,E | Exam | |
| 29 Jan. | ..9:00-12:00 | 2.99 | Exam |
A calculator will be needed. It is allowed to use books, handouts, notes, old exams, computer, tablet, software, internet and chatGPT during the exam. Obviously, it is not allowed disturb others or to communicate with others during the exam. It is not allowed to use a phone. It is not allowed to use software with which one can communicate.
29 January 2025 (657 KB)
7 November 2024 (430 KB)
1 February 2024 (532 KB)
9 November 2023 (242 KB)
18 July 2023 (435 KB)
27 June 2023 (206 KB)
12 July 2022 (329 KB)
21 June 2022 (207 KB)
10 August 2021 (252 KB) different due to lockdown rules
28 June 2021 (263 KB) different due to lockdown rules
11 August 2020 (154 KB)
29 June 2020 (290 KB)
13 August 2019 (440 KB)
1 July 2019 (125 KB) results
14 August 2018 (121 KB) results
2 July 2018 ( 61 KB) results
15 August 2017 (408 KB)
3 July 2017 (112 KB)
9 August 2016 (239 KB)
27 June 2016 (219 KB)
11 August 2015 (1.06 MB) results
29 June 2015 (342 KB) results
12 August 2014 (159 KB) results
30 June 2014 ( 91 KB) results
13 August 2013 (187 KB) results
1 July 2013 (104 KB) results
28 August 2012 ( 95 KB) results
2 July 2012 ( 95 KB) results
24 August 2011 (352 KB) results
30 June 2011 (302 KB) results
25 August 2010 (168 KB) results
1 July 2010 (153 KB) results
29 June 2009 ( 64 KB) results
25 August 2008 ( 58 KB) results
30 June 2008 (102 KB) results
27 August 2007 (102 KB)
2 July 2007 (141 KB) results
3 July 2006 ( 59 KB)
23 January 2006 (399 KB) results
27 June 2005 (247 KB)
24 January 2005 (233 KB)
(not for the exam)
- Youtube: Building a masonry shell roof
- Youtube: Reinforced concrete dome with baloon formwork
- Youtube: Yoshimura pattern
- Movie: Membrane stress in the hoop direction (8.51 MB) Membrane stress in the axial direction (9.11 MB), F. te Loo, "Knik van schalen met de eindige-elementenmethode; Filmpjes die het gedrag in beeld brengen", Bacheloreindwerk TU Delft faculteit Civiele Techniek, 18 januari 2021, online: https://phoogenboom.nl/BSc_projects/eindrapport_te_loo.pdf
- Youtube: MythBusters, continued
- Book: J. Blaauwendraad, J.H. Hoefakker, Structural Shell Analysis, Understanding and Application, Springer, 2014, 305 pp.
- Report: Y. Yoshimura, "On the Mechanism of Buckling of a Circular Shell under Axial Compression", NACA Technical memorandum No. 1390, Sec 7.7, 1955
- Report: W.T. Koiter, A consistent first approximation in the general theory of thin elastic shells, Part 1, Foundations and linear theory, report, Delft University of Technology, 1959, 47 pp. Koiter's notation
- Report: J.L. Sanders, An improved first-approximation theory for thin shells, NASA technical report R-24, 1959, 16 pp.
- Paper: Th. von Kármán, H.S. Tsien, "The buckling of thin cylindrical shells under axial compression", Jounal of the Aeronautical Sciences Vol. 8, 1941, pp. 303-312
- Paper: G. Kirchhoff, "Über das Gleichgewicht und die Bewegung einer elastischen Scheibe", Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 1850 Issue 40, 1850