CIE4150 Plastic Analysis of Structures, course 2021/2022

Dr.ir. P.C.J. Hoogenboom (Email p.c.j.hoogenboom@tudelft.nl)
Prof.ir. A.C.W.M. Vrouwenvelder

Period 3, February 7 - April 7, 2022
4 credits (ECTS)

The only reason why structural designers sleep soundly is the second theorem of plasticity theory. This theorem says that no matter how I designed my structures, they are safe because everything was in equilibrium, nowhere the stresses were too large and I used ductile components and joints. In this course we study the first and second theorem and apply them to frames and plates.

Objectives

After completing this course you will know how plastic hinges develop in concrete and steel beams and plates. You will understand commonly used material yield criteria and cross-section interaction diagrams. You will be able to calculate the ultimate load of beams, frames and plates. You will understand redistribution of the force flow in structures and you will understand the limitations of plasticity theory.

Scope

Plastic material behaviour and the consequences for structural behaviour. Incremental computations where the load is gradually increased from zero until the collapse limit (suitable for computer implementation). Upper- and lower-bound approximations (suitable for hand calculations). Discussion on the theory and its application to beams, portals, frames and in plane and laterally loaded plates. Fundamental aspects of yield criteria (Von Mises, Tresca, reinforced concrete). Interaction of bending moment, shear force and normal force. Normality rule. Upper- and lower-bound theorems and deformation capacity.

Schedule

8 Feb. No lecture Chapter Collegerama
10 Feb. Paradoxes in elasticity, Historical overview, Material behaviour, Behaviour of an indetermined bar structure (handout 0) Unloading and residual stresses, Bauschinger effect. 1
15 Feb. Yielding in steel beams and in reinforced concrete beams, Incremental analysis of a portal, Prager's theorems. 2, 3
17 Feb. Example of an upperbound analysis, Example of a lowerbound analysis. Overview of the conditions for upperbound and lowerbound analysis. Virtual work equation (handout 1), Strange but correct applications. 3 18 Feb. 2020
22 Feb. More strange but correct applications, Proof of the upper bound theorem, Proof of the lower bound theorem, Exam example (21 January 2016). 3 20 Feb. 2020
24 Feb. Exam example continued, Rotation centre, Distributed load, Temperature loading, Support settlements, Misfits, Prestress. 25 Feb. 2020
1 March Yield contour of a simple truss, Yield contour of a simple frame, Normality of deformation (associated flow), Convexity of the yield contour, Importance of convexity for structural design, Yield criteria of Rankine, De Saint Venant, Tresca, Beltrani, Von Mises, Mohr-Coulomb and Drucker-Prager (handout 3a, handout 3b). 5, 6 27 Feb. 2020
3 March Interaction diagram of moment and axial force, Rectangular cross-section, I-sections and tubular sections (handout 4a), Reinforced concrete cross-sections, Virtual work in a plastic hinge for rhombic M-N interaction (handout 4b), Mending math, Moment-curvature diagram, Effect of rolling stresses on strength and buckling load, Cable action in steel beams, Arch action in concrete beams. 7 3 March 2020
8 March Exam example upper bound calculation with M-N interaction (21 January 2016), Interaction diagram of moment and shear force, Shear interaction for rectangular cross-sections, I-sections. 8 5 March 2020
10 March Selecting a strut-and-tie model (handout 5b), Detailing with a strut-and-tie model (handout 5c), Plate boundary conditions, Plate collapse (picture 1, 2, 3, 4, 5, 6, 7, 8, 9, 10), Upperbound analysis of the example (handout 5a), Other yield line patterns, Positive plate moments (handout 6), Impossible yield line patterns. Introduction to the analysis competition (handout 7). 1, 2, 3, 4 10 March 2020
15 March Exam example of a plate upper bound analysis (22 January 2020), Lower bound solution of a rectangular plate, Comparison of the upper and lower bound, Twistless equilibrium (Hillerborg strip method). Sign up for the analysis competition. 5 12 March 2020
not 28:00 to 40:40
17 March Exam example of a plate lower bound analysis (22 January 2020), Rectangular plate with clamped edges, Refined yield pattern in corners, Comparison of the elastic, upper bound and lower bound solutions, Square plate with free edges, Circular plates with distributed loading and point loading. Point load on a square plate, Edges loads, Column supports. 5, 7, 8, 9, 10 17 March 2020
until 39:04
17 March 2020
22 March No lecture
24 March Analysis competition. Which will be faster, the elastic team or the plastic team? Results. Example movie (77 MB) of Miss Z. Procházková performing a plastic frame analysis in 5 min. 4 sec.
7 April Exam 13:30-16:30 hours
16 June Exam 13:30-16:30 hours

Lecture Books

A.C.W.M. Vrouwenvelder and J. Witteveen, "Plastic Analysis of Structures, The plastic behaviour and the calculation of beams and frames (2.31 MB) subjected to bending", Lecture book Delft University of Technology, March 2003.
A.C.W.M. Vrouwenvelder and J. Witteveen, "Plastic Analysis of Structures, The plastic behaviour and the calculation of plates (1.81 MB) subjected to bending", Lecture book Delft University of Technology, March 2003.

Course Relations

CT3109 Structural Mechanics 4
CT5144 Stability
CT5142 Computational Methods

Handouts

0. Simple bar structure example (139 KB)
1. Virtual work equation (2.32 MB) A. Verruijt, "Soil Mechanics" Delft University of Technology, 2001, pp 322-324, online
3a. Transparancies on yield criteria (170 KB)
3b. Material yield contours (49.9 KB)
4a. Interaction diagrams for steel sections (21.8 KB)
4b. Virtual work in a rhombic interaction diagram (21.7 KB)
5b. Strut-and-tie model selection (111 KB)
5c. strut-and-tie model detailing (487 KB)
... Picture 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (15.6 MB). Made by Miss. C.E. Hammer, 2006
5a. Plate experiment (29.4 KB).
6. Plate moments (48.2 KB)
7. Competition (423 KB)

Exam

During the exam, it is allowed to use readers, handouts, books, notes, calculator, laptop computer, tablet and internet.
All software is allowed with the exception of the programs Mp (icozct.tudelft.nl/TUD_CT/software/mp/) or Slab (www.limitstate.com).
Obviously, during the exam it is not allowed to communicate information with other people than the teacher.

Previous Exams

29 June 2023 (0.42 MB)
13 April 2023 (1.48 MB)
16 June 2022 (1.39 MB)
7 April 2022 (1.98 MB)
different due to lockdown rules results 24 June 2021 (0.18 MB)
different due to lockdown rules results 8 April 2021 (0.97 MB)
13 August 2020 (1.01 MB)
27 May 2020 (1.56 MB)
22 January 2020 (0.76 MB)
11 April 2019 (0.49 MB) results
22 January 2019 (0.61 MB) results
12 April 2018 (0.69 MB)
25 January 2018 (0.95 MB)
13 April 2017 (0.61 MB) results
26 January 2017 (0.38 MB) results
7 April 2016 (0.37 MB)
21 January 2016 (0.59 MB) results
9 April 2015 (0.62 MB) results
22 January 2015 (0.69 MB) results
10 April 2014 (0.58 MB) results
23 January 2014 (0.68 MB) results
11 April 2013 (0.67 MB) results
24 January 2013 (0.95 MB) results, more results, more results
12 April 2012 (0.81 MB) results
26 January 2012 (1.44 MB) results
7 April 2011 (1.19 MB) results
20 January 2011 (1.14 MB) results
8 April 2010 (1.13 MB) results, results
21 January 2010 (1.14 MB) results
17 June 2009 (0.79 MB) results
15 January 2009 (1.00 MB) results
19 August 2008 (0.95 MB) results
18 June 2008 (1.09 MB) results
17 January 2008 (1.57 MB) results
21 August 2007 (1.86 MB) results
20 June 2007 (0.99 MB) results
2 February 2007 (1.27 MB) results
18 January 2007 (1.98 MB) results
22 August 2006 (1.52 MB) results
21 June 2006 (1.49 MB) results
23 August 2005 (1.80 MB)
15 June 2005 (1.67 MB)
19 August 2004 (1.38 MB)
9 June 2004 (1.67 MB)
21 August 2003 (2.37 MB)
11 June 2003 (2.75 MB)
30 August 2002 (622 KB)
18 December 2001 (696 KB)
31 August 2001 (768 KB)