Technical University Delft Faculty of Civil Engineering and Geosciences PLASTICITY Ct 4150 The plastic behaviour and the calculation of plates subjected to bending Prof. ir. A.C.W.M. Vrouwenvelder Prof. ir. J. Witteveen March 2003 Ct 4150 Preface Course CT4150 is a Civil Engineering Masters Course in the field of Structural Plasticity for building types of structures. The course covers both plane frames and plates. Although most students will already be familiar with the basic concepts of plasticity, it has been decided to start the lecture notes on frames from the very beginning. Use has been made of rather dated but still valuable course material by Prof. J. Stark and Prof, J. Witteveen. After the first introductory sections the notes go into more advanced topics like the proof of the upper and lower bound theorems, the normality rule and rotation capacity requirements. The last chapters are devoted to the effects of normal forces and shear forces on the load carrying capacity, both for steel and for reinforced concrete frames. The concrete shear section is primarily based on the work by Prof. P. Nielsen from Lyngby and his co-workers. The lecture notes on plate structures are mainly devoted to the yield line theory for reinforced concrete slabs on the basis of the approach by K. W. Johansen. Additionally also consideration is given to general upper and lower bound solutions, both for steel and concrete, and the role plasticity may play in practical design. From the theoretical point of view there is ample attention for the correctness and limitations of yield line theory for reinforced concrete plates on the one side and von Mises and Tresca type of materials on the other side. This, however, is not intended for examination. I would like to thank ir Cox Sitters for his translation of the original Dutch text into English as well as for his many suggestions for improvements. A. Vrouwenvelder Delft, 2003 Table of contents Preface Notation ................................................................................................................................. 2 1 Introduction....................................................................................................................... 4 2 Elastic-plastic behaviour of a plate Lower- and upper-bound theorems................................................................................... 5 2.1 Behaviour of a plate under increasing load............................................................. 5 2.2 The upper-bound theorem ....................................................................................... 6 2.3 The lower-bound theorem ....................................................................................... 6 2.4 Validity of the theorems.......................................................................................... 6 3 Yield-line theory............................................................................................................... 7 3.1 Material behaviour .................................................................................................. 7 3.2 Yield-line theory...................................................................................................... 7 3.3 Yield-line pattern..................................................................................................... 7 3.4 The work equation................................................................................................... 8 4 Simply-supported rectangular plate................................................................................ 11 4.1 Rectangular plate with length twice the width (b = 2a) ........................................ 11 4.2 Additional formulae.............................................................................................. 12 4.3 Rectangular plate with arbitrarily chosen dimensions (b = a) ............................ 14 4.4 Some examples...................................................................................................... 16 5 Lower-bound calculation and design methods ............................................................... 18 5.1 Equilibrium equation and conditions .................................................................... 18 5.2 The twistless case .................................................................................................. 22 5.3 Design in accordance with the theory of plasticity ............................................... 22 6 Alternative upper-bound calculation (direct formulation of the equilibrium of the plate parts) ............................................... 25 6.1 Equivalent nodal forces and moments................................................................... 25 6.2 Minimisation of the load factor............................................................................. 27 7 The rectangular restrained plate...................................................................................... 29 7.1 Upper-bound solution............................................................................................ 29 7.2 Lower-bound solution........................................................................................... 30 7.3 Approximation of yield zones ............................................................................... 31 8 Simply supported square plate with two free edges ....................................................... 35 8.1 Some upper-bound solutions................................................................................. 35 8.2 Elastic solution...................................................................................................... 38 9 Circular plates................................................................................................................. 39 9.1 Uniform load on a simply supported circular plate............................................... 39 9.2 Uniform load on a restrained circular plate........................................................... 42 9.3 Point load in the centre of a simply supported circular plate ................................ 43 9.4 Point load in the centre of a restrained circular plate............................................ 44 10 Point loads and simple supports on columns.................................................................. 46 10.1 Point load in the centre of a simply supported square plate.................................. 46 10.2 Point load in the centre of a restrained square plate.............................................. 46 10.3 Infinitely long simply supported plate................................................................... 47 10.4 Point load on free edges and free corners.............................................................. 49 10.5 Plate on columns ................................................................................................... 51 11 Yield criteria of the largest principal moment, Tresca and von Mises ........................... 54 11.1 General formulation of yield criterion................................................................... 54 11.2 The yield criterion of the largest principal moment (square yield criterion) ........ 54 11.3 The yield criterion of Tresca ................................................................................. 56 11.4 The yield criterion of von Mises (Huber, Hencky) ............................................... 57 11.5 Lower-bound calculation of a torsion panel.......................................................... 58 12 Yield criterion for reinforced concrete slabs .................................................................. 60 12.1 Yield line in x- or y-direction ................................................................................ 60 12.2 Yield line under an angle with the y direction....................................................... 61 12.3 Yield line calculation of reinforced concrete torsion panel .................................. 63 12.4 Yield criterion formulated in moments with respect to the x-y system................. 64 12.5 Example: lower-bound calculation of reinforced concrete torsion panel.............. 69 12.6 Example: design calculation.................................................................................. 71 13 General background on plastic calculation of plates ...................................................... 74 13.1 Further description of ideal-plastic material behaviour ........................................ 74 13.2 General procedure for the upper-bound calculation.............................................. 76 13.3 Yield criterion for reinforced concrete slabs – additional considerations............. 83 14 Final considerations ........................................................................................................ 93 Literature......................................................................................................................... 95 Appendix A: Formulae for plates ................................................................................... 97 Appendix B: Transformation formulae for plate moments .......................................... 100 Questions ...................................................................................................................... 101 Answers to question 1 – 16........................................................................................... 111 1 0 Notation Notation of symbols a,b - plate dimensions e1, e2, … - strain parameters