Technical University Delft Faculty of Civil Engineering and Geosciences PLASTICITY Ct 4150 The plastic behaviour and the calculation of beams and frames subjected to bending Prof. ir. A.C.W.M. Vrouwenvelder 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 1 Introduction ................................................................................................................... 4 1.1 Computational models in structural mechanics .................................................... 4 1.2 Modelling of the material behaviour..................................................................... 5 1.3 The elastic-plastic behaviour of simple bar structures .......................................... 7 1.4 Elasticity versus plasticity approach ................................................................... 10 1.5 Historical overview............................................................................................. 11 2 The elastic-plastic calculation ..................................................................................... 13 2.1 The moment-curvature distribution for the rectangular cross-section ................ 13 2.2 Arbitrary cross-sections....................................................................................... 17 2.3 The moment-curvature distribution of reinforced concrete ................................ 21 2.4 The elastic-plastic behaviour of a statically indeterminate beam ....................... 23 2.5 The elastic-plastic behaviour of a frame ............................................................. 26 3 Plastic limit state analysis ........................................................................................... 29 3.1 Introduction ......................................................................................................... 29 3.2 Upper-bound theorem ......................................................................................... 29 3.2.1 Systematic application................................................................................. 32 3.2.2 Special cases................................................................................................ 35 3.2.3 Uniformly distributed load.......................................................................... 36 3.2.4 Proof of the upper-bound theorem .............................................................. 37 3.3 Lower-bound theorem ......................................................................................... 38 3.3.1 Application.................................................................................................. 38 3.3.2 Uniformly distributed load.......................................................................... 40 3.3.3 Proof of the lower-bound theorem .............................................................. 41 3.4 Combination of upper- and lower-bound theorems ............................................ 41 3.5 Some consequences of the lower- and upper-bound theorems ........................... 43 4 Rotation capacity......................................................................................................... 44 4.1 Introduction ......................................................................................................... 44 4.2 Restrained steel beams ........................................................................................ 44 4.3 Experiments by Stüssi and Kollbrunner.............................................................. 48 4.4 Reinforced concrete............................................................................................. 49 5 The yield contour........................................................................................................ 52 5.1 Plane truss........................................................................................................... 52 5.2 Yield contour of a portal frame........................................................................... 54 5.3 Normality ............................................................................................................ 57 5.4 Plastic potential, convexity ................................................................................. 59 6 Yield criteria................................................................................................................ 63 6.1 Introduction ......................................................................................................... 63 6.2 The yield criterion of Tresca (steel) .................................................................... 63 6.3 The yield criterion of von Mises (steel) .............................................................. 65 6.4 The yield criterion of Mohr-Coulomb (concrete, rock / soil) ............................. 67 7 Effects of normal forces on plastic frame behaviour .................................................. 70 7.1 The influence of the normal force on the fully plastic moment.......................... 70 7.1.1 Introduction ................................................................................................. 70 7.1.2 Rectangular cross-section............................................................................ 70 7.1.3 Arbitrary double-symmetric cross-section .................................................. 72 7.2 The moment-curvature diagram in the presence of a normal force .................... 76 7.2.1 Rectangular cross-sections .......................................................................... 76 7.2.2 I-shaped cross-sections................................................................................ 78 7.3 Reinforced concrete cross-section....................................................................... 79 7.4 Yield function and normality .............................................................................. 84 8 The influence of the transverse force on the full-plastic moment............................... 98 8.1 Steel cross-sections ............................................................................................. 98 8.1.1 Rectangular cross-sections .......................................................................... 99 8.1.2 I-sections ................................................................................................... 103 8.2 Reinforced concrete cross-sections ................................................................... 105 8.2.1 Introduction ............................................................................................... 105 8.2.2 Yield lines................................................................................................. 106 8.2.3 Beam with stirrup reinforcement / upper-bound calculation .................... 109 8.2.4 Beam with stirrup reinforcement / lower-bound calculation .................... 112 Reading list................................................................................................................ 118 1 Introduction 1.1 Computational models in structural mechanics In structural mechanics the following basic quantities appear: • Displacements: ux, uy, uz • Strains: εxx, εyy, εzz, εxy, εyz, εxz • Stresses: σxx, σyy, σzz, σxy, σyz, σxz • Forces: fx, fy, fz. For static situations the relations between these quantities are given by three sets of equations: 1) Kinematical equations 2) Constitutive equations 3) Equilibrium equations Strains Stresses Displacements Forces εxx, εyy, εzz, σxx, σyy, σzz, ux, uy, uz Fx, Fy, Fz εxy, εxz, εyz σxy, σxz, σyz Kinematical Constitutive Equilibrium equations equations equations (6 eq.) (6 eq.) (3 eq.) Fig. 1.1: Relations between the basic quantities in structural mechanics. As indicated in Figure 1.1, the kinematical equations relate the strain components to the displacements and the stress components are related to internal and external forces