Postgraduate Course: Plastic Analysis of Frames and Slabs (PGEE10004)
Course Outline
School  School of Engineering 
College  College of Science and Engineering 
Credit level (Normal year taken)  SCQF Level 10 (Postgraduate) 
Availability  Not available to visiting students 
SCQF Credits  10 
ECTS Credits  5 
Summary  In this module, two segments extend the student's knowledge and understanding of the theory of structures to plastic behaviour. The first presents a deeper understanding of the plastic analysis of frames: the second covers yield line analysis of reinforced concrete slabs. 
Course description 
LECTURES: TITLES & CONTENTS
Segment 1 Plastic collapse of frame structures
L1 Introduction
Structure of the course. Aims of the course. References with comments. The theorems of plastic analysis: upper and lower bound theorems, their basis and assumptions. Ductility requirements for plastic collapse in steel members: plastic and compact sections.
L2 Full plastic moments of crosssections
Stressstrain relationships for materials and their simplification into a 2 parameter model. Models for hand and for computer analysis. The analysis of crosssections of any complexity to determine the full plastic moment about an axis.
L3 Axial loads and cross sections in different materials
The effect of axial load on the plastic moment. The interaction diagram for simple sections: all four quadrants of the interaction diagram and its significance in structures. Ultimate moment interaction diagrams in reinforced concrete or composite steelconcrete sections.
L4 Plastic collapse of continuous beams
Review of plastic collapse of beam structures. Changes of section at supports and within spans. Rules for locations of plastic hinges and number of mechanisms. Minimisation of collapse loads when hinge locations are not preordained.
L5 Portal frames
Plastic collapse of a simple single bay portal frame. Locations of hinges, types of mechanisms illustrated in this simple example. Effect of pinned bases. The interaction diagram. Overcomplete collapse and its significance. Nonproportional loading.
L6 General rules on collapse of frames
Application of plastic analysis to multistorey and multibay frames. Elementary and combined mechanisms. Joint rotation as an elementary mechanism, simple beam mechanisms, simple sway mechanisms, combining mechanisms.
Rules for the locations of plastic hinges. Rules for assessing numbers of redundancies.
Rules to determine the number of sway modes. Rules to determine the numbers and types of elementary and combined mechanisms. Application of the rules.
L7 Single storey portal frames
Analysis of a multibay portal frame. Application of the upper bound theorem. Identifying hinge locations. Determining the number of independent elementary mechanisms. Identifying the elementary mechanisms. Combining mechanisms: compatibility requirements and methodology. Application of the lower bound theorem to verify the collapse load. Use of the lower bound theorem on the wrong mechanism.
L8 Multistorey portal frames
Identifying hinge locations. Determining the number of independent elementary mechanisms. Identifying the number of sway modes and their forms. Identifying possible elementary mechanisms: alternative choices. Analysis of the elementary mechanisms. Combination of mechanisms. Compatibility requirements and methodology. Lower bound theorem in the presence of multiple sway modes. Sway equilibrium equations.
L9 Upper and lower bound theorems and their significance
Full statement of the two theorems. Uniqueness. Demonstration of outcome of applying each theorem to modes that are not the correct collapse mode. Use of the upper bound theorem and minimisation. Use of the lower bound theorem and safe design. Exploitation of lower bound theorem in elastic analysis.
Requirements for the theorems to be valid. Ductility and stability effects.
L10 Other factors and aspects
Modifications of the evaluated collapse loads caused by different phenomena. Effect of axial loads on full plastic moment, and on ultimate moments in reinforced concrete. Effect of instability on collapse loads. Geometric nonlinearity and its outcomes for different loadings and geometries. Brittle materials and the effects of shrinkage, creep, lack of fit, settlement etc.
Segment 2 Yield line analysis of reinforced concrete slabs
L1 Introduction
Introduction to yield line analysis: behaviour of rigid plastic material, fundamentals of yield line theory and methods of analysis, equilibrium and virtual work methods.
L2 Simple example of one way bending
Simple calculation of a collapse load for one way bending and its relationship to plastic collapse of beams.
L3 The yield line: bending and twisting moments
Calculations of bending and twisting moments on the yield lines for isotropic and orthotropic reinforcement; calculation of normal rotation on a yield line. Compatibility requirements of yield line patterns.
L4 Collapse mechanisms
Fundamentals and assumptions for collapse mechanisms. Collapse mechanisms for slabs with different boundary conditions based on these assumptions.
L5 Example problems
Orthotropic slabs of different geometries and load cases. Determination of collapse loads using the upper bound theorem. Discussion of the reasons for examining alternative collapse mechanisms. Derivation of formulae for the analysis of slabs of various shapes under different loading conditions (point load, line load and distributed load) and different boundary conditions.
L6 Lower bound theorem and other phenomena
The yield line as an upper bound method: upper and lower bound theorems of plasticity for slabs. Use of finite element analysis with the lower bound theorem. Compressive membrane action and its causes. Relationship of yield line load to true collapse. Tensile membrane action. Geometrically nonlinearity and its effect on behaviour. The meaning of a collapse load. Punching shear.
L7 Revision
Review of the whole module. Significance of collapse load evaluation. Lower bound theorem and its importance in elastic analysis and design. Importance of ductility, and warnings about brittle materials. Relationship between hand calculations and computer calculations. Material and geometric nonlinearity.
TUTORIALS: TITLES & CONTENTS
Segment 1 Plastic analysis of frames
Tutorial 1 Plastic moments of crosssections
This tutorial covers the determination of the full plastic moment of various crosssections, followed by the development of interaction diagrams for crosssections.
Tutorial 2 Plastic collapse of multibay and multistorey frames
This tutorial covers problems involving interaction diagrams for simple frames, combined mechanisms for multistorey and multibay frames.
Segment 2 Yield line analysis of slabs
Tutorial 3 Yield line analysis of slabs
A single tutorial sheet with many questions, beginning with simple problems and progressing to complex yield line mechanisms.
These tutorials should all be completed and handed in as they provide an excellent preparation for a professional career as well as the examination.

Entry Requirements (not applicable to Visiting Students)
Prerequisites 

Corequisites  
Prohibited Combinations  
Other requirements  None 
Course Delivery Information
Not being delivered 
Learning Outcomes
On completion of this course, the student will be able to:
 demonstrate the ability to calculate the plastic collapse loads of complex two dimensional frame structures, to identify the independent mechanisms and combine them, to use the upper and lower bound theorems to find the true collapse load, and to produce engineering designs of frame structures based on plastic collapse analysis;
 demonstrate the ability to calculate the yield line collapse load of reinforced concrete slabs of complex geometry with isotropic and orthotropic reinforcement using the upper bound theorem, and to apply the method to the proportioning of reinforcement in a slab.

Reading List
Segment 1 Plastic analysis of frames
Course reference
 Plastic Design of Frames
J.F. Baker and J. Heyman
Cambridge University Press 1969
Suggested further reading
 Plastic Theory of Structures
M.R. Horne
Pergamon Press 1981
 The Steel Skeleton Volume II
J.F. Baker, M.R. Horne and J. Heyman
Cambridge University Press 1956
 Plastic methods for steel and concrete structures
S.S.J. Moy
Macmillan 1996
Segment 2 Yield line analysis of slabs
Course reference
 Reinforced and prestressed concrete
F.H. Kong and R.H. Evans
van Nostrand Reinhold (UK) 1987
Suggested further reading
 Yield line analysis of slabs
L.L. Jones and R.H. Wood
Thames and Hudson, Chatto and Windus, 1967
 Structural Concrete
R.P. Johnson
McGraw Hill
 Yield line analysis of slabs
K.W. Johanson
Cement and Concrete Association, London 1972
 Plastic methods for steel and concrete structures
S.S.J. Moy
Macmillan 1996

Additional Information
Graduate Attributes and Skills 
Not entered 
Keywords  Not entered 
Contacts
Course organiser  Dr Hwa Kian Chai
Tel:
Email: Hwakian.Chai@ed.ac.uk 
Course secretary  Mr Craig Hovell
Tel: (0131 6)51 7080
Email: c.hovell@ed.ac.uk 

