- ARCHIVE for reference only

University Homepage
DRPS Homepage
DRPS Search
DRPS Contact
DRPS : Course Catalogue : School of Engineering : Civil

Undergraduate Course: Plastic Analysis of Frames and Slabs 4 (CIVE10003)

Course Outline
School School of Engineering College College of Science and Engineering
Course type Standard Availability Available to all students
Credit level (Normal year taken) SCQF Level 10 (Year 4 Undergraduate) Credits 10
Home subject area Civil Other subject area None
Course website None Taught in Gaelic? No
Course description 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.
Entry Requirements
Pre-requisites Students MUST have passed: Theory of Structures 3 (CIVE09015)
Prohibited Combinations Other requirements None
Additional Costs None
Information for Visiting Students
Pre-requisites None
Displayed in Visiting Students Prospectus? Yes
Course Delivery Information
Delivery period: 2010/11 Semester 1, Available to all students (SV1) WebCT enabled:  Yes Quota:  None
Location Activity Description Weeks Monday Tuesday Wednesday Thursday Friday
King's BuildingsLecture1-11 14:00 - 15:50
King's BuildingsTutorial1-11 16:10 - 17:00
First Class Week 1, Thursday, 14:00 - 15:50, Zone: King's Buildings. Lecture Theatre 1, Sanderson Building
Exam Information
Exam Diet Paper Name Hours:Minutes Stationery Requirements Comments
Main Exam Diet S1 (December)Plastic Analysis of Frames and Slabs 42:0012 sides / 2 x graph
Summary of Intended Learning Outcomes
By the end of the course, the student should 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.
Assessment Information
The assessment will be made on the basis of:
Degree examination 100%
Special Arrangements
Additional Information
Academic description Not entered
Syllabus 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 cross-sections
Stress-strain relationships for materials and their simplification into a 2
parameter model. Models for hand and for computer analysis. The analysis of
cross-sections of any complexity to determine the full plastic moment about an
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 steel-concrete 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. Over-complete collapse and its significance. Nonproportional
L6 General rules on collapse of frames
Application of plastic analysis to multi-storey and multi-bay 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.
C:\AA\Teaching\YEAR4-5\M13 U00408-Plas+YldLine\CorGuide\2010\U00408 Course Guide to Plastic Analysis of Frames and Slabs 4-JMR V10.docLast Saved: 16
August 2010
L7 Single storey portal frames
Analysis of a multi-bay 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 Multi-storey 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.
C:\AA\Teaching\YEAR4-5\M13 U00408-Plas+YldLine\CorGuide\2010\U00408 Course Guide to Plastic Analysis of Frames and Slabs 4-JMR V10.docLast Saved: 16
August 2010
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.
C:\AA\Teaching\YEAR4-5\M13 U00408-Plas+YldLine\CorGuide\2010\U00408 Course Guide to Plastic Analysis of Frames and Slabs 4-JMR V10.docLast Saved: 16
August 2010

Tutorials: Titles & Contents
Segment 1 Plastic analysis of frames
Tutorial 1 Plastic moments of cross-sections
This tutorial covers the determination of the full plastic moment of various
cross-sections, followed by the development of interaction diagrams for crosssections.
Tutorial 2 Plastic collapse of multi-bay and multi-storey frames
This tutorial covers problems involving interaction diagrams for simple frames,
combined mechanisms for multi-storey and multi-bay 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.
Transferable skills Not entered
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 pre-stressed 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
Study Abroad Not entered
Study Pattern Not entered
Keywords Not entered
Course organiser Prof Michael Rotter
Tel: (0131 6)50 5718
Course secretary Mrs Laura Smith
Tel: (0131 6)50 5690
Help & Information
Search DPTs and Courses
Degree Programmes
Browse DPTs
Humanities and Social Science
Science and Engineering
Medicine and Veterinary Medicine
Other Information
Important Information
copyright 2011 The University of Edinburgh - 31 January 2011 7:28 am