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Theory of plasticity and its application on steel and reinforced concrete structures. Extremum Principles (2014)

Name of course:

Theory of plasticity and its application on steel and reinforced concrete structures. Extremum Principles

ECTS credits:


Course parameters:

Language: English

Level of course: PhD course

Time of year: Fall 2014 (one week)

No. of contact hours: 30

No. of hours in total incl. preparation, assignments and the like: 150

Capacity limits: None

Objectives of the course:

The main objective of this course is to give the basis of the theory of plasticity. To master the principle given by the theory of plasticity is very essential for the structural engineer. By mastering this principle it is relatively simple to analyze and  design  complex structures. The main objective of the course is to give the student the capability to:

  • Achieve a good physical understanding of the behavior of even more complex structural elements and to be able to design and make calculations on these elements
  • Be able to model complicated structures and their individual parts of principally arbitrary type 

Learning outcomes and competences:

At the end of the course, the student should be able to:

  • Explain the principle of v. Mises associated flow rule
  • Explain different material models and to argue for the suitability of these models on the individual materials
  • Explain the theoretical background for the upper and lower bound theorems
  • Give an account of the fundamental concepts of lines of discontinuity and planes of discontinuity including strains in lines of discontinuity
  • Argue for the difference between plane stress field and plane strain field
  • Set up the basic equations of the theory of plasticity and to be able to describe the contents of these equations
  • Argue for the reasonableness of using the theory of plasticity on concrete structures
  • Argue for the need of constructing yield surfaces
  • Reproduce the principle of constructing yield surfaces
  • Use yield surfaces on design cases
  • Based on the upper or lower bound theorems, argue for a well-qualified choice of designing tool to model structures/structural elements)
  • Applying upper and lower bound theorems to determined strength of structures/structural elements

Compulsory programme:

Active participation at the theoretical lessons is required to obtain the course diploma. Approved oral presentation and report.

Course contents:

Introduction to static and geometrical conditions

Physical conditions, that means a) yield conditions b) constitutive equations for ideal plastic materials

Extremum principles (upper and lower bound theorems)

Description of physical behavior of reinforced concrete (unconfined, confined, cracked, reinforcement, bond, strength reductions/effectiveness factors)

Applications of upper and lower bound theorems (discs, beams, plates, cracks/joints, anchorage, three dimensional modelling)


Master degree in Structural or Mechanical Engineering

Name of lecturer:

Lars German Hagsten

Type of course/teaching methods:

Combined lectures and exercises


L. G. Hagsten: “Mechanics/Theory of Plasticity - Basis for the Modeling of Structures” 2012

M. P. Nielsen & L. C. Hoang: “Limit Analysis and Concrete Plasticity”. 3rd Edition. CRC Press. 2011

Course homepage:


Course assessment:

Oral examination at the end of the course week + approval of a subsequent submitted assignment (Passed/not Passed). It should be noted that the course/the written assignment requires a great deal of work in the weeks/months following the course week and until delivery of the assignment


Department of Engineering

Special comments on this course:


27 – 3 October 2014 (please note the subsequent assignment)


Aarhus University


For registration: Please contact Lars German Hagsten, e-mail


If you have any questions, please contact Lars German Hagsten, e-mail



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Revised 20.06.2016