# Nonlinear Behavior of Steel Structures

## Course Description:

Objective of this course is to enable students to understand the fundamental concepts of nonlinear behavior of metal structures and its consequences on the choice of appropriate methods of structural analysis and design. Geometric and material nonlinearities and their interaction are covered, as well as influence of initial imperfections. Through the course a better understanding of the theoretical background of steel structures design rules is achieved, and the necessary abilities are developed in order to carry out numerical calculations for obtaining the ultimate strength of unconventional steel members and structures, which are not covered by the pertinent codes.

### Prerequisite Knowledge

It is recommended that students have basic knowledge of structural mechanics, strength of materials, structural analysis and steel structures. Knowledge of basic principles of the finite element method is also desirable.

### Course Units

# Title Description Hours
1 Introduction Objectives and organization of the course, prerequisites of linear behavior, fundamental concepts of nonlinear behavior, types of nonlinear behavior, material nonlinearity, geometric nonlinearity, interaction between nonlinearities, influence of imperfections, examples of nonlinear behavior, (compressed cylindrical shell, simply-supported compressed bar, triangular von Mises truss, triangular truss apse), overview of types of nonlinear behavior, importance of imperfections and types of analysis. 2Χ3=6
2 Material nonlinearity Constitutive behavior of steel (actual and idealized), failure criteria under combined stress field, elastic-plastic behavior of cross-sections in bending, the concept of plastic hinge, elastic-plastic behavior of cross-sections in bending and axial force, elastic-plastic behavior of cross-sections in bending and shear force, elastic-plastic behavior of simply-supported beam, clamped beam, 2-span continuous beam, simply-supported frame, clamped frame. 1Χ3=3
3 Geometric nonlinearity – 1DOF systems Concept of geometric nonlinearity, linear and nonlinear buckling theory, equilibrium or Euler method, energy method (equilibrium and stability theorems), dynamic method (phase diagrams, bounded and unbounded motion, relation between eigenfrequencies and stability, influence of initial conditions, influence of damping), examples of perfect and imperfect 1-DOF systems failing via a symmetric stable, symmetric unstable o asymmetric bifurcation point or via a limit point, recommended analysis methods, influence of initial imperfections, correlation of 1DOF models with actual structural systems. 2Χ3=6
4 Geometric nonlinearity – MDOF systems Equilibrium or Euler method, energy method, dynamic method, linear and nonlinear buckling theory, buckling modes, shape and size of initial imperfections, buckling mode interaction, influence of ratio between critical buckling loads and imperfection amplitudes on the nonlinear response.
5 Numerical solution of nonlinear problems Features of the finite element method for nonlinear problems, solution algorithms of the resulting system of nonlinear equations, step-wise load incrementation, full and modified Newton-Raphson method, convergence criteria, selection of appropriate analysis method, number of load steps, maximum number of iterations per step, convergence limits, load versus displacement control, arc-length methods, case studies in nonlinear finite element software, von Mises truss, elastic and elastic-plastic buckling of bars in compression, buckling of frames, cylindrical shells, unstiffened and stiffened plates, local buckling, design methodology of steel structures employing nonlinear finite element analyses. 4Χ3=12
6 Applications from research Nonlinear in-plane behavior of arches and corresponding design methodology, local buckling of wind turbine towers near the man-hole and evaluation of strengthening methods, interaction of global buckling, local buckling and material yielding in built-up columns, fatigue of wind turbine tower connections. 1Χ3=3
7 Applications from structural engineering practice Design of beams with varying cross-section for the steel roof of Panathinaikos stadium in Votanikos, design of pylons and main girders with varying cross-section for the steel roof of Aristotle’s Lyceum protection shelter, design of buried oil pipeline between Thessaloniki and Skpje at crossings of active seismic faults, design of façade and dome of Oval Tower in Limassol, Cyprus. 1Χ3=3
8 Presentation of term projects Oral presentation of term projects. 1Χ3=3

### Learning Objectives

In this course it is attempted to provide balance between acquiring knowledge of the theoretical background of nonlinear structural behavior and expertise in applied analysis and design methods. Initially simple models are treated analytically, aiming at qualitative understanding of concepts and problems. This is followed by a numerical treatment with the finite element method to confront these issues in realistic complex structures. Upon the successful completion of the course, the students will be able to:

• Understand the theoretical background of ULS checks of steel members according to Eurocode 3.
• Identify cases in which nonlinear analyses are requird for the design of a steel structure.
• Design unconventional steel structure by means of nonlinear numerical analyses.

### Teaching Methods

 Teaching methods - In-class lectures. - Solution of simple examples and case studies in class. There is continuous interconnected flow of theoretical and applied lectures and exercises. For carrying out their term prject, which is an indispensable part of the course, students are assisted by PhD candidates who are available at times outside lectures hours. PowerPoint presentations Additional material presented on the black board. Yes, learning of finite element software Adina, and applying it for solving nonlinear structural problems in general and for the term project in particular. Yes Term projects (either individual or in groups of two) including in-class PowerPoint presentation and preparation of technical report. Yes

### Student Assessment

• Final written exam: 50%
• Assignments (projects, reports): 50%

### Textbooks - Bibliography

• Gantes, C.J., “Nonlinear Structural Behavior – Emphasis on Metal Structures”, Hellenic Academic EBooks, 2016 (in Greek).
• Kounadis, A.N., “Introduction in the Nonlinear Theory of Elastic Stability”, Symeon Editions, Athens, 1998 (in Greek).
• Educational material from the course website.

## Lecture Time - Place:

• Thursday, 13:45 – 15:30,
Rooms:
• Ζ. Κτ. 1 Πολ., Αιθ. 1
• Friday, 13:45 – 15:30,
Rooms:
• Αμφ. Μετ. Κατασκ.

© 2017 School of Civil Engineering, ΕΜΠ