Steel Structures

Design using FEM

Rolf Kindmann, Matthias Kraus

• Publisher: Ernst & Sohn
• Year: 2012
• Print ISBN: 9783433029787
• eBook ISBN: 9783433601266
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Civil Engineering
• Index: Technology & Engineering

This book presents the design of steel structures using finite element methods (FEM) according to the current state of the art in Germany and the rest of Europe. After a short introduction on the basics of the design, this book illustrates the FEM with a focus on internal forces, displacements, critical loads and modal shapes. Next to finite element procedures for linear calculations considering the stress states of normal force, biaxial bending and warping torsion, non-linear calculations and the stability cases of flexural buckling, lateral torsional buckling and plate buckling are concentrated on significantly. In this context, design procedures for stability according to the standard Eurocode 3 is introduced and discussed. In addition, important fundamental issues are covered, such as the determination of cross-section properties as well as the elastic and plastic cross-section resistance. Complementary, finite element procedures for cross sections are dealt with, which will have an increasing importance in future.
This book has evolved within the teaching activities of the authors in the lecture Computer-oriented Design of Steel Structures on the Master?s Program Computational Engineering at the University of Bochum. It covers the total variety of demands needed to be discussed for the safe, economic and modern design of steel structures.

1

Introduction

1.1 Verification Methods

For civil engineering structures the ultimate limit state (structural safety) and serviceability limit state have to be verified, see for example DIN 18800 Part 1. Since components for steel constructions are usually rather slender and thin-walled, structural safety verifications for constructions susceptible to losing stability regarding flexural, lateral torsional and plate buckling are of major significance and therefore constitute a main focal subject in static calculations. In this context, the determination of internal forces and moments, deformations and critical loads is a central task. Its solution is treated in this book using the finite element method (FEM).

The calculations and verifications have to meet the legal requirements as well as the state of the art. For steel structures the basic standard DIN 18800 and corresponding engineering standards, or Eurocode 3, have to be taken into consideration. Table 1.1 contains a compilation of the verification methods according to DIN 18800 and the verifications as they are generally applied. Eurocode 3 contains similar regulations.

Table 1.1 Verification procedures according to DIN 18800 and common verifications

The use of a verification method implies that the individual cross section parts (webs and flanges) can carry the compression stresses, so that no buckling occurs and a sufficient rotation capacity is provided. Assistance for the checking of the b/t relations can be found in profile tables; see for example [29]. If only longitudinal axial stresses and shear stresses occur, it is

. The verification of the equivalent stress (verification method Elastic-Elastic) is only required for σ/σ_R,d and τ/τ_R,d > 0.5. Perfectly plastic internal forces and moments for rolled sections can be found in profile tables [29], interaction conditions and verifications using the partial internal forces method in [29] and [25].

The subscript “d” for S_d and R_d in Table 1.1 indicates that the stresses must be determined using the design parameters of the loads and that the design resistance has to be applied; see Section 1.7. Section 1.4 “Linear and Nonlinear Calculations” includes specifications concerning the calculation of stress and resistance.

1.2 Methods to Determine the Internal Forces and Moments

As it is generally known, internal forces and moments in statically determinate systems may be calculated with the help of equilibrium conditions and intersection methods. This is not possible with statically indeterminate systems and thus another solution procedure is used, such as the force method, which is the classical method of structural analysis. It is appropriate for hand calculation and very straightforward since it is easy to understand in engineering terms. However, the disadvantage is that for differing structural systems many approaches must be developed and, moreover, it is completely inappropriate for many tasks.

Figure 1.1 Unknown values of the force, displacement and reduction method for a selected example

Figure 1.1 exemplifies a singlely indeterminate girder. Hence, when using the force method, one unknown force value must be defined. After this, the moment distribution can be determined using the equilibrium conditions. The basis of the method is always the choice of a statically determinate structure (primary structure). Since there are several possibilities for doing so, the two systems in Figure 1.1 are selected examples.

Generally, there are three methods for determining the internal forces and moments:

• Force method
• Displacement method → FEM
• Reduction method → FEM

Moreover, there are numerous variations within these three methods, which cannot be discussed in detail here. Whereas when using the force method, the forces are the unknown variables of the emerging equation system, when using the displacement method, the unknown variables are the displacements, i.e. the displacements and rotations. If the structural system is divided into finite elements (e.g. beam elements or segments), the displacement method is extremely appropriate for a generalised formulation and so is applicable in many different situations. The ideas involved are not simple in engineer terms but are very mathematical because large amounts of data must be handled with sizable equation systems solved. The actual amount of data and the size of the equation system will, of course, depend on the system under consideration, but it will certainly be more than would be needed for the force method.

Figure 1.1 shows the application of the displacement method. Using this method, the unknown values are the deformations at the nodes, i.e. for the examined beam the displacement w and the rotation φ. Thus, there are two unknowns per node, so depending on the geometric boundary conditions, there will be between one and 19 unknowns in each example. Using the FE model with 10 elements, a relatively large number of unknowns (19) occur, but there is no need of further hand calculation, which is an advantage. For procedural reasons, all state variables (bending moments, shear forces, displacements, rotations) at the nodes, i.e. virtually in the entire system, are determined.

Due to the numeric complexity, the widespread use of the FEM with the displacement method is closely connected to the rapid development of high-capacity computers. Until about 1985, it was important to model structures using finite elements in such a way that the limited memory capacity was sufficient and that computing times did not escalate. Nowadays, these considerations are only important for exceptional structures and calculations. Then again, it is often seen that in static calculations exaggeratedly fine FE-modelling or the use of inappropriate finite elements create reams of paper. As shown in Figure 1.1, it can be very reasonable to calculate a single-span beam using an FEM program, since all values for the necessary verifications are directly obtained by the program and the corresponding pages for the static calculation can be printed out with minimal effort.

The third method mentioned above is the reduction method, which is suitable for continuous beams including instance sharp bends. The unknowns of the resulting equation system are the unknown internal forces and displacements at the beginning of the beam structure (see Figure 1.1), so that for beams, a maximum of seven unknowns results. Accordingly, the require...

Table of contents

• Cover
• Title Page
• Copyright
• Preface
• 1: Introduction
• 2: Cross Section Properties
• 3: Principles of FEM
• 4: FEM for Linear Calculations of Beam Structures
• 5: FEM for Nonlinear Calculations of Beam Structures
• 6: Solution of Equation Systems and Eigenvalue Problems
• 7: Stresses According to the Theory of Elasticity
• 8: Plastic Cross Section Bearing Capacity
• 9: Verifications for Stability and according to Second Order Theory
• 10: FEM for Plate Buckling
• 11: FEM for Cross Sections
• References
• Index