Contents
1 Introduction
1.1 Verification Methods
1.2 Methods to Determine the Internal Forces and Moments
1.3 Element Types and Fields of Application
1.4 Linear and Nonlinear Calculations
1.5 Designations and Assumptions
1.6 Fundamental Relationships
1.7 Limit States and Load Combinations
1.8 Introductory Example
1.9 Content and Outline
1.10 Computer Programs
2 Cross Section Properties
2.1 Overview
2.2 Utilisation of Symmetry Properties
2.3 Standardisation Part I: Centre of Gravity, Principal Axes and Moments of Inertia
2.4 Calculation of Standardised Cross Section Properties Part I
2.5 Standardisation Part II: Shear Centre, Warping Ordinate and Warping Constant
2.6 Warping Ordinate
2.7 Shear Centre M
3 Principles of FEM
3.1 General Information
3.2 Basic Concepts and Methodology
3.3 Progress of the Calculations
3.4 Equilibrium
3.5 Basis Functions for the Deformations
4 FEM for Linear Calculations of Beam Structures
4.1 Introduction
4.2 Beam Elements for Linear Calculations
4.3 Nodal Equilibrium in the Global Coordinate System
4.4 Reference Systems and Transformations
4.5 Systems of Equations
4.6 Calculation of the Deformations
4.7 Determination of the Internal Forces and Moments
4.8 Determination of Support Reactions
4.9 Loadings
4.10 Springs and Shear Diaphragms
4.11 Hinges
5 FEM for Nonlinear Calculations of Beam Structures
5.1 General
5.2 Equilibrium at the Deformed System
5.3 Extension of the Virtual Work
5.4 Nodal Equilibrium with Consideration of the Deformations
5.5 Geometric Stiffness Matrix
5.6 Special Case: Bending with Compression or Tension Force
5.7 Initial Deformations and Equivalent Geometric Imperfections
5.8 Second Order Theory Calculations and Verification Internal Forces
5.9 Stability Analysis / Critical Loads
5.10 Eigenmodes / Buckling Shapes
5.11 Plastic Hinge Theory
5.12 Plastic Zone Theory
6 Solution of Equation Systems and Eigenvalue Problems
6.1 Equation Systems
6.2 Eigenvalue Problems
7 Stresses According to the Theory of Elasticity
7.1 Preliminary Remarks
7.2 Axial Stresses due to Biaxial Bending and Axial Force
7.3 Shear Stresses due to Shear Forces
7.4 Stresses due to Torsion
7.5 Interaction of All Internal Forces and Verifications
7.6 Limit Internal Forces and Moments on the Basis of the Theory of Elasticity
8 Plastic Cross Section Bearing Capacity
8.1 Effect of Single Internal Forces
8.2 Limit Load-Bearing Capacity of Cross Sections
8.3 Limit Load-Bearing Capacity of Doubly-Symmetric I-Cross Sections
8.4 Computer-Oriented Methods
9 Verifications for Stability and according to Second Order Theory
9.1 Introduction
9.2 Definition of Stability Cases
9.3 Verification according to Second Order Theory
9.4 Verifications for Flexural Buckling with Reduction Factors
9.5 Calculation of Critical Forces
9.6 Verifications for Lateral Torsional Buckling with Reduction Factors
9.7 Calculation of Critical Moments
9.8 Verifications with Equivalent Imperfections
9.9 Calculation Examples
10 FEM for Plate Buckling
10.1 Plates with Lateral and In-Plane Loading
10.2 Stresses and Internal Forces
10.3 Displacements
10.4 Constitutive Relationships
10.5 Principle of Virtual Work
10.6 Plates in Steel Structures
10.7 Stiffness Matrix for a Plate Element
10.8 Geometric Stiffness Matrix for Plate Buckling
10.9 Plates with Longitudinal and Transverse Stiffeners
10.10 Verifications for Plate Buckling
10.11 Determination of Buckling Values and Eigenmodes with FEM
10.12 Calculation Examples
11 FEM for Cross Sections
11.1 Tasks
11.2 Principle of Virtual Work
11.3 One-Dimensional Elements for Thin-Walled Cross Sections
11.4 Two-Dimensional Elements for Thick-Walled Cross Sections
11.5 Calculation Procedure
11.6 Calculation Exa mples
References
Index
Prof. Dr.-Ing. Rolf Kindmann
Ruhr-Universität Bochum
Lehrstuhl für Stahl-, Holz-und Leichtbau
Universitätsstraße 150
44801 Bochum
Dr.-Ing. Matthias Kraus
Ingenieursozietät SKP
Prinz-Friedrich-Karl-Str. 36
44135 Dortmund
Language Polishing by Paul Beverley, London
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Preface
Steel structures are usually beam or plate structures consisting of thin-walled cross sections. For their design, deformations, internal forces and moments as well as stresses are needed, and the stability of the structures is of great importance. Generally, the finite element method (FEM) is used for structural analysis and as a basis for the verification of sufficient load-bearing capacity.
This book presents the relevant procedures and methods needed for calculations, computations and verifications according to the current state of the art in Germany and the rest of Europe. In doing so the following topics are treated in detail:
The basis of the calculations and verifications are the German standard DIN 18800 and the German version of Eurocode 3. They are widely comparable, however, the final version of Eurocode 3 has just been published and the corresponding national annexes have to be considered.
This book has evolved from the extensive experience of the authors in designing and teaching steel structures. It is used as lecture notes for the lecture “Computer-oriented Design of Steel Structures” on the Masters’ programme “Computational Engineering” at the Ruhr-University Bochum, Germany. Large parts of the contents were taken from German books – see [25], [31] and [42] – and therefore, the references at the end of the book contain many publications in the German language. Further information can be found at www.kindmann.de, www.rub.de/stahlbau and www.skp-ing.de.
The authors would like to thank Mr Florian Gerhard for the translations, Mr Paul Beverley for language polishing and Mr Peter Steinbach for the drawing of figures.
Rolf Kindmann
Matthias Kraus
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/σR,d and τ/τR,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 Sd and Rd 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:
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 requirements for disc space and calculating time are low, which was, as already mentioned above, of vital importance until about 1985. The reduction method was often used to design plate-girder bridges, since even for multi-span girders only two unknowns arise (main beam, transfer of vertical loads). Computer programs using the reduction method are rare these days. However, the procedure can definitely be found in current FEM programs for beams and frameworks, though here it is first calculated with a relatively rough division into finite elements according to the displacement method. Subsequently, the individual beams are generally divided into five to ten elements in order to be analysed more closely using the reduction method. Further details on the reduction method can be found in [31].
1.3 Element Types and Fields of Application
For FEM calculations structures are idealised using structural systems (beams, frameworks, plates, etc.) and are then appropriately divided into finite elements – see Figure 1.3. A distinction is drawn between:
Figure 1.2 Element types and possible nodal degrees of freedom
Figure 1.3 Examples for the discretisation of different problems of steel structures using finite elements
In Figure 1.2, corresponding elements are exemplified. If beams and frameworks are to be analysed, it may in some cases be useful to examine the cross section with the help of the FEM. Depending on the task, the following elements are used:
For the calculation of steel structures almost exclusively beam elements are used (see Figure 1.3a). These are often part of the following structural systems:
The quoted static systems mainly appear in structural, industrial and plant engineering. Due to different stresses, beam elements with up to seven deformation variables in each node (degrees of freedom) are required. The number of required deformations per node is discussed in more detail in the Chapters 3 and 5.
Finite beam elements are also commonly used for the calculation of bridges. Area elements (plates, shells) are rarely used, whether for plate, beam-framework, bow or cable-stayed bridges. An essential reason for this is that the current standards and codes are almost exclusively designed to suit the needs of calculating beam structures. Moreover, apart from a few exceptions, the accuracy of these calculations is quite sufficient.
An interesting field of application for finite area elements in steel structures is plate buckling. For example, Figure 1.3b shows the upper flange of a beam which has been divided into finite elements in order to perform an analysis of plate buckling. This topic is discussed in Chapter 10, where a rectangular plate element for the determination of eigenvalues and modal shapes is derived. Apart from that, area elements are of course used for specific scientific research and development. Since, as has been mentioned, area elements are not often used, and volume elements even less so, for steel structures, the following can be stated:
Finite elements for the analysis of cross sections are covered in Chapter 11. As an example, Figure 1.3c shows the finite element modelling of a rolled I-section using area elements with curvilinear boundaries.
1.4 Linear and Nonlinear Calculations
Theoretically and numerically, linear calculations (first order theory) constitute the starting point. The following assumptions are the basis:
Nonlinear calculations usually require a higher effort than linear calculations. Concerning the nonlinearity, we need to distinguish between physical and geometric nonlinearities. Regarding physical nonlinearity, the assumption of a linear elastic material behaviour is renounced and the plastifications in parts of the construction are considered in order to obtain more economic structures, i.e. structures of less weight. As far as the plastification is only considered regarding the bearing capacity of the cross sections, this approach is to be assigned to the verification method Elastic-Plastic in Table 1.1. Internal forces and moments are determined according to the elastic theory (elastic calculation of the system) and only load cases are permitted where a maximum of one plastic hinge occurs. In comparison to that, the plastic bearing capacity of the cross sections and the system are utilised with the verification method Plastic-Plastic, i.e. the spread of plastic zones or the development of several plastic hinges is permitted.
While the physical nonlinearity is mainly considered for economic reasons, the geometrical nonlinearities for structures susceptible to losing stability are indispensable for safety reasons. In comparison to linear calculations, relatively large deformations lead to higher internal forces and moments. For that reason, verifications against flexural, lateral torsional or plate buckling have to be executed.
In conjunction with geometric nonlinear calculations, it should be mentioned that the verifications according to the valid standards and codes, as for instance DIN 18800 Part 2, rely on a linearisation according to second order theory. This approximation is therefore the basis for the determination of deformations, internal forces and moments as well as critical loads (eigenvalues) in conformity with the codes. As a general rule, the accuracy of calculations according to second order theory is sufficient in terms of applications in engineering practice since deformations for steel structures are usually relatively small. In exceptional cases, it may be necessary to perform precise geometric nonlinear calculations. This is always the case when large or even very large deformations occur.
Summing up, the following can be stated:
1.5 Designations and Assumptions
In this section, descriptions and assumptions are compiled which are needed for beam and frame structures. Some also apply for plates and the FE analysis of cross sections. In the Chapters 10 and 11, other terms and assumptions are added relating to these topics. The basis for the designations is found in DIN 1080 and DIN 18800.
Abbreviations
| ODE | ordinary differential equation |
| COS | coordinate system |
| LCC | load case combination |
| SMI | self moment of inertia |
| PIF-method | partial internal forces method |
| tot | total |
| ult | ultimate |
| cr | critical |
Variables in the global X-Y-Z coordinate system
Beam structures are divided into beam elements, which are connected to each other at the nodes. As shown in Figure 1.2, nodes can also be arranged on the inside of an element (internal nodes). Nodes are defined in the global X-Y-Z coordinate system (COS) by using the coordinates Xk, Yk and Zk as shown in Figure 1.4. Moreover, all global deformations and loads at the nodes relate to this COS. For reasons of clarity, the subscript k has been neglected for these values in Figure 1.4.
The deformations in the global COS are marked by an overbar (horizontal line above the variable). This designation will also be used for vectors and matrices if they apply to the global COS.
Figure 1.4 Definition of deformations and loads in the global X-Y-Z coordinate system
Variables in the local x-y-z coordinate system
Coordinates, ordinates and reference points
| X | longitudinal direction of the local COS |
| y, z | principal axes in the cross section plane (local COS) |
| ω | standardised warping ordinate |
| S | centre of gravity |
| M | shear centre |
Figure 1.5 Beam in the local coordinate system with displacements, internal forces and moments
Beam elements apply to a local x-y-z COS and, as longitudinal beam axis, the x-axis is defined through the centre of gravity S. The axes y and z are the principal axes of the cross section. According to Figure 1.5, some of the displacements and internal forces and moments apply to the centre of gravity S, others to the shear centre M (y = yM, z = zM). For warping torsion a standardised warping ordinate ω is used.
Deformation variables
| u, v, w | displacements in x, y and z-direction (local COS) |
φx =  |
rotation about the x-axis (twist) |
φy  – W' |
rotation about the y-axis |
| φy – V' |
rotation about the z-axis |
| Ψ' |
derivative of the angle of twist |
Figure 1.6 Definition of positive deformations in the local -C"OS
Loads
| qx, qy, qz | distributed loads |
| mx | distributed torsional moment |
| MωL | single load warping moment |
Figure 1.7 Positive directions and application points of local loads
Internal forces and moments
| N | longitudinal, axial force |
| Vy, Vz | shear forces |
| My, Mz | bending moments |
| Mx | torsional moment |
| Mxp, Mxs | primary and secondary torsional moment |
| Mω | warping bimoment |
| Mrr | see Table 5.1 (page 172) |
| Subscript el: | Limit internal forces and moments according to the elastic theory |
| Subscript pl: | Limit internal forces according to the plastic theory |
| Subscript d: | design value |
If the common definition of positive internal forces and moments (internal force definition I) is used, the forces at the negative beam intersection act in directions opposite to the ones specified in Figure 1.8. With the sign definition II, the directions of actions at both beam intersections are in compliance with the ones in Figure 1.8. In Figure 1.9, both definitions are shown for uniaxial bending with axial force.
Figure 1.8 Internal forces and moments at the positive intersection of a beam
According to custom, further subscripts are used to distinguish beam elements and nodes.
Figure 1.9 Internal forces and moments of the beam element “e” for uniaxial bending with axial force and sign definitions I and II
Stresses
| σx, σy, σz | normal stresses |
| τxy, τxz, τyz | shear stresses |
| σv | equivalent stress |
Figure 1.10 Stresses at the positive intersection of a beam
Cross section properties
| A | Area |
| Iy, Iz | principal moments of inertia |
| Iω | warping constant |
| IT | torsion constant (St Venant) |
| Wy, Wz | section modulus |
| Sy, Sz | static moments |
| iM, ry, rz, rω | values for second order theory and stability; see Table 5.1 |
 |
polar radius of gyration (inertia) |
Further symbols and assumptions
Material properties (isotropic material)
| E | modulus of elasticity, Young’s modulus |
| G | shear modulus |
| Q | transverse contraction, Poisson’s ratio |
| fy | yield strength, yield stress |
| fu | ultimate tensile strength |
| εu | ultimate strain |
Partial safety factors
| γM | factor for resistances (material) |
| γF | factor for loads (force) |
Figure 1.11 Assumptions for material behaviour
Matrices and vectors
| S | vector of internal forces and moments |
| K | stiffness matrix |
| G | geometric stiffness matrix |
| v | vector of deformations |
| p | load vector subscript |
| e: | element |
An overbar above the matrices and vectors indicates that they refer to the global coordinate system (X, Y, Z).
As long as nothing else is stated, the following assumptions and conditions apply:
1.6 Fundamental Relationships
Displacements (linear beam theory)
As is common for beams, y and z are the principal axes of the cross section and Z is the standardised warping ordinate – see Chapter 2. The longitudinal displacement uS refers to the centre of gravity S and the displacements vM and wM describe the displacement of the shear centre M. For the longitudinal displacement u of an arbitrary point of the cross section the following formula applies:
(1.1)
The first component is the displacement due to an axial force load. The second and the third components result from the bending moments and describe the displacements as a consequence of cross section rotations φy and φz. Here Formula (1.1) only covers displacements for which the cross section remains plane. The fourth component comprises the longitudinal displacement due to torsional loads depending on the derivative of the angle of twist ψ.
The displacements v und w in the cross section plane result from the displacement of the shear centre M and from additional components deriving from the rotation about the longitudinal axis (twist):
(1.2)
(1.3)
Strains
The strains are linked to the displacements by geometric relationships. According to [25], the following relations are valid for the linear beam theory. For the displacements, Formulas (1.1) to (1.3) are considered and in addition, by neglecting secondary shear deformations, it is v'M φz, w'M =φy and ψ ='.
(1.4a)
(1.4b, c)
(1.4d)
(1.4e)
(1.4f)
Constitutive equations and stresses
The constitutive equations describe the correlation between stresses and strains. Neglecting the transverse strain, with the use of Hooke’s law, a material law describing isotropic, linear elastic material behaviour, and the strains defined in Formulas (1.4), the following stresses can be stated:
(1.5)
(1.6)
(1.7)
Internal forces and moments
Stresses can be summarised to resulting internal forces and moments. It must be pointed out that the axial force and the bending moments act at the centre of gravity, while shear forces, the torsional moments as well as the warping bimoment are related to the shear centre – see Figure 1.8.
Table 1.2 Internal forces and moments as resultants of stresses
Division of linear beam theory (infinite shear stiffness) into four subproblems
Table 1.3 shows four subproblems – biaxial bending with axial force and torsion – associated with the linear theory of beams with infinite shear stiffness. The table contains an allocation of loads, displacements and internal forces/moments as well as information concerning the equilibrium of a beam element and the stress σx.
Table 1.3 Division of the linear beam theory according to [25]
1.7 Limit States and Load Combinations
Limit states
The limit states of structures to be analysed and the corresponding load combinations are defined in “load standards” such as DIN 1055 [7] and EC 1 [9]. For the application additional information is given in the standards (e.g. DIN 18800 [8], EC 3 [10]). In this context, the bearing capacity of a structure characterises the ability of the carrying members to resist all loadings which may occur during the erection work and the service life. The ultimate limit state describes a load situation of the structure where a violation of the limit would lead to a calculative collapse or a comparable failure, for example a rupture or a loss of stability and stable equilibrium, respectively. The demands on the ultimate limit state are related to the safety of people and the safety of the building including its equipment and facilities. In general, the states which may have to be observed cover the loss of the position stability (lifting, overturning, buoying upwards), the failure of the structure or its members including the foundation (rupture, changeover in a kinematic chain, loss of stability) and the failure due to fatigue influences on the material and other time-related effects. With regard to steel structures, the ultimate limit state to be verified depends on the verification method (see Table 1.1):
Other limit states that may be relevant are: flexural buckling, lateral torsional buckling, plate and shell buckling as well as fatigue. In general, it has to be verified, for the entire structure and its members, that the design value of the internal forces and moments or stresses Sd due to the design loading Fd is smaller than the design resistance Rd:
(1.8)
The servicability limit state describes the conditions of a building beyond which it can no longer be used for its designated purpose. The demands on the serviceability are related to the function of the building, the safety of people and the structural appearance. It has to be verified that the design value of stress at the serviceability limit state does not exceed the design value of a serviceability criterion (e.g. tolerable deformations). Limit states for the serviceability are not specified in DIN 18800 and they are usually arranged and agreed on individually if they are not specified in other basic or engineering standards.
Since the ultimate limit state is the basis of a safe design, ensuring that the structure and its parts do not fail, is primary focus of this book.
Design loads and resistances
The safety concepts of the German and European standards are very similar. Both use so-called partial safety factors γF and γM for the determination of the design loads and resistances. These factors increase the “actual” loads to the design level and decrease the resistances accordingly. The factor γF considers a possible unfavourable deviation of the load in terms of the statistical spatiotemporal spread and, in addition, possible insecurities in the mechanical and stochastic model. The factor γM includes the spread of the particular resistance value and also covers inaccuracies in the mechanical model related to the calculation of the resistances.
The design value of a load Fd is determined by:
(1.9)
Here, γF is the partial safety factor which is associated with the particular load and Fk is the characteristic value of the load. If necessary, a combination factor ψ as stated in Eq. (1.9) may be considered.
The design value of the resistance parameters Md is calculated by dividing the characteristic value of the resistance Mk (e.g. strength of the material fy,k and fu,k) with the partial safety factor γM:
(1.10)
Load combinations and resistance at the ultimate limit state
For the verification of the bearing capacity of a steel structure at the ultimate limit state different load combinations have to be examined which are mainly classified as follows:
For the basic combinations two separate cases with corresponding loads F have to be considered. According to DIN 18800, this results in the following combinations:
(1.11a)
(1.11b)
To clarify that the loads in the combination are rather combined and not necessarily directly added to each other, possibly due to acting in different directions or even at different positions of the structure, the symbol “
” is used.
The design value of the permanent loads Gd is determined by:
(1.12)
If the permanent load reduces the stress due to the variable loads, the partial safety for the permanent load has to be set to γF = 1.0. It should be mentioned that additional rules are specified in the standards concerning the reduction of stress due to parts of the permanent loads.
The design value of the variable loads Qi,d of the combinations with one unfavourable variable load at a time is
(1.13a)
and for all variable loads acting unfavourably it is:
(1.13b)
For exceptional combinations, design values of the permanent loads Gd, all variable loads Qi,d and one exceptional load FA,d have to be considered. In contrast to Formulas (1.12) and (1.13b), the partial safety factor is used with γF = 1.0 here. The design value for the exceptional load FA,d is determined with a partial factor of γF = 1.0 as well.
At the ultimate limit state, the partial safety factor for the resistance is usually taken with:
(1.14)
The factor is not only used for the determination of the design material strength but has to be used for the design stiffness as well, which is determined with the nominal values of the cross section properties and the characteristic values of the elasticity modulus or the shear modulus, respectively. If the stability of members is not decisive, the factor JM may be taken as 1.0.
Load combinations and resistance at the serviceability limit state
The safety factors γF, combination factors ψ and load combinations to be considered for the verification have to be arranged individually if they are not specified in different basic or engineering standards. At the serviceability limit state a partial safety factor of γM = 1.0 is usually valid.
1.8 Introductory Example
The following example is aimed to give a first overview of the verification methods according to DIN 18800 given in Table 1.1. In doing so, the main focus is set to the ultimate limit state. Due to the significance of this state as the basis of a safe design, as mentioned previously, it is the main focus of this book. Figure 1.12 illustrates a two-span girder with a uniformly distributed load to be verified. The distributed load is considered to consist of two components, one due to the dead load and one component including the snow loads, as shown in the figure.
Figure 1.12 Structural system of the introductory example
The calculation of the design load values follows with the load combination of Eq. (1.11b) regarding the partial safety factors γF = 1.35 for the permanent load and γF = 1.50 for one variable load according to Eq. (1.12) and (1.13a). This leads to the following design load qd:
With the partial safety factor of JM = 1.1, the design yield strength of steel S 235 is:
Verification method Elastic-Elastic
First of all, the stress in the system is determined by calculating the internal forces and moments. The mid support plays a key role for the verification of the bearing capacity since here the internal forces and moments are at maximum (see Figure 1.13). Using the internal forces and moments, maximum stresses can be calculated, leading to the following verifications:
⇒ verification is not successful!
The (necessary) verification of the equivalent stress
cannot be successful due to max σ > fy,d.
Figure 1.13 Bending moment and shear force according to the elastic theory
Verification method Elastic-Plastic
In order to verify the system in Figure 1.12, the plastic capacities of the cross section bearing capacity can be taken into consideration. Using the Elastic-Plastic procedure, the internal forces and moments are calculated according to the elastic theory – see Figure 1.13. For the verification of a sufficient load-bearing capacity the interaction conditions (e.g. DIN 18800) or the partial internal forces method can now be applied (see Chapter 8).
The use of the interaction conditions according to DIN 18800 requires knowledge of the internal forces and moments at the perfectly plastic state. By using the profile tables [29], Mpl,d = 285.2 kNm and Vpl,d = 419 kN can directly be obtained. This leads to the following verification:
Verification method Plastic-Plastic
As shown with the previous verification, it is not possible to verify the bearing capacity of the system in Figure 1.12 if the plastic reserves of the cross section are regarded at one position of the beam, which is, in this case, at the mid support. However, after the bearing capacity is reached at that position, a plastic hinge will develop and the system will still be able to carry additional loads since it will not be kinematic at that load stage. With the development of the plastic hinge (cross section in a perfectly plastic state) at the mid support due to MB and VB, the interaction condition used with the Elastic-Plastic procedure has to be fulfilled exactly (“= 1” instead of “d 1”). With V/Vpl,d > 0.33, it is:
This formula describes what maximum bending moment the cross section is able to carry at B with regard to the acting shear force.
Figure 1.14 illustrates the structural system regarding the symmetry after the formation of the plastic hinge. For reasons of clarity, the subscript “d” to point out the design loads is neglected here. With regard to the equilibrium of the beam, the following formulas can be stated for the internal forces depending on the position x:
Figure 1.14 Structural system after insertion of a plastic hinge at the mid support
With the equilibrium, the shear force at the support VB can be determined in terms of MB, as shown above. By regarding this relationship in the previous equation for MB, which was gained from the interaction condition, a formula for the calculation of the moment can be stated, which is now independent of VB:
The formation of the plastic hinge at the mid support, i.e. the full plastification, corresponds to MB = 251.3 kNm. This moment is smaller than Mpl,d = 285.2 kN due to the action of the shear force. It now has to be checked whether the arising internal forces and moments within the beam span can be carried by the cross section. The decisive stress in the span is caused by the internal bending moment. It reaches its maximum at the position V(x) = 0. Using the equilibrium formulation for V(x), this leads to:
At that position, the bending moment can now be calculated with the equilibrium equation of M(x):
For the verification within the beam span the internal forces and moments are considered with V = 0 and max MF = 204 kNm, leading to the following condition:
If the condition is fulfilled, there is no development of a plastic hinge within the beam span. Therefore, the system will not form a plastic mechanism (chain) as failure mode and the load-bearing capacity can be verified using the Plastic-Plastic method. However, it should be mentioned that additional verifications are necessary:
1.9 Content and Outline
Figure 1.15 contains an overview of the chapters of this book showing their interrelationship. The aim of the figure is to show which chapters are based on one another. At the same time, it gives information about which basic knowledge is of advantage for the understanding of a given chapter.
Figure 1.15 Chapter structure and dependencies
As shown in Figure 1.15, Chapters 2 and 3 are of foundational character. In Chapter 2 the cross section properties arising in beam theory are discussed. Their knowledge is of fundamental importance for the application of beam theory (Chapters 4, 5, 6 and 9) and for a further treatment of cross sectional issues (Chapters 7, 8, 11). Chapter 3 gives information about the principles of the finite element method (FEM). The basic idea of the method is needed for the understanding of Chapters 4, 5, 10 and 11 dealing with the numerical approach for beams and frameworks, for plates and for cross sections of beams.
Chapters 4, 5, 6 and 9 deal exclusively with the topic “beams, frames and members”. Here, the numerical backgrounds and procedures, the solution methods and the verification of bearing capacity are dealt with in detail. Since beams have a special importance in steel construction, these chapters are a central part of the book. With regard to the formulation of finite beam elements, the cross section properties (Chapters 2 and 11) are of significance and for the verification of members the resistance of the cross sections (Chapter 7 and 8).
Figure 1.15 shows that Chapters 2, 7, 8 and 11 can be described by the umbrella term “cross sections”. While in Chapter 2 the cross section properties arising from beam theory are discussed, Chapters 7 and 8 give information about the bearing capacity of cross sections. It is shown how stresses are determined and how the plastic resistance of cross sections can be analysed. The contents of chapters 2 and 7 are very helpful for the understanding of Chapter 11 where the cross sections are treated using the finite element method (FEM).
Chapter 10 deals with the finite element method for plane load-bearing structures. The main focus is set on deriving a finite element for plate buckling and on the verification procedures against this failure mode. Besides Chapter 3 about the FEM-principles, this chapter is also closely related to Chapter 6 referring to the solution methods for equation systems and eigenvalue problems.
1.10 Computer Programs
Calculation examples in this book and additional examinations have mainly been performed using the following computer programs:
These are programs of the Institute of Steel and Composite Structures at the Ruhr-University Bochum, Germany. Information on FE-Beams (for beams, columns etc.), FE-Frames, FE-Plate Buckling, CSP-Three Plates, CSP-Table (CSP: Cross Section Properties) and a multitude of further programs can be taken from [33]; see www.ruhr-uni-bochum.de/stahlbau as well. Details on the programs CSP-FE and CSP-FE ML are included in [54].
For the purpose of comparison and for further analysis, calculations with the following programs have also been carried out:
| •RSTAB | Ing.-Software Dlubal GmbH, Tiefenbach, Germany |
| •RFEM | Ing.-Software Dlubal GmbH, Tiefenbach, Germany |
| •BT II | Friedrich + Lochner GmbH, Stuttgart, Germany |
| •DRILL | FIDES DV-Partner GmbH, München, Germany |
| •ABAQUS | ABAQUS, Inc., Providence, Rhode Island, USA |
| •ANSYS | ANSYS, Inc., Canonsburg, Pennsylvania, USA |