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2010 | Buch

Kapitel lesen Erstes Kapitel lesen

Plates and FEM

Surprises and Pitfalls

verfasst von: J. Blaauwendraad

Verlag: Springer Netherlands

Buchreihe : Solid Mechanics and Its Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The Finite Element Method, shortly FEM, is a widely used computational tool in structural engineering. For basic design purposes it usually suf ces to apply a linear-elastic analysis. Only for special structures and for forensic investigations the analyst need to apply more advanced features like plasticity and cracking to account for material nonlinearities, or nonlinear relations between strains and displacements for geometrical nonlinearity to account for buckling. Advanced analysis techniques may also be necessary if we have to judge the remaining structural capacity of aging structures. In this book we will abstain from such special cases and focus on everyday jobs. Our goal is the worldwide everyday use of linear-elastic analysis, and dimensioning on basis of these elastic computations. We cover steel and concrete structures, though attention to structural concrete prevails. Structural engineers have access to powerful FEM packages and apply them intensively. Experience makes clear that often they do not understand the software that they are using. This book aims to be a bridge between the software world and structural engineering. Many problems are related to the correct input data and the proper interpretation and handling of output. The book is neither a text on the Finite Element Method, nor a user manual for the software packages. Rather it aims to be a guide to understanding and handling the results gained by such software. We purposely restrict ourselves to structure types which frequently occur in practise.

Inhaltsverzeichnis

Frontmatter

Theory of Plates

Frontmatter

Chapter 1. Plate Membrane Theory

Abstract

The word plate is a collective term for systems in which transfer of forces occurs in two directions; walls, deep beams, floors and bridge slabs are all plates. We distinguish two main categories, plates that are loaded in their plane, and plates loaded perpendicularly to their plane. For both categories we give an approach with differential equations, such that a basic understanding is provided and for certain characteristic cases an exact solution can be determined. We follow the displacement method, working with differential equations. In plates that are loaded in their plane, the plane stress state is called the membrane state. All stress components are parallel to the mid- plane of the plate. In special cases we can simply determine the stresses.

Johan Blaauwendraad

Chapter 2. Applications of the Plate Membrane Theory

Abstract

In this chapter we will give solutions for plates, which are loaded only on their edges. This implies that no distributed forces px and py occur, and the fourth-order bi-harmonic equation (1.23) reduces to the simple form

$$\nabla^2\nabla^2 u_x=0$$

(2.1)

When a general solution has been found for u _x, the solution for u _y can be derived from the relation between u _x and u _y as given in Eq. (1.17). If we choose the first equation, the relation is (P _x = P _y = 0)

$$\left(\frac{\partial^2}{\partial x^2}+\frac{1-\nu}{2}\frac{\partial^2}{\partial y^2}\right)u_x+\left(\frac{1+\nu}{2}\frac{\partial^2}{\partial x\partial y}\right)u_y=0$$

(2.2)

We will demonstrate two types of solution. In the first type, solutions for the displacements u _x and u _y will be tried, which are polynomials in x and y. We will see that interesting problems can be solved through this ’inverse method‘. The second type of solution is found by assuming a periodic distribution (sine or cosine) in one direction. Then in the other direction an ordinary differential equation has to be solved. This approach is suitable for deep beams or walls.

Johan Blaauwendraad

Chapter 3. Thick Plates in Bending and Shear

Abstract

A plate subjected to a load perpendicular to its plane is in a state of bending and transverse shear. If the plate is of concrete, it is called a slab. Plates are generalizations of beams. A beam spans one direction, but a plate is able to carry loads in two directions. Figure 3.1 shows an example of a plate on four supports under a point load Fz. The mid-plane of the plate is in the x-y plane and Fz is acting in z-direction perpendicular to the plate. The plate will undergo deflections, and moments and shear forces can be expected. The aim of of this chapter is to explain how these stress resultants can be determined. If both bending moments and shear forces occur, in general bending deformations and shear deformations have to be accounted for. For beams, it is known that shear deformation can be neglected only if the beam is slender. Similarly we must distinguish between thin plates and thick plates. We will start so that the theory applies for both categories. In Chapter 4 we reduce the complexity and restrict ourselves to the theory for thin plates and its application. An important reason for starting in a general way, including thick plates is, that many computer programs also offer options for thick plates.

Johan Blaauwendraad

Chapter 4. Thin Plates in Bending

Abstract

Classical text books on thin plate bending theory are due to Girkmann [2] and Timoshenko and Gere [9]. Szilard [10] and Reddy [11] are authors of more recent books on the subject.We will not copy their derivation of theory rather we follow a different approach. We make thick plate theory of Chapter 3 our starting point, and derive thin plate theory from it as a limit case. Apart of the different approach, also the aim of this book is different. Plate theory need to be covered only as far as necessary to explain traps and surprises in applications of the Finite Element Method.

Johan Blaauwendraad

Chapter 5. Rectangular Plate Examples

Abstract

We focus on special aspects of the theory of thin plates by discussing a state of constant bending curvature in Section 5.1 and a panel of constant torsion in Section 5.2. In Section 5.3 we show the effectiveness of a square simplysupported plate subject to a distributed load. In Section 5.4 we discuss the special case of a twist-less plate. Finally, we devote Section 5.5 to a viaduct subject to an edge load.

Johan Blaauwendraad

Chapter 6. Circular Membrane Plates

Abstract

In this chapter circular plates in a membrane state require our attention. In Section 6.1 we study plates for axisymmetric load. Section 6.2 is devoted to non-axisymmetric load. Plate bending will be the subject of Chapter 7.

Johan Blaauwendraad

Chapter 7. Circular Thin Plates in Bending

Abstract

In this chapter we elaborate on the theory of thin plates for circular plates resisting an axisymmetric load. We can do that in a compact way by using the derivation of the bi-harmonic equation for rectangular coordinates in Chapter 4 and its transformation to polar coordinates in Chapter 6.

Johan Blaauwendraad

Didactical Discrete Models

Frontmatter

Chapter 8. Discrete Model for Membrane Analysis

Abstract

This chapter and the next chapter (Chapter 9) are intended as an intermediate step between the classical approach with differential equations and the current computational Finite Element analysis. In pre-FE days, differential equations were solved approximately by Finite Difference (FD) analysis. In that method a grid is chosen over the area of the plate, and the differential equation at each grid point (node) is displaced by an algebraic equation. Solving the set of linear equations leads to an approximate solution of the problem, a solution which becomes more accurate as the mesh is chosen finer. The Finite Element Method (FEM) is the successor of the Finite Difference Method (FDM), in a way which makes it much easier to model plates of any shape and to satisfy boundary conditions. The model for membrane states to be discussed in the present chapter serves two educational goals. First, the discussion is a simple preparation to the stiffness method. The concept of stiffness matrix is introduced, boundary effects are easily accounted for, and often the same solution is obtained as in a classic FD-analysis. Second, the structural engineer sees that different transfer mechanisms are present in a plane stress state. Some members carry horizontal axial forces in the horizontal direction, other members vertical axial forces and special members shear forces. In Chapter 9 comparable members will occur for bending in two directions, and a special member for torsion.

Johan Blaauwendraad

Chapter 9. Discrete Model for Plate Bending

Abstract

In Chapter 8 we discussed an approximation for trusses. The elastic deformations were lumped in springs. We may apply a similar model to beams in bending as a first step to a discrete model for plate bending. A rigid element that has rotational springs at its ends replaces a beam element of length a. This model is depicted in Figure 9.1. The rotational spring is considered to be composed of two parallel springs, for the compression and tension zones, respectively. The rotational rigidity at each end is D. It is required that the beam-ends in both the model and the actual beam have the same rotation e for a constant moment M. This requirement is met if D = 2EI/a. When two beam elements are linked together, the two rotational end springs are connected in series. The rigidity of the resulting rotational spring is D = EI/a.

Johan Blaauwendraad

FE-Based Design in Daily Practice

Frontmatter

Chapter 10. FEM Essentials

Abstract

In Chapters 8 and 9 the stiffness method was introduced in the framework of discrete models out of a pre-FEM era. In those models we had to apply different element types to model the complete membrane behaviour: springs for normal forces and panels for shear forces. The same was necessary in modeling plates in bending; there separate rotational springs and torsion panels were used. Compared to this approach the Finite ElementMethod (FEM) has been a major step forward. One and the same membrane element accounts for normal forces in two directions and shear forces, and one and the same plate bending element accounts for bending in two directions and torsion. The present Chapter is an overview of the main features of FEM codes and comments on its practical use. The aim of this book is not an in-depth presentation of FEM theory. For that purpose we refer to standard text books of Zienkiewicz et al. [16] and Hughes [17].

Johan Blaauwendraad

Chapter 11. Handling Membrane FEM Results

Abstract

We consider a box-shaped steel bridge as shown in Figure 11.1. The box has two horizontal walls, at the top and bottom respectively, and two vertical walls, one at each outer side. There are no inner vertical longitudinal webs in the bridge. The structure is a new bridge on an existing pier that had been used for a narrower structure. Therefore, the two bearings at each bridge end do not coincide with the vertical side walls of the box structure, but are in a position more inward. It makes the end diaphragm behave as a beam with a four-point loading as drawn in Figure 11.1. The diaphragm consists of three parts, two squares outside the bearings and one between the bearings which has a length over depth ratio of two. The outer vertical walls are schematized as vertical stiffeners at the ends of the diaphragm. These walls carry forces F to the two ends of the diaphragm. These point loads cause support reactions F in the bearings. In order to introduce these concentrated support reactions in the web of the diaphragm, two vertical stiffeners have been added at the position of the bearings.

Johan Blaauwendraad

Chapter 12. Understanding FEM Plate Bending

Abstract

This is the first chapter on FE-based analysis of plate bending. We include this for in-depth understanding of the nature of the approximation. The Finite Element Method does not violate the kinematic and constitutive relationships, but satisfies equilibrium conditions only in an average sense. We will investigate to what extent equilibrium is maintained. The shape of the plate is purposely chosen so that all kinds of boundary conditions can be included. For the goal of this Chapter we choose a coarse mesh. When discussing real structural problems in subsequent chapters, we will apply finer meshes. The chosen concrete structure is shown in Figure 12.1, with a set of axes x,y in the horizontal plane, shown in the left-hand top corner. The slab has 40 elements and 55 nodes. The nodes and elements are numbered. Element numbers are put in a frame throughout the whole chapter, to distinguish them from node numbers.

Johan Blaauwendraad

Chapter 13. FE Analysis for Different Supports

Abstract

In Chapter 4 we became acquainted with various edge conditions in thin plate theory; in Chapter 5 we applied this knowledge to square plates with three different support conditions. Here we meet the three cases again, now they appear in a FE analysis. In Chapter 5 we considered a two-way sine load and a homogeneous distributed load. That was done in order to be able to solve the differential equation. Here we need not make that difference in load type and will subject the plate to a homogenous load in all cases, as we did earlier for the discrete model in Section 9.4. In Chapter7 we became acquainted with the behaviour of circular plates subjected to both distributed load and a point load. Here we consider the behaviour of a square plate due to a central point load. It will appear that the response near the point load is of the same nature as occurs for the point load on a circular plate. In all analyses we choose Kirchhoff theory.

Johan Blaauwendraad

Chapter 14. Handling Peak Moments

Abstract

This chapter is devoted to the subject of local peaks in moment distributions. These occur on top of columns and at receding walls, and for more than one reason structural engineers do not know how to handle them. It is not clear how seriously such peaks must be taken, and how they can be smoothed. An additional problem is the dependency on the mesh fineness; the finer the mesh, the higher the peak. So, the engineer may think to be punished for being serious. This chapter intends to provide practical hints on choosing mesh fineness and designing reinforcement.

Johan Blaauwendraad

Chapter 15. Sense and Nonsense of Mindlin

Abstract

In Chapters 3 and 4 we presented the theories of Mindlin (more properly Mindlin and Reissner) and Kirchhoff, without explaining which theory must be used in a particular practical case. Commercially provided FEM software usually offers both options and even may have chosen one of them as the default option. The goal of this chapter is to help users make a proper choice.

Johan Blaauwendraad

Chapter 16. Reinforcement Design Using Linear Analysis

Abstract

Design of reinforced concrete structures can be described by the following consecutive steps: 1. Select the initial dimensions of all the structural elements using simple rules of thumb or experience. These dimensions should be able to satisfy the serviceability and ultimate limit states, and should fulfill the requirements for adequate site execution and any other applicable requirement (e.g. acoustic isolation, fire protection, etc.). 2. Perform a global structural analysis to calculate the internal forces (or stresses) due to the combination of loads defined in the codes. The method almost used exclusively today is the finite element method and the behaviour of the structure is assumed to be linear elastic at this stage.

Johan Blaauwendraad

Chapter 17. Special Slab Systems

Abstract

A wide-slab floor consists of a prefabricated and a cast-in-situ component. The prefabricated component is a thin floor unit of 2.4 m width. This is the dark shaded part in Figure 17.1. The thickness is between 50 and 70 mm. These wide elements are placed side by side without connection and are supported by scaffolding. They are the formwork for the second component, concrete which will be cast in situ on top of these floor units. This is the light shaded part in Figure 17.1. The total slab thickness is between 150 and 250 mm. On top of the prefabricated floor units there is a truss-shaped reinforcement (hat-shaped cross-section). Jointly the floor unit and reinforcement truss act as an integrated girder to provide sufficient stiffness during transport. Furthermore, the truss reinforcement assures sufficient interlock with the concrete above the floor units. The units are either reinforced by mild steel (passive reinforcement) or pre-tensioned.

Johan Blaauwendraad

Chapter 18. Special Topics and Trends

Abstract

We devote this chapter to four special subjects, two for membrane plates and two for plates in bending. In Section 18.1 we address the stringer-panel model for membrane calculations, and in Section 18.2 we show provisional results of membrane calculations with concrete compression stresses only. An advanced approach for orthotropic plate bending is the subject of Section 18.3. Finally, we discuss plates on soil foundations in Section 18.4.

Johan Blaauwendraad

Chapter 19. Case History of Cable-Stayed Wide-Box Bridge

Abstract

In this book we focus on two-dimensional plate structures. In Chapter 10 we already stated that plate elements also may be assembled spatially. Here an example of such structure will be discussed. We consider a critical erection phase of a cable-stayed steel bridge, and will model the structure with only membrane plate elements. The case study is included in the book with the consent of Rijkswaterstaat, the national governmental agency in the Netherlands responsible for infrastructural works [35].

Johan Blaauwendraad

Shape Orthotropy

Frontmatter

Chapter 20. Shape-Orthotropic Membrane Rigidities

Abstract

Many plate structures in buildings and bridges cannot be handled as isotropic plates. Stiffeners may occur, and can be different in two orthogonal directions. The multi-cell plate in Figure 20.1 is an example. The figure shows that FE models are possible on different detailed levels. Model 1, consisting of a spatial assemblage of isotropic volume elements, explains the behaviour best. However, the input is complicated and the output massive and complex. The calculation results appear as stresses at nodes and are not easy to interpret or to translate into dimensioning of pre-stressing and reinforcement.

Johan Blaauwendraad

Chapter 21. Orthotropic Plates in Bending and Shear

Abstract

The constitutive laws for bending and transverse shear in isotropic homogeneous plates were discussed in Chapter 3. We refer to this chapter for definitions of moments and curvatures. There bending moments mxx and myy occur, and equal twisting moments mxy and myx, all defined per unit length (therefore have unit of force). The constitutive relationship between moments and curvatures is

$$\left\{\begin{array}{l}m_{xx}\\ m_{yy}\\ m_{xy} \end{array}\right\} = \frac{Et^3}{12(1-\nu^2)} \left[\begin{array}{llc} 1 & \nu & 0\\ \nu & 1 & 0\\ 0 & 0 & \frac{1}{2}(1-\nu) \end{array}\right] \left\{\begin{array}{l}\kappa_{xx}\\ \kappa_{yy}\\ \rho_{xy} \end{array}\right\}$$

(21.1)

Johan Blaauwendraad

Backmatter

Titel: Plates and FEM
verfasst von: J. Blaauwendraad
Verlag: Springer Netherlands
Electronic ISBN: 978-90-481-3596-7
Print ISBN: 978-90-481-3595-0
DOI: https://doi.org/10.1007/978-90-481-3596-7

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