Beam Structures: Classical and Advanced Theories

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Beam theories are exploited worldwide to analyze civil, mechanical,automotive, and aerospace structures. Many beam approaches havebeen proposed during the last centuries by eminent scientists suchas Euler, Bernoulli, Navier, Timoshenko, Vlasov, etc. Most ofthese models are problem dependent: Would you like to tell us about a lower price?

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It is important for everyone to know these tricks as technology continues to improve. From the Back Cover Beam theories are exploited worldwide to analyze civil, mechanical,automotive, and aerospace structures. Wiley; 1 edition September 26, Language: Be the first to review this item Amazon Best Sellers Rank: Related Video Shorts 0 Upload your video. International Journal of Solids and Structures, 41 7 , Librescu L and Song O On the static aeroelastic tailoring of composite aircraft swept wings modelled as thin-walled beam structures. Composites Engineering, 2, Journal of Mechanics, 22 1 , Journal of Mathematical Analysis and Applications, , Mucichescu DT Bounds for stiffness of prismatic beams.

Journal of Structural Engi- neering, , International Journal of Solids and Structures, 36 , International Journal of Solids and Structures, 37, Qin Z and Librescu L On a shear-deformable theory of anisotropic thin-walled beams: Composite Structures, 56, Schardt R Eine Erweiterung der technischen Biegetheorie zur berechnung prismatis- cher Faltwerke. Der Stahlhau, 35, Schardt R Verallgemeinerte technische Biegetheorie. Schardt R Generalized beam theory-an adequate method for coupled stability prob- lems.

Thin-Walled Structures, 19 , Silvestre N Second-order generalised beam theory for arbitrary orthotropic materials. Thin-Walled Structures, 40 9 , Silvestre N Generalised beam theory to analyse the buckling behaviour of circular cylindrical shells and tubes. Thin-Walled Structures, 45 2 , Silvestre N and Camotim D First-order generalised beam theory for arbitrary or- thotropic materials. Timoshenko SP On the corrections for shear of the differential equation for transverse vibrations of prismatic bars.

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Philosophical Magazine , 41 , Timoshenko SP On the transverse vibrations of bars of uniform cross section. Philo- sophical Magazine, 43, Journal of Applied Mechanics, 67, International Journal of Solid Structures, 36, Wagner W and Gruttmann F A displacement method for the analysis of flexural shear stresses in thin-walled isotropic composite beams. Washizu, K Variational methods in elasticity and plasticity. Yu W and Hodges DH Elasticity solutions versus asymptotic sectional analysis of homogeneous, isotropic, prismatic beams. Journal of Applied Mechanics, 71 , Journal of the American Helicopter Society, 50 1 , International Journal of Solids and Structures, 39, AIAA Journal, 40 , Symbols and reference systems that will be used throughout the book are also introduced.

The following relevant variables are introduced: Classical and Advanced Theories, First Edition. The calculation of such a deformed state remains the fundamental problem of 3D elasticity. On the boundary of the body C, the stress vector components at a generic point must fulfill the follow- ing conditions: These are known as mechanical boundary conditions.

In this book both symbols s and y are used to indicate the engineering strain components. Compliances can be also used: Other variables could be introduced, such as stress functions as well as compatibility conditions on strains. Appropriate variational statements can be introduced to establish the variational form of the 3 D equations, well-known examples being: These are not discussed in this book, details can be found in the book by Washizu The 3D equations can be formulated in different ways according to the choice made for the unknown variables, that is, displacement, stress, or mixed formulations can be used.

This book only refers to displacement unknowns. In the displacement-formulated theory of structures, the displacement variables are expressed in terms of other displacement values or variables. If a 3D problem is considered, the generic displacement variable, s, will be written as where F T x, y, z are base functions exploited to approximate s above V; S T are the problem unknowns, which can be displacement values, their derivatives, or generic higher-order terms.

The meaning of S T depends on the adopted expansion. PVD is used to obtain governing equations according to the S T variables. The role of F t is related to the model to be approximated. In the case of plates, for instance, base functions are introduced along the thickness direction, see Figure 1. In the case of beams, base functions are introduced above the cross-section, Q, see Figure 1. Governing equations can be formulated in strong and weak forms. Both for- mulations will be utilized in this book. Strong form solutions typically have two contributions: Weak form solutions in this book are obtained by means of finite elements FEs.

Only one governing equation is obtained in this case: Boundary conditions are imposed by acting on the stiffness matrix. The following considerations about strong and weak forms should be noted: This means that the only error source is related to the model assumptions. On the other hand, closed form solutions are usually available for a limited number of geometries and boundary conditions. The error in the solution is not due to the model assumptions alone, more factors, such as the number of elements used in the case of FE models, play a role.

The computational cost is also related to the discretiza- tion refinement level: A detailed description of both formulations will be given in the following chapters of this book. This book represents a contribution on the use of generalized expansions such as Equation 1. Hierarchical beam theories or, better, ID models will be introduced for the analysis of any type of structure by referring to generalized displacement variables only.

Reference Washizu, K Variational methods in elasticity and plasticity Pergamon. Beam geometry, displacements, strains, and stresses are referred to an orthonormal Cartesian reference coordinate frame as shown in Figure 2. Unless stated differently, this reference system is adopted. The mechanics of a beam under bending was first understood and described by Leonardo da Vinci as stated in Reti and Ballarini They are the reference models to analyze slender homogeneous structures under bending loads.

Starting from the a priori hypotheses for the kinematics of a beam under bending, the displacement field, strains, stresses, and resulting forces will be derived. The limitations and the differences between the two models will be pointed out by means of some practical examples. I the cross-section is rigid on its plane; Beam Structures: II the cross-section rotates around a neutral surface remaining plane; III the cross-section remains perpendicular to the neutral surface during deformation.

On the basis of the third hypothesis and according to the definition of shear strains, shear deformations y yz and y yx are disregarded: The EBBT presents three unknown variables. I axial force N y: The minus sign in Equation 2. A is the area of the cross-section measure of the geometric entity Q , S x and S z are its static momenta, and I xx , I xl , and are the cross-section momenta of inertia.

It should be noted that no assumptions for the Cartesian reference system have been introduced. The tensor of inertia and, as a consequence. A refer- ence system is called principal for a given cross-section when the product momen- tum computed according to that reference system is equal to zero. With reference to Figure 2. In this ref- erence system, the tensor of inertia is diagonal and Equations 2.

The cross-section is still rigid on its plane, it rotates around a neutral surface remaining plane, but it is no longer constrained to remain perpendicular to it, see Figure 2. Shear deformations e xy and e yz are now accounted for. Only the non-null strain components are reported: The shear predicted by the TBT should be corrected since the model yields a constant value above the cross-section, whereas it is at least parabolic in order to satisfy the stress-free boundary conditions on the unloaded edges of the cross-section.

The shear correction factor is mainly related to the cross-section geometry. In the literature there are many methods to compute k, see, for instance, Timoshenko , Cowper , Pai and Schulz , Gruttmann et al. A discussion on the shear correction factor is beyond the scope of this book. It will be shown that higher-order models yield shear stresses that are compliant with the mechanical boundary conditions.

The stress resultants are obtained by integrating the axial stress on the cross- section: I axial force N: Similar considerations are still valid when the classical laminated theory CLT and the first order shear deformability theory FSDT for plates are considered Carrera and Petrolo, A slender and a moderately thick beam are considered. The following parameters are computed first: Carrera E and Petrolo M Guidelines and recommendations to construct theories for metallic and composite plates.

AIAA Journal, 48 1 2 , International Journal of Solids and Structures, 36, Reti, L The unknown Leonardo. Philosophical Magazine, 41, — The adoption of classical beam theories only allows one to deal with a linear distribution of the out-of-plane displacement component, while a constant in-plane stretching is provided.

The complete linear expansion case CLEC , overcomes this limitation by assuming a full linear expansion of the variables above the cross-section. After a theoretical description of CLEC, practical examples will be given to underline the importance of the linear stretching terms. Once the displacement variables are available, the strain components can be obtained straightforwardly: This limitation cannot be neglected in many cases, especially when thin-walled structures are considered.

However, the analysis of compact beams can also be inadequate if conducted with classical models. CLEC can provide a linear distribution of in-plane stretching that implies the deformed configuration shown in Figure 3.

In the case of a free-vibration analysis, similar considerations are still valid. An isotropic material is used with E — 70 GPa and v — 0. Two load cases are considered: In both cases a Figure 3. Analyses are conducted by means ofEBBT. Results are evaluated in terms of vertical displacements of two points: The deformed free-tip cross- sections are shown in Figures 3. The role of CLEC is clear in detecting the torsion of the beam whereas classical models consider the bending behavior only.

Full text of "Beam Structures Classical And Advanced Theories"

Classical models detect exactly the same solution of the first load case since the absence of linear in-plane terms in the displacement field does not permit us to include the torsion in the EBBT and TBT models. For the sake of clarity, the explicit expression for two components is reported hereafter: International Journal of Applied Mechanics , 2 1 , Shock and Vibrations, 18 3 , Novozhilov VV Theory of elasticity.

Schardt R Verallgemeinerte technische biegetheorie. Thin-Walled Structures, 19, Washizu K Variational methods in elasticity and plasticity. The aim of the present chapter is to introduce the models mentioned above in a unified manner via a condensed notation that represents a basic step towards the Carrera Unified Formulation CUF.

Finally, a discussion on Poisson locking and its correction will be given. The introduction of the CLEC finite element allowed us to introduce the stiffness matrix as in Equation 3. It is important to point out that the matrix can be considered as being composed of nine sub-matrices of dimension 3 x 3, as in Beam Structures: It is extremely important to point out that the formal expression of each component of the sub-matrices does not depend on the expansion functions. That is, corre- sponding components of different sub-matrices have the same formal expression, as shown in the following: This implies that the sub-matrix can be considered as a fundamental invariant nucleus which can be used to build the global stiffness matrix in an automatic way.

Let us introduce the following notation for the expansion functions, F x: Coherently with the introduced notation, the matrix in Equation 4. The exploitation of the four indexes in the formal expression of the fundamental nucleus makes it possible to compute the stiffness matrix by means of four nested FOR- cycles, that is, one for each index. Starting from the CLEC case, two possible techniques can be used to obtain TBT; 1 the rearranging of rows and columns of the stiffness matrix; 2 penalization of the stiffness terms related to u X2 , u X3 , u Z2 , and u Z3 the latter is preferred here in the numerical applications.

This condition can be imposed by using a penalty value x in the following constitutive equations: Thin and moderately thick structures are therefore considered. The cross-section edge dimension is 0. The Poisson ratio, v, is equal to 0. A concentrated load, P z , is applied at the mid-span center point, and is equal to 50 N. A benchmark solution is obtained by means of the EBBT: The analysis is conducted by using AOfour-node B4 , elements.

The results in Table 4. It is clear that a contradiction exists between kinematics and constitutive laws: In the following chap- ters, we will present a tool that can be used to overcome this important limitation. The second remedy illustrated here is based on a proper modification of such laws. Classical plate theories correct PL by imposing that the out-of-plane normal stress is zero.

PL correction is obtained for the beam theory in the same manner: The cross-section edge, h, is 0.

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The results are obtained via a FE model having a A0 four-node element B4 , mesh. The second column indicates the results obtained with the PL correction activated, while the third column reports the results with the PL correction deactivated. It can be seen how the correction of PL enhances the flexibility of the structure and the convergence rate.

Similar results are obtained also in the case offree-vibration analyses Carrera et al. It is clear that: We will see in the next chapters how more refined models than the bilinear model are able to detect the exact solution with no need for PL correction. References Carrera E and Brischetto S a Analysis of thickness locking in classical, refined and mixed multilayered plate theories. Composite Structures, 82 4 , Carrera E and Brischetto S b Analysis of thickness locking in classical, refined and mixed theories for layered shells.

Composite Structures, 85 1 , Journal of Applied Mechanics, 78 2. Advanced beam formulations for free vibration analysis of conventional and joined wings. Journal of Aerospace Engineering, In Press. The bend- ing of slender beams, for instance, is well described by the Euler-Bernoulli model, which has three unknowns, whereas, torsion or thin-walled beam anal- ysis requires more sophisticated theories with a larger number of variables.

This modeling approach is therefore problem dependent since a beam theory is ad hoc built to face a particular structural analysis and, therefore, there is no guarantee it can be extended to other cases. The problem dependency of a beam model limits its application field, and its extension to enhanced models is not straightforward.

This chapter describes the theoretical layout of a novel unified approach that overcomes the limits of classical modeling techniques. Displacement fields are in fact obtained in a unified manner, regardless of the order of the theory, which is considered as an input of the analysis. The overcoming step, from a basic to a higher-order model, is immediate and does not require any ad hoc implementa- tions.

The unified formulation will be presented and then exploited to derive the governing equations in both strong- and weak forms. According to the Einstein notation, the repeated subscript r indicates summation. The choice of F z and M is arbitrary, that is, different base functions of any order can be taken into account to model the kinematic field of a beam above the cross-section. One possible choice is related to the use of Taylor-like polynomials consisting of the 2D base x' zj , where i and j are positive integers. Such a model is referred to as full since all the terms of the expansion are used.

Starting from the unified form of the displacement field in Equation 5. The closed form solution is addressed first, then the finite element formulation is described.

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S stands for a virtual variation. A normal vector with the same orientation as the x- or z-axis identifies a positive lateral surface. Its explicit terms are K? The assembly procedure for the stiffness matrix is based on the use of r and s which are opportunely exploited to implement the FORTRAN statements. The core indexes are those related to the expansion functions F z and F s ; the stiffness matrix is computed by varying r and ,v as shown in Figure 5. We assume that the external loadings vary towards y in the following manner: Term a is ttlJT 5.

The displacement field in Equations 5. For a fixed approximation order, the algebraic system has to be assembled accord- ing to the summation indexes r and s. Its solution yields the maximal displacement amplitudes. The strains are retrieved by the geometric relations, Equations 5.

The beam model order is given by the expansion on the cross-section, and the number of nodes per element is related to the approximation along the longitudinal axis. The fundamental nucleus is a 3 x 3 array which is formally independent of the order of the beam model. The core indexes are those related to the expansion functions F z and F s , and the fundamental nucleus is computed by varying r and s, as shown in Figure 5. Any-order beam theory can be computed since the definition of the order acts on the r.

Global Stiffness Matrix Figure 5. The last equation can be rewritten in the following compact manner: The virtual variation of the external work due to p u p is given by By introducing the F z expansions and the nodal displacements, we obtain where a p stands for the loading application coordinate. A generic line load, l a p y , can be treated similarly. The virtual variation of the external work due to l a p y is given by By introducing the F r expansions and the nodal displacements, we obtain where a p , fi p stand for the loading application coordinates above the cross-section.

References Carrera E and Giunta G. Carrera E and Giunta G. Further reading Carrera E and Nali P Mixed piezoelectric plate elements with direct evaluation of transverse electric displacement. Developments in Computational Structures Technology, Ch. International Journal of Applied Mechanics. While classical models are able to describe problems such as the bending of a compact cross-section beam, refined models are mandatory to describe the mechanical response in case of different loadings e.

A detailed description of various refined beam models is given in this chapter. Generic A' -order models will be addressed finally. The beam model given by Equation 6. The shear components can be similarly derived. The normal strain components have to he computed at node 1 i. The total number of displacement variables of the Figure 6. International Joournal for Numerical Methods in Engineering, 80 4 , Material Science and Engineering, 10 1. Mechanics of Advanced Materials and Structures, 17 8 , This is due to the hierarchical nature of the CUF matrices, which allows us to deal with arbitrary models in an automatic way.

Some of them are typical of every type of FE model, others are specific to the CUF, and the adopted solution represents one of the successful points of the entire formulation. The following main issues will be discussed: A set of numerical problems will be given in order to offer references for com- parison purposes.

Moreover, the given numerical examples will permit us to carry Beam Structures: It should be mentioned that this chapter does not represent a detailed and comprehensive guide to implementing FE codes; its purpose is instead to describe details related to the CUF beam models adopted here to obtain the results shown in this book.

Attention is given here to all the data needed for the CUF FE model analysis, which can be divided into two groups: The former indicate all those data that are commonly used in most FE codes. The latter indicate specific inputs needed for CUF models. As L is fixed, the structure has to be discretized, that is, a mesh has to be created. A mesh is usually defined according to: The type of element parameter indicates the number of nodes per element, that is, the order of the shape functions. In the framework of the CUF, three beam elements are implemented: What the FE code receives as input is generally a set of two files: The first one contains the spanwise coordinates of each node, the second one defines the set of nodes of each beam element, Example 1 shows different mesh generations in CUF FEM models together with some comments concerning some critical issues.

A convergence analysis is usually performed to define the mesh size of the problem. Node ID v-coordinate 1 0. The first column indicates the identification number ID of each node, the second column shows the spanwise locations. The first column shows the ID of the element, the other two columns define which of the nodes of the FE model is the first and second local node of an element. The order of the local nodes is related to the shape functions they are related to and, therefore, cannot be arbitrary.

Node ID y-coordinate 1 0.

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  • A detailed analysis of loading models is not the aim of this book, but can be found in the excellent books by Bathe and Onate The static FE analysis conducted in the examples given in this book uses point loads which, especially for thin-walled structures, represent severe test cases since they provoke a number of local effects that are generally hard to detect with ID formulations.

    Only a few parameters are needed to define a point load: The procedure necessary to transfer the load to the FE model will be described in more detail in the next sections. Boundary conditions are defined at a nodal level; this means that once a node is constrained, all the cross-section points in correspondence to the node will be constrained. Different types of constraints can be applied in the CUF beam model, the most important being clamped point, hinged, and hinged with free horizontal translation, all of which are graphically shown in Figures 7.

    All the displacement components are locked in a clamped point; rotations are only allowed in a hinged point. The constraining technique in the CUF will be discussed in the following sections. In the present book two types of structural analysis are conducted: The definition of the analysis will automatically determine the FE matrices that have to be computed as well as the output files.

    In the case of linear static analysis the stiffness matrix and the loading force will be computed and as a result we obtain the displacement, strain, and stress fields. As far as the free-vibration analysis in concerned, stiffness and mass matrices will be computed for the eigenvalue analysis.

    Natural frequencies and modal shapes will be provided as output data. The order, N, is set by the input and the analysis is then conducted by building all the FE matrices related to the chosen order. As an example, if N is set equal to two, the analysis will be conducted by means of the following beam model: This means that the set of displacement variables that has to be retained can be inserted as input, thus making it possible to deal with beam models such as the following one: The proper choice of the order is, in general, problem dependent.

    A convergence analysis is usually conducted to establish which order of the beam model has to be used, as commonly done to define the mesh size of the problem. The structure is loaded with a vertical force equal to —50 N. A convergence analysis on the loading point vertical displacement regarding both mesh and beam order has to be performed for two values of the beam length L, namely, and 10; that is, in the case of a slender and a moderately thick beam. Rows are related to differently refined meshes while columns are related to increasing order beam models.

    This convergence study highlights some fundamental and typical aspects related to the used of refined models: This is due to the Poisson locking correction that artificially enhances the flexibility of the beam. In the case of free -vibration analyses, convergence studies can be performed on natural frequencies. Such studies lead to qualitatively similar results Carrera et al. As far as the related input data are concerned, the numer- ical computation of the surface integral needs the cross-section to be discretized into a number of triangular elements.

    Such a discretization can be performed using a common FE preprocessor. The data needed in the present FE code are the coor- dinates of each node above the cross-section and the connectivity of the triangular elements. An example of a numerical mesh for a three-cell airfoil- shaped beam cross-section is shown in Figure 7.

    Examples of convergence studies of the numerical cross-section mesh will be given in the next sections. Typical input data are then composed of the set of node coordinates and the connectivity of the elements. Such data have to be provided for all the elements of the cross-section numerical mesh.

    As in the previous sections of this chapter, attention is focused on particular issues related to the CUF implementation; the aim is not to give a comprehensive FE programming guide. Critical issues related to the numerical computation of the integrals involved in the matrices are also discussed. They vary from one to the number of nodes per element two for B2, three for B3, and four for B4. It should be pointed out that the number of components of the nucleus is related to the fact that the unknowns of the problems are the generalized displacement variables.

    If other primary unknowns are cho- sen, the dimension of the nucleus could in general change Carrera and Nali, The outer cycles show the beam mesh charac- teristics, whereas the inner ones are related to the beam order features. The core of the procedure is represented by the computation of the nucleus components, which involves line and surface integrals These integrals are computed numerically; the adopted computing techniques are described in the following sections.

    This method permits us to integrate polynomial functions precisely Bathe ; Onate, and is based on the following approximation: A quadrature of order k integrates a polynomial of order 2k — 1 precisely. Line integrals are computed over each element, that is, each integral domain is defined by the element boundaries. In order to be able to use the formula in Equation 7. Another useful formula is related to the derivatives of the functions expressed in the global and natural domains: Locking is due to an overestimation of the shear stiffness of the structures, which tends to be infinite as the thickness tends to zero.

    Many techniques are able to attenuate this effect. This formulation is extremelysuitable for computer implementations and can deal with mosttypical engineering challenges. It will also be of great interest to graduate andpostgraduate students of mechanical, civil and aerospaceengineering. About the Authors ix Preface xi Introduction xiii References xvii 1 Fundamental equations of continuous deformable bodies 1 1. Hooke's law 4 1.