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A Novel Stiffness/Flexibility-Based Method for Euler-Bernoulli

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A Novel Stiffness/Flexibility-Based Method for Euler-Bernoulli Applied Mathematical Modelling 40 (2016) 7627–7655 Contents lists available at ScienceDirect Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm A novel stiffness/flexibility-based method for Euler–Bernoulli/Timoshenko beams with multiple discontinuities and singularities ∗ Reza Khajavi Earthquake Research Center, Ferdowsi University of Mashhad, VakilAbad Blvd., Mashhad, Iran a r t i c l e i n f o a b s t r a c t Article history: Nonprismatic beams have vast and various applications in mechanical and structural sys- Received 17 July 2015 tems; thus, much research is dedicated to develop well-performed stiffness matrices for Revised 28 February 2016 beams with different forms of section changes and singularities. This paper introduces Accepted 23 March 2016 stiffness matrices by the use of flexibility and stiffness methods. For this purpose, the Available online 1 April 2016 spring model of the beam element is introduced. This model provides an innovative phys- Keywords: ical interpretation for the beam element. It can also be used to develop finite elements Euler–Bernoulli beam for both Euler–Bernoulli and Timoshenko beams with different combinations of tapering, Discontinuity singularity and discontinuity. In this model, beam sections are represented by some ap- Flexibility propriate virtual springs; their connection in series/parallel gives the flexibility/stiffness Stiffness matrix of the beam element. The obtained matrices are in general forms and applicable to Timoshenko beam different beam conditions. ©2016 Elsevier Inc. All rights reserved. 1. Introduction Beam member, as one of the most applicable components in building construction, has attracted much research interest in the structural engineering community. Before using computers in structural analysis, engineers had to use 1D simplified beam models. Today, in spite of the availability of high-performance computing processors, analysis of 2- or 3-D structural building models are still computationally expensive, and thus not available for every-day engineering routines. Decades of engineering experience, indeed, have proved the efficiency and practicality of 1D beam theories for structural modeling. To this date, almost all design codes and standards for steel and concrete buildings are developed according to 1D beam theories. The two Euler–Bernoulli (EB) and Timoshenko (T) beam theories are the most practical ones used for 1D beam modeling. EB formulation assumes that cross sections remain planar after deformation, due to negligible shear strains. This assumption is only valid for long beams. T formulation accounts for shear strains; it is specially used for the analysis of short beams, where shear strains could not be ignored. Researchers have developed finite element solutions for the differential equations of EB and T beam theories. Finite Element Method (FEM) is a powerful numerical tool which has greatly made use of the fast-advancing performance of computers. When FEM is applied to any of beam differential equations, an algebraic system of equations with a well-defined stiffness matrix is developed for the beam element. ∗ Tel.: + 98 51 38804374; fax: + 98 51 38791191. E-mail address: [email protected] , [email protected] http://dx.doi.org/10.1016/j.apm.2016.03.029 S0307-904X(16)30154-8/© 2016 Elsevier Inc. All rights reserved. 7628 R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655 Alongside the primary efforts to find stiffness matrices for the beam elements, the concept of stiffness and flexibility methods emerged [1] . In the stiffness method, interpolating functions are defined for element displacement, out of which other unknown fields (e.g. deformation and internal moment) are derived. In the flexibility method, moment field is as- sumed as independent, and is represented by appropriate interpolating functions. In the stiffness (flexibility) method, the equilibrium (kinematic) equations are defined in their weak forms and the kinematic and constitutive (equilibrium and constitutive) equations are used in their strong forms. The stiffness method is typically used for computer implementation; however, for nonprismatic (and inelastic) beams, the stiffness matrix obtained by the stiffness method is not a suitable option. In fact, displacement interpolating functions cannot follow complex displacement variations along nonprismatic beams. To overcome such problem, some authors ex- amined improved interpolation functions [2,3] . Stiffness matrices were developed for beams with certain tapering patterns by integrating exact governing differential equations [4–9] . These stiffness matrices have exact explicit forms, but they are complicating in their implementation and representation (especially for complicated patterns of tapering and discontinuity). As an alternative, some other authors developed the stiffness matrix by the flexibility method [10–14] . As mentioned before, in the flexibility method, one considers interpolation functions for the independent field of internal forces. In most applied cases, variations in the internal forces are much simpler than displacements; thus, the stiffness matrices obtained by flexibility methods turn out more efficient than those obtained from stiffness methods. However, no absolute judgment could be made about the supremacy of one over the other; as it seems that each is more efficient in estimating a specific unknown field [15] . In the literature, stiffness- and flexibility-based finite element methods are characterized by their independent fields as mentioned before; however, this paper presents a new interpretation of the two methods by introducing the spring model of the beam element. Engineers are well accustomed to the concept of the spring; they often use it as a handy tool to simplify complex structural behaviors and models. It is characterized for its simple configuration, versatility, adaptability, and its ability to represent any combination of springs with a single one. In spring modeling, the connection of the springs can be categorized into parallel or series configurations. Two or more springs are in series if the external force/moment applied to the ensemble is exactly transferred to each spring. In such a configuration, the deformation of the ensemble thus equals the sum of those of the individual springs. If the deformation of the ensemble is the same for all of the constituent springs and the force/moment of the ensemble is the sum of those of the individual springs, they are said to be parallel. In this paper, it is shown that stiffness/flexibility-based finite element methods, used for the EB and T beam theories, could be redefined according to the concept of cross-sectional springs. Some preliminaries are introduced in Section 2 . The stiffness matrices for EB and T beam elements with multiple tapering and abrupt discontinuities are then obtained in Sections 3 and 4 respectively. The proposed method can efficiently deal with singularities (slope discontinuities); this will be verified in Section 5 by several numerical experiments. In short, the proposed method is characterized by its simplicity, generality (in dealing with different types of tapering, discontinuity and singularity), and new physical understanding of the beam theory. 2. Formulation In the following, a linear elastic finite element in the domain ⊂ R d with d ∈ {1, 2, 3} is considered under static loading. The unknown fields of the element are assumed to be displacement, u˜ ∈ C k (R d ) , k ∈ { 0 , 1 } , strain, ε˜ ∈ C k (R d ) , k ∈ {−1 , 0 , 1 } , and stress, σ˜ ∈ C 1 (R d ) , k ∈ {−1 , 0 , 1 } ( C k (R d ) is the real function space with k -order continuity). In the stiffness method, the independent field of approximate displacement u is described as a linear combination of k d some generalized functions Nˆ q . They constitute a basis for the degree- n polynomial space P n () ⊂ C (R ) : u = Nˆ q qˆ . (2.1) Here, qˆ is the vector of generalized coordinates. The u -dependent strain ε u is derived from displacement according to the kinematic law: u ε = L Nˆ q qˆ = Bˆ q qˆ , (2.2) where L is the kinematic differential operator and Bˆ q is the strain-generalized coordinates matrix. Since rigid body motions of geometrically-linear finite elements do not produce any deformation [16] , Bˆ q can be partitioned as: Bˆ q = [ B qr | B q ] = [ 0 nr | B q ] , (2.3a) u qr ε = [ 0 | B ] , (2.3b) nr q q with nr representing the number of rigid body motions. q r involves generalized coordinates for the rigid body motions, and q represents energy-conserving strain states. The elastic constitutive law is indicated as: σu = C . ε u, (2.4) R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655 7629 T with C being the constitutive matrix. Pre-multiplying the above equation by B q and after substituting from Eqs. (2.2) and ( 3.3 b), an integration over the domain gives: T σu = T . , Bq d Bq C Bq d q (2.5) or < σ> = , Pq Kqs q (2.6) < σ> where Pq and Kqs are the generalized internal force vector and the generalized stiffness matrix, respectively. The subscript s in Kqs implies that the matrix is obtained by the stiffness method. Note that in Eq. (2.5), Bq is a basis for the space of the approximate strain field. In the flexibility method, the independent field of stress σ is written as a linear combination of generalized functions b q . 1 d These functions constitute a basis for the polynomial space of P n () ⊂ C (R ) : σ = σ , bq q (2.7) σ where q is generalized kinetic coordinates. The constitutive law is: σ − ε = C 1 . σ, (2.8) σ T where ɛ is the σ-dependent strain field. Pre-multiplying the above equation by b q and integrating over the spatial domain gives: σ − T ε = T 1 . σ , bq d bq C bq d q (2.9) or < ε > = σ , Eq Fq q (2.10) = T −1 < ε > = T ε σ where Fq bq C bq d is the flexibility matrix, and Eq bq d is the generalized strain vector.
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