An efficient procedure to find shape functions and stiffness matrices of nonprismatic Euler–Bernoulli and Timoshenko beam elements

Shooshtari, Ahmad; Khajavi, Reza

doi:10.1016/j.euromechsol.2010.04.003

Cited by 46 publications

(18 citation statements)

References 38 publications

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“…Marques et al (2012) developed a consistent buckling design procedure for the taper columns based on the finite element scheme. An efficient procedure to find shape functions and stiffness matrices of non-prismatic elements was obtained by Shooshtari and Khajavi (2010). Based on the finite element method, Meng et al (2011) presented exact stiffness matrix of tapered beam.…”

Section: Introductionmentioning

confidence: 99%

Stability of non-prismatic frames with flexible connections and elastic supports

2015

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An accurate formulation is obtained to determine critical load, and corresponding equivalent effective length factor of a simple frame. The presented methodology is based on the exact solutions of the governing differential equations for buckling of a frame with tapered and/or prismatic columns. Accordingly, the influences of taper ratio, shape factor, flexibility of connections, and elastic supports on the critical load, and corresponding equivalent efficient length factor of the frame will be investigated. The authors' findings can be easily applied to the stability design of general non-prismatic frames. Moreover, comparing the results with the accessible outcomes demonstrate the accuracy, efficiency and capabilities of the proposed formulation.

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Section: Introductionmentioning

confidence: 99%

Stability of non-prismatic frames with flexible connections and elastic supports

2015

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“…Specifically, the proposed approach exploits the Timoshenko kinematics and develops a simple and effective beam model that differs from the Timoshenko-like homogeneous beam model proposed by Balduzzi et al (2016) mainly by a more complex description of the cross-section stress distribution. In particular, within the proposed model the horizontal stress distribution is determined through homogenization techniques, usually adopted also for non-homogeneous prismatic beams (Li and Li 2002;Shooshtari and Khajavi 2010;Frese and Blaß 2012) and successfully applied also to functionally graded materials (Murin et al 2013a, b), whereas the non-trivial shear distribution is recovered through a generalization of the Jourawsky theory (Jourawski 1856;Bruhns 2003). As a consequence, the present paper not only relaxes the hypothesis on beam geometry but provides also an alternative, more rigorous, and more effective strategy for the reconstruction of the cross-section stress distribution.…”

Section: Paper Aims and Outlinementioning

confidence: 99%

“…A diffused approach for non-prismatic beam modeling consists in using prismatic beam Ordinary Differential Equations (ODEs) and assuming that the cross-section area and inertia vary along the beam axis (Portland Cement Associations 1958;Timoshenko and Young 1965;Romano and Zingone 1992;Friedman and Kosmatka 1993;Shooshtari and Khajavi 2010;Trinh and Gan 2015;Maganti and Nalluri 2015), neglecting the effects of boundary equilibrium on stress distributions and the resulting non trivial constitutive relations. The so far introduced approach received criticisms since the sixties of the past century (Boley 1963;Tena-Colunga 1996) and, as a conse-quence, several researchers propose alternative strategies trying to improve the non-prismatic beam modeling (El-Mezaini et al 1991;Vu-Quoc and Léger 1992;Tena-Colunga 1996).…”

Section: Literature Reviewmentioning

confidence: 99%

“…As an example, the technologies for the manufacturing of wooden or composite beams allow to produce bodies made of materials with different mechanical properties (Frese and Blaß 2012). Furthermore, existing elementary model assumptions include that steel and aluminum beams with I or H cross-section behave under the hypothesis of plane stress whereas the variable beam depth is considered by proportional variation of the different mechanical properties within the crosssection (Schreyer 1978;Li and Li 2002;Shooshtari and Khajavi 2010). In both cases, a planar model capable to tackle multilayer non-prismatic beams i.e., the object of this document, represents a necessary tool for the modeling and first design of such bodies as well as the starting point for the development of more refined 3D beam models.…”

Section: Introductionmentioning

confidence: 99%

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Planar Timoshenko-like model for multilayer non-prismatic beams

Balduzzi

Aminbaghai

Auricchio

et al. 2017

Int J Mech Mater Des

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This paper aims at proposing a Timoshenkolike model for planar multilayer (i.e., non-homogeneous) non-prismatic beams. The main peculiarity of multilayer non-prismatic beams is a non-trivial stress distribution within the cross-section that, therefore, needs a more careful treatment. In greater detail, the axial stress distribution is similar to the one of prismatic beams and can be determined through homogenization whereas the shear distribution is completely different from prismatic beams and depends on all the internal forces. The problem of the representation of the shear stress distribution is overcame by an accurate procedure that is devised on the basis of the Jourawsky theory. The paper demonstrates that the proposed representation of cross-section stress distribution and the rigorous procedure adopted for the derivation of constitutive, equilibrium, and compatibility equations lead to Ordinary Differential Equations that couple the axial and the shear bending problems, but allow practitioners to calculate both analytical and numerical solutions for almost arbitrary beam geometries. Specifically, the numerical examples demonstrate that the proposed beam model is able to predict displacements, internal forces, and stresses very accurately and with moderate computational costs. This is also valid for highly heterogeneous beams characterized by thin and extremely stiff layers.

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“…In fact, also nowadays, the displacement analysis of nonprismatic beams are based on Euler-Bernoulli or Timoshenko beam ODEs in which cross-section area and inertia are tackled as parameters varying along the beam axis [7] [11] [12] [13]. Unfortunately, these modeling approaches are not able to tackle the complex stress distribution's effects and lead to unsatisfactory results as noticed since the sixties of past century [14] [15] [16].…”

Section: Introductionmentioning

confidence: 99%

Serviceability Analysis of Non-Prismatic Timber Beams: Derivation and Validation of New and Effective Straightforward Formulas

Balduzzi¹,

Hochreiner²,

Füssl³

et al. 2017

OJCE

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This paper provides innovative and effective instruments for the simplified analysis of serviceability limit states for pitched, kinked, and tapered GLT beams. Specifically, formulas for the evaluation of maximal horizontal and vertical displacements are derived from a recently-proposed Timoshenko-like nonprismatic beam model. Thereafter, the paper compares the proposed serviceability analysis formulas with other ones available in literature and with highlyrefined 2D FE simulations in order to demonstrate the effectiveness of the proposed instruments. The proposed formulas lead to estimations that lie mainly on the conservative side and the errors are smaller than 10% (exceptionally up to 15%) in almost all of the cases of interest for practitioners. Conversely, the accuracy of the proposed formulas decreases for thick and highly-tapered beams since the beam model behind the proposed formulas cannot tackle local effects (like stress concentrations occurring at bearing and beam apex) that significantly influence the beam behavior for such geometries. Finally, the proposed formulas are more accurate than the ones available in literature since the latter ones often provide non-conservative estimations and errors greater than 20% (up to 120%).

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An efficient procedure to find shape functions and stiffness matrices of nonprismatic Euler–Bernoulli and Timoshenko beam elements

Cited by 46 publications

References 38 publications

Stability of non-prismatic frames with flexible connections and elastic supports

Stability of non-prismatic frames with flexible connections and elastic supports

Planar Timoshenko-like model for multilayer non-prismatic beams

Serviceability Analysis of Non-Prismatic Timber Beams: Derivation and Validation of New and Effective Straightforward Formulas

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