OJCE  Vol.7 No.1 , March 2017
Serviceability Analysis of Non-Prismatic Timber Beams: Derivation and Validation of New and Effective Straightforward Formulas
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 non-prismatic beam model. Thereafter, the paper compares the proposed serviceability analysis formulas with other ones available in literature and with highly-refined 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%).
Cite this paper: Balduzzi, G. , Hochreiner, G. , Füssl, J. and Auricchio, F. (2017) Serviceability Analysis of Non-Prismatic Timber Beams: Derivation and Validation of New and Effective Straightforward Formulas. Open Journal of Civil Engineering, 7, 32-62. doi: 10.4236/ojce.2017.71003.

[1]   Piazza, M., Tomasi, R. and Modena, R. (2005) Strutture in legno—Materiale, calcolo e progetto secondo le nuove normative europee. Hoepli.

[2]   Bruhns, O.T. (2003) Advanced Mechanics of Solids. Springer, Berlin Heidelberg.

[3]   Timoshenko, S. and Goodier, J.N. (1951) Theory of Elasticity. 2nd Edition, McGraw-Hill, New York, Toronto, London.

[4]   Lekhnitskii, S.G. (1968) Anisotropic Plates. Gordon and Breach, New York, London, Paris, Montreaux, Tokyo, Melbourne.

[5]   Krahula, J.L. (1975) Shear Formula for Beams of Variable Cross Section. AIAA (American Institute of Aeronautics and Astronautics) Journal, 13, 1390-1391.

[6]   Riberholt, H. (1979) Tapered Timber Beams. Proceedings of the CIB-W18, Meeting 11, Vienna, 1-14.

[7]   Ozelton, E.C. and Baird, J.A. (2002) Timber Designers’ Manual. Blackwell Science Ltd., Oxford.

[8]   Flaig, M. and Blass, H.J. (2013) Tapered Beams Made of Cross Laminated Timber. In: Aicher, S., Reinhardt, H.-W. and Garrecht, H., Eds., Materials and Joints in Timber Structures, Springer, Berlin, 667-676.

[9]   EN 1995 (2004) Eurocode 5: Design of Timber Structures.

[10]   CNR DT 206/2006 (2006) Istruzioni per il progetto, l’esecuzione ed il controllo delle strutture in legno.

[11]   Timoshenko, S.P. and Young, D.H. (1965) Theory of Structures. McGraw-Hill, New York, Toronto, London.

[12]   Shooshtari, A. and Khajavi, R. (2010) An Efficient Procedure to Find Shape Functions and Stiffiness Matrices of Nonprismatic Euler-Bernoulli and Timoshenko Beam Elements. European Journal of Mechanics, A/Solids, 29, 826-836.

[13]   Porteous, J. and Kermani, A. (2013) Structural Timber Design to Eurocode 5. Wiley, Oxford.

[14]   Boley, B.A. (1963) On the Accuracy of the Bernoulli-Euler Theory for Beams of Variable Section. Journal of Applied Mechanics, 30, 374-378.

[15]   Vu-Quoc, L. and Léger, P. (1992) Efficient Evaluation of the Exibility of Tapered i-Beams Accounting for Shear Deformations. International Journal for Numerical Methods in Engineering, 33, 553-566.

[16]   Hodges, D.H., Rajagopal, A., Ho, J.C. and Yu, W.B. (2010) Stress and Strain Recovery for the In-Plane Deformation of an Isotropic Tapered Strip-Beam. Journal of Mechanics of Materials and Structures, 5, 963-975.

[17]   Rao, S.S. and Gupta, R.S. (2001) Finite Element Vibration Analysis of Rotating Timoshenko Beams. Journal of Sound and Vibration, 242, 103-124.

[18]   Li, G.-Q. and Li, J.-J. (2002) A Tapered Timoshenko-Euler Beam Element for Analysis of Steel Portal Frames. Journal of Constructional Steel Research, 58, 1531-1544.

[19]   El-Mezaini, N., Balkaya, C. and Citipitioglu, E. (1991) Analysis of Frames with Nonprismatic Members. Journal of Structural Engineering, 117, 1573-1592.

[20]   Paglietti, A. and Carta, G. (2009) Remarks on the Current Theory of Shear Strength of Variable Depth Beams. The Open Civil Engineering Journal, 3, 28-33.

[21]   Rubin, H. (1999) Analytische Berechnung von Staben mit linear veranderlicher Hohe unter Berücksichtigung von M-, Q- und N-Verformungen. Stahlbau, 68, 112-119.

[22]   Hodges, D.H., Ho, J.C. and Yu, W.B. (2008) The Effect of Taper on Section Constants for In-Plane Deformation of an Isotropic Strip. Journal of Mechanics of Materials and Structures, 3, 425-440.

[23]   Auricchio, F., Balduzzi, G. and Lovadina, C. (2015) The Dimensional Reduction Approach for 2D Non-Prismatic Beam Modelling: A Solution Based on Hellinger-Reissner Principle. International Journal of Solids and Structures, 15, 264-276.

[24]   Beltempo, A., Balduzzi, G., Alfano, G. and Auricchio, F. (2015) Analytical Derivation of a General 2D Non-Prismatic Beam Model Based on the Hellinger-Reissner Principle. Engineering Structures, 101, 88-98.

[25]   Balduzzi, G., Aminbaghai, M., Sacco, E., Füssl, J., Eberhardsteiner, J. and Auricchio, F. (2016) Non-Prismatic Beams: A Simple and Effective Timoshenko-Like Model. International Journal of Solids and Structures, 90, 236-250.

[26]   Balduzzi, G., Hochreiner, G., Füssl, J. and Auricchio, F. (2016) Performance Evaluation of New Straightforward Formula for the Serviceability Analysis of Cambered Timber Beams. In Proceedings of WCTE2016—World Conference on Timber Engineering, Vienna, August 2016, 1-8.

[27]   Bauchau, O. and Craig, J. (2009) Structural Analysis With Applications to Aerospace Structures. In: Solid Mechanics and Its Applications, Vol. 163, Springer, The Netherlands.

[28]   Schneider, K.J. and Albert, A. (2014) Bautabellen für Ingenieure: mit Berechnungshinweisen und Beispielen. Bundesanzeiger Verlag GmbH.

[29]   De Angelis, A. (2006) Strutture in legno lamellare: progettazione e calcolo. DEI tipografia del genio civile.

[30]   BS EN 1194:1999 (1999) Timber Structures. Glued Laminated Timber. Strength Classes and Determination of Characteristic Values.

[31]   Frese, M. and Blaβ, H.J. (2012) Asymmetrically Combined Glulam Simplified Verification of the Bending Strength. Proceedings of the CIB-W18, Meeting 45, Vaxjo, 1-10.

[32]   (2011) ABAQUS User’s and Theory Manuals—Release 6.11. Simulia, Providence, RI, USA.

[33]   Ladeveze, P. and Simmonds, J. (1998) New Concepts for Linear Beam Theory with Arbitrary Geometry and Loading. European Journal of Mechanics, A/Solids, 17, 377-402.