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O R I G I NA L

Xingzhuang Zhao

Analytical solution of deflection of multi-cracked beams on elastic foundations under arbitrary boundary conditions using a diffused stiffness reduction crack model

Received: 29 February 2020 / Accepted: 28 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract Crack can significantly affect the performance of structures and is one of the crucial indicators of damage in structural health monitoring. In this paper, the deflection behaviors of Euler–Bernoulli beams with arbitrary open edge cracks under arbitrary elastic boundary conditions are investigated. A continuous diffused stiffness reduction crack model is implemented to simulate the cracks in beams, which can incorporate multiple cracks and consider the stiffness reduction effect in the vicinity of a crack. With the proposed diffused stiffness reduction model, the fourth-order differential equation governing the deflection behavior of the multicracked Euler–Bernoulli beam is constructed. The powerful variational iteration method is applied to obtain the analytical solution of the multi-cracked beams on elastic foundations. Five shape functions are introduced, based on which the deflection of the multi-cracked beam is proposed. Both the solutions corresponding to the general elastic boundary conditions and the conventional boundary conditions are presented explicitly. The deflection solution is benchmarked and verified against the literature, and encouraging agreements are obtained. Parametric studies are carried out to investigate the influences of crack position, crack ratio, stiffness of the elastic foundation, and boundary conditions on the deflection of the cracked beams. The proposed crack model and the deflection solution overcome some of the limitations in the literature. Keywords Laplace transform · Shape function · Elastic boundary condition · Structural health monitoring · Variational iteration method

List of symbols dci E0 E(x)I (x) E 0 I0 h kf k0r , k Lr k0t , k Lt K 0r , K 1r K 0t , K 1t L q Q

Depth of the ith crack Young’s modulus Variable flexural stiffness Constant flexural stiffness Height of beam Stiffness of the foundation Rotational spring stiffness Translational spring stiffness Non-dimensional rotational spring stiffness Non-dimensional translational spring stiffness Length of beam Uniform load Shear force

X. Zhao (B) Department of Civil and Environmental Engineering, University of Maryland, College Park, MD 20740, USA E-mail: [email protected]

X. Zhao

rc (x), rc (ξ ) rci Si (ξ )(i = 0, 1, 2, 3 and 4) x xci y(x) β γ0 εci ηci λ(τ ; ξ, β) ξ ξci σci φ(ξ ) Ψi (ξ )(i = 0, 1, 2, 3 and 4)

Stiffness reduction function The crack ratio of the ith crack Shape functions of intact beam Coordinate along the beam Location of the ith crack Deflection Non-dimensional stiffness of the elastic foundation Non-dimensional uniform load Non-dimensional nominal width of the ith crack Stiffness reduction factor Generalized Lagrange’s multiplier Non-dimensional coordinate along the beam Non-dimensi