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Abstract
This paper presents geometrically nonlinear behavior of cracked fiber reinforced composite beams by using finite element method with and the first shear beam theory. Total Lagrangian approach is used in the nonlinear kinematic relations. The crack model is considered as the rotational spring which separate into two parts of beams. In the nonlinear solution, the Newton-Raphson is used with incremental displacement. The effects of fibre orientation angles, the volume fraction, the crack depth and locations of the cracks on the geometrically nonlinear deflections of fiber reinforced composite are examined and discussed in numerical results. Also, the difference between geometrically linear and nonlinear solutions for the cracked fiber reinforced composite beams.

Key Words
fiber reinforced composite; geometrically nonlinear analysis; beams; crack; finite element method; total Lagrangian

Address
Department of Civil Engineering, Bursa Technical University, Yıldırım Campus, Yıldırım, Bursa 16330, Turkey.

Abstract
In this study, we investigated the effect of fatigue crack propagation of the beams which have a vital importance in engineering applications, on the natural frequency of the system. Beams which have a wide range of applications, are used as fundamental structural elements in engineering structures. Therefore, early detection of any damages in these structures is of vital importance for the prevention of possible destructive damages. One of the widely used methods of early detection of damages is the vibration analysis of the structure. Hence, it is of vital importance to detect and monitor any changes in the natural frequencies of the structure. From this standpoint, in this study we experimentally investigated the effect of fatigue crack propagation on beams produced from 4140 steel, of the natural frequency of the beam. A crack was opened on the 8

Key Words
vibration analysis; cracked beam; fatigue crack propagation; natural frequency

Address
(1) Habibullah Bilge, Murat Pakdil: Department of Mechanical Engineering, Abant Izzet Baysal University, 14280, Bolu, Turkey; (2) Habibullah Bilge: Institute of Natural Sciences, Sakarya University, 54187, Sakarya, Turkey; (3) Emre Doruk: R&D Department, TOFAS-FIAT, 16369, Bursa, Turkey; (4) Emre Doruk: Department of Mechanical Engineering, Yalova University, 77200, Yalova, Turkey; (5) Fehim F

Abstract
This paper presents a theoretical and finite element (FE) study on the stress intensity factors of double-edged cracked steel beams strengthened with carbon fiber reinforced polymer (CFRP) plates. By simplifying the tension flange of the steel beam using a steel plate in tension, the solutions obtained for the stress intensity factors of the double-edged cracked steel plate strengthened with CFRP plates were used to evaluate those of the steel beam specimens. The correction factor α1 was modified based on the transformed section method, and an additional correction factor φ was introduced into the expressions. Threedimensional FE modeling was conducted to calculate the stress intensity factors. Numerous combinations of the specimen geometry, crack length, CFRP thickness and Young's modulus, adhesive thickness and shear modulus were analyzed. The numerical results were used to investigate the variations in the stress intensity factor and the additional correction factor φ. The proposed expressions are a function of applied stress, crack length, the ratio between the crack length and half the width of the tension flange, the stiffness ratio between the CFRP plate and tension flange, adhesive shear modulus and thickness. Finally, the proposed expressions were verified by comparing the theoretical and numerical results.

Key Words
stress intensity factor; carbon fiber reinforced polymer (CFRP); strengthening; cracked steel beam; finite element models

Address
(1) Hai-Tao Wang, Habeeb M. Zakari: College of Civil and Transportation Engineering, Hohai University, Nanjing, China; (2) Gang Wu, Yu-Yang Pang: Key Laboratory of Concrete and Prestressed Concrete Structures of the Ministry of Education, Southeast University, Nanjing, China.

Abstract
Cracks and defects may occur anywhere in a plate under tension. Cracks can affect the buckling stability performance and even the failure mode of the plate. A search of the literature reveals that the reported research has mostly focused on the study of plates with central and small cracks. Considering the effectiveness of cracks on the buckling behavior of plates, this study intends to investigate the effects of some key parameters, i.e., crack size and location as well as the plate aspect ratio and support conditions, on the buckling behavior, stress intensity factor (SIF), and the failure mode (buckling or fracture) in cracked plates under tension. To this end, a sophisticated mathematical code was developed using MATLAB in the frame-work of extended finite element method (XFEM) in order to analyze the buckling stability and collapse of numerous plate models. The results and findings of this research endeavor show that, in addition to the plate aspect ratio and support conditions, careful consideration of the crack location and size can be quite effective in buckling behavior assessment and failure mode prediction as well as SIF evaluation of the cracked plates subjected to tensile loading.

Key Words
cracked plates; tensile loading; buckling; stress intensity factor; extended finite element method

Address
Parham Memarzadeh, Sayedmohammad Mousavian and Mohammad Hosseini Ghehi: Department of Civil Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran Tadeh Zirakian: Department of Civil Engineering and Construction Management, California State University, Northridge, CA, USA

Abstract
The steel-concrete double composite girder in the negative flexural region combines an additional concrete slab to the steel bottom flange to prevent the local steel buckling, however, the additional concrete slab may lower down the neutral axis of the composite section, which is a sensitive factor to the tensile stress restraint on the concrete deck. This is actually of great importance to the structural rationality and durability, but has not been investigated in detail yet. In this case, a series of 5.5 m-long composite girder specimens were tested by negative bending, among which the bottom slab configuration and the longitudinal reinforcement ratio in the concrete deck were the parameters. Furthermore, an analytical study concerning about the influence of bottom concrete slab thickness on the cracking and sectional bending-carrying capacity were carried out. The test results showed that the additional concrete at the bottom improved the composite sectional bending stiffness and bending-carrying capacity, whereas its effect on the concrete crack distribution was not obvious. According to the analytical study, the additional concrete slab at the bottom with an equivalent thickness to the concrete deck slab may provide the best contributions to the improvements of crack initiation bending moment and the sectional bending-carrying capacity. This can be applied for the design practice.

Key Words
double composite action; negative flexural region; cracking moment; bending-carrying capacity; structural rationality

Address
Chen Xu, Boyu Zhang and Qingtian Su: Department of Bridge Engineering, Tongji University, Shanghai, China Siwei Liu: Shanghai Municipal Engineering Design Institute(Group) Co., Ltd., Shanghai, China

Abstract
This work aims to study effects of the crack and the surface energy on the free longitudinal vibration of axially functionally graded nanorods. The surface energy parameters considered are the surface stress, the surface density, and the surface Lamé constants. The cracked nanorod is modelled by dividing it into two parts connected by a linear spring in which its stiffness is related to the crack severity. The surface and bulk material properties are considered to vary in the length direction according to the power law distribution. Hamilton\'s principle is implemented to derive the governing equation of motion and boundary conditions. Considering the surface stress causes that the derived governing equation of motion becomes non-homogeneous while this was not the case in works that only the surface density and the surface Lamé constants were considered. To extract the frequencies of nanorod, firstly the non-homogeneous governing equation is converted to a homogeneous one using an appropriate change of variable, and then for clamped-clamped and clamped-free boundary conditions the governing equation is solved using the harmonic differential quadrature method. Since the present work considers effects of all the surface energy parameters, it can be claimed that this is a comprehensive work in this regard.

Key Words
functionally graded materials; free axial vibration; cracked nanorod; surface energy; harmonic differential quadrature method

Address
Reza Nazemnezhad: School of Engineering, Damghan University, Damghan, Iran Hassan Shokrollahi: Department of Mechanical Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran

Abstract
Although detailed shell analysis is suitable to predict the ductile crack initiation life of steel members, such detailed method adds time expense and complexity. In order to simply predict the ductile crack initiation life of stiffened steel bridge piers, a total of 33 cases are simulated to carry out the parametric analyses. In the analysis, the effects of the width-to-thickness ratio, slenderness ratio, plate thickness and so on are considered. Both shell analyses and beam analyses about these 33 cases are conducted. The plastic strain and damage index obtained from shell and beam analyses are compared. The modified factor Bs is determined based on the predicted results obtained from both shell and beam analyses in order to simulate the strain concentration at the base corner of the steel bridge piers. Finally, three experimental results are employed to verify the validity of the proposed method in this study.

Key Words
stiffened steel bridge pier; ductile crack initiation; evaluation method; cyclic loading

Address
Wataru Fujie, Miki Taguchi and Hanbin Ge: Department of Civil Engineering, Meijo University, Nagoya 468-8502, Japan Lan Kang: School of Civil Engineering and Transportation, South China University of Technology, Guangzhou, Guangdong Province, 510641, People's Republic of China; State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou, Guangdong Province, 510641, People's Republic of China Bin Xu: College of Civil Engineering, Huaqiao University, Xiamen, Fujian 361021, People' s Republic of China Key Laboratory for Intelligent Infrastructure and Monitoring of Fujian Province (Huaqiao University), Xiamen, Fujian 361021, People's Republic of China

Abstract
This paper proposes a novel topology optimization method generating multiple materials for external linear plane crack structures based on the combination of IsoGeometric Analysis (IGA) and eXtended Finite Element Method (X-FEM). A so-called eXtended IsoGeometric Analysis (X-IGA) is derived for a mechanical description of a strong discontinuity state's continuous boundaries through the inherited special properties of X-FEM. In X-IGA, control points and patches play the same role with nodes and sub-domains in the finite element method. While being similar to X-FEM, enrichment functions are added to finite element approximation without any mesh generation. The geometry of structures based on basic functions of Non-Uniform Rational B-Splines (NURBS) provides accurate and reliable results. Moreover, the basis function to define the geometry becomes a systematic p-refinement to control the field approximation order without altering the geometry or its parameterization. The accuracy of analytical solutions of X-IGA for the crack problem, which is superior to a conventional X-FEM, guarantees the reliability of the optimal multi-material retrofitting against external cracks through using topology optimization. Topology optimization is applied to the minimal compliance design of two-dimensional plane linear cracked structures retrofitted by multiple distinct materials to prevent the propagation of the present crack pattern. The alternating active-phase algorithm with optimality criteria-based algorithms is employed to update design variables of element densities. Numerical results under different lengths, positions, and angles of given cracks verify the proposed method's efficiency and feasibility in using X-IGA compared to a conventional X-FEM.

Key Words
multi-material; topology optimization; crack problem; X-IGA; IGA; X-FEM; Non-Unifrom Rational B-spline

Address
Thanh T. Banh, Jaehong Lee and Dongkyu Lee: Department of Architectural Engineering, Sejong University, Seoul 05006, Korea Joowon Kang: Department of Architecture, Yeungnam University, Gyeongsan 38541, Korea

Abstract
In this study free vibration analysis of a cracked Goland composite wing is investigated. The wing is modelled as a cantilevered beam based on Euler- Bernoulli equations. Also, composite material is modelled based on lamina fiber-reinforced. Edge crack is modelled by additional boundary conditions and local flexibility matrix in crack location, Castigliano's theorem and energy release rate formulation. Governing differential equations are extracted by Hamilton's principle. Using the separation of variables method, general solution in the normalized form for bending and torsion deflection is achieved then expressions for the cross-sectional rotation, the bending moment, the shear force and the torsional moment for the cantilevered beam are obtained. The cracked beam is modelled by separation of beam into two interconnected intact beams. Free vibration analysis of the beam is performed by applying boundary conditions at the fixed end, the free end, continuity conditions in the crack location of the beam and dynamic stiffness matrix determinant. Also, the effects of various parameters such as length and location of crack and fiber angle on natural frequencies and mode shapes are studied. Modal analysis results illustrate that natural frequencies and mode shapes are affected by depth and location of edge crack and coupling parameter.

Key Words
cracked wing; dynamic stiffness matrix; two interconnected Euler–Bernoulli beams

Address
Ali Reza Torabi: Fracture Research Laboratory, Faculty of New Science and Technologies, University of Tehran, Tehran, Iran Shahrokh Shams and Mahdi Fatehi-Narab: Faculty of New Sciences and Technologies, University of Tehran, Tehran, Iran, P.C:1439957131

Abstract
In this paper, a fracture criterion for predicting the failure of the cracked composite specimens under mixed mode I/II loading is provided. Various tests performed on composite components reveal that cracks always grow along the fibers in the isotropic media. Using a new material model called reinforcement isotropic solid (RIS) concept, it is possible to extend the isotropic mixed mode fracture criteria into composite materials. In the proposed criterion, maximum shear stress (MSS) theory which is widely used for failure investigation of un-cracked isotropic materials will be extended to composite materials in combination with RIS concept. In the present study, cracks are oriented along the fibers in the isotropic material. It is assumed that at the onset of fracture, crack growth will be in a path where the shear stress has the highest value according to the MSS criterion. Investigating the results of this criterion and comparing with the available experimental data, it is shown that, both the crack propagation path and the moment of crack growth are well predicted. Available mixed mode I/II fracture data of various wood species are used to evaluate and verify the theoretical results.

Key Words
extended maximum shear stress path; fracture criterion; mixed mode I/II loading; composite materials; reinforcement isotropic solid model; RIS concept; crack growth

Address
Sadra Shahsavar and Mahdi Fakoor: Faculty of New Sciences and Technologies, University of Tehran, Tehran, Iran Filippo Berto: Norwegian University of Science and Technology, Norway

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