# Modeling of the Bending Stiffness of a Bimaterial Beam by the Approximation of One-Dimensional of Laminated Theory

Save this PDF as:

Size: px
Start display at page:

Download "Modeling of the Bending Stiffness of a Bimaterial Beam by the Approximation of One-Dimensional of Laminated Theory"

## Transcription

1 . Flores-Domínguez Int. Journal of Engineering Research and Applications RESEARCH ARTICLE OPEN ACCESS odeling of the Bending Stiffness of a Bimaterial Beam by the Approimation of One-Dimensional of Laminated Theory. Flores-Domínguez Institut de écanique et d Ingénierie de Bordeau, Arts et étiers ParisTech, 3345 Talence, France ABSTRACT In this paper, a mathematical model representing the bending stiffness of a bimaterial beam is proposed. The classical laminated plate theory specialized to 1D is used for modeling the stiffness bending of a beam. Different material configurations (metal-polymer, metal-composite material and metal-metal) with three different ratios of layer thickness were evaluated by the analytical model were proposed. Virtual eperiments by finite element analysis were carried out to verify the accuracy of the proposed approach. Finite element models of each arrangement were built and the recommendations for the AST three-point bending test were followed in the numerical simulation. The average difference of the stiffness results calculated by the analytical model and by finite element simulation was less than.11%. Keywords - Bending stiffness, finite element analysis, laminated theory, multi-material. I. INTRODUCTION During the last decades, engineers have tried to optimize their product design by integrating increasing numbers of functions in the properties of a material. This tendency has enabled comple assemblies to be replaced by simpler structures, involving the design of new materials when none of the classical monolithic materials embodies all the required functions. Thus, in the domain of most applications, various types of multi-materials have been proposed. A multi-material or an assembly system of materials can be defined as a combination of two or more materials in a predetermined geometry and scale [1, ]. To define a multi-material answering a set of requirements, the designer is confronted with an infinity of potential solutions among which he has to make as objective as possible choices. The composite laminate materials are an alternative design solution in terms of specific strength and stiffness and they offer significant freedom to the designer by allowing, the strength and stiffness optimization of a component or structure for a particular application [3]. Designing a multi-material involves the determination of all the characteristic parameters. The most used method begins by complete description of the set of requirements, the selection of the geometry of the assembly, the load type and the materials selection to the choice of the multimaterials components in order to allow a quantified evaluation of its performance [4-6]. The mechanical properties of the multimaterial are a function of the choice of materials that form it and its geometrical arrangement within the structure. For laminated materials the mathematical models describe their behavior [7]. The fleural behavior of multi-material beam has been studied etensively by many investigators [8-1]. A model based on classical laminated plate theory reduced to one-dimension was developed to obtain the elastic modulus of a bimaterial [11]. The aim of this study is to obtain a simple mathematical model that describes the behavior of a multi-material beam subjected to a three-point bending. The interest is to determine the bending stiffness of the beam formed by two different materials. In this work, a one-dimensional (1D) bending model of a bimaterial structure is developed in a convenient way to obtain the bending stiffness of the bimaterial. The model is validated with a virtual eperimental analysis by finite element analysis where the recommendations of the AST for three-point bending tests were followed. 49 P a g e

2 . Flores-Domínguez Int. Journal of Engineering Research and Applications II. LAINATE ANALYSIS A perfectly bonded bimaterial subject to a moment is analyzed, and is based on classical laminated plate theory (CLPT), specialized 1D. The bimaterial is of length L, width b, and thicknesses h 1 and h for each material, see Fig. 1. The -coordinate is the aial coordinate and the z-coordinate is the through-thickness coordinate, with z = at the midplane ((h 1 + h )/). A concentrated load P is applied at mid span. In this manner, the bimaterial system may be modeled using the first-order laminated theory [1], here specialized to 1D. Within the linear elastic region, the stresses () are proportional to strains (), i i E (1) Here, subscripts i = 1, correspond to each material, and E is the elastic modulus for isotropic materials, or the effective modulus for composite materials. Figure. Deformed geometry of the section of a beam in the theory of laminated. where is the mid-plane strain, and is the midplane curvature given by: d (4) d In general, the force and moment resultants, N and, are defined as [14], N h/ dz (5a) h/ h/ zdz (5b) h/ Figure 1. Schematic representation of a bimaterial system According to the Kirchhoff hypothesis [1, 13] (see Fig. ), the aial displacement u of a point at (, z) may be calculated using the mid-plane aial displacement u and the rotation of the cross section, this is: where u u, z (), and is the rotation of a cross section at, originally plane and perpendicular to the specimen ais. The corresponding variation of strain through the thickness is given by: z (3) Substitution of Equation 1 and 3 into Equation 5 yields, after integration and simplification N A B (6a) B D (6b) where A, B, and D are, respectively, the 1D etensional, coupling and bending stiffness, given by: A E1h 1 Eh (7a) 1 B h1 h E E (7b) D [ E1 h1 3 h1 h E h 3 h1 h ] (7c) 1 It is assumed that both materials have a moderate inter-laminar shear modulus and a large (>1) length-to-thickness ratio. Thus, shear deformation is epected to be minor and can be neglected. 493 P a g e

3 . Flores-Domínguez Int. Journal of Engineering Research and Applications For the case of bending loading eamined herein, the only applied load is the concentrated load P at mid span. The moment associated with the simply supported beam with a concentrated load is given as: thus, Equation 6a with P L, b P L L, L b yield (8) B (9a) A Substituting Equation 9a into Equation 6b yields the mid-plane curvature Since B D A (9b) /, the deflection of w the mid-plane can be found by integrating the Equation 9b, where the conditions boundary w/ (slope of the bending) at L/ and w (deflection) at are evaluated for the first and second integral constants, respectively. The deflection equation is given by the following: w 3 P A L b AD B (1) The maimum deflection is given in L/, thus the maimum deflection displacement is given by: 3 3 PL A PL m 48b AD B 48b (11) where is the deflection measure at the center, P is the concentrated load, L is the spam between the supports and m is termed as the fleural rigidity (EI for a homogeneous beam) of the beam. The slope of the load-deflection curve is termed as the bending stiffness, that is: P 48b K m (1) 3 L Note that if E1 E, the maimum defection 3 3 equation PL /4Ebh of a simply supported beam with rectangular cross section subjected to concentrated load P in mid-span is recovered. III. ANALYSIS OF BENDING STIFFNESS In order to ascertain the accuracy of the obtained analytical epression of bending stiffness for the bi-material beam, a virtual eperimental analysis by finite element analysis was carried out. The tests were performed using the commercial FE software ANSYS echanical APDL v14.5. In this analysis, the recommendations of the AST for three-point bending tests [14] to characterize the stiffness of the beam were followed. The principal scheme of three-point bending test is described in Fig. 1. As previously discussed in section, the bending stiffness can be determined in terms of the Young's modulus of the materials, the layers thickness, the span between supports and the width of the beam. In order to validate the Equation (1), nine different models with dimensions of mm were proposed. The total thickness h of the beam was maintained constant and three different thickness ratios of the layers (h1/h) were used: 3, 1, and 1/3. Four materials were used for the layers. The arrangements of the layers for the system were: metal-polymer, metal-composite material and metalmetal, where aluminum was used as material. The mechanical properties of the materials are presented in Table 1. Table 1. aterial properties used in this study. aterial E [GPa] Property Aluminum (Al) Steel (St).3 Polymer (PC) CFRP Finite Element Analysis The 3D models of the different bimaterial beams proposed were constructed. The models consist of two laminates perfectly bonded at the 494 P a g e

4 . Flores-Domínguez Int. Journal of Engineering Research and Applications interface. The models were discretized with 3-D - node solid element for both materials. These elements have quadratic displacement behavior and they have three degrees of freedom per node, translated into the nodal, y and z directions [15]. For each material, the nonlinear behavior was considered and isotropic hardening rule for multipoint material model was used. To specify the three-point bending test, the nodes at the end left bottom of the beam were constrained in the z translational displacement and the nodes at the end right bottom of the beam were fully constrained. The load was applied in negative z direction of the beam at half the length along the entire width of the beam and it was applied in steps from zero to the von ises stress of any of the two materials was equal to the ultimate stress of its material. Eample of the discretized models is shown in Fig. 3. The model for material configurations CFRP-Aluminum with 1/3 thickness ratio is presented. The mesh was formed by 585 nodes and 14 elements. Detail of the material thickness can be observed. stiffness was calculated using the equation K= P/. Fig. 4 presents the load-deflection curves obtained by simulation of the different proposed models. Figure 4. Load-deflection curves, obtained by simulation of three-point bending test. The slopes are the stiffness of different models. Eample of the contour of displacement and stress obtained of the finite element analysis are shown in Fig. 5 and 6 respectively. In Fig. 5, the transverse displacement uz of the material configuration CFRP-Al beam with h1/h=1 is depicted. The maimum displacement is observed at mid spam. The Von ises stress of the same beam is observed in Fig. 6. The obtained values by the laminated plate theory specialized for 1D and by finite element simulation are listed in Table. The stiffness determined by analytical model and numerical simulation, are compared for each of the proposed models. Figure 3. FE model of three-point bending test. IV. RESULTS AND DISCUSSION The evaluation of the stiffness of the bimaterial beam was conducted. The bending stiffness K of the beam was calculated using the analytical model and was compared with the results of numerical simulation. In the finite element analysis, the transverse deflection uz was evaluated from the successful eecution of the ANSYS software after conducting several convergence tests. From the ultimate load and the maimum displacement in z direction that each model of the beam presents, the bending In both analyses were observed, that for PC- Al arrangement, the stiffness increased with decreasing of the ratio thickness. On the other hand, for CFRP-Al and St-Al arrangements, the stiffness decreased with decreasing of the ratio thickness. That is, the stiffness bending increases with increasing the thickness of the material with higher Young's modulus. Table shows that the analytical results are close to finite element simulation results. The values of bending stiffness obtained by the analytical model are higher than the values obtained by finite element simulation. The differences in the stiffness calculated by analytical model and simulation were less than 3.5% and the average difference was less than.11%. The results obtained by the analytical 495 P a g e

5 . Flores-Domínguez Int. Journal of Engineering Research and Applications model yielded a good agreement with the finite element results obtained. The proposed analytical model equation illustrates the interaction between different variables. In this study, the combination of the different Young's modulus of the materials and the thickness ratio were observed. Accordingly, two response surface graphs were generated. Figure 5. Deflection of the CFRP-Aluminum beam with thickness ratio 1. Al-St Al-St 1/ The behavior of the analytical model obtained by the laminated plate theory specialized to 1D as function of the Young's modulus of the laminated materials, is presented in Fig. 7. In this figure, the response surface reveals that an increase in Young's modulus of any layers cause an increase in the bending stiffness and localized the optimum values of each Young s modulus for maimum response. The representation of the analytical model of the bending stiffness considering the total thickness of the beam and the thickness ratio of the layers can be observed in Fig. 8. This figure shown that the total thickness variable has a greater effect on the response to the increasing the stiffness of beam to a greater etent. This fact can be eplained by the greater amount of beam material, with the same ratio of thickness. a. Stress Figure 6. Stress presented in the CFRP-Aluminum beam with thickness ratio 1. Table. Bending Stiffness calculated by analytical model and finite element simulation Figure 7. Response surface described by the model analytical, which represents bending stiffness as a function of the Young's modulus of the laminated materials. aterial configurations Ratio h1/h K [1 3 N/m] Analytical Simulation % Al-CP Al-CP Al-CP 1/ Al-CFRP Al-CFRP Al-CFRP 1/ Al-St Figure 8. Response surface for the bending stiffness as a function of the total thickness of the beam and the thickness ratio of the layers. 496 P a g e

6 . Flores-Domínguez Int. Journal of Engineering Research and Applications V. CONCLUSIONS A simplified analytical model to characterize the bending stiffness of a bimaterial beam was presented. Analysis based on classical laminated plate theory (CLPT) specialized to 1D was carried out to obtain the mathematical model of the stiffness. odels of different material configurations and different thickness ratio of the layer were proposed to calculate the bending stiffness. These models were analyzed using the mathematical model and by eperiments performed by finite element simulation. The fleural response of the bimaterial beam by numerical simulation was studied for three-point bending configuration. The differences in the stiffness calculated by analytical model and simulation were less than 3.5% and the average difference was less than.11%. It was found that the analytical solution provided good agreement with the eperimental results. This mathematical model can be used with different configurations material and layer thicknesses. The analysis could be readily etended to multilayers. ACKNOWLEDGEENTS Flores wishes to epress their gratitude to the Council of Science and Technology of eico State (COECYT), eico for their financial support. REFERENCES [1] Kromm FX, Quenisset J, Lorriot T, Harry R, Wargnier H. Definition of a multimaterials design method, aterials & Design, 8(1), 7, [] Ashby F, Brechet YJ. Designing hybrid materials, Acta aterialia, 51(19), 3, [3] oon Chang-Kwon. The effect of interfacial microstructure on the interfacial strength of glass fiber/polypropylene resin composites, Journal of applied polymer science, 54(1), 1994, 73-8 [4] Ashby F. aterials selection in conceptual design, aterials science and technology, 5(6), 1989, [5] Bréchet Y, Ashby F, Dupeu, Louchet F. Choi et usage des matériau, Techniques de l ingénieur, 5(1), 1997, 1 3. [6] Bréchet Y, Ashby, Salvo L. Sélection des matériau et des proceeds (Presses Universitaires de Lausanne; ). [7] Reddy, J. N. echanics of laminated composite plates: theory and analysis, Vol. 1 (Boca Raton: CRC press, 1997): [8] Xiao, X., Hsiung, C. K., & Zhao, Z. Analysis and modeling of fleural deformation of laminated steel, International Journal of echanical Sciences, 5(1), 8, [9] Reis, P. N. B., Ferreira, J. A.., Antunes, F. V., & Costa, J. D.. Fleural behaviour of hybrid laminated composites, Composites Part A: Applied Science and anufacturing, 38(6), 7, [1] Reddy, B. S., Reddy, A. R., Kumar, J. S., & Reddy, K. V. K. Bending analysis of laminated composite plates using finite element method, International Journal of Engineering, Science and Technology, 4(), 1, [11] F.Avilés, A.I. Oliva, A. ay-pat. Determination of Elastic odulus in a Bimaterial Through a One-dimensional Laminated odel, Journal of aterials Engineering and Performance, 17(4), 8, [1] J.. Whitney. Structural analysis of laminated anisotropic plates (CRC Press, 1987). [13].W. Hyer, Stress analysis of fiberreinforced composite materials (assachusetts, cgraw-hill, 1998). [14] AST D 79 Standard Test ethods for Fleural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating aterials, American Society for Testing and aterials, Annual Book of AST Standards, 8.1, [15] ANSYS, Inc. ANSYS 14.5 anual, P a g e

### 7.4 The Elementary Beam Theory

7.4 The Elementary Beam Theory In this section, problems involving long and slender beams are addressed. s with pressure vessels, the geometry of the beam, and the specific type of loading which will be

### A HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS

A HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS A. Kroker, W. Becker TU Darmstadt, Department of Mechanical Engineering, Chair of Structural Mechanics Hochschulstr. 1, D-64289 Darmstadt, Germany kroker@mechanik.tu-darmstadt.de,

### Effect of Specimen Dimensions on Flexural Modulus in a 3-Point Bending Test

Effect of Specimen Dimensions on Flexural Modulus in a 3-Point Bending Test M. Praveen Kumar 1 and V. Balakrishna Murthy 2* 1 Mechanical Engineering Department, P.V.P. Siddhartha Institute of Technology,

### INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 2, No 1, 2011

Interlaminar failure analysis of FRP cross ply laminate with elliptical cutout Venkateswara Rao.S 1, Sd. Abdul Kalam 1, Srilakshmi.S 1, Bala Krishna Murthy.V 2 1 Mechanical Engineering Department, P. V.

### Bending of Simply Supported Isotropic and Composite Laminate Plates

Bending of Simply Supported Isotropic and Composite Laminate Plates Ernesto Gutierrez-Miravete 1 Isotropic Plates Consider simply a supported rectangular plate of isotropic material (length a, width b,

### Chapter 12 Plate Bending Elements. Chapter 12 Plate Bending Elements

CIVL 7/8117 Chapter 12 - Plate Bending Elements 1/34 Chapter 12 Plate Bending Elements Learning Objectives To introduce basic concepts of plate bending. To derive a common plate bending element stiffness

### Flexural Analysis of Deep Aluminum Beam

Journal of Soft Computing in Civil Engineering -1 (018) 71-84 journal homepage: http://www.jsoftcivil.com/ Fleural Analysis of Deep Aluminum Beam P. Kapdis 1, U. Kalwane 1, U. Salunkhe 1 and A. Dahake

### Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 11

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Module - 01 Lecture - 11 Last class, what we did is, we looked at a method called superposition

### Level 7 Postgraduate Diploma in Engineering Computational mechanics using finite element method

9210-203 Level 7 Postgraduate Diploma in Engineering Computational mechanics using finite element method You should have the following for this examination one answer book No additional data is attached

### Longitudinal buckling of slender pressurised tubes

Fluid Structure Interaction VII 133 Longitudinal buckling of slender pressurised tubes S. Syngellakis Wesse Institute of Technology, UK Abstract This paper is concerned with Euler buckling of long slender

### 1.105 Solid Mechanics Laboratory Fall 2003

1.105 Solid Mechanics Laboratory Fall 2003 Eperiment 6 The linear, elastic behavior of a Beam The objectives of this eperiment are To eperimentally study the linear elastic behavior of beams under four

### SERVICEABILITY OF BEAMS AND ONE-WAY SLABS

CHAPTER REINFORCED CONCRETE Reinforced Concrete Design A Fundamental Approach - Fifth Edition Fifth Edition SERVICEABILITY OF BEAMS AND ONE-WAY SLABS A. J. Clark School of Engineering Department of Civil

### PLAT DAN CANGKANG (TKS 4219)

PLAT DAN CANGKANG (TKS 4219) SESI I: PLATES Dr.Eng. Achfas Zacoeb Dept. of Civil Engineering Brawijaya University INTRODUCTION Plates are straight, plane, two-dimensional structural components of which

### American Society for Testing and Materials (ASTM) Standards. Mechanical Testing of Composites and their Constituents

Mechanical Testing of Composites and their Constituents American Society for Testing and Materials (ASTM) Standards Tests done to determine intrinsic material properties such as modulus and strength for

### Effect of magnetostrictive material layer on the stress and deformation behaviour of laminated structure

IOP Conference Series: Materials Science and Engineering PAPER OPEN ACCESS Effect of magnetostrictive material layer on the stress and deformation behaviour of laminated structure To cite this article:

### P. M. Pankade 1, D. H. Tupe 2, G. R. Gandhe 3

ISSN: 78 7798 Volume 5, Issue 5, May 6 Static Fleural Analysis of Thick Beam Using Hyperbolic Shear Deformation Theory P. M. Pankade, D. H. Tupe, G. R. Gandhe P.G. Student, Dept. of Civil Engineering,

### Fracture Behaviour of FRP Cross-Ply Laminate With Embedded Delamination Subjected To Transverse Load

Fracture Behaviour of FRP Cross-Ply Laminate With Embedded Delamination Subjected To Transverse Load Sriram Chintapalli 1, S.Srilakshmi 1 1 Dept. of Mech. Engg., P. V. P. Siddhartha Institute of Technology.

### 3.032 Problem Set 2 Solutions Fall 2007 Due: Start of Lecture,

3.032 Problem Set 2 Solutions Fall 2007 Due: Start of Lecture, 09.21.07 1. In the beam considered in PS1, steel beams carried the distributed weight of the rooms above. To reduce stress on the beam, it

### ME 7502 Lecture 2 Effective Properties of Particulate and Unidirectional Composites

ME 75 Lecture Effective Properties of Particulate and Unidirectional Composites Concepts from Elasticit Theor Statistical Homogeneit, Representative Volume Element, Composite Material Effective Stress-

### Module III - Macro-mechanics of Lamina. Lecture 23. Macro-Mechanics of Lamina

Module III - Macro-mechanics of Lamina Lecture 23 Macro-Mechanics of Lamina For better understanding of the macromechanics of lamina, the knowledge of the material properties in essential. Therefore, the

### FLEXURAL RESPONSE OF FIBER RENFORCED PLASTIC DECKS USING HIGHER-ORDER SHEAR DEFORMABLE PLATE THEORY

Asia-Pacific Conference on FRP in Structures (APFIS 2007) S.T. Smith (ed) 2007 International Institute for FRP in Construction FLEXURAL RESPONSE OF FIBER RENFORCED PLASTIC DECKS USING HIGHER-ORDER SHEAR

### Theories of Straight Beams

EVPM3ed02 2016/6/10 7:20 page 71 #25 This is a part of the revised chapter in the new edition of the tetbook Energy Principles and Variational Methods in pplied Mechanics, which will appear in 2017. These

### Nonlinear bending analysis of laminated composite stiffened plates

Nonlinear bending analysis of laminated composite stiffened plates * S.N.Patel 1) 1) Dept. of Civi Engineering, BITS Pilani, Pilani Campus, Pilani-333031, (Raj), India 1) shuvendu@pilani.bits-pilani.ac.in

### MECHANICS OF MATERIALS Sample Problem 4.2

Sample Problem 4. SOLUTON: Based on the cross section geometry, calculate the location of the section centroid and moment of inertia. ya ( + Y Ad ) A A cast-iron machine part is acted upon by a kn-m couple.

### Post Graduate Diploma in Mechanical Engineering Computational mechanics using finite element method

9210-220 Post Graduate Diploma in Mechanical Engineering Computational mechanics using finite element method You should have the following for this examination one answer book scientific calculator No

### ESE TOPICWISE OBJECTIVE SOLVED PAPER I

C I V I L E N G I N E E R I N G ESE TOPICWISE OBJECTIVE SOLVED PAPER I FROM 1995-018 UPSC Engineering Services Eamination, State Engineering Service Eamination & Public Sector Eamination. Regd. office

### Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 13

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras (Refer Slide Time: 00:25) Module - 01 Lecture - 13 In the last class, we have seen how

### Modelling the nonlinear shear stress-strain response of glass fibrereinforced composites. Part II: Model development and finite element simulations

Modelling the nonlinear shear stress-strain response of glass fibrereinforced composites. Part II: Model development and finite element simulations W. Van Paepegem *, I. De Baere and J. Degrieck Ghent

### Finite element modelling of infinitely wide Angle-ply FRP. laminates

www.ijaser.com 2012 by the authors Licensee IJASER- Under Creative Commons License 3.0 editorial@ijaser.com Research article ISSN 2277 9442 Finite element modelling of infinitely wide Angle-ply FRP laminates

### Problem d d d B C E D. 0.8d. Additional lecturebook examples 29 ME 323

Problem 9.1 Two beam segments, AC and CD, are connected together at C by a frictionless pin. Segment CD is cantilevered from a rigid support at D, and segment AC has a roller support at A. a) Determine

### STRENGTH OF MATERIALS-I. Unit-1. Simple stresses and strains

STRENGTH OF MATERIALS-I Unit-1 Simple stresses and strains 1. What is the Principle of surveying 2. Define Magnetic, True & Arbitrary Meridians. 3. Mention different types of chains 4. Differentiate between

### An Increase in Elastic Buckling Strength of Plate Girder by the Influence of Transverse Stiffeners

GRD Journals- Global Research and Development Journal for Engineering Volume 2 Issue 6 May 2017 ISSN: 2455-5703 An Increase in Elastic Buckling Strength of Plate Girder by the Influence of Transverse Stiffeners

### SANDWICH COMPOSITE BEAMS for STRUCTURAL APPLICATIONS

SANDWICH COMPOSITE BEAMS for STRUCTURAL APPLICATIONS de Aguiar, José M., josemaguiar@gmail.com Faculdade de Tecnologia de São Paulo, FATEC-SP Centro Estadual de Educação Tecnológica Paula Souza. CEETEPS

### Analysis of Thick Cantilever Beam Using New Hyperbolic Shear Deformation Theory

International Journal of Research in Advent Technology, Vol.4, No.5, May 16 E-ISSN: 1-967 Analysis of Thick Cantilever Beam Using New Hyperbolic Shear Deformation Theory Mr. Mithun. K. Sawant 1, Dr. Ajay.

### Bending Load & Calibration Module

Bending Load & Calibration Module Objectives After completing this module, students shall be able to: 1) Conduct laboratory work to validate beam bending stress equations. 2) Develop an understanding of

### The stiffness tailoring of megawatt wind turbine

IOP Conference Series: Materials Science and Engineering OPEN ACCESS The stiffness tailoring of megawatt wind turbine To cite this article: Z M Li et al 2013 IOP Conf. Ser.: Mater. Sci. Eng. 52 052008

### FINITE ELEMENT ANALYSIS OF COMPOSITE MATERIALS

FINITE ELEMENT ANALYSIS OF COMPOSITE MATERIALS Ever J. Barbero Department of Mechanical and Aerospace Engineering West Virginia University USA CRC Press Taylor &.Francis Group Boca Raton London New York

### Parametric study on the transverse and longitudinal moments of trough type folded plate roofs using ANSYS

American Journal of Engineering Research (AJER) e-issn : 2320-0847 p-issn : 2320-0936 Volume-4 pp-22-28 www.ajer.org Research Paper Open Access Parametric study on the transverse and longitudinal moments

### UNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich

UNIVERSITY OF SASKATCHEWAN ME 313.3 MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich A CLOSED BOOK EXAMINATION TIME: 3 HOURS For Marker s Use Only LAST NAME (printed): FIRST

### Finite element analysis of FRP pipelines dynamic stability

Boundary Elements and Other esh Reduction ethods XXXVIII 39 Finite element analysis of FRP pipelines dynamic stability D. G. Pavlou & A. I. Nergaard Department of echanical and Structural Engineering and

### Multi Linear Elastic and Plastic Link in SAP2000

26/01/2016 Marco Donà Multi Linear Elastic and Plastic Link in SAP2000 1 General principles Link object connects two joints, i and j, separated by length L, such that specialized structural behaviour may

### AN EFFECTIVE SOLUTION OF THE COMPOSITE (FGM S) BEAM STRUCTURES

Engineering MECHANICS, Vol. 15, 2008, No. 2, p. 115 132 115 AN EFFECTIVE SOLUTION OF THE COMPOSITE (FGM S) BEAM STRUCTURES Justín Murín, Vladimír Kutiš* The additive mixture rules have been extended for

### 4.5 The framework element stiffness matrix

45 The framework element stiffness matri Consider a 1 degree-of-freedom element that is straight prismatic and symmetric about both principal cross-sectional aes For such a section the shear center coincides

### 3-D Finite Element Analysis of Instrumented Indentation of Transversely Isotropic Materials

3-D Finite Element Analysis of Instrumented Indentation of Transversely Isotropic Materials Abstract: Talapady S. Bhat and T. A. Venkatesh Department of Material Science and Engineering Stony Brook University,

### Section 6: PRISMATIC BEAMS. Beam Theory

Beam Theory There are two types of beam theory aailable to craft beam element formulations from. They are Bernoulli-Euler beam theory Timoshenko beam theory One learns the details of Bernoulli-Euler beam

### Finite elements for plates and shells. Advanced Design for Mechanical System LEC 2008/11/04

Finite elements for plates and shells Sstem LEC 008/11/04 1 Plates and shells A Midplane The mid-surface of a plate is plane, a curved geometr make it a shell The thickness is constant. Ma be isotropic,

### ANALYSIS OF LOAD FLOW AND STRESS CONCENTRATIONS IN TEXTILE COMPOSITES

16 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS ANALYSIS OF LOAD FLOW AND STRESS CONCENTRATIONS IN TEXTILE COMPOSITES Deepak Goyal*, John D. Whitcomb*, Julian Varghese* *Department of Aerospace Engineering,

### Computational Analysis for Composites

Computational Analysis for Composites Professor Johann Sienz and Dr. Tony Murmu Swansea University July, 011 The topics covered include: OUTLINE Overview of composites and their applications Micromechanics

### Lecture #10: Anisotropic plasticity Crashworthiness Basics of shell elements

Lecture #10: 151-0735: Dynamic behavior of materials and structures Anisotropic plasticity Crashworthiness Basics of shell elements by Dirk Mohr ETH Zurich, Department of Mechanical and Process Engineering,

### Finite Element Modelling with Plastic Hinges

01/02/2016 Marco Donà Finite Element Modelling with Plastic Hinges 1 Plastic hinge approach A plastic hinge represents a concentrated post-yield behaviour in one or more degrees of freedom. Hinges only

### Rigid and Braced Frames

RH 331 Note Set 12.1 F2014abn Rigid and raced Frames Notation: E = modulus of elasticit or Young s modulus F = force component in the direction F = force component in the direction FD = free bod diagram

### Micromechanical analysis of FRP hybrid composite lamina for in-plane transverse loading

Indian Journal of Engineering & Materials Sciences Vol. 15, October 2008, pp. 382-390 Micromechanical analysis of FRP hybrid composite lamina for in-plane transverse loading K Sivaji Babu a *, K Mohana

### An efficient analytical model to evaluate the first two local buckling modes of finite cracked plate under tension

278 An efficient analytical model to evaluate the first two local buckling modes of finite cracked plate under tension Abstract The analytical approach is presented for both symmetric and anti-symmetric

### Using MATLAB and. Abaqus. Finite Element Analysis. Introduction to. Amar Khennane. Taylor & Francis Croup. Taylor & Francis Croup,

Introduction to Finite Element Analysis Using MATLAB and Abaqus Amar Khennane Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business

### Bending of Shear Deformable Plates Resting on Winkler Foundations According to Trigonometric Plate Theory

J. Appl. Comput. ech., 4(3) (018) 187-01 DOI: 10.055/JAC.017.3057.1148 ISSN: 383-4536 jacm.scu.ac.ir Bending of Shear Deformable Plates Resting on Winkler Foundations According to Trigonometric Plate Theor

### International Journal of Advanced Engineering Technology E-ISSN

Research Article INTEGRATED FORCE METHOD FOR FIBER REINFORCED COMPOSITE PLATE BENDING PROBLEMS Doiphode G. S., Patodi S. C.* Address for Correspondence Assistant Professor, Applied Mechanics Department,

### ISSN: ISO 9001:2008 Certified International Journal of Engineering Science and Innovative Technology (IJESIT) Volume 2, Issue 4, July 2013

Delamination Studies in Fibre-Reinforced Polymer Composites K.Kantha Rao, Dr P. Shailesh, K. Vijay Kumar 1 Associate Professor, Narasimha Reddy Engineering College Hyderabad. 2 Professor, St. Peter s Engineering

### Methods of Analysis. Force or Flexibility Method

INTRODUCTION: The structural analysis is a mathematical process by which the response of a structure to specified loads is determined. This response is measured by determining the internal forces or stresses

### APPLICATION OF DIRECT VARIATIONAL METHOD IN THE ANALYSIS OF ISOTROPIC THIN RECTANGULAR PLATES

VOL. 7, NO. 9, SEPTEMBER 0 ISSN 89-6608 006-0 Asian Research Publishing Network (ARPN). All rights reserved. APPLICATION OF DIRECT VARIATIONAL METHOD IN THE ANALYSIS OF ISOTROPIC THIN RECTANGULAR PLATES

Universit of Liège Aerospace & Mechanical Engineering Aircraft Structures Structural & Loading Discontinuities Ludovic Noels Computational & Multiscale Mechanics of Materials CM3 http://www.ltas-cm3.ulg.ac.be/

### 7. Hierarchical modeling examples

7. Hierarchical modeling examples The objective of this chapter is to apply the hierarchical modeling approach discussed in Chapter 1 to three selected problems using the mathematical models studied in

### Application of Finite Element Method to Create Animated Simulation of Beam Analysis for the Course of Mechanics of Materials

International Conference on Engineering Education and Research "Progress Through Partnership" 4 VSB-TUO, Ostrava, ISSN 156-35 Application of Finite Element Method to Create Animated Simulation of Beam

### Chapter 5 Structural Elements: The truss & beam elements

Institute of Structural Engineering Page 1 Chapter 5 Structural Elements: The truss & beam elements Institute of Structural Engineering Page 2 Chapter Goals Learn how to formulate the Finite Element Equations

### Prediction of Micromechanical Behaviour of Elliptical Frp Composites

Prediction of Micromechanical Behaviour of Elliptical Frp Composites Kiranmayee.Nerusu Dept. of Mechanical Engg. P. V. P. Siddhartha Institute of Technology, Vijayawada 520 007, A.P, India. P. Phani Prasanthi

### Module-6: Laminated Composites-II. Learning Unit-1: M6.1. M 6.1 Structural Mechanics of Laminates

Module-6: Laminated Composites-II Learning Unit-1: M6.1 M 6.1 Structural Mechanics of Laminates Classical Lamination Theory: Laminate Stiffness Matrix To this point in the development of classical lamination

### INTERFACE CRACK IN ORTHOTROPIC KIRCHHOFF PLATES

Gépészet Budapest 4-5.Ma. G--Section-o ITERFACE CRACK I ORTHOTROPIC KIRCHHOFF PLATES András Szekrénes Budapest Universit of Technolog and Economics Department of Applied Mechanics Budapest Műegetem rkp.

### 1 Static Plastic Behaviour of Beams

1 Static Plastic Behaviour of Beams 1.1 Introduction Many ductile materials which are used in engineering practice have a considerable reserve capacity beyond the initial yield condition. The uniaxial

ANALYSIS RADIAL FOLDED PLATE Ms.Pranoti Satish Bhamare 1, Prajakta S. Bramhankar 2, Pooja G. Baviskar 3 1,2,3 Assistant Professor, Department of Civil Engineering, Guru Gobind Singh College of Engineering

### Accepted Manuscript. R.C. Batra, J. Xiao S (12) Reference: COST Composite Structures. To appear in:

Accepted Manuscript Finite deformations of curved laminated St. Venant-Kirchhoff beam using layerwise third order shear and normal deformable beam theory (TSNDT) R.C. Batra, J. Xiao PII: S0263-8223(12)00486-2

### Fig. 1. Circular fiber and interphase between the fiber and the matrix.

Finite element unit cell model based on ABAQUS for fiber reinforced composites Tian Tang Composites Manufacturing & Simulation Center, Purdue University West Lafayette, IN 47906 1. Problem Statement In

### Passive Damping Characteristics of Carbon Epoxy Composite Plates

Journal of Materials Science and Engineering A 6 (-) 35-4 doi:.765/6-63/6.-.5 D DAVID PUBLISHING Passive Damping Characteristics of Carbon Epoxy Composite Plates Dileep Kumar K * and V V Subba Rao Faculty

### Finite Element Method

Finite Element Method Finite Element Method (ENGC 6321) Syllabus Objectives Understand the basic theory of the FEM Know the behaviour and usage of each type of elements covered in this course one dimensional

### Mechanical and Thermal Properties of Coir Fiber Reinforced Epoxy Composites Using a Micromechanical Approach

Mechanical and Thermal Properties of Coir Fiber Reinforced Epoxy Composites Using a Micromechanical Approach Sandhyarani Biswas Department of Mechanical Engineering, N.I.T Rourkela, INDIA Abstract: Now-a-days,

### MITOCW MITRES2_002S10linear_lec07_300k-mp4

MITOCW MITRES2_002S10linear_lec07_300k-mp4 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources

### Plane Strain Test for Metal Sheet Characterization

Plane Strain Test for Metal Sheet Characterization Paulo Flores 1, Felix Bonnet 2 and Anne-Marie Habraken 3 1 DIM, University of Concepción, Edmundo Larenas 270, Concepción, Chile 2 ENS - Cachan, Avenue

### ELASTOPLASTIC STEEL BEAM BENDING ANALYSIS BY USING ABAQUS

11 ELASTOPLASTIC STEEL BEAM BENDING ANALYSIS BY USING ABAQUS Biswajit Jena * * Corresponding author: biswa.tech88@gmail.com Abstract: In recent years tremendous efforts have been made in development of

### 4.MECHANICAL PROPERTIES OF MATERIALS

4.MECHANICAL PROPERTIES OF MATERIALS The diagram representing the relation between stress and strain in a given material is an important characteristic of the material. To obtain the stress-strain diagram

. Stresses in Pavements.1. Stresses in Fleible Pavements.1.1. Stresses in Homogeneous Mass Boussinesq formulated models for the stresses inside an elastic half-space due to a concentrated load applied

### Institute of Structural Engineering Page 1. Method of Finite Elements I. Chapter 2. The Direct Stiffness Method. Method of Finite Elements I

Institute of Structural Engineering Page 1 Chapter 2 The Direct Stiffness Method Institute of Structural Engineering Page 2 Direct Stiffness Method (DSM) Computational method for structural analysis Matrix

### Optimum Height of Plate Stiffener under Pressure Effect

The st Regional Conference of Eng. Sci. NUCEJ Spatial ISSUE vol., No.3, 8 pp 459-468 Optimum Height of Plate Stiffener under Pressure Effect Mazin Victor Yousif M.Sc Production Engineering University of

### University of Sheffield The development of finite elements for 3D structural analysis in fire

The development of finite elements for 3D structural analysis in fire Chaoming Yu, I. W. Burgess, Z. Huang, R. J. Plank Department of Civil and Structural Engineering StiFF 05/09/2006 3D composite structures

### CHAPTER 7 FINITE ELEMENT ANALYSIS OF DEEP GROOVE BALL BEARING

113 CHAPTER 7 FINITE ELEMENT ANALYSIS OF DEEP GROOVE BALL BEARING 7. 1 INTRODUCTION Finite element computational methodology for rolling contact analysis of the bearing was proposed and it has several

### Kirchhoff Plates: Field Equations

20 Kirchhoff Plates: Field Equations AFEM Ch 20 Slide 1 Plate Structures A plate is a three dimensional bod characterized b Thinness: one of the plate dimensions, the thickness, is much smaller than the

### Thermal buckling and post-buckling of laminated composite plates with. temperature dependent properties by an asymptotic numerical method

hermal buckling and post-buckling of laminated composite plates with temperature dependent properties by an asymptotic numerical method F. Abdoun a,*, L. Azrar a,b, E.M. Daya c a LAMA, Higher School of

### UNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES

UNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES A Thesis by WOORAM KIM Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the

### Chapter 11. Displacement Method of Analysis Slope Deflection Method

Chapter 11 Displacement ethod of Analysis Slope Deflection ethod Displacement ethod of Analysis Two main methods of analyzing indeterminate structure Force method The method of consistent deformations

### Due Tuesday, September 21 st, 12:00 midnight

Due Tuesday, September 21 st, 12:00 midnight The first problem discusses a plane truss with inclined supports. You will need to modify the MatLab software from homework 1. The next 4 problems consider

### Nonlinear RC beam element model under combined action of axial, bending and shear

icccbe 21 Nottingham University Press Proceedings of the International Conference on Computing in Civil and Building ngineering W Tiani (ditor) Nonlinear RC beam element model under combined action of

### EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 4 Pure Bending

EA 3702 echanics & aterials Science (echanics of aterials) Chapter 4 Pure Bending Pure Bending Ch 2 Aial Loading & Parallel Loading: uniform normal stress and shearing stress distribution Ch 3 Torsion:

### December 10, PROBLEM NO points max.

PROBLEM NO. 1 25 points max. PROBLEM NO. 2 25 points max. B 3A A C D A H k P L 2L Given: Consider the structure above that is made up of rod segments BC and DH, a spring of stiffness k and rigid connectors

### Mechanics of Inflatable Fabric Beams

Copyright c 2008 ICCES ICCES, vol.5, no.2, pp.93-98 Mechanics of Inflatable Fabric Beams C. Wielgosz 1,J.C.Thomas 1,A.LeVan 1 Summary In this paper we present a summary of the behaviour of inflatable fabric

### Basic Energy Principles in Stiffness Analysis

Basic Energy Principles in Stiffness Analysis Stress-Strain Relations The application of any theory requires knowledge of the physical properties of the material(s) comprising the structure. We are limiting

### Steps in the Finite Element Method. Chung Hua University Department of Mechanical Engineering Dr. Ching I Chen

Steps in the Finite Element Method Chung Hua University Department of Mechanical Engineering Dr. Ching I Chen General Idea Engineers are interested in evaluating effects such as deformations, stresses,

### STRUCTURAL ANALYSIS OF FIBER REINFORCED COMPOSITE MATERIALS CESAR AUGUSTO ROJAS. Presented to the Faculty of the Graduate School of

STRUCTURAL ANALYSIS OF FIBER REINFORCED COMPOSITE MATERIALS by CESAR AUGUSTO ROJAS Presented to the Faculty of the Graduate School of The University of Teas at Arlington in Partial Fulfillment of the Requirements

### Passive Damping Characteristics of Carbon Epoxy Composite Plates

Journal of aterials Science and Engineering A 6 (1-2) (2016) 35-42 doi: 10.17265/2161-6213/2016.1-2.005 D DAVID PUBLISHIG Passive Damping Characteristics of Carbon Epoxy Composite Plates Dileep Kumar K

### Due Monday, September 14 th, 12:00 midnight

Due Monday, September 14 th, 1: midnight This homework is considering the analysis of plane and space (3D) trusses as discussed in class. A list of MatLab programs that were discussed in class is provided

### Nomenclature. Length of the panel between the supports. Width of the panel between the supports/ width of the beam

omenclature a b c f h Length of the panel between the supports Width of the panel between the supports/ width of the beam Sandwich beam/ panel core thickness Thickness of the panel face sheet Sandwich