Bending of Simply Supported Isotropic and Composite Laminate Plates
|
|
- Shon Boone
- 6 years ago
- Views:
Transcription
1 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, thickness t, - with t << a, b -, elastic modulus E, Poisson s ratio ν) under a uniform pressure q acting normal to the surface of the plate. Let the x axis be aligned with the length of the plate, the y axis with its width and the z axis with its thickness with z = located in the middle of the plate thickness. For isotropic plates, the (engineering) stress-strain relationships are given by Q 11 Q 1 Q 1 Q 11 Q 66 ǫ x ǫ y ǫ xy where Q is the reduced stiffness matrix with components Q 66 = Q 11 = Q 1 = E 1 ν νe 1 ν E (1 + ν) = G The Kirchoff-Love hypothesis is commonly used to simplify the analysis of plates. If the thickness of a plate is much smaller that its length and its width plate straight normals to the plane of the plate remain approximately straight, of constant length and normal to the plane of the plate when the plate deforms. This is equivalent to neglecting the shearing strains in planes perpendicular to the middle surface (z = z = ) as well as the normal strain normal to the plate (ǫ z = ). Under the hypothesis, the strain components in the 1
2 plate are given in terms of the strains and curvatures along the middle surface of the plate, respectively ǫ x, ǫ y, ǫ xy, κ x, κ y, κ xy, i.e. ǫ x ǫ y ǫ xy ǫ x κ ǫ x y + z κ ǫ y xy κ xy So that the stress-strain relationships become Q 11 Q 1 Q 1 Q 11 Q 66 κ x κ y ǫ x ǫ y + z ǫ xy κ xy The mid plane strains and curvatures are given respectively by ǫ x ǫ y ǫ xy u / x v / y u / y + v / x and κ x w / x κ y w / y κ xy w / x y and where u, v, w = w are the components of displacement at the middle surface of the plate. The normal and shear forces acting on the plate are given by N x N y N xy t t dz The bending and twisting moments acting on the plate are given by M x M y M xy t t zdz Combination and rearrangement of the above yields finally N x N y N xy Et 1 ν νet 1 ν νet 1 ν Et 1 ν 1 ν Et 1 ν ǫ x ǫ y ǫ xy
3 and M x M y M xy Et 3 1(1 ν ) νet 3 1(1 ν ) νet 3 1(1 ν ) Et 3 1(1 ν ) 1 ν Et 3 1(1 ν ) κ x κ y κ xy Under the Kirchoff-Love hypothesis, commonly used when analyzing the deformation of plates and shells, and assuming small deflections (linear elasticity), it can be shown that the condition of mechanical equilibrium assuming that the edges of the plate are free to move in the plane of the plate has the form M x x + M y y M xy x y = q Combining with the above yields the governing equation in terms of the deflection of the mid-surface of the plate: where D 4 w x 4 + D 4 w x y + D 4 w y 4 = q D = Et 3 1(1 ν ) is the flexural rigidity (bending stiffness) of the plate. This is a bi-harmonic equation that must be solved subject to specific boundary conditions. For simply supported edges, deflections and normal moments are zero along the edges, i.e. for x = and x = a. w =, w =, w x = w y = for y = and y = b. An exact solution of this problem was obtained by Navier in 18 and it is w = 16q π 6 D with m = 1, 3, 5,... and n = 1, 3, 5,... m=1 n=1 sin( mπx a )sin(nπy b ) mn( m a + n b ) 3
4 The bending and twisting moments are given by M x = D( w x + ν w y ) M y = D( w y + ν w x ) M xy = D(1 ν) w x y Finally, the elastic strain energy of the plate V is given by V = D a b ( w x + w y ) (1 ν)[ w x w y ( w x y ) ]dxdy Composite Laminate Plates Composite plates are produced by stacking thin sheets of fiber reinforced polymer called plies. Consider simply a supported rectangular ply (ply thickness h, ply length a, width b) consisting of unidirectionally aligned reinforcing fibers embedded in a polymer matrix, (elastic moduli E 1 (longitudinal, parallel to the fibers), E (transverse to the fibers), shear modulus G 1, longitudinal Poisson ratio ν 1, transverse Poisson ratios ν 1, ν 3 ) under a uniform load q acting normal to the surface of the ply. If a ply is aligned with the fiber direction coinciding with the x axis and the transverse direction with the y axis, the material is called specially orthotropic and the stress-strain relations are Q 11 Q 1 Q 1 Q Q 66 where components of the reduced stiffness matrix are κ x κ y ǫ x ǫ y + z ǫ xy κ xy Q 11 = E 1 1 ν 1 ν 1 Q 1 = ν 1E 1 ν 1 ν 1 Q = E 1 ν 1 ν 1 4
5 Q 66 = G 1 An important design feature of composite plies is that the longitudinal and transverse directions of any individual ply can be oriented at any angle θ with respect the a x y axis system used as reference. Hence for a ply where the fiber axis is oriented at an angle θ with respect to the reference x axis the stress-strain relationships are instead given by Q 11 Q 1 Q 16 Q 1 Q Q 6 Q 16 Q6 Q66 where Q is the transformed reduced stiffness matrix with components given by Q 11 = Q 11 cos 4 θ + (Q 1 + Q 66 )sin θ cos θ + Q sin 4 θ ǫ x ǫ y ǫ xy Q 1 = (Q 11 + Q 4Q 66 )sin θ cos θ + Q 1 (sin 4 θ + cos 4 θ) Q = Q 11 sin 4 θ + (Q 1 + Q 66 )sin θ cos θ + Q cos 4 θ Q 16 = (Q 11 Q 1 Q 66 )sin θ cos 3 θ + (Q 1 Q + Q 66 )sin 3 θ cos θ Q 6 = (Q 11 Q 1 Q 66 )sin 3 θ cosθ + (Q 1 Q + Q 66 )sinθ cos 3 θ Q 66 = (Q 11 + Q Q 1 Q 66 )sin θ cos θ + Q 66 (sin 4 θ + cos 4 θ) Another very important design feature of composites is that plies can be stacked to form thicker sections called laminates. Moreover, the stacking pattern can be selected to optimize the properties of the laminate. Consider a composite laminate (thickness t) formed by stacking N composite plies each with thickness h k, k = 1,, 3,..., N such that t = N h k, following a suitably selected stacking pattern. The normal and shear forces acting on the laminate are given by N x N y N xy t t dz = N zk z k 1 dz And the bending and twisting moments acting on the laminate are given by t N zk zdz = zdz M x M y M xy t 5 z k 1
6 and Performing the indicated integrations and rearranging yields N x A 11 A 1 A 16 ǫ x B 11 B 1 B 16 N y A 1 A A 6 ǫ y + B 1 B B 6 N xy A 16 A 6 A 66 B 16 B 6 B 66 M x M y M xy ǫ xy B 11 B 1 B 16 ǫ x B 1 B B 6 ǫ y B 16 B 6 B 66 ǫ xy + κ x κ y κ xy D 11 D 1 D 16 κ x D 1 D D 6 κ y D 16 D 6 D 66 κ xy where A, B and D are, respectively, the extensional, coupling and bending stiffnesses of the laminate with components given by N A ij = ( Q ij ) k (z k z k 1 ) B ij = 1 N ( Q ij ) k (zk zk 1) D ij = 1 N ( 3 Q ij ) k (zk 3 zk 1) 3 In practice, for the computation of all the A ij, B ij, D ij, the mid-plane of the laminate is selected as the origin of the z axis (z = = z (N+1)/ ) and the locations z, z 1, z,..., z N (except z (N+1)/ ) representing the boundaries of each ply. For instance for a three ply laminate formed with plies of thickness (in mm) h 1 =.1, h =., h 3 =.1, one has z =., z 1 =.1, z =., z 3 =.1, z 4 =.. Again, under the Kirchoff-Love and small deflection hypotheses, the equation governing the deflection w(x, y) of the plate in this case is 4 w D 11 x + 4D 4 w 4 16 x 3 y + (D 4 w 1 + D 66 ) x y + 4D 4 w 6 x y + D 4 w 3 y = q 4 where D 11, D 16, D 1, D 66, D, D 6 are the bending stiffnesses of the composite plate. The boundary conditions are, for x = and x = a. w =, w =, w D 11 x + D w 1 y + D 16 w D 1 x + D w y + D 6 6 w x y = w x y =
7 for y = and y = b. The elastic strain energy of the plate V is given by V = 1 a b [D 11 ( w x ) + D 1 w x w y + D ( w y ) + 4D 66 ( w x y ) + w w 4D 16 x x y + 4D w w 6 y x y ]dxdy A specially kind of laminate is obtained when specially orthotropic plies are symmetrically arranged about the laminate middle surface. A laminate without shear or twist coupling nor bending-extension coupling is obtained (i.e. B ij = and D 16 = D 6 = ). The deflection equation in this case becomes 4 w D 11 x + (D 4 w D 66 ) x y + D 4 w y = q 4 An exact solution of this problem can be obtained by an approach similar to the one used by Navier in 18 for the isotropic plate and it is w = 16q π 6 m=1 n=1 ( 1 mn )sin(mπx a )sin(nπy b ) D 11 ( m a )4 + (D 1 + D 66 )( m a ) ( n b ) + D ( n b )4 with m = 1, 3, 5,... and n = 1, 3, 5,... However, for many laminates, the stiffness components B ij, D 16 and D 6 may be non-zero, an exact solution cannot be obtained and approximation methods are required. 3 The Ritz Method The Ritz method is a technique discovered by Ritz to determine approximate solutions to the partial differential equations encountered in plate theory. The method is an application of the principle of minimum potential energy. The total potential energy of a loaded plate consists of the internal elastic strain energy of the plate V minus the potential energy of the external forces. In the case considered here, the potential energy associated with the external forces is given by Ω = So that the total potential energy E is a B E = V Ω qw dxdy 7
8 The Ritz method is then implemented by assuming that the deflection can be expressed as a linear combination of simple orthogonal functions (basis functions) satisfying the specified boundary conditions. For the situation at hand, a suitable expression is then w (x, y) = I J i=1 j=1 c k sin( iπx a )sin(jπy b ) where the c k are unknown coefficients (k = i + (j 1)I = 1,, 3,..., K where K = I J) and the numbers I and J will determine the accuracy of the approximation. The assumed expression for w is then substituted into V and Ω and the integrations performed. The result of this is a more or less complicated looking equation involving all the unknown coefficients. The principle of minimum potential energy is implemented by differentiating the total potential energy with respect to each coefficient and equation the result to zero, i.e. E = E =... = E = c 1 c c K This operation produces a system of K simultaneous linear algebraic equations whence the values of the unknown coefficients can be determined using standard methods. Substitution of the obtained values into the assumed equation for w yields the desired approximate solution. 4 The Finite Element Method The finite element method can be regarded as a generalization of the Ritz method. First, the computational domain is subdivided into a collection of contiguous, non-overlapping sub-domains connected at nodes. As in the Ritz method, one writes down an expression for the approximated quantity as a linear combination of the basis functions and then minimizes the total potential energy of the system with respect to a set of unknown coefficients. The unknown coefficients in the case of the finite element method turn out to be the nodal values of the dependent variable whose approximation is being sought. The basis functions selected for the finite element approximation are simple functions of compact support. Perhaps the best known example are the linear roof functions with value of one at a node, zero at all next neighbor nodes and varying linearly in between according to a simple first order Lagrange interpolating polynomial. In implementing the method one proceeds as with the classic Ritz approach and minimizes the total potential energy. The result is a set of linear algebraic equations for the unknown coefficients. Most importantly, since the basis functions are of compact support, the resulting system is sparse. Solution of the system produces the nodal values and substitution into the assumed expression for the approximated quantity yields the finite element approximation. The accuracy of the approximation can be increased either by increasing the number of elements in the subdivision or by increasing the order of the interpolating polynomial or both. 8
9 5 References 1.- S.P. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, nd ed. McGraw-Hill, New York, L.P. Kollar and G.S. Springer, Mechanics of Composite Structures, Cambridge U.P. Cambridge, 3. 9
Lecture 15 Strain and stress in beams
Spring, 2019 ME 323 Mechanics of Materials Lecture 15 Strain and stress in beams Reading assignment: 6.1 6.2 News: Instructor: Prof. Marcial Gonzalez Last modified: 1/6/19 9:42:38 PM Beam theory (@ ME
More informationPresented By: EAS 6939 Aerospace Structural Composites
A Beam Theory for Laminated Composites and Application to Torsion Problems Dr. BhavaniV. Sankar Presented By: Sameer Luthra EAS 6939 Aerospace Structural Composites 1 Introduction Composite beams have
More informationAnalytical Strip Method for Thin Isotropic Cylindrical Shells
IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE) e-issn: 2278-1684,p-ISSN: 2320-334X, Volume 14, Issue 4 Ver. III (Jul. Aug. 2017), PP 24-38 www.iosrjournals.org Analytical Strip Method for
More informationON THE NUMERICAL ANALYSIS OF COMPOSITE MATERIAL
The 4th International Conference Advanced Composite Materials Engineering COMAT 8- October, Brasov, Romania O THE UMERICAL AALYSIS OF COMPOSITE MATERIAL D.D. icoara Transilvania University, City Brasov,
More informationComposites Design and Analysis. Stress Strain Relationship
Composites Design and Analysis Stress Strain Relationship Composite design and analysis Laminate Theory Manufacturing Methods Materials Composite Materials Design / Analysis Engineer Design Guidelines
More informationTABLE OF CONTENTS. Mechanics of Composite Materials, Second Edition Autar K Kaw University of South Florida, Tampa, USA
Mechanics of Composite Materials, Second Edition Autar K Kaw University of South Florida, Tampa, USA TABLE OF CONTENTS 1. INTRODUCTION TO COMPOSITE MATERIALS 1.1 Introduction... 1.2 Classification... 1.2.1
More informationModule-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
More informationThe stiffness of plates
The stiffness of plates 1. Introduction The word plate is a collective term for elements in which forces can be transferred in two directions. Floors, walls, bridge slabs and laminates are all plates.
More informationCHAPTER THREE SYMMETRIC BENDING OF CIRCLE PLATES
CHAPTER THREE SYMMETRIC BENDING OF CIRCLE PLATES * Governing equations in beam and plate bending ** Solution by superposition 1.1 From Beam Bending to Plate Bending 1.2 Governing Equations For Symmetric
More informationChapter 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
More informationVIBRATION CONTROL OF RECTANGULAR CROSS-PLY FRP PLATES USING PZT MATERIALS
Journal of Engineering Science and Technology Vol. 12, No. 12 (217) 3398-3411 School of Engineering, Taylor s University VIBRATION CONTROL OF RECTANGULAR CROSS-PLY FRP PLATES USING PZT MATERIALS DILEEP
More informationChapter 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
More informationComparison of Ply-wise Stress-Strain results for graphite/epoxy laminated plate subjected to in-plane normal loads using CLT and ANSYS ACP PrepPost
Comparison of Ply-wise Stress-Strain results for graphite/epoxy laminated plate subjected to in-plane normal loads using CLT and ANSYS ACP PrepPost 1 Mihir A. Mehta, 2 Satyen D. Ramani 1 PG Student, Department
More informationLAMINATED COMPOSITE PLATES
LAMINATED COMPOSITE PLATES David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 02139 February 10, 2000 Introduction This document is intended
More informationFig. 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
More informationModule 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
More informationNomenclature. 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
More informationFinite 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
More informationComputational 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
More informationBilinear Quadrilateral (Q4): CQUAD4 in GENESIS
Bilinear Quadrilateral (Q4): CQUAD4 in GENESIS The Q4 element has four nodes and eight nodal dof. The shape can be any quadrilateral; we ll concentrate on a rectangle now. The displacement field in terms
More informationCOMPOSITE PLATE THEORIES
CHAPTER2 COMPOSITE PLATE THEORIES 2.1 GENERAL Analysis of composite plates is usually done based on one of the following the ries. 1. Equivalent single-layer theories a. Classical laminate theory b. Shear
More informationEffect 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,
More informationA NEW REFINED THEORY OF PLATES WITH TRANSVERSE SHEAR DEFORMATION FOR MODERATELY THICK AND THICK PLATES
A NEW REFINED THEORY OF PLATES WITH TRANSVERSE SHEAR DEFORMATION FOR MODERATELY THICK AND THICK PLATES J.M. MARTÍNEZ VALLE Mechanics Department, EPS; Leonardo da Vinci Building, Rabanales Campus, Cordoba
More informationANALYSIS OF COMPOSITE RECTANGULAR PLATES BASED ON THE CLASSICAL LAMINATED PLATE THEORY
The 4th International Conference Advanced Composite Materials Engineering COMAT 2012 18-20 October 2012, Brasov, Romania ANALYSIS OF COMPOSITE RECTANGULAR PLATES BASED ON THE CLASSICAL LAMINATED PLATE
More informationInternational 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,
More informationLAMINATED COMPOSITE PLATES
LAMINATED COMPOSITE PLATES David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 02139 February 10, 2000 Introduction This document is intended
More informationEFFECT OF LAMINATION ANGLE AND THICKNESS ON ANALYSIS OF COMPOSITE PLATE UNDER THERMO MECHANICAL LOADING
Journal of MECHANICAL ENGINEERING Strojnícky časopis, VOL 67 (217), NO 1, 5-22 EFFECT OF LAMINATION ANGLE AND THICKNESS ON ANALYSIS OF COMPOSITE PLATE UNDER THERMO MECHANICAL LOADING Arnab Choudhury 1,
More informationA 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,
More informationLinear elastic analysis of thin laminated beams with uniform and symmetric cross-section
Applied and Computational Mechanics 2 (2008) 397 408 Linear elastic analysis of thin laminated beams with uniform and symmetric cross-section M. Zajíček a, a Faculty of Applied Sciences, UWB in Pilsen,
More information3D Elasticity Theory
3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.
More informationAPPLICATION OF THE GALERKIN-VLASOV METHOD TO THE FLEXURAL ANALYSIS OF SIMPLY SUPPORTED RECTANGULAR KIRCHHOFF PLATES UNDER UNIFORM LOADS
Nigerian Journal of Technology (NIJOTECH) Vol. 35, No. 4, October 2016, pp. 732 738 Copyright Faculty of Engineering, University of Nigeria, Nsukka, Print ISSN: 0331-8443, Electronic ISSN: 2467-8821 www.nijotech.com
More informationDynamic Analysis of Laminated Composite Plate Structure with Square Cut-Out under Hygrothermal Load
Dynamic Analysis of Laminated Composite Plate Structure with Square Cut-Out under Hygrothermal Load Arun Mukherjee 1, Dr. Sreyashi Das (nee Pal) 2 and Dr. A. Guha Niyogi 3 1 PG student, 2 Asst. Professor,
More informationMechanical Behavior of Composite Tapered Lamina
International Journal of Engineering Research and Development e-issn: 2278-067X, p-issn: 2278-800X, www.ijerd.com Volume 10, Issue 8 (August 2014), PP.19-27 Mechanical Behavior of Composite Tapered Lamina
More informationSTRESSES WITHIN CURVED LAMINATED BEAMS OF DOUGLAS-FIR
UNITED STATES DEPARTMENT OF AGRICULTURE. FOREST SERVICE - FOREST PRODUCTS LABORATORY - MADISON, WIS. STRESSES WITHIN CURVED LAMINATED BEAMS OF DOUGLAS-FIR NOVEMBER 1963 FPL-020 STRESSES WITHIN CURVED LAMINATED
More informationAnalysis of Non-Rectangular Laminated Anisotropic Plates by Chebyshev Collocation Method
146 Analysis of Non-Rectangular Laminated Anisotropic Plates by Chebyshev Collocation Method Chih-Hsun LIN and Ming-Hwa R. JEN The purpose of this work is to solve the governing differential equations
More informationOptimum 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
More informationVIBRATION AND DAMPING ANALYSIS OF FIBER REINFORCED COMPOSITE MATERIAL CONICAL SHELLS
VIBRATION AND DAMPING ANALYSIS OF FIBER REINFORCED COMPOSITE MATERIAL CONICAL SHELLS Mechanical Engineering Department, Indian Institute of Technology, New Delhi 110 016, India (Received 22 January 1992,
More informationBuckling Behavior of 3D Randomly Oriented CNT Reinforced Nanocomposite Plate
Buckling Behavior of 3D Randomly Oriented CNT Reinforced Nanocomposite Plate Outline Introduction Representative Volume Element (RVE) Periodic Boundary Conditions on RVE Homogenization Method Analytical
More informationModule 5: Laminate Theory Lecture 17: Laminate Constitutive Relations. The Lecture Contains: Laminate Constitutive Relations
Lecture 17: Laminate Constitutive Relations The Lecture Contains: Laminate Constitutive Relations Classification of Laminates Cross-Ply Laminates Specially Orthotropic Laminates Examples Homework References
More informationEFFECT OF RETROFITTING ON FLEXURAL RIGIDITY OF RECTANGULAR FRP PLATES
EFFECT OF RETROFITTING ON FLEXURAL RIGIDITY OF RECTANGULAR FRP PLATES STEVEN J. MAKONIS, JR. Department of Civil & Environmental Engineering Old Dominion University Norfolk, VA 23528-0241, USA smako001@odu.edu
More informationFinite 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
More informationCOPYRIGHTED MATERIAL. Index
Index A Admissible function, 163 Amplification factor, 36 Amplitude, 1, 22 Amplitude-modulated carrier, 630 Amplitude ratio, 36 Antinodes, 612 Approximate analytical methods, 647 Assumed modes method,
More informationChapter 3. Load and Stress Analysis
Chapter 3 Load and Stress Analysis 2 Shear Force and Bending Moments in Beams Internal shear force V & bending moment M must ensure equilibrium Fig. 3 2 Sign Conventions for Bending and Shear Fig. 3 3
More informationPLAT 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
More informationNONCLASSICAL MODELS IN THE SHELL THEORY WITH APPLICATIONS TO MULTILAYERED NANOTUBES
COMPDYN 0 3 rd ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering M. Papadrakakis M. Fragiadakis V. Plevris eds. Corfu Greece 5-8 May 0 NONCLASSICAL
More informationStructural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian
Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: Mode-Superposition Method Mode-Superposition Method:
More informationComposite Structures. Indian Institute of Technology Kanpur
Mechanics of Laminated Composite Structures Nachiketa Tiwari Indian Institute of Technology Kanpur Lecture 23 Analysis of an Orthotropic Ply Lecture Overview Introduction Engineering constants for an 2
More informationKirchhoff 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
More informationINTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 3, No 4, 2013
INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 3, No 4, 2013 Copyright by the authors - Licensee IPA- Under Creative Commons license 3.0 Research article ISSN 0976 4399 Pure bending analysis
More informationUnit 18 Other Issues In Buckling/Structural Instability
Unit 18 Other Issues In Buckling/Structural Instability Readings: Rivello Timoshenko Jones 14.3, 14.5, 14.6, 14.7 (read these at least, others at your leisure ) Ch. 15, Ch. 16 Theory of Elastic Stability
More informationMECHANICS OF COMPOSITE STRUCTURES
MECHANICS OF COMPOSITE STRUCTURES LÁSZLÓ P. KOLLÁR Budapest University of Technology and Economics GEORGE S. SPRINGER Stanford University PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE
More informationChapter 5 Plane-Stress Stress-Strain Relations in a Global Coordinate System
Chapter 5 Plane-Stress Stress-Strain Relations in a Global Coordinate System One of the most important characteristics of structures made of fiber-reinforced materials, and one which dictates the manner
More informationHygrothermal stresses in laminates
Hygrothermal stresses in laminates Changing environment conditions (temperature and moisture) have an important effect on the properties which are matrix dominated. Change in temperaturet and moisture
More informationFLEXIBILITY METHOD FOR INDETERMINATE FRAMES
UNIT - I FLEXIBILITY METHOD FOR INDETERMINATE FRAMES 1. What is meant by indeterminate structures? Structures that do not satisfy the conditions of equilibrium are called indeterminate structure. These
More informationTable of Contents. Preface... 13
Table of Contents Preface... 13 Chapter 1. Vibrations of Continuous Elastic Solid Media... 17 1.1. Objective of the chapter... 17 1.2. Equations of motion and boundary conditions of continuous media...
More informationApplication of Laplace Iteration method to Study of Nonlinear Vibration of laminated composite plates
(3) 78 795 Application of Laplace Iteration method to Study of Nonlinear Vibration of laminated composite plates Abstract In this paper, free vibration characteristics of laminated composite plates are
More informationTHEORY OF PLATES AND SHELLS
THEORY OF PLATES AND SHELLS S. TIMOSHENKO Professor Emeritus of Engineering Mechanics Stanford University S. WOINOWSKY-KRIEGER Professor of Engineering Mechanics Laval University SECOND EDITION MCGRAW-HILL
More informationFree Vibration Response of a Multilayer Smart Hybrid Composite Plate with Embedded SMA Wires
11(2014) 279-298 Free Vibration Response of a Multilayer Smart Hybrid Composite Plate with Embedded SMA Wires Abstract In this paper, free vibration response of a hybrid composite plate was studied. Effects
More informationLevel 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
More informationPart D: Frames and Plates
Part D: Frames and Plates Plane Frames and Thin Plates A Beam with General Boundary Conditions The Stiffness Method Thin Plates Initial Imperfections The Ritz and Finite Element Approaches A Beam with
More informationCHAPTER -6- BENDING Part -1-
Ishik University / Sulaimani Civil Engineering Department Mechanics of Materials CE 211 CHAPTER -6- BENDING Part -1-1 CHAPTER -6- Bending Outlines of this chapter: 6.1. Chapter Objectives 6.2. Shear and
More informationAdvanced Structural Analysis EGF Section Properties and Bending
Advanced Structural Analysis EGF316 3. Section Properties and Bending 3.1 Loads in beams When we analyse beams, we need to consider various types of loads acting on them, for example, axial forces, shear
More informationUNCONVENTIONAL 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
More informationPractical expressions for the design of laminated glass
Practical expressions for the design of laminated glass Laura Galuppi (1), Giampiero Manara (2), Gianni Royer Carfagni (1) (1) Department of Civil-Environmental Engineering and Architecture, University
More informationFree vibration analysis of beams by using a third-order shear deformation theory
Sādhanā Vol. 32, Part 3, June 2007, pp. 167 179. Printed in India Free vibration analysis of beams by using a third-order shear deformation theory MESUT ŞİMŞEK and TURGUT KOCTÜRK Department of Civil Engineering,
More informationMaterials and Structures. Indian Institute of Technology Kanpur
Introduction to Composite Materials and Structures Nachiketa Tiwari Indian Institute of Technology Kanpur Lecture 34 Thermal Stresses in Plates Lecture Overview Introduction Mechanical and Thermal Strains
More informationPURE BENDING. If a simply supported beam carries two point loads of 10 kn as shown in the following figure, pure bending occurs at segment BC.
BENDING STRESS The effect of a bending moment applied to a cross-section of a beam is to induce a state of stress across that section. These stresses are known as bending stresses and they act normally
More informationParametric 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
More informationFREE VIBRATION ANALYSIS OF THIN CYLINDRICAL SHELLS SUBJECTED TO INTERNAL PRESSURE AND FINITE ELEMENT ANALYSIS
FREE VIBRATION ANALYSIS OF THIN CYLINDRICAL SHELLS SUBJECTED TO INTERNAL PRESSURE AND FINITE ELEMENT ANALYSIS J. Kandasamy 1, M. Madhavi 2, N. Haritha 3 1 Corresponding author Department of Mechanical
More informationANALYSIS OF THE INTERACTIVE BUCKLING IN STIFFENED PLATES USING A SEMI-ANALYTICAL METHOD
EUROSTEEL 2014, September 10-12, 2014, Naples, Italy ANALYSIS OF THE INTERACTIVE BUCKLING IN STIFFENED PLATES USING A SEMI-ANALYTICAL METHOD Pedro Salvado Ferreira a, Francisco Virtuoso b a Polytechnic
More informationFLEXURAL 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
More informationBasic Equations of Elasticity
A Basic Equations of Elasticity A.1 STRESS The state of stress at any point in a loaded bo is defined completely in terms of the nine components of stress: σ xx,σ yy,σ zz,σ xy,σ yx,σ yz,σ zy,σ zx,andσ
More informationTHERMAL STRESS ANALYSIS OF A FIBER-EPOXY COMPOSITE MATERIAL
HERMAL SCIENCE, Year 2011, Vol. 15, No. 2, pp. 559-563 559 HERMAL SRESS ANALYSIS OF A FIBER-EPOXY COMPOSIE MAERIAL by Radoljub P. OMI] a, Aleksandar S. SEDMAK b*, Dobrivoje M. ]AI] c, Marko V. MILOŠ b,
More informationConsider an elastic spring as shown in the Fig.2.4. When the spring is slowly
.3 Strain Energy Consider an elastic spring as shown in the Fig..4. When the spring is slowly pulled, it deflects by a small amount u 1. When the load is removed from the spring, it goes back to the original
More informationTheories 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
More informationMathematical model of static deformation of micropolar elastic circular thin bar
Mathematical model of static deformation of micropolar elastic circular thin bar Mathematical model of static deformation of micropolar elastic circular thin bar Samvel H. Sargsyan, Meline V. Khachatryan
More informationMIXED RECTANGULAR FINITE ELEMENTS FOR PLATE BENDING
144 MIXED RECTANGULAR FINITE ELEMENTS FOR PLATE BENDING J. N. Reddy* and Chen-Shyh-Tsay School of Aerospace, Mechanical and Nuclear Engineering, University of Oklahoma, Norman, Oklahoma The paper describes
More informationDESIGN OF LAMINATES FOR IN-PLANE LOADING
DESIGN OF LAMINATES FOR IN-PLANOADING G. VERCHERY ISMANS 44 avenue F.A. Bartholdi, 72000 Le Mans, France Georges.Verchery@m4x.org SUMMARY This work relates to the design of laminated structures primarily
More informationCellular solid structures with unbounded thermal expansion. Roderic Lakes. Journal of Materials Science Letters, 15, (1996).
1 Cellular solid structures with unbounded thermal expansion Roderic Lakes Journal of Materials Science Letters, 15, 475-477 (1996). Abstract Material microstructures are presented which can exhibit coefficients
More informationSupplementary figures
Supplementary figures Supplementary figure 1: The fang model (longitudinal section) used for the stiffness analysis. The base of the fang is considered fixed (i.e. zero displacement constraint) and the
More informationPassive 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
More informationConstitutive Equations
Constitutive quations David Roylance Department of Materials Science and ngineering Massachusetts Institute of Technology Cambridge, MA 0239 October 4, 2000 Introduction The modules on kinematics (Module
More informationComb resonator design (2)
Lecture 6: Comb resonator design () -Intro Intro. to Mechanics of Materials School of Electrical l Engineering i and Computer Science, Seoul National University Nano/Micro Systems & Controls Laboratory
More informationSmart Materials, Adaptive Structures, and Intelligent Mechanical Systems
Smart Materials, Adaptive Structures, and Intelligent Mechanical Systems Bishakh Bhattacharya & Nachiketa Tiwari Indian Institute of Technology Kanpur Lecture 19 Analysis of an Orthotropic Ply References
More informationCOMPUTER AIDED DESIGN IN CASE OF THE LAMINATED COMPOSITE MATERIALS
6 th International Conference Computational Mechanics and Virtual Engineering COMEC 15 15-16 October 15, Braşov, Romania COMPUER AIDED DESIGN IN CASE OF HE LAMINAED COMPOSIE MAERIALS Camelia Cerbu ransilvania
More informationPlate analysis using classical or Reissner- Mindlin theories
Plate analysis using classical or Reissner- Mindlin theories L. Palermo Jr. Faculty of Civil Engineering, State Universiv at Campinas, Brazil Abstract Plates can be solved with classical or Reissner-Mindlin
More informationLecture 8. Stress Strain in Multi-dimension
Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]
More informationUnit 13 Review of Simple Beam Theory
MIT - 16.0 Fall, 00 Unit 13 Review of Simple Beam Theory Readings: Review Unified Engineering notes on Beam Theory BMP 3.8, 3.9, 3.10 T & G 10-15 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics
More informationNonlinear 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
More informationME 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-
More informationApplication of piezoelectric actuators to active control of composite spherical caps
Smart Mater. Struct. 8 (1999 18. Printed in the UK PII: S964-176(991661-4 Application of piezoelectric actuators to active control of composite spherical caps Victor Birman, Gareth J Knowles and John J
More informationIraq Ref. & Air. Cond. Dept/ Technical College / Kirkuk
International Journal of Scientific & Engineering Research, Volume 6, Issue 4, April-015 1678 Study the Increasing of the Cantilever Plate Stiffness by Using s Jawdat Ali Yakoob Iesam Jondi Hasan Ass.
More informationDYNAMIC RESPONSE OF SYNTACTIC FOAM CORE SANDWICH USING A MULTIPLE SCALES BASED ASYMPTOTIC METHOD
ECCM6-6 TH EUROPEAN CONFERENCE ON COMPOSITE MATERIALS, Seville, Spain, -6 June 4 DYNAMIC RESPONSE OF SYNTACTIC FOAM CORE SANDWICH USING A MULTIPLE SCALES BASED ASYMPTOTIC METHOD K. V. Nagendra Gopal a*,
More informationThe CR Formulation: BE Plane Beam
6 The CR Formulation: BE Plane Beam 6 Chapter 6: THE CR FORMUATION: BE PANE BEAM TABE OF CONTENTS Page 6. Introduction..................... 6 4 6.2 CR Beam Kinematics................. 6 4 6.2. Coordinate
More information3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1
Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is
More informationClassical Lamination Theory: The Kirchhoff Hypothesis
212 Chapter 6 Classical Lamination Theory: The Kirchhoff Hypothesis In the preceding chapters we developed the tools needed to understand the elastic response of a small volume of fiber-reinforced material
More informationHIGHER-ORDER THEORIES
HIGHER-ORDER THEORIES THIRD-ORDER SHEAR DEFORMATION PLATE THEORY LAYERWISE LAMINATE THEORY J.N. Reddy 1 Third-Order Shear Deformation Plate Theory Assumed Displacement Field µ u(x y z t) u 0 (x y t) +
More informationUNIVERSITY OF HAWAII COLLEGE OF ENGINEERING DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING
UNIVERSITY OF HAWAII COLLEGE OF ENGINEERING DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING ACKNOWLEDGMENTS This report consists of the dissertation by Ms. Yan Jane Liu, submitted in partial fulfillment
More informationThe problem of isotropic rectangular plate with four clamped edges
Sādhanā Vol. 32, Part 3, June 2007, pp. 181 186. Printed in India The problem of isotropic rectangular plate with four clamped edges C ERDEM İMRAK and ISMAIL GERDEMELI Istanbul Technical University, Faculty
More information2008 by authors and 2008 Springer Science+Business Media
Antti H. Niemi, Harri Hakula, and Juhani Pitkäranta. 28. Point load on a shell. In: Karl Kunisch, Günther Of, and Olaf Steinbach (editors). Numerical Mathematics and Advanced Applications. Proceedings
More information