Stiffness Matrix Method Beam Example
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Easychair Preprint Design and Analysis of Cantilever Beam
EasyChair Preprint № 4804 Design and Analysis of Cantilever Beam Sridhar Parshana, Manoj Kumar Dandu and Narendhar Kethavath EasyChair preprints are intended for rapid dissemination of research results and are integrated with the rest of EasyChair. December 25, 2020 MallaReddy College of Engineering Maisammaguda Group 1 Group members Parshana sridhar (17Q91A0334) Dandu Manoj kumar (17Q91A0331) Kethavath Narendhar (17Q91A0323) Guide : V Ramya MODELING AND FE ANALYSIS OF CANTILEVER BEAM ABSTRACT This study investigates the deflection and stress distribution in a long, slender cantilever beam of uniform rectangular cross section made of linear elastic material properties that are homogeneous and isotropic. The deflection of a cantilever beam is essentially a three dimensional problem. An elastic stretching in one direction is accompanied by a compression in perpendicular directions. In this project ,static and Modal analysis is a process to determine the stress ,strain and deformation. vibration characteristics (natural frequencies and mode shapes) of a structure or a machine component while it is being designed. It has become a major alternative to provide a helpful contribution in understanding control of many vibration phenomena which encountered in practice. In this work we compared the stress and natural frequency for different material having same I, C and T cross- sectional beam. The cantilever beam is designed and analyzed in ANSYS. The cantilever beam which is fixed at one end is vibrated to obtain the natural frequency, mode shapes and deflection with different sections and materials. INTRODUCTION BEAM A beam is a structural element that is capable of withstanding load primarily by resisting against bending. The bending force induced into the material of the beam as a result of the external loads, own weight, span and external reactions to these loads is called a bending moment. -
4. an Analytical Model of Buckling Panels of Steel Beams at Elevated
A COMPONENT-BASED APPROACH TO MODELLING BEAM-END BUCKLING ADJACENT TO BEAM-COLUMN CONNECTIONS IN FIRE A thesis submitted for the degree of Doctor of Philosophy In the Faculty of Engineering of The University of Sheffield by GUAN QUAN Department of Civil and Structural Engineering The University of Sheffield August, 2016 ACKNOWLEGEMENTS My special thanks go to Professor Ian Burgess and Dr Shan-Shan Huang for their guidance, inspiration and dedication in supervising me throughout my research project. Their continuous awareness and rich experience made this PhD project nowhere near a lofty mountain, but near a highway. My thanks also go to their financial support on my international conferences, which are great opportunities to present my own work and to communicate with worldwide researchers. The financial support provided by CSC program and by the University of Sheffield is also greatly appreciated. My thanks also go to my colleagues and friends in the Structural Fire Engineering Research Group for the precious friendship we developed; and to Dr Ruirui Sun for his continuous advice on the development of Vulcan. I would like to express my gratefulness to my family for their love, understanding and sharing during the course of my PhD. Thanks to my boyfriend Dr Jun Ye. We have been together for more than six years from Wuhan University to Zhejiang University in China and to the University of Sheffield in the UK. Thank you for your love, accompany and support. I’m glad that we are going to put an end to our PhD courses as well as our single states together. -
A New Stiffness Matrix for Analysis of Flexural Line Continuum 1, O
The International Journal Of Engineering And Science (Ijes) ||Volume|| 2 ||Issue|| 2 ||Pages|| 57-62 ||2013|| Issn: 2319 – 1813 Isbn: 2319 – 1805 A New Stiffness Matrix for Analysis Of Flexural Line Continuum 1, 2, 3, O. M. Ibearugbulem, L. O. Ettu, J. C. Ezeh 1,2,3,Civil Engineering Department, Federal University of Technology, Owerri, Nigeria ---------------------------------------------------------Abstract------------------------------------------------------ The analysis of line continuum by equilibrium approach to obtain exact results requires direct integration of the governing equations and satisfying the boundary conditions. This procedure is cumbersome and 4x4 stiffness matrix system with 4x1 load vector has been used as practical simplification for a long time. The problem with this system is that it is inefficient and does not give exact results, especially for classical studies that analyse only one element as a whole. This paper presents 5x5 stiffness matrix system as a new stiffness system to generate exact results. The stiffness matrix was developed using energy variational principle. The approach differs from the 4x4 stiffness matrix approach by considering a deformable node at the centre of the continuum which brings the number of deformable nodes to five. By carrying out direct integration of the governing differential equation of the line continuum, a five-term shape function was obtained. This was substituted into the strain energy equation and the resulting functional was minimized, resulting in a 5x5 stiffness matrix used herein. The five term shape function was also substituted into load-work equation and minimized to obtain load vector for uniformly distributed load on the continuum. Classical studies of beams of four different boundary conditions under uniformly distributed load and point load were performed using the new 5x5 stiffness matrix with 5x1 load vector for one element analysis. -
Euler-Bernoulli Beam Theory
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 February 2021 doi:10.20944/preprints202102.0559.v1 EULER−BERNOULLI BEAM THEORY USING THE FINITE DIFFERENCE METHOD Article EULER−BERNOULLI BEAM THEORY: FIRST-ORDER ANALYSIS, SECOND-ORDER ANALYSIS, STABILITY, AND VIBRATION ANALYSIS USING THE FINITE DIFFERENCE METHOD Valentin Fogang Abstract: This paper presents an approach to the Euler−Bernoulli beam theory (EBBT) using the finite difference method (FDM). The EBBT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations (or partial differential equations). The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. The model developed in this paper consists of formulating partial differential equations with finite differences and introducing new points (additional points or imaginary points) at boundaries and positions of discontinuity (concentrated loads or moments, supports, hinges, springs, brutal change of stiffness, etc.). The introduction of additional points permits us to satisfy boundary conditions and continuity conditions. First-order analysis, second-order analysis, and vibration analysis of structures were conducted with this model. Efforts, displacements, stiffness matrices, buckling loads, and vibration frequencies were determined. Tapered beams were analyzed (e.g., element stiffness matrix, second-order analysis). Finally, a direct -
Structural Analysis Program Using Direct Stiffness Method for the Design of Reinforced Concrete Column
Journal of Electrical Engineering and Computer Sciences ISSN : 2528 - 0260 Vol. 2, No.1, June 2017 STRUCTURAL ANALYSIS PROGRAM USING DIRECT STIFFNESS METHOD FOR THE DESIGN OF REINFORCED CONCRETE COLUMN AHMAD FAZA AZMI Engineering Faculty, Bhayangkara University Of Surabaya Jalan Ahmad Yani 114, Surabaya 60231, Indonesia E-mail: [email protected] ABSTRACT Most of the analyzes and calculations in civil engineering have been using computer. But many commercial structural analysis programs are relatively expensive. Hence one of the alternative way is using a self developed structural analysis program. The computer program is constructed using Visual Basic 6 programming language. Using direct stiffness method and theory of ACI 318-02, a numerical procedure along with a computer program was developed for the structural analysis and design of reinforced concrete member. From the direct stiffness method,in terms of structural analysis, the output that obtained from the program are internal element forces and support reactions node displacements. From ACI 318-02 codes, in terms of reinforced concrete member analysis, the program generate the amount of longitudinal and lateral reinforcement of concrete member based on the output in the previous structure analysis program. The program also reveals the biaxial bending-axial interaction diagrams in each direction of rectangular and circular column. The interaction diagram of the columns is a visual representation to determine the maximum axial load and moment exceeded the capacity of the column. Keywords: Structural, Analysis, Reinforced, Concrete, Column 1. INTRODUCTION Structural analysis and design is known to human beings since early civilizations. Structural analysis and design of a reinforced concrete column with biaxial bending and axial force is complicated without the assistance of a computer program. -
DIRECT STIFFNESS METHOD for ANALYSIS of SKELETAL STRUCTURES Dr. Suresh Bhalla
DIRECT STIFFNESS METHOD FOR ANALYSIS OF SKELETAL STRUCTURES http://web.iitd.ac.in/~sbhalla/cvl756.html Dr. Suresh Bhalla Professor Department of Civil Engineering Indian Institute of Technology Delhi 1 The audio of this lecture is based on classroom recording from CVL 756 class for batch 2019-20. Use earphone/ amplifier for best audio experience. 2 SLOPE DEFLECTION METHOD Applied moment M θ Carry over moment 4EI 2EI L 6EI 6EI L 2 L2 L 12EI 3 6EI L 2 L 6EI 12EI L2 L3 3 Generalized slope deflection equations 2EI 3( ) θ 2 1 M1 2 M1 21 2 2 L L θ1 2EI 3( 2 1) 1 M2 M 2 1 2 2 L L F F1 2 M M F F 1 2 1 2 L 2EI 6( ) 2EI 6( 2 1) 2 1 F 3 3 F2 31 3 2 1 2 1 2 2 L L L L 4 PUT IN MATRIX FORM 12EI 6EI 12EI 6EI 3 2 3 2 F L L L L 1 6EI 4EI 6EI 2EI 1 M1 2 2 1 • Symmetric L L L L F 12EI 6EI 12EI 6EI 2 2 • K +ive 3 2 3 2 ii M L L L L 2 6EI 2EI 6EI 4EI 2 L2 L L2 L Force Stiffness Displacement vector matrix vector (symmetric) 5 F1 K11 K12 . K1n 1 1 2 3 F K K . K 2 21 22 2n 2 . . . Fn Kn1 Kn1 . Knn n n = Degrees of freedom (DKI) th Fi = Force along i degree of freedom th i = Displacement along i degree of freedom th j col. -
Direct Stiffness Method: Beams
Module 4 Analysis of Statically Indeterminate Structures by the Direct Stiffness Method Version 2 CE IIT, Kharagpur Lesson 27 The Direct Stiffness Method: Beams Version 2 CE IIT, Kharagpur Instructional Objectives After reading this chapter the student will be able to 1. Derive member stiffness matrix of a beam element. 2. Assemble member stiffness matrices to obtain the global stiffness matrix for a beam. 3. Write down global load vector for the beam problem. 4. Write the global load-displacement relation for the beam. 27.1 Introduction. In chapter 23, a few problems were solved using stiffness method from fundamentals. The procedure adopted therein is not suitable for computer implementation. In fact the load displacement relation for the entire structure was derived from fundamentals. This procedure runs into trouble when the structure is large and complex. However this can be much simplified provided we follow the procedure adopted for trusses. In the case of truss, the stiffness matrix of the entire truss was obtained by assembling the member stiffness matrices of individual members. In a similar way, one could obtain the global stiffness matrix of a continuous beam from assembling member stiffness matrix of individual beam elements. Towards this end, we break the given beam into a number of beam elements. The stiffness matrix of each individual beam element can be written very easily. For example, consider a continuous beam ABCD as shown in Fig. 27.1a. The given continuous beam is divided into three beam elements as shown in Fig. 27.1b. It is noticed that, in this case, nodes are located at the supports. -
Use of Mathcad in Computing Beam Deflection by Conjugate Beam Method
Session 2649 Use of Mathcad in Computing Beam Deflection by Conjugate Beam Method Nirmal K. Das Georgia Southern University Abstract The four-year, ABET-accredited Civil Engineering Technology curriculum at Georgia Southern University includes a required, junior-level course in Structural Analysis. One of the topics covered is the conjugate beam method for computing slope and deflection at various points in a beam. The conjugate beam method is a geometric method and it relies only on the principles of statics. The usefulness of this method lies in its simplicity. The students can utilize their already acquired knowledge of shearing force and bending moment to determine a beam’s slope and deflection. An approach to teaching this important method of structural analysis that complements the traditional lecturing through inclusion of a powerful, versatile and user-friendly computational tool, is discussed in this paper. Students will learn how to utilize Mathcad to perform a variety of calculations in a sequence and to verify the accuracy of their manual solutions. A Mathcad program is developed for this purpose and examples to illustrate the computer program are also included in this paper. The integration of Mathcad will enhance students’ problem-solving skills, as it will allow them to focus on analysis while the software performs routine calculations. Thus it will promote learning by discovery, instead of leaving the student in the role of a passive observer. Introduction With the objective of enhanced student learning, adoption of various instructional technology and inclusion of computer-aided problem-solving modules into the curriculum has been a trend for civil engineering and civil engineering technology programs. -
Beam Element
Finite Element Method: Beam Element Beam Element ¨ Beam: A long thin structure that is subject to the vertical loads. A beam shows more evident bending deformation than the torsion and/or axial deformation. ¨ Bending strain is measured by the lateral deflection and the rotation -> Lateral deflection and rotation determine the number of DoF (Degree of Freedom). Mechanics and Design SNU School of Mechanical and Aerospace Engineering Finite Element Method: Beam Element ¨ Sign convention 1. Positive bending moment: Anti-clock wise rotation. 2. Positive load: y$ -direction. 3. Positive displacement: y$ -direction. Mechanics and Design SNU School of Mechanical and Aerospace Engineering Finite Element Method: Beam Element ¨ The governing equation dVdV -w-wdxdx$ +ˆ -dV dV= 0 =0or or w w= =- - dxdx$ ˆ dMdM VVdxdx$ ˆ+-dM dM= =0 0or or VV == dxdx$ˆ Mechanics and Design SNU School of Mechanical and Aerospace Engineering Finite Element Method: Beam Element ¨ Relation between beam curvature (k ) and bending moment 1 M d 2v$ M k = = or = r EI dx$ 2 EI d 2v$ Curvature for small slope (q = dv$ / dx$ ): k = ¨ dx$ 2 r : Radius of the deflection curve, v$: Lateral displacement function along the y$ -axis direction E : Stiffness coefficient, I : Moment of inertia along the z$-axis Mechanics and Design SNU School of Mechanical and Aerospace Engineering Finite Element Method: Beam Element Solving the equation with M , d 2 F d 2v$I EI = - wax$f dx$ 2 HG dx$ 2 KJ When EI is constant, and force and moment are only applied at nodes, d 4v$ EI = 0 dx$ 4 Mechanics and Design SNU School of Mechanical and Aerospace Engineering Finite Element Method: Beam Element Step 1: To select beam element type Step 2: To select displacement function Assumption of lateral displacement 3 2 vˆ() xˆ = a1 xˆ + a 2 xˆ + a 3 xˆ + a 4 - Complete 3-order displacement function is suitable, because it has four degree of freedom (one lateral displacement and one small rotation at each node) - The function is proper, because it satisfies the fundamental differential equation of a beam. -
Structural Analysis.Pdf
Contents:C Ccc Module 1.Energy Methods in Structural Analysis.........................5 Lesson 1. General Introduction Lesson 2. Principle of Superposition, Strain Energy Lesson 3. Castigliano’s Theorems Lesson 4. Theorem of Least Work Lesson 5. Virtual Work Lesson 6. Engesser’s Theorem and Truss Deflections by Virtual Work Principles Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method................................................................107 Lesson 7. The Force Method of Analysis: An Introduction Lesson 8. The Force Method of Analysis: Beams Lesson 9. The Force Method of Analysis: Beams (Continued) Lesson 10. The Force Method of Analysis: Trusses Lesson 11. The Force Method of Analysis: Frames Lesson 12. Three-Moment Equations-I Lesson 13. The Three-Moment Equations-Ii Module 3. Analysis of Statically Indeterminate Structures by the Displacement Method..............................................................227 Lesson 14. The Slope-Deflection Method: An Introduction Lesson 15. The Slope-Deflection Method: Beams (Continued) Lesson 16. The Slope-Deflection Method: Frames Without Sidesway Lesson 17. The Slope-Deflection Method: Frames with Sidesway Lesson 18. The Moment-Distribution Method: Introduction Lesson 19. The Moment-Distribution Method: Statically Indeterminate Beams With Support Settlements Lesson 20. Moment-Distribution Method: Frames without Sidesway Lesson 21. The Moment-Distribution Method: Frames with Sidesway Lesson 22. The Multistory Frames with Sidesway Module 4. Analysis of Statically Indeterminate Structures by the Direct Stiffness Method...........................................................398 Lesson 23. The Direct Stiffness Method: An Introduction Lesson 24. The Direct Stiffness Method: Truss Analysis Lesson 25. The Direct Stiffness Method: Truss Analysis (Continued) Lesson 26. The Direct Stiffness Method: Temperature Changes and Fabrication Errors in Truss Analysis Lesson 27. -
Eml 4500 Finite Element Analysis and Design
CHAP 4 FINITE ELEMENT ANALYSIS OF BEAMS AND FRAMES 1 INTRODUCTION • We learned Direct Stiffness Method in Chapter 2 – Limited to simple elements such as 1D bars • we will learn Energy Method to build beam finite element – Structure is in equilibrium when the potential energy is minimum • Potential energy: Sum of strain energy and potential of applied loads UV Potential of • Interpolation scheme: applied loads Strain energy vx()4HXGWNq () x {} Beam Interpolation Nodal deflection function DOF 2 BEAM THEORY • Euler-Bernoulli Beam Theory – can carry the transverse load – slope can change along the span (x-axis) – Cross-section is symmetric w.r.t. xy-plane – The y-axis passes through the centroid – Loads are applied in xy-plane (plane of loading) y y Neutral axis Plane of loading x z A L F F 3 BEAM THEORY cont. • Euler-Bernoulli Beam Theory cont. – Plane sections normal to the beam axis remain plane and normal to the axis after deformation (no shear stress) – Transverse deflection (deflection curve) is function of x only: v(x) – Displacement in x-dir is function of x and y: u(x, y) dv udvdu 2 dv uxy(, ) u () x y 0 y 0 dx xx xdxdx2 dx y y(dv/dx) Neutral axis x y = dv/dx L F v(x) 4 BEAM THEORY cont. udvdu 2 • Euler-Bernoulli Beam Theory cont. 0 y xx 2 – Strain along the beam axis: 00 du/ dx xdxdx – Strain xx varies linearly w.r.t. y; Strain yy = 0 – Curvature: dv22/ dx – Can assume plane stress in z-dir basically uniaxial status dv2 EEEy xx xx 0 dx2 • Axial force resultant and bending moment dv2 P dA E dA E ydA PEA OOOxx 0 dx2 0 AAA dv2 dv2 MEI M y dA E ydA E y2 dA dx2 OOOxx 0 dx2 AAA EA: axial rigidity Moment of inertia I(x) EI: flexural rigidity 5 BEAM THEORY cont. -
DNVGL-CG-0128 Buckling
CLASS GUIDELINE DNVGL-CG-0128 Edition October 2015 Buckling The electronic pdf version of this document, available free of charge from http://www.dnvgl.com, is the officially binding version. DNV GL AS FOREWORD DNV GL class guidelines contain methods, technical requirements, principles and acceptance criteria related to classed objects as referred to from the rules. © DNV GL AS Any comments may be sent by e-mail to [email protected] If any person suffers loss or damage which is proved to have been caused by any negligent act or omission of DNV GL, then DNV GL shall pay compensation to such person for his proved direct loss or damage. However, the compensation shall not exceed an amount equal to ten times the fee charged for the service in question, provided that the maximum compensation shall never exceed USD 2 million. In this provision "DNV GL" shall mean DNV GL AS, its direct and indirect owners as well as all its affiliates, subsidiaries, directors, officers, employees, agents and any other acting on behalf of DNV GL. CHANGES – CURRENT This is a new document. Changes - current Class guideline — DNVGL-CG-0128. Edition October 2015 Page 3 Buckling DNV GL AS CONTENTS Changes – current...................................................................................................... 3 Contents Section 1 Introduction............................................................................................... 7 1 Objective.................................................................................................7 2 Buckling methods..................................................................................