The EST Method Structural Analysis

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This book proposes a method, devised by the author, of solving boundary value problems for the structural analysis of frames, beams, and trusses.

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  • 2

  • Andres Lahe

    The EST MethodStructural analysis

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  • 2Published by Tallinn University of Technology Press

    Copyright: Andres Lahe, 2014

    ISBN 978-9949-23-677-0 (publication)ISBN 978-9949-23-678-7 (PDF)

    Printed in Estonia by Tallinna Raamatutrukikoda

    This book is licensed under a Creative Commons Attribution-ShareAlike 3.0 UnportedLicense. To view a copy of this license, visit http://creativecommons.org/licenses/

    by-sa/3.0/ or send a letter to Creative Commons, 543 Howard Street, 5th Floor, San Francisco,California, 94105, USA.

    Program excerpts in the book are subject to the terms of the GNU GPL programs License2.0 or later. To view a copy of the GNU General Public License license, visit http://www.gnu.org/licenses/old-licenses/gpl-2.0.html or write to the Free Software

    Foundation, Inc. 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.

  • Mente et manu

    The development of sparse matrix algorithms has led to changes in computa-tional structural mechanics and in the corresponding methods of analysis. The ESTmethod is a most general method of analysis for framed structures: beams, trusses,and frames. In this method, compatibility equations for displacements and equilib-rium equations for the member end forces at joints are incorporated. This bookconsiders the solving of boundary value problems of frames, beams and trusses withthe EST method. The boundary value problem (differential equations together witha set of boundary conditions) is well posed in the method. Solutions to differen-tial equations are represented with the initial parameter method (see also the uni-versal equation of elastic curve of a beam [KL09] and the transfer matrix method).Kinematic and static boundary conditions are composed by a computer program. Aframe member has 6 initial parameters and 6 end variables (3 displacements and 3forces). There are 12+ unknowns in the system of sparse equations for a frame( is the number of elements, that of support reactions). To assemble and solveboundary problem equations, the following steps are to be made:

    1. writing the basic equations of a frame,

    2. writing the compatibility equations of displacements at joints,

    3. writing the joint equlibrium equations,

    4. writing the side conditions,

    5. writing the restrictions on support displacements,

    6. solving the compiled system of sparse equations,

    7. output: initial parameter vectors for element displacements and forces; supportreactions.

    Round-off errors are reduced by scaling (multiplying) the displacements and rota-tions by the basic stiffness (scaling multiplier). After solving a system of equationsfor boundary value problem, the displacements and rotations found are divided by thebasic stiffness.

    In Chapter 1 of the book, a brief introduction to the fundamental relations of a frameelement is given. Chapter 2 deals with the derivations of the EST method equations.The GNU Octave function LaheFrameDFIm.m of assembling and solving the boundaryproblem has been tested1 with different input data (frames) shown in Chapter 3. The

    1./EST_method_examples.pdf. Can be found on the CD attached to the book.

  • 4procedure of computing statically determinate frames with or without the displacementsand rotations (with or without the compatibility equations of the displacements atjoints) is considered in Chapter 4. Second-order structural analysis and the EST methodfor a second-order theory are treated in Chapters 5 and 6. Chapters 7 and 8 deal withthe incremental loading method of plastic analysis for framed structures. A full plasticmoment at a plastic hinge is described by the side conditions of the EST method.

    The book deals with equations in sparse matrix form. Appendix A contains asummary of sparse matrices. Descriptions and links to the GNU Octave programs ofthe EST method are presented in Appendix D.

    The motivation to compose the EST method has come from the books [PL63],[Kra90], [Kra91a], [Kra91b], [KW90], and [Kra91c]. The method outlined in [Lah97a],[Lah97b], and [Lah98a] differs from the transfer matrix method [PL63], [LT80] andboundary element method [Str89], [Har87], [BW80].

    I am obliged to Aime-Rutt Hall for correcting my English and Tiia Eikholm for offeringthe cover design.

    This book is dedicated to the memory of my wife Lilja.

    Andres Lahe

  • Table of Contents

    List of Figures 9

    List of Tables 13

    I First-order structural analysis 15

    1 Introduction 17

    1.1 Fundamental relations of a frame element . . . . . . . . . . . . . . . . . 17

    1.2 Basic equations for a beam/frame . . . . . . . . . . . . . . . . . . . . . . 22

    1.2.1 The Euler-Bernoulli beam equation . . . . . . . . . . . . . . . . . 22

    1.2.2 Basic equations for a frame element . . . . . . . . . . . . . . . . . 25

    1.2.3 The basic system of equations for a frame element . . . . . . . . . 28

    1.3 The system equations of a frame . . . . . . . . . . . . . . . . . . . . . . . 30

    2 Equations of the EST method 33

    2.1 Basic equations of the system . . . . . . . . . . . . . . . . . . . . . . . . 33

    2.2 Compatibility equations of displacements . . . . . . . . . . . . . . . . . . 34

    2.2.1 Compatibility conditions at beam joints . . . . . . . . . . . . . . 39

    2.3 Equations of joint equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 40

    2.3.1 Equilibrium equations of beam joints . . . . . . . . . . . . . . . . 45

    2.4 Side conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

    2.5 Restrictions on support displacements . . . . . . . . . . . . . . . . . . . . 47

    2.6 Initial parameter vector for an element . . . . . . . . . . . . . . . . . . . 49

    2.7 Element displacements and forces . . . . . . . . . . . . . . . . . . . . . . 50

    2.8 The truss element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

    3 Statically indeterminate problems 57

    3.1 Illustrative frame problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . 57

    3.2 Illustrative frame problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . 65

    3.3 Illustrative frame problem 3 . . . . . . . . . . . . . . . . . . . . . . . . . 73

    3.4 Illustrative continuous beam problem . . . . . . . . . . . . . . . . . . . . 82

    3.5 Illustrative truss problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

    5

  • 6 TABLE OF CONTENTS

    4 Statically determinate problems 1034.1 Illustrative frame problem . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.2 Illustrative beam problem . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.3 Illustrative truss problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

    II Second-order structural analysis 123

    5 Second-order structural analysis 1255.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.2 The governing equation for a beam-column . . . . . . . . . . . . . . . . . 1255.3 Solutions of a beam-column governing equation . . . . . . . . . . . . . . 130

    5.3.1 The set of solutions to a homogeneous differential equation . . . . 1315.3.2 Transfer matrix for a beam element with axial force . . . . . . . . 1335.3.3 The particular solution of non-homogeneous differential equation . 1365.3.4 Transformation of a transfer matrix . . . . . . . . . . . . . . . . 141

    6 The EST method for a second-order theory 1476.1 System equations for a frame . . . . . . . . . . . . . . . . . . . . . . . . 1476.2 Illustrative frame problem 1 . . . . . . . . . . . . . . . . . . . . . . . . . 1496.3 Illustrative frame problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . 157

    III Plastic analysis of frames 167

    7 Plasticity and limit design 1697.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

    7.1.1 The plastic moment . . . . . . . . . . . . . . . . . . . . . . . . . . 1707.1.2 Work done in a plastic hinge . . . . . . . . . . . . . . . . . . . . . 171

    7.2 Methods of plastic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 173

    8 The EST method. Limit design 1778.1 Illustrative problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

    8.1.1 The n=3 times statically indeterminate frame . . . . . . . . . . . 1818.1.2 The n=2 times statically indeterminate frame . . . . . . . . . . . 1878.1.3 The n=1 times statically indeterminate frame . . . . . . . . . . . 1948.1.4 The n=0 times statically indeterminate frame . . . . . . . . . . . 200

    IV Appendices 209

    A Matrices 211A.1 Sparse matrices and GNU Octave . . . . . . . . . . . . . . . . . . . . . . 211

    A.1.1 Introduction to sparse matrices . . . . . . . . . . . . . . . . . . . 211A.1.2 Creating sparse matrices . . . . . . . . . . . . . . . . . . . . . . 214

  • TABLE OF CONTENTS 7

    A.1.3 Sparse matrix functions in the EST method . . . . . . . . . . . . 216A.2 Transformation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

    B Work and work-energy theorem 221B.1 Work done by internal and external forces . . . . . . . . . . . . . . . . . 221

    C Transfer matrices 225C.1 Transfer matrices of first-order analysis . . . . . . . . . . . . . . . . . . . 225C.2 Transfer matrices of second-order analysis . . . . . . . . . . . . . . . . . 228

    D Computer programs for the EST method 235D.1 Programs for first-order analysis . . . . . . . . . . . . . . . . . . . . . . . 235D.2 Programs for second-order analysis . . . . . . . . . . . .