ANALYSIS OF JEROME J. CONNOR, Sc.D., Massachusetts Institute of Technology, is Professor of Civil Engineering at Massa-chusetts Institute of Technology. He has been active in STRUCTURAL MEMBER teaching and research in structural analysis and mechanics at the U.S. Army Materials and Mechanics Research Agency and for some years at M.I.T. His primary inter-est is in computer based analysis methods, and his current SYSTEMSresearch is concerned with the dynamic analysis of pre-stressed concrete reactor vessels and the development of finite element models for fluid flow problems. Dr. Connor is one of the original developers of ICES-STRUDL, and has published extensively in the structural field. JEROME J. CONNOR i, Massachusetts Institute of Technology ' THE RONALD PRESS COMPANY· NEW YORK : ':,:':,;:::::,,:":I:..'~: '.:::':Copyright © 1976 by THE RONALD PRESS COMPANY All Rights Reserved No part of this book may be reproduced in any form without permission in writing from the publisher. Library of Congress Catalog Card Number: 74-22535 PRINTED IN THE UNITED STATES OF AMERICA Preface With the development over the past decade of computer-based analysis methods, the teaching of structural analysis subjects has been revolutionized. The traditional division between structural analysis and structural mechanics became no longer necessary, and instead of teaching a preponderance of solu-tion details it is now possible to focus on the underlying theory. What has been done here is to integrate analysis and mechanics in a sys-tematic presentation which includes the mechanics of a member, the matrix formulation of the equations for a system of members, and solution techniques. The three fundamental steps in formulating a problem in solid mechanics-enforcing equilibrium, relating deformations and displacements, and relating forces and deformations-form the basis of the development, and the central theme is to establish the equations for each step and then discuss how the com-plete set of equations is solved. In this way, a reader obtains a more unified view of a problem, sees more clearly where the various simplifying assumptions are introduced, and is better prepared to extend the theory. The chapters of Part I contain the relevant topics for an essential back-ground in linear algebra, differential geometry, and matrix transformations. Collecting this material in the first part of the book is convenient for the con-tinuity of the mathematics presentation as well as for the continuity in the following development. Part II treats the analysis of an ideal truss. The governing equations for small strain but arbitrary displacement are established and then cast into matrix form. Next, we deduce the principles of virtual displacements and virtual forces by manipulating the governing equations, introduce a criterion for evaluating the stability of an equilibrium position, and interpret the gov-erning equations as stationary requirements for certain variational principles. These concepts are essential for an appreciation of the solution schemes de-scribed in the following two chapters. Part III is concerned with the behavior of an isolated member. For com-pleteness, first are presented the governing equations for a deformable elastic solid allowing for arbitrary displacements, the continuous form of the princi-ples of virtual displacements and virtual forces, and the stability criterion. Unrestrained torsion-flexure of a prismatic member is examined in detail and then an approximate engineering theory is developed. We move on to re-strained torsion-flexure of a prismatic member, discussing various approaches for including warping restraint and illustrating its influence for thin-walled iiiiv PREFACE open and closed sections. The concluding chapters treat the behavior of planar and arbitrary curved members. How one assembles and solves the governing equations for a member sys-tem is discussed in Part IV. First, the direct stiffness method is outlined; Contents then a general formulation of the governing equations is described. Geo-metrically nonlinear behavior is considered in the last chapter, which dis-cusses member force-displacement relations, including torsional-flexural coupling, solution schemes, and linearized stability analysis. I-MATHEMATICAL PRELIMINARIES The objective has been a text suitable for the teaching of modern structural member system analysis, and what is offered is an outgrowth of lecture notes 1 Introduction to Matrix Algebra 3 developed in recent years at the Massachusetts Institute of Technology. To 3 the many students who have provided the occasion of that development, I am 1-1 Definition of a Matrix deeply appreciative. Particular thanks go to Mrs. Jane Malinofsky for her 1-2 Equality, Addition, and Subtraction of Matrices patience in typing the manuscript, and to Professor Charles Miller for his 1-3 Matrix Multiplication 5 5 1-4 Transpose of a Matrix 8 encouragement. 1-5 Special Square Matrices 10 JEROME J. CONNOR 1-6 Operations on Partitioned Matrices 12 1-7 Definition and Properties of a Determinant 16 Cambridge, Mass. 1-8 Cofactor Expansion Formula 19 1-9 Cramer's Rule 21 January, 1976 1-10 Adjoint and Inverse Matrices 22 1-11 Elementary Operations on a Matrix 24 1-12 Rank of a Matrix 27 1-13 Solvability of Linear Algebraic Equations 30 2 Characteristic-Value Problems and Quadratic Forms 46 2-1 Introduction 46 2-2 Second-Order Characteristic-Value Problem 48 2-3. Similarity and Orthogonal Transformations 52 2-4 The nth-Order Symmetrical Characteristic-Value Problem 55 2-5 Quadratic Forms 57 3 Relative Extrema for a Function 66 3-1 Relative Extrema for a Function of One Variable 66 3-2 Relative Extrema for a Function of n Independent Variables 71 3-3 Lagrange Multipliers 75 4 Differential Geometry of a Member Element 81 4-1 Parametric Representation of a Space Curve 81 4-2 Arc Length 82 vvi . CONTENTS CONTENTS Vii 4-3 4-4 4-5 Unit Tangent Vector Principal Normal and Binormal Vectors Curvature, Torsion, and the Frenet Equations 85 86 88 8-4 8-5 Incremental Formulation; Classical Stability Criterion Linearized Stability Analysis 191 200 4-6 Summary of the Geometrical Relations for a Space 4-7 Curve Local Reference Frame for a Member Element 91 92 9 Force Method-Ideal Truss 210 4-8 Curvilinear Coordinates for a Member Element 94 9-1 General 210 9-2 Governing Equations-Algebraic Approach 211 5 Matrix Transformations for a Member Element 5-1 Rotation