# STRUCTURAL ENGINEERING AND GEOMECHANICS Liner Analysis of Structural System Aslam Kassimali LINEAR ANALYSIS OF STRUCTURAL SYSTEMS

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6 engineering structures, the equations of condition are usually used to satisfy the conditions of zero bending moments at the locations of internal hinges. Consider, for example, a beam composed of two members AB and BC connected by an internal hinge at B, and supported by four external support reactions, as shown in Figure 2. As the beam is in equilibrium under the action of applied loads and support reactions, it must satisfy the three equations of equilibrium ( F X = 0, F = Y 0, and M = 0 ). Furthermore, since the internal hinge at B cannot transmit moment, it provides one AB independent equation of condition that the bending moment at B is zero ( M = 0 or B BC M B = 0 ). Since all four unknown reactions of the beam of Figure 2 can be determined by solving the three equilibrium equations plus one equation of condition, the beam is considered to be statically determinate. Thus, in general, if a beam is supported by r (number of) support reactions and has e c (number of) equation of conditions, and if r = 3+ ec, then the beam is statically determinate because all of its unknown reactions can be determined by solving the three equations of equilibrium plus e c equations of condition. However, if r > 3+ ec, then all of the unknown reactions ( r ) cannot be determined from the available equations ( 3+ ec ) and the beam is considered to be statically indeterminate. For such beams, the degree of indeterminacy is given by i = r ( 3 + ec ). Finally, if r < 3+ ec, then the beam is not supported by a sufficient number of reactions to prevent all possible movements in its plane, and is therefore referred to as statically unstable. Figure 2. Statically determinate beam The conditions of static determinacy and indeterminacy stated in the preceding paragraph, although necessary, are not sufficient in the sense that a beam may be supported by enough reactions ( r 3+ ec ), but may still be unstable due to improper arrangement of supports or internal hinges. Such a beam is referred to as geometrically unstable. In order for a beam to be geometrically stable, it must be supported by at least three support reactions, all of which must neither be parallel nor concurrent. Also, any part of the beam containing three hinges in a row should be supported at a minimum of one intermediate point to avoid geometric instability.

7 3.2. Plane Frames A frame is considered to be statically determinate, if all of its support reactions and internal member end forces can be determined by applying the equations of equilibrium and condition. Consider an arbitrary plane frame composed of m members and j joints. The frame is supported by r support reactions, and has e c equations of condition. The analysis of the frame involves evaluation of r unknown external reactions and six internal end forces for each member (i.e., axial force, shear and bending moment, at each member end). Thus, a total of r+ 6m unknown quantities needs to be determined. As three equilibrium equations can be written for each member and each joint, the total number of equations (equilibrium plus condition) available is 3 m+ j + e. By comparing the number of unknowns to be determined with the ( ) c number of available equations, the conditions for static determinacy, indeterminacy, and instability of plane frames can be expressed as: 3 m+ r = 3 j+ ec statically determinate frame 3 m+ r > 3 j+ ec statically indeterminate frame 3 m+ r < 3 j+ e statically unstable frame c For statically indeterminate frames, the degree of indeterminacy is given by i = ( 3 m+ r) ( 3j+ ec ). As in the case of beams, each internal hinge located between the ends of a plane frame member provides one equation of condition. If a frame contains a hinged joint connecting several members, then the number of equations of condition at the joint equals the number of members meeting at the joint minus one Bibliography TO ACCESS ALL THE 39 PAGES OF THIS CHAPTER, Visit: (1) ASCE Standard Minimum Design Loads for Buildings and Other Structures, ASCE/SEI 7-05 (2006). American Society of Civil Engineers, Virginia, USA. [This standard specifies minimum design loads for structures] International Building Code (2006). International Code Council, Virginia, USA. [This code contains minimum design loads for structures] Kassimali A. (2009). Structural analysis, Fourth Edition, Cengage Learning. [This book presents the classical methods of structural analysis] Kassimali A. (1999). Matrix Analysis of Structures, Brooks/Cole, USA. [This book covers the matrix (computer-based) methods of structural analysis]

8 Meriam, J.L. and Kraige, L.G. (1986). Engineering Mechanics, Volume 1, Statics, Second Edition, John Wiley and Sons, New York, USA. [Chapter 5 of this book covers analysis of cables subjected to various types of Loads] Biographical Sketch Aslam Kassimali was born in Karachi, Pakistan. He received his Bachelor of Engineering (B.E.) degree in civil engineering from the University of Karachi (N.E.D. College) in Pakistan in In 1971, he earned a Master of Engineering (M.E.) degree in civil engineering from the Iowa State University in Ames, Iowa, USA. After completing further studies and research at the University of Missouri at Columbia in the USA, he received Master of Science (M.S.) and Ph.D. degrees in civil engineering in 1974 and 1976, respectively. His practical experience includes working as a Structural Design Engineer for Lutz, Daily and Brain, Consulting Engineers (USA), from January to July 1973, and as a Structural Engineering Specialist and Analyst for Sargent & Lundy Engineers in Chicago (USA) from 1978 to He joined Southern Illinois University Carbondale (USA) as an Assistant Professor in 1980, was promoted the rank of Professor in 1993, and was awarded the title of Distinguished Teacher in He is currently a Professor and Distinguished Teacher in the Department of Civil & Environmental Engineering at Southern Illinois University in Carbondale, Illinois (USA). He has authored and coauthored four textbooks on structural analysis and mechanics, and a number of papers in the area of nonlinear structural analysis. Dr. Kassimali is a member of the American Society of Civil Engineers (ASCE).

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