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A New Approach To Solve Beam Deflection Problems Using The Method Of Segments

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Conference

2006 Annual Conference & Exposition

Location

Chicago, Illinois

Publication Date

June 18, 2006

Start Date

June 18, 2006

End Date

June 21, 2006

ISSN

2153-5965

Conference Session

What's New in Mechanics of Materials?

Tagged Division

Mechanics

Page Count

21

Page Numbers

11.79.1 - 11.79.21

DOI

10.18260/1-2--98

Permanent URL

https://peer.asee.org/98

Download Count

5250

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Paper Authors

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Hartley T. Grandin Worcester Polytechnic Institute

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Hartley T. Grandin, Jr. is a Professor Emeritus of Engineering Mechanics and Design in the Mechanical Engineering Department at Worcester Polytechnic Institute. He has authored the textbook Fundamentals of the Finite Element Method that was published by Macmillan in 1986. Since his retirement from WPI in 1996, he teaches a mechanics of materials course each year and is currently writing the fifth draft of an introductory textbook with the co-author. In 1983 he received the WPI Board of Trustees’ Award for Outstanding Teaching. He received his B.S. in 1955 and an M.S. in 1960 in Mechanical Engineering from Worcester Polytechnic Institute and a Ph.D. in Engineering Mechanics from the Department of Metallurgy, Mechanics and Materials Science at Michigan State University in 1972. E-mail: hgrandin@rcn.com and hgrandin@wpi.edu.

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Joseph Rencis University of Arkansas

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Abstract
NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract

A New Approach to Solve Beam Deflection Problems using the Method of Segments

Abstract

This paper presents a new approach to solving beam deflection problems. The approach involves the direct application of derived force-deformation formulas, a procedure commonly used with axial and torsion bar problems. This direct application of derived force-deformation formulas, referred to by the authors as Method of Segments, is extended to beam deflection analysis in order to provide a solution procedure for beams that is consistent in philosophy and application with that presented in most mechanics of materials textbooks for axially loaded bars and torsionally loaded shafts. The beam force-deformation formulas, involving slope and displacement, are derived by double integration for a beam of uniform cross-section, material and distributed loading with end shear forces and couples. Application of the formulas is direct and requires no integration or continuity equations. Furthermore, by identifying segments of uniform geometry, material and distributed loading, this approach can easily be applied to beams of discontinuous geometry and material that supports both concentrated and distributed loading. Introduction

The great majority of undergraduate mechanics of materials textbooks1-50 directly apply previously derived force-deformation formulas to problems involving the straight bar subjected to centric axial loading and the straight circular cross-section bar (shaft) subjected to twisting couples. In both cases, the bars are uniform in cross-section and material, and the concentrated loads are applied at the ends and distributed loads are continuous along the full length. The force-deformation formulas are shown in Figures 1 and 2 for bars subjected to centric axial loading and twisting couples, respectively. These formulas, referred to as Material Law Formulas by the authors, are commonly found in mechanics of materials textbooks1-50.

y L ua u(x) ub p, force/length x F F a b a b x

Figure 1. Material Law Formulas for a uniform bar with end centric axial and centric uniform distributed loads plus temperature change.

Grandin, H. T., & Rencis, J. (2006, June), A New Approach To Solve Beam Deflection Problems Using The Method Of Segments Paper presented at 2006 Annual Conference & Exposition, Chicago, Illinois. 10.18260/1-2--98

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