Analysis of Inelastic Deformations

Barry T. Rosson

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Conference: 2015 ASEE Annual Conference & Exposition
Location: Seattle, Washington
Publication Date: June 14, 2015
Start Date: June 14, 2015
End Date: June 17, 2015
ISBN: 978-0-692-50180-1
ISSN: 2153-5965
Conference Session: Mechanics of Materials
Tagged Division: Mechanics
Page Count: 25
Page Numbers: 26.212.1 - 26.212.25
DOI: 10.18260/p.23551
Permanent URL: https://peer.asee.org/23551
Download Count: 468

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Barry T. Rosson P.E. Florida Atlantic University

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Dr. Barry Rosson is a Professor of Structural Engineering at Florida Atlantic University. Prior to joining FAU, he was a professor at the University of Nebraska for 16 years. His areas of research interest are in nonlinear structural mechanics, numerical methods, structural dynamics and steel structures. He has received numerous campus and college-wide teaching and service awards. At the national level of the American Society of Civil Engineers, he has served as the Chair of the society-wide Committee on Professional Practice, President of the Architectural Engineering Institute, and Associate Editor of the Journal of Structural Engineering. He is a Fellow of both ASCE and AEI and is a Registered Professional Engineer. Prior to working as a professor, he was a consulting engineer at Ellisor and Tanner, Inc. in Dallas, Texas.

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Abstract

Analysis of Inelastic Deformations Using Closed-Form Reduced Rigidity EquationsStudents learning about inelastic deformations in structural mechanics courses are oftenpresented with detailed information about the distribution of stress in the spread of plasticityzones, but straightforward relationships that effectively deal with the reduced rigidity and how itultimately affects the deformations of the structure have been lacking up until now. This paperpresents an approach whereby the instruction of a broad range of elasto-plastic deformationconditions can be taught to students where in the spread of plasticity zones the actual conditionsof stress and reduced rigidity remain apparent and are completely accounted for whendetermining member deformations. Typically when presenting this subject for instructionalpurposes, textbook authors use elastic perfectly-plastic material behavior and rectangular cross-sections for flexural members and solid circular shafts for torsional members. This sameapproach is taken here because, as it will be shown, these shapes lend themselves to closed-formed expressions and are thus ideal for introducing inelastic deformations in a straightforwardmanner.The development of the following closed-form reduced rigidity equation for elasto-plasticrectangular beams is discussed first. = 3−2 M is the bending moment where the reduced flexural rigidity EIep is to be evaluated, My is theyield moment (Sσy) and EI is the elastic flexural rigidity.Several examples are presented to illustrate the ease with which this closed-form equation can beused to determine elasto-plastic beam deflections. When using this equation with the virtualwork method, convenient closed-form area and centroid formulas of the M/EIep curvaturediagrams in the spread of plasticity zones are developed for both the linear moment condition(refer to the table below) and quadratic moment condition. When using the virtual workrelationship ∆ = ΣAiδMi (over a specified region i of the beam, Ai is the area of the curvaturediagram and δMi is the virtual moment at the centroid of the curvature diagram), these area andcentroid formulas provide the students with a powerful new tool to explore elasto-plasticdeterminate beams using a wide range of loading and support conditions.The development of closed-form reduced rigidity equations for elasto-plastic rectangular beam-columns is discussed next. Under moment-thrust conditions that cause an inelastic response, twoseparate stress states are possible – (Case I ) where yielding occurs only on one end of the cross-section and (Case II ) where yielding occurs on both ends. Using these reduced rigidityequations, examples are presented that illustrate how these closed-form expressions can be usedto analyze elasto-plastic beam-columns under varying conditions of moment and thrust. 3 ⁄ 1 ⁄ = 1− + 4 1− ⁄ 4 1− ⁄ 3 3 = − −2 Py is the yield thrust (Aσy).Finally, the closed-form reduced rigidity equation is developed for a solid circular shaft underelasto-plastic torsional response. Examples are given to illustrate the ease with which thisrelationship can be used directly to determine the angle of twist of elasto-plastic determinateshafts with multiple torsional loads. = 4−3 !T is the torque where the reduced torsional rigidity JGep is to be evaluated, Ty is the yield torque(Jτy/c) and JG is the elastic torsional rigidity.The equations and methodologies presented use closed-form reduced rigidity equations toillustrate the effects of inelastic material behavior over a broad range of structural conditions andallow the subject to be taught as straightforward natural extensions of the elastic analysismethods.Table 1. Area and centroid formulas for the case of linear moment variation over the elasto-plastic response region Lep (Mmax ≤ Mp)

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Rosson, B. T. (2015, June), Analysis of Inelastic Deformations Paper presented at 2015 ASEE Annual Conference & Exposition, Seattle, Washington. 10.18260/p.23551

TY  - CPAPER
AB  -                        Analysis of Inelastic Deformations Using                       Closed-Form Reduced Rigidity EquationsStudents learning about inelastic deformations in structural mechanics courses are oftenpresented with detailed information about the distribution of stress in the spread of plasticityzones, but straightforward relationships that effectively deal with the reduced rigidity and how itultimately affects the deformations of the structure have been lacking up until now. This paperpresents an approach whereby the instruction of a broad range of elasto-plastic deformationconditions can be taught to students where in the spread of plasticity zones the actual conditionsof stress and reduced rigidity remain apparent and are completely accounted for whendetermining member deformations. Typically when presenting this subject for instructionalpurposes, textbook authors use elastic perfectly-plastic material behavior and rectangular cross-sections for flexural members and solid circular shafts for torsional members. This sameapproach is taken here because, as it will be shown, these shapes lend themselves to closed-formed expressions and are thus ideal for introducing inelastic deformations in a straightforwardmanner.The development of the following closed-form reduced rigidity equation for elasto-plasticrectangular beams is discussed first.                                          =       3−2       M is the bending moment where the reduced flexural rigidity EIep is to be evaluated, My is theyield moment (Sσy) and EI is the elastic flexural rigidity.Several examples are presented to illustrate the ease with which this closed-form equation can beused to determine elasto-plastic beam deflections. When using this equation with the virtualwork method, convenient closed-form area and centroid formulas of the M/EIep curvaturediagrams in the spread of plasticity zones are developed for both the linear moment condition(refer to the table below) and quadratic moment condition. When using the virtual workrelationship ∆ = ΣAiδMi (over a specified region i of the beam, Ai is the area of the curvaturediagram and δMi is the virtual moment at the centroid of the curvature diagram), these area andcentroid formulas provide the students with a powerful new tool to explore elasto-plasticdeterminate beams using a wide range of loading and support conditions.The development of closed-form reduced rigidity equations for elasto-plastic rectangular beam-columns is discussed next. Under moment-thrust conditions that cause an inelastic response, twoseparate stress states are possible – (Case I ) where yielding occurs only on one end of the cross-section and (Case II ) where yielding occurs on both ends. Using these reduced rigidityequations, examples are presented that illustrate how these closed-form expressions can be usedto analyze elasto-plastic beam-columns under varying conditions of moment and thrust.                                               3   ⁄                1   ⁄                         = 1−                    +                                               4 1− ⁄               4 1− ⁄                                                                 3           3                        =                     −       −2 Py is the yield thrust (Aσy).Finally, the closed-form reduced rigidity equation is developed for a solid circular shaft underelasto-plastic torsional response. Examples are given to illustrate the ease with which thisrelationship can be used directly to determine the angle of twist of elasto-plastic determinateshafts with multiple torsional loads.                                                   =       4−3                                                              !T is the torque where the reduced torsional rigidity JGep is to be evaluated, Ty is the yield torque(Jτy/c) and JG is the elastic torsional rigidity.The equations and methodologies presented use closed-form reduced rigidity equations toillustrate the effects of inelastic material behavior over a broad range of structural conditions andallow the subject to be taught as straightforward natural extensions of the elastic analysismethods.Table 1. Area and centroid formulas for the case of linear moment variation over the         elasto-plastic response region Lep (Mmax ≤ Mp)
AU  - Barry T. Rosson P.E.
CY  - Seattle, Washington
DA  - 2015/06/14
PB  - ASEE Conferences
TI  - Analysis of Inelastic Deformations
UR  - https://peer.asee.org/23551
DO  - 10.18260/p.23551
ER  -