Third Edition MECHANICS OF 10 MATERIALS - site. OF MATERIALS Third Edition Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University CHAPTER

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MECHANICS OF MATERIALSThird EditionFerdinand P. BeerE. Russell Johnston, Jr.John T. DeWolfLecture Notes:J. Walt OlerTexas Tech UniversityCHAPTER 2002 The McGraw-Hill Companies, Inc. All rights reserved. 10ColumnsPresented by:Dr. Mohammed Arafa 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf10 - 2ColumnsStability of StructuresEulers Formula for Pin-Ended BeamsExtension of Eulers FormulaSample Problem 10.1Eccentric Loading; The Secant FormulaSample Problem 10.2Design of Columns Under Centric LoadSample Problem 10.4Design of Columns Under an Eccentric Loadcontents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf10 - 3Stability of Structures In the design of columns, cross-sectional area is selected such that- allowable stress is not exceededallAP - deformation falls within specificationsspecAEPL After these design calculations, may discover that the column is unstable under loading and that it suddenly becomes sharply curved or buckles. contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf10 - 4Stability of Structures Consider model with two rods and torsional spring. After a small perturbation, moment ingdestabiliz 2sin2moment restoring 2LPLPK Column is stable (tends to return to aligned orientation) if LKPPKLPcr422 contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf10 - 5Stability of Structures Assume that a load P is applied. After a perturbation, the system settles to a new equilibrium configuration at a finite deflection angle. sin42sin2crPPKPLKLP Noting that sin < , the assumed configuration is only possible if P > Pcr. contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf10 - 6Eulers Formula for Pin-Ended Beams Consider an axially loaded beam. After a small perturbation, the system reaches an equilibrium configuration such that02222yEIPdxydyEIPEIMdxyd Solution with assumed configuration can only be obtained if 2222222rLEALArEAPLEIPPcrcrcontents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf10 - 7Eulers Formula for Pin-Ended Beams s ratioslendernesrLtresscritical srLEALArEAPAPLEIPPcrcrcrcr 2222222 The value of stress corresponding to the critical load, Preceding analysis is limited to centric loadings.contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf10 - 8Extension of Eulers Formula A column with one fixed and one free end, will behave as the upper-half of a pin-connected column. The critical loading is calculated from Eulers formula, length equivalent 22222LLrLELEIPeecrecrcontents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf10 - 9Extension of Eulers Formulacontents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf10 - 10Sample Problem 10.1An aluminum column of length L and rectangular cross-section has a fixed end at B and supports a centric load at A. Two smooth and rounded fixed plates restrain end A from moving in one of the vertical planes of symmetry but allow it to move in the other plane.a) Determine the ratio a/b of the two sides of the cross-section corresponding to the most efficient design against buckling.b) Design the most efficient cross-section for the column.L = 20 in.E = 10.1 x 106 psiP = 5 kipsFS = 2.5contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf10 - 11Sample Problem 10.1 Buckling in xy Plane:127.01212,231212aLrLaraabbaAIrzzezzz Buckling in xz Plane:12/21212,231212bLrLbrbababAIryyeyyy Most efficient design:27.012/2127.0,,babLaLrLrLyyezze35.0baSOLUTION:The most efficient design occurs when the resistance to buckling is equal in both planes of symmetry. This occurs when the slenderness ratios are equal.contents.pptcontents.pptcontents.ppt 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSThirdEditionBeer Johnston DeWolf10 - 12Sample Problem 10.1L = 20 in.E = 10.1 x 106 psiP = 5 kipsFS = 2.5a/b = 0.35 Design: 26226222crcr6.138psi101.100.35lbs 125006.138psi101.100.35lbs 12500kips 5.12kips 55.26.13812in 202122bbbbrLEbbAPPFSPbbbLrLecrcryein. 567.035.0in. 620.1babcontents.pptcontents.pptcontents.ppt

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