The following will be discussed in this chapter: Stability of structures, euler’s formula for pin-ended beams, extension of euler’s formula, eccentric loading, the secant formula, design of columns under centric load, design of columns under an eccentric load. | Third Edition CHAPTER MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Columns Lecture Notes: J. Walt Oler Texas Tech University © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Columns Stability of Structures Euler’s Formula for Pin-Ended Beams Extension of Euler’s Formula Sample Problem Eccentric Loading; The Secant Formula Sample Problem Design of Columns Under Centric Load Sample Problem Design of Columns Under an Eccentric Load © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 10 - 2 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Stability of Structures • In the design of columns, cross-sectional area is selected such that - allowable stress is not exceeded σ= P ≤ σ all A - deformation falls within specifications δ= PL ≤ δ spec AE • After these design calculations, may discover that the column is unstable under loading and that it suddenly becomes sharply curved or buckles. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 10 - 3 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Stability of Structures • Consider model with two rods and torsional spring. After a small perturbation, K (2∆θ ) = restoring moment L L P sin ∆θ = P ∆θ = destabilizing moment 2 2 • Column is stable (tends to return to aligned orientation) if L P ∆θ Pcr. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 10 - .
