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Column Design (4.11-4.13). MAE 316 – Strength of Mechanical Components NC State University Department of Mechanical and Aerospace Engineering. Real world examples. Continue…. Jack. Column failure. Quebec Bridge designed by Theodore Cooper. It collapsed in 1907. . Introduction.
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Column Design(4.11-4.13) MAE 316 – Strength of Mechanical Components NC State University Department of Mechanical and Aerospace Engineering Column Design
Real world examples Column Design
Continue… Jack Column Design
Column failure Quebec Bridge designed by Theodore Cooper. It collapsed in 1907. Column Design
Introduction • “Long” columns fail structurally (buckling failure) • “Short” columns fail materially (yielding failure) • We will examine buckling failure first, and then transition to yielding failure. P P “Long” “Short” P Column Design P
Buckling of Columns (4.12) • For a simply supported beam (pin-pin) y, v v(x) y, v F.B.D The solution to the above O.D.E is: x Column Design
Buckling of Columns (4.12) • Apply boundary conditions: v(0)=0 & v(l)=0 • Either A = 0 (trivial solution, no displacement) or • The critical load is then • The lowest critical load (called the Euler buckling load) occurs when n=1. where n = 1, 2, … Column Design
Buckling of Columns (4.12) • The corresponding deflection at the critical load is • The beam deflects in a half sine wave. • The amplitude A of the buckled beam cannotbe determined by this analysis. It is finite, butnon-linear analysis is required to proceedfurther Pcr A Column Design
Buckling of Columns (4.12) • Define the radius of gyration as • Pcr can be written as • If l/k is large, the beam is long and slender. • If l/k is small, the beam is short an stumpy. • To account for different boundary conditions (other than pin-pin), use the end-condition constant, C. Column Design
Buckling of Columns (4.12) • Theoretical end condition constants for various boundary conditions (Figure 4-18 in textbook) • (a) pin-pin • (b) fixed-fixed • (c) fixed-free • (d) fixed-pin • Values of C for design are given in Table 4-2 in the textbook Column Design
Buckling of Columns (4.12) 1.Stability (1) Unstable equilibrium (2) Stable equilibrium (3) Neutral equilibrium Column Design
F F Buckling of Columns (4.12) Column Design
Example 4-16 Column Design
Example Column AB carries a centric load P of magnitude 15 kips. Cables BC and BD are taut and prevent motion of point B in the xz-plane. Using Euler’s formula and a factor of safety of 2.2, and neglecting the tension in the cables, determine the maximum allowable length L.Use E = 29 x 106 psi. Column Design
How to prevent buckling? • 1) Choose proper cross-sections • 2) End conditions • 3) Change compression to tension (cable stayed bridges) Column Design
Critical Stress (4.12) Note: Sy = yield strength = σy • Plot σcr=Pcr/A vs. l/k • Long column: buckling occurs elastically before the yield stress is reached. • Short column: material failure occurs inelastically beyond the yield stress. • A “Johnson Curve” can be used to determine the stress for an intermediate column Johnson Curve Sy l/k Failure No failure l/k ? Column Design
Critical Stress (4.13) • Johnson equation • Where • Notice asl/k 0, Pcr/A Sy Column Design
Critical Stress (4.13) • To find the transition point (l/k)1between the Johnson and Euler regions Column Design
Design of Columns (4.13) • Procedure • Calculate (l/k)1and l/k • If l/k ≥ (l/k)1use Euler’s equation • If l/k≤ (l/k)1use Johnson’s equation • Notice we haven’t included factor of safety yet… Column Design
Example 4-19 Column Design
Example Find the required outer diameter, with F.S.=3, for the column shown below. Assume P = 15 kN, L = 50 mm, t = 5 mm, Sy = 372 MPa, E = 207,000 MPa, and the material is 1018 cold-drawn steel. t do Column Design
Example Find the maximum allowable force, Fmax, that can be applied without causing pipe to buckle. Assume L = 12 ft, b = 5 ft, do = 2 in, t = 0.5 in, and the material is low carbon steel. F t L do b Column Design
Struts or short compression members • Strut - short member loaded in compression • If eccentricity exists, maximum stress is at B with axial compression and bending. • Note that it is not a function of length • If bending deflection is limited to 1 percent of e, then from Eq. (4-44), the limiting slenderness ratio for strut is Column Design