Civil Engineering Reference
In-Depth Information
• Hooke's law is applied
• Member deflections are small,
the solution of the differential equation of the deflection curve (Equation 6.14), where
U
k 2 e and e is the eccentricity of load ( Figure 6.5c) , is the secant formula (Chen
and Lui, 1987)
=−
P y
P cr =
(ec/r 2 ) sec (
2 EI ,
(6.22)
/ 2 ) P cr L 2 /
1
+
π
π
where P y =
A g F y , c is the distance from the neutral axis to the extreme fiber of the
member cross section, and r is the radius of gyration of the member cross section.
The secant formula was considered appropriate for inelastic buckling of mem-
bers from initial curvature and load eccentricity. However, it does not include the
consideration of residual stresses, which are of considerable importance in modern
steel structures. Therefore, Equation 6.22 is no longer used to determine the critical
buckling force of compression members. The equation was used in the AREMA rec-
ommended practice prior to 1969, but was discontinued as a basis for compression
member design because of the difficulty associated with its use and indication that
Euler-type formulas are appropriate for eccentrically loaded compression members
(AREMA, 2008).
6.3.1.1.3 Elastic Buckling of Members with Geometric Imperfections
(Initial Out-of-Straightness)
Assuming that
• The member is concentrically loaded
• Plane sections remain plane after deformation
• Flexural deflection is considered only (shear deflection is neglected)
• Hooke's law is applied
• Member deflections are small,
the solution of the differential equation of the deflection curve (Equation 6.14), where
U
k 2
=−
δ 0 sin (
π
x/L) ( Figure 6.5d), is the Perry-Robertson formula (Chen and Lui,
1987)
P y
P cr =
2 EI))) ,
(6.23)
1
+
(
δ 0 c/r 2 )( 1 /( 1
(P cr L 2 /
π
where
δ 0 is the out-of-straightness at the middle of the member (Figure 6.5d).
For low values of L/r (generally less than about 60), out-of-straightness geometric
imperfections are usually not an important design consideration.
6.3.1.2
Inelastic Compression Members
Steel railway bridge members of the usual length and slenderness will buckle at
loads above the proportional limit, F p , ( Figure 6.3) when some cross-section fibers
 
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