Civil Engineering Reference
In-Depth Information
6.3.1.1.1 Elastic Buckling with Load, P, Applied along the Centroidal
Axis of the Member
Assuming that
• The member has no geometric imperfections (perfectly straight)
• 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 differential equation of the deflection curve is
d 2 y(x)
d x 2
k 2 y(x)
+
=
U ,
(6.14)
where y(x) is the lateral deflection of compression member, U depends on the effects
on load, P , of the compression member end conditions, and
P
EI ,
k 2
=
where P is the load applied at the end and along the centroidal axis of the compression
member. The solution of Equation 6.14 for the elastic critical buckling force, P cr ,is
readily accomplished by consideration of the appropriate boundary conditions (Wang
et al., 2005). The elastic critical buckling force for various member end conditions is
shown in Table 6.3. Figures 6.4 and 6.5 illustrate the various compression member
end conditions in Table 6.3. The critical buckling force can be expressed as
2 EI
(KL) 2 .
π
P cr =
(6.15)
TABLE 6.3
Elastic Critical Buckling Force for Concentrically Loaded Members
with Various End Conditions
End Condition
U
P cr
2 EI/L 2
Both ends pinned (Figure 6.4a)
0
π
2 EI/L 2
Both ends fixed (Figure 6.4b)
(V/EI)x + M/EI
4 π
k 2
2 EI/ 4 L 2
One end fixed and other end free (Figure 6.4c)
Δ
π
2 EI/L 2 )
One end hinged and other end fixed (Figure 6.4d)
(M/EIL)x
2.046 ( π
2 EI/L 2
One end guided and other end fixed (Figure 6.5a)
P Δ / 2 EI
π
2 EI/ 4 L 2
π
One end hinged and other end guided (Figure 6.5b)
0
L
Bending
moment at end of member, Δ = Lateral deflection at free or guided end of member with other
end fixed.
=
Length of member between end supports, V
=
Shear force in member, M
=
 
 
Search WWH ::




Custom Search