YIELDING OF COLUMN SECTION SUBJECTED TO BIAXIAL BENDING
57
y
+σ s
h
o
x y-f(x)
−σ s
b Figure 5.4 Ultimate yielding state of the rectangular section
where f ðxÞ is an undetermined function and can be determined with functional extremum theory. Assume that N and My are constant and then f ðxÞ should make Mx maximum. This problem is actually a definite integral inflexion problem with restrain of the definite integral. According to the Lagrange multiplicator method, the functional of the problem can be expressed as H ¼ s ½h2 =4 f 2 ðxÞ l1 ½2 s xf ðxÞ l2 ½2 s f ðxÞ :
ð5:13Þ
From the Euler equation, @H d @H ¼ 0; @f ðxÞ dx @f 0 ðxÞ
ð5:14Þ
f ðxÞ ¼ l1 x l2 :
ð5:15Þ
one obtains
Equation (5.15) indicates that the neutral curve of the rectangular section is a straight line (see Figure 5.5). Substituting Equation (5.15) into Equation (5.12) yields ð5:16aÞ
N ¼ 2 s bl2 ; " # bh2 l21 b 3 2 b Mx ¼ 2 s l2 ; 8 3 2 2
ð5:16bÞ
1 My ¼ s b3 l1 : 6
ð5:16cÞ
y
y
o
(a)
x
o
(b)
y
x
o
(c)
Figure 5.5 Neutral axial positions of the rectangular section
x