Double Modulus 2j2z1v

57

Buckling of Columns

theory and agrees better than the double-modulus theory with test results. These inelastic buckling analyses using effective modulus are just academic history today. The present-day finite element codes capable of conducting incremental analyses of the geometric and material nonlinearities, as refined in their final form in the 1980s, can correctly evaluate the inelastic column strengths, including the effects of initial imperfections, inelastic material properties including strain hardening, and residual stresses.

1.13.1. Double-Modulus (Reduced Modulus) Theory Assumptions 1) Small displacement theory holds. 2) Plane sections remain plane. This assumption is called Bernoulli, or Euler, or Navier hypothesis. 3) The relationship between the stress and strain in any longitudinal fiber is given by the stress-strain diagram of the material (compression and tension, the same relationship). 4) The column section is at least singly symmetric, and the plane of bending is the plane of symmetry. 5) The axial load remains constant as the member moves from the straight to the deformed position. Loading Et governs

P

P = Pcr

cr p

Unloading E governs

NA

h2

h1 dx e

R

dx

cg

d z2

1

= E

2

= Et

2

s2

1

Inside concave

2

Et: the slope of stress-strain curve at = cr

z1

s1 cr

Compressive stress

Figure 1-28 Reduced modulus model

1

Outside convex

58

Chai Yoo

In small displacement theory, the curvature of the bent column is 1 d2y df ^ 2 ¼ R dx dx

(1.13.1)

From a similar triangle relationship, the flexural strains are computed 31 ¼ z1 y00

(1.13.2)

32 ¼ z2 y00

(1.13.3)

and the corresponding stresses are s1 ¼ Eh1 y00

(1.13.4)

s2 ¼ Et h2 y00

(1.13.5)

where Et ¼ tangent modulus, s1 ðtensionÞ ¼ Ez1 y00 and s2 ðcompressionÞ ¼ Et z2 y00 . The pure bending portion (no net axial force) requires Z

h1

h2

Z

s1 dA þ 0

s2 dA ¼ 0

(1.13.6)

0

Equating the internal moment to the external moment yields Z

h1

s1 z1 dA þ

0

Z

h2

s2 z2 dA ¼ Py

Equation (1.13.6) is expanded to Z h1 Z 00 00 Ey z1 dA þ Et y 0

Let

Q1 ¼

Z

h2

z2 dA ¼ 0

(1.13.8)

0

h1

z1 dA and

(1.13.7)

0

Q2 ¼

0

Z

h2

z2 dA0EQ1 þ Et Q2 ¼ 0 0

(1.13.9) Equation (1.13.7) is expanded to ! Z y E

h1

00

0

z21 dA þ Et

Z

0

h2

"

z22 dA

¼ Py

(1.13.10)

Buckling of Columns

59

EI1 þ Et I2 (1.13.11) I which is called the reduced modulus that depends on the stress-strain relationship of the material and the shape of the cross section. I1 is the moment of inertia of the tension side cross section about the neutral axis and I2 is the moment of inertia of the compression side cross section such that Z h1 Z h2 2 I1 ¼ z1 dA and I2 ¼ z22 dA (1.13.12) Let

E ¼

0

0

Equation (1.13.10) takes the form EIy00 þ Py ¼ 0

(1.13.13)

Equation (1.13.13) is the differential equation of a column stressed into the inelastic range identical to Eq. (1.3.3) except that E has been replaced by E, the reduced modulus. If it can be assumed that E is constant, then Eq. (1.13.13) is a linear differential equation with constant coefficients, and its solution is identical to that of Eq. (1.3.3), except that E is replaced by E. Corresponding critical load and critical stress based on the reduced modulus are p2 EI (1.13.14) Pr;cr ¼ 2 ‘ and p2 E (1.13.15) sr;cr ¼ ! "2 ‘ r Introducing sr ¼

E Et I 2 I 1 þ < 1:0 ¼ E I I E

and

s ¼

Et < 1:0 E

(1.13.16)

the differential equation based on the reduced modulus becomes EIsr y00 þ Py ¼ 0

(1.13.17)

and sr ¼ s

I2 I1 þ I I

and

sr;cr ¼

Pr;cr p2 Esr ¼ ! "2 A ‘ r

(1.13.18)

60

Chai Yoo

The procedure for determining sr;cr may be summarized as follows: 1) For s 3 diagram, prepare s s diagram. 2) From the result of step 1, prepare sr s curve. 3) From the result of step 2, prepare sr ð‘=rÞ curve.

1.13.2. Tangent-Modulus Theory Assumptions The assumptions are the same as those used in the double-modulus theory, except assumption 5. The axial load increases during the transition from the straight to slightly bent position, such that the increase in average stress in compression is greater than the decrease in stress due to bending at the extreme fiber on the convex side. The compressive stress increases at all points; the tangent modulus governs the entire cross section. If the load increment is assumed to be negligibly small such that DP <<< P

(1.13.19)

Et Iy00 þ Py ¼ 0

(1.13.20)

then and the corresponding critical stress is st;cr ¼

Pt;cr p2 :Es ¼ ! A ‘ 2 r

with s ¼

Et E

(1.13.21)

Hence, st vs ‘=r curve is not affected by the shape of the cross section. The procedure for determining the st ð‘=rÞ curve may be summarized as follows: 1) From s 3 diagram, establish s s curve. 2) From the result of step 1, prepare st ð‘=rÞ. P+ P

Et

P

s

y cr

M

p r

E P+ P

P

M = – EtIy

Figure 1-29 Tangent-modulus model

1