Deformation capacity of steel structures

J. Construct. Steel Research 17 (1990) 33-94

Deformation Capacity of Steel Structures

Ben Kato Toyo University, Kujirai, Kawagoe-City, 350, Japan

ABSTRACT In ultimate limit state design and seismic design, the formation of plastic hinges is assumed in the frame analysis, postulating an unlimited rotation capacity of structural members. However, the plastic deformation capacity of steel structural members with a stress gradient along their length is closely related to the stress-strain relationship of the steel, and if the yield ratio (the ratio of yield stress to tensile strength) of the steel is too high, one cannot expect any substantial plastic deformation. In this paper, the plastic deformation capacities of tension members, columns, beam-columns and beams are evaluated as functions of the yield ratio. For beam-columns and beams, the deformation capacities are evaluated more precisely in terms of the complementary energy of the steel. The deformation capacity of steel members subjected to local buckling is also discussed.

NOTATION a A Ag

Half-length of the cell A r e a of either flange Gross area of section A n Net area of section At Sectional area of stub-column Aw A r e a of web b Half-width of flange, half-width of the cell 33 J. Construct. Steel Research 0143-974X/91/$03.50 (~) 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain

34 d D E Er Est h i I ! Me Mi Mp P Py PE PR Q

R Rb s tf tw Y

ot af aw 6 eo est eu ~y r/0 77, 0u 0y t¢ A /x o-u try

~bp

Ben Kato

Depth of web ( = Dst = 2Est/) Flexural rigidity in strain-hardening region Young's modulus ( = try/ep) Reduced modulus Strain-hardening modulus Distance between centres of gravity of two-flange model Radius of gyration Geometrical moment of inertia Length of member External moment Internal moment Full plastic moment Axial thrust Axial yield load Eulerload Reduced modulus load Horizontal shear force ( = 1/Y = tr.u/try) Inverse of the yield ratio of material ( = o-cr/O'y) Normalized maximum local buckling stress Slenderness Flange thickness Web thickness ( = try/O'u) Yield ratio of material (= P/Py = O'o/try ) Axial force ratio ( = E/ftry(tf/b) 2) Slenderness parameter of flange (= E/wtry(tw/d) 2) Slenderness parameter of web Deflection of column, beam--column and beam ( = est - ey) Plateau length of tr-e curve Strain at strain-hardening point Strain at maximum strength point Yield strain ( = 0u/0y) Rotation capacity in terms of slope ( = ~bu/$y) Rotation capacity in terms of deflection angle Slope of member at ultimate strength state Slope of member at yielding ( = Est ep/try = Est/Er) Material constant Normalized length of plastic region Normalized length of plastic region Maximum stress of steel material Yield stress (= ep/h = ~bst- thy) Curvature at strain-hardening point in rigidplastic-strain-hardening model

Deformation capacity of steel structures

~bu ~u ~by

35

Curvature at ultimate strength state Deflection angle of member at ultimate strength state Deflection angle of member at yielding

1 INTRODUCTION In assessment of the ultimate load-carrying capacity of statically indeterminate structures, the plastic deformation capacity of the constituent members becomes the key parameter because the capability for stress redistribution within structures is highly dependent on the members' deformability. For example, in the plastic analysis of a braced frame, a sufficient rotation capacity of so-called plastic hinges and a sufficient plastic elongation of the bracing members are postulated. In seismic design, a much greater plastic deformation capacity is sometimes required for energy dissipation. Generally, steel structures are reputed to be ductile structures. However, such a reputation cannot be accepted without qualification, as steel structural members are susceptible to buckling, and the plastic deformation capacity itself is highly dependent on the yield ratio of the steel. From this point of view, the post-yielding or post-buckling behaviours of tension members, compression members, beam-columns and beams, and the post local buckling behaviour of plate elements of the member sections, are analysed. It will be shown that the deformation capacities of all these members are highly dependent on the yield ratio of the base material. The deformation capacity of beam-columns, beams and locally buckled members can be evaluated more precisely by using the complementary energy of the stress-strain relationship of the steel. In this paper, efforts are made to solve these problems by means of a mathematical approach to obtain closed-form solutions by using simplified structural models, because it is believed that clear insight cannot be achieved through numerical analysis.

2 TENSION MEMBER Usually, the main bodies of tension members such as diagonal bracings and truss members have a uniform section along their length and therefore they seem to have sufficient deformation capacity. However, these tension members have bolt holes at their end connections and at intersections with bracings, and at these joint regions the sectional area varies monotonically along the length.

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Ben Kato

T X

Y A(x)"~ (7

0

T Fig. 1. Tapered plate.

y

~

st

~'u

Fig. 2. Stress-strain curve.

T h e inelastic behaviour of a tapered plate as shown in Fig. 1 is analysed, in which plate width varies monotonically, with a m i n i m u m sectional area at a-a, and the plate is subjected to tension, T. Figure 2 is the stress-strain relationship of the steel. T h e m a x i m u m tensile strength of the plate, Tmax, is reached w h e n the stress in the m i n i m u m section a - a attains the tensile strength, o-u, and is given as T m a x ~-

O-u.A(0)

(1)

where A (0) is the sectional area at a-a. T h e stress at a section at distance x from the origin u n d e r the applied tension, aTmax(0 -< a -< 1"0), is o'Tmax O'(x) -

A (x)

A(0) -

ao'u A (x)

(2)

where A (x) is the sectional area of the plate at coordinate x. T h e extension of the plastic region, X, u n d e r a given load, aTmax, is obtained by replacing o-(x) in eqn (2) by the yield stress, O'y, as oru

A(X) = a--.A(O), O-y

A(X)>A(O)

Introducing a term for the yield ratio of the material, Y = (ry/(ru, the above equation is rewritten as A(X) = ---~A(0),

A(X) ->A(0)

(3)

Deformation capacity of steel structures

37

W h e n the slope of a tapered plate is given as a function of x, the extension of the plastic region X under an arbitrary tensile force, aTmax, is obtained from eqn (3). If the steel is elastic-perfectly plastic, i.e. Y = 1 as shown by the dashed line in Fig. 2, it can be seen from eqn (3) that A (X) = A (0) even at the ultimate strength state of the specimen (a = 1). This means that there will be no extension of the plastic region and the specimen will be broken off as soon as the stress in the minimum section attains the yield stress. It is also obvious that the smaller the yield ratio, Y, the larger will be the extension of the plastic region. The elongation of the specimen, 4, under the tensile force, a T m a x , is calculated as 1

-~ 3, = Xest +

e u - - 6st

1-----~-~

fox[

]

A (0) Y dx + a e Y [ ; A (0) a A (x----ff Y Jx ~ ox

(4)

Introducing a = 1 into eqn (4), the maximum elongation t~ma x at the ultimate strength state is obtained as

1

"~ ~max

=

[

Sst

1 -- r (Su - e~t)

- - -

+ ey [ t A(O)

]

X +

eu - e~t

1-Y

x A (0) •

ax

(5)

A(X)- 1A(0) = 0 The m a x i m u m elongation according to eqn (5) is compared with the experimental value in Fig. 3.1 Test specimens as shown in Fig. 3(a), which were made of steel grades SM41, SM50Y and SM58, with different yield ratios, were tested and the maximum elongations are plotted in relation to the yield ratios in Fig. 3(b). These test results are compared with the theoretical prediction given by eqn (5). The correlation between test and theory is satisfactory at the stress level o" = 0.95O-u, although the test results at the stress level o- = o-u are somewhat scattering. This seems to be caused by the occurrence of uncontrolled flow of material at the m a x i m u m stress state, which makes exact measurement very difficult. It can be seen from this figure that the yield ratio of structural steels increases with

38

Ben Kato 20.C

i

Y'
a

-)\,

flo.o

;ss 5.0

SM 50 ¥ / I\ nSM 58 I le\t

oSM ~OY I /

I I Defo~r~tio~t

---F----41,~t.....t....

0.2 0.4 0.6 0.8 1.0

( a ) in mm

(b)

~

Y

Fig. 3. Correlation between test and theory (tapered plate). (a) Tapered plate. (b) Prediction and test result.

increase in their yield stress, resulting in a decrease of deformation capacity. Tapering of the cross-section in actual tension members occurs at bolt holes where the change of cross-section is much steeper, and as the effect of stress concentration cannot be ignored, another series of test as shown in Fig. 4(a) were carried out, and test results were compared with the theoretical predictions based on the finite element technique applicable to the plastic range. 2 Figure 4(b) shows the finite element subdivision used in this analysis, and Fig. 4(c) is the comparison between test results and theoretical predictions. The simple prediction according to eqn (5) is shown by the dashed line in Fig. 4(c). The correlation between test results and the prediction by numerical analysis is satisfactory. In the domain on the right-hand side of line a-a, the simple prediction by eqn (5) underestimates the deformation capacity. In this domain, the extension of the plastic region remains within the diameter of the bolt hole, and the strain disturbance that occurs in this region might have a favourable effect on the deformation capacity. To secure sufficient plastic elongation of the diagonal bracings within a frame subjected to a major earthquake, the ultimate strength of a joint with bolt holes must be greater than the yield strength of the gross section of the diagonal bracings. This condition is written as An-o-u>Ag-ory

or

A.>-YAg

(6)

where A . is the net area of the member and Ag is the gross sectional area of the member.

39

Deformation capacity of steel structures Finite element Approximate

....

80mm

,

a=0"95au vE

T 16

1~

0

I0

ttttt • SS4I

XL\

x SM50 o SMSOY a

t = 10mm GG=200mm

(a)

012

(b)

i~ ~

SM58

I

0.4

01.6

01.8

1.0

(c)

Fig. 4. Correlation between test and theory (plate with a hole). (a) Plate with a hole. (b) Finite element subdivision. (c) Prediction and test result. In fact, diagonal bracings of many factory buildings and warehouses had been cut off at their end joints without developing substantial elongation on the occasions of the Tokachioki earthquake (1968) and the Miyagikenoki earthquake (1978). It was found that the yield ratio of the steel material of these bracing members had been unusually high. This experience was reflected in the revision of the seismic design regulations of the Japanese building code which became effective in June 1981.

3 COMPRESSION M E M B E R The theory of flexural buckling of a centrally loaded steel column with common slenderness has been thoroughly established. In this section, the post-buckling behaviour of a very short steel column is discussed. A more detailed description of this problem is given in Ref. 3. The post-buckling behaviour of short centrally loaded columns depends greatly on the stress-strain characteristics of the material. For structural steel, the elastic, plastic-fow and strain-hardening regions define the main characteristics involved. When a column of this material is subjected to monotonically increasing axial deformation, its flexural rigidity reduces to zero as the axial stress builds up to the yield value. When this point is

40

Ben Kato

reached, the column will immediately buckle laterally, indicating that the bifurcation point has been reached. The column will maintain static equilibrium with increasing axial deformation but the axial load will gradually decrease from its initial yield value. After a certain amount of axial deformation, strain hardening in the material will occur, causing the axial load to increase with further increase in axial deformation. If the column is very short, the earlier loss in load-carrying capacity will be completely restored as the axial deformation is increased and the ultimate load-carrying capacity eventually reached will be considerably larger than the initial yield load. During post-buckling of very short columns, the lateral displacement pattern is complex. For example, after a limited amount of buckling, the column can actually return to its original straight position before buckling a second time to complete failure. To provide a better understanding of the post-buckling behaviour described above, analytical relations for axial load vs lateral deflection up to complete failure will be derived below using a simplified mathematical model similar to that of Shanley. 4 The characteristic behaviour obtained from this model is then compared with the results of tests conducted on H-section steel columns. It is believed that the findings of this study will shed considerable light on other types of inelastic buckling, e.g. the coupled flexural-torsional buckling of steel beams and beam-columns which is such an important consideration in plastic design or seismic design. Based on experimental observation, Galambos and coworkers 5'6 first pointed out that the initiation of flexural-torsional buckling of an H-section member does not disturb the development of its in-plane plastic moment provided the slenderness of the member is sufficiently small. Lay 7 and Lay and Galambos 8 then treated this problem theoretically and introduced the intuitive concept of a 'dynamic jump of strain in the plastic region'. As lateral buckling can be thought of as the flexural buckling of a flange in its own plane, the present analysis explains the physical meaning of Lay's intuitive assumption. Another type of inelastic buckling which can be better understood with the present analysis is the local flange buckling of H-sections. In this case, a phenomenon similar to the post-buckling behaviour described above is observed, i.e. a flange will start to wrinkle as soon as the compressive stress reaches the yield point but the amplitude of the wrinkle will not continue to grow with increases in loading of the member; thus, the overall load-carrying capacity of the member will not be reduced provided the width-to-thickness ratio of the flange is within certain limits. Although this is a problem of plate instability, the fundamental mechanism of equilibrium is similar to that shown by the present analysis.

Deformation capacity of steel structures

41

3.1 Analytical model and initial critical load

As pointed out above, the post-buckling behaviour of short centrally loaded columns depends greatly on the stress-strain characteristics of the material. Structural steel is characterized by the elastic, plastic-flow, and strain-hardening regions of the stress-strain relationship. In the present study, this relationship is simplified by assuming the rigid-plastic-flowstrain-hardening relation shown in Fig. 5 by a bold line, where O-yis the yield stress, ey is the corresponding yield strain, and est is the strain at which strain hardening begins. The strain-hardening modulus Est is assumed to remain constant in this relation. A centrally loaded steel column with both ends hinged is modelled mathematically using the Shanley model shown in Fig. 6. This model has ff

~u

0

i

i

Ey

Est ~p = Est -- Ey

Fig. 5. Stress-strain relation.

#P

~F_-

T

--

2

-

c.

1;.

tp

(a)

4---b

Fig. 6. Shanley's model.

÷ b~

(b)

42

Ben Kato

two rigid arms interconnected by a cell consisting of two flange elements, each of cross-sectional area A. It is assumed that buckling takes place in the web plane only and that shear deformations can be ignored. The initial critical load of the column is easily determined from the model by equating the external and internal bending moments. As indicated in Fig. 6(b), ec and e t denote the changes in length of cell elements c and t, respectively, which occur after the start of bending. The -esulting geometry changes cause a lateral deflection as given by =

,

0

=

(ec+e l

=

a,

+

where ec = e J 2 a and e t = et/2a are the strains in flange elements c and t, 2a is the cell length, and b is the half-width of the cell, equal to the radius of gyration and also equal to the core radius in the model. The external bending m o m e n t at this hinge is alP m~ = P6 = ~ f f - (ec + et)

and the internal bending m o m e n t about the hinge point is Mi = Ab(Ecec + Etet)

where Ec and Et are the effective moduli of their respective elements. Equating internal and external bending moments, one obtains (7)

where s = l/i = l/b is the slenderness and a = a/l. When buckling occurs in the elastic range, Ec = Et = E, in which case eqn (7) reduces to the Euler form 4AE

PE-

(8)

~.s 2

If PE is larger than the axial yield load for the column, elastic buckling cannot occur; however, inelastic buckling can be initiated as soon as the stress in the flange elements of the cell reaches the yield value. Buckling is initiated at this instant of loading as no bending stiffness is present in the m e m b e r (Ec = Et = 0). T h u s , the initial critical load of a short column is always equal to the axial yield load for the section, i.e. Py = 2Ao-y. Of principal interest here, however, is the post-buckling behaviour of such

Deformation capacity of steel structures

43

members. Therefore, the subsequent analytical treatment considers the post-yield behaviour of columns of slenderness s <- 2 V ' ( A E / a t r y ) .

3.2 Post-buckling behaviour 3.2.1 R e g i o n I - - s t r a i n in e l e m e n t c increases f r o m ey to est while elastic strain reversal takes place in element t

Immediately after the column starts to deflect laterally under the action of an axial yield thrust, strain reversal must take place in element t to maintain static equilibrium. Clearly, if the strain in both elements were to increase into the plastic flow range, there could be no change in the internal bending moment to balance the change in external bending moment caused by deflection of the column. Thus, strain reversal in element t must take place. Let us consider the increment of hinge rotation A0 from the critical straight state as shown in Fig. 7. The increment of strain in element t, which is the elastic strain reversal, is shown equal to zero in this figure, consistent with the infinite modulus of the material as represented by the rigid portion of the rigid-plastic-flow-strain-hardening relationship. The increment of lateral deflection of the column is given by l

al

~2

A6= ~ A 0 = .-:-. Ae¢= ~ b A e c 4b 4 where Aec is the inelastic strain increment in element c. The increment of internal bending moment about the hinge point is A M i = - AP tb = - APb

where AP t is the increment of axial force in element t and Ap is the increment of external axial thrust. These increments of force are taken as

Est ~p

A

--Aec -~

-- r---I C.

//

b b ~_ r---i t. A¸

_

---~AP

/\ 41 APt

AM,

///

~

Limit S t a t e of Region 1.

Fig. 7. Region 1.

44

Ben Kato

positive when they represent increasing compressive forces. T h e increm e n t of external bending m o m e n t at the hinge is AMe = (Py + Ap) A8 Equating the increments of internal and external bending m o m e n t s , one obtains Ap -

- PyA6 b+A6

thus, the axial thrust P for a given deflection A6 b e c o m e s p = py+Ap

=

~

y

(9)

Introducing the non-dimensional expression A8

A8 . . . . b

as 2

4

Aec

eqn (9) can be written as P =

(1)

1 + Ate' Py

(10)

E q u a t i o n s (9) and (10) are valid until Aec reaches the value e0 = est - Ey. At this terminal point, the lateral deflection 61 (or 6~), the i n c r e m e n t of axial thrust AP 1, and the total axial thrust P1 are given by the relations O~S2

61 = ( A 6 ) ~

= ep --

4

bep

(11)

or

~S 2

6~ = - - T - % { -6] ]p Ap I = Ap, = \ 1 + 6~ ,/

(12)

y

(13)

Deformation capacity of steel structures

45

where ts~ = tsl/b 3.2.2 R e g i o n 2--strain in element c increases into the strain-hardening region while the strain in element t remains in the elastic region The increments of strain, deflection and forces are m e a s u r e d from the terminal state of Region 1 as shown in Fig. 8. In this case, o n e obtains the relations ~.S 2

Ats=--4--bAec,

O~S2

or

A6' = - ~ - A e c

(15)

APc = AEstAec

(16)

A P = APc+APt

(17)

Ami = (APc - APt)b = (2AEstAe~ - A e ) b

(18)

Ame = (P, + AP)(ts~ + Ats) - P, ts1 = Pl At5 + APt51 + APAt5

(19)

w h e r e APe d e n o t e s the increment of force in element c. Equating the a b o v e increments of internal and external bending m o m e n t s gives 2AE~tAec - AP = PlAts' + APts] + APAts'

from which AP = (PR -- PI) At5' 1 + ts] +Ats'

--~Aec I--- i

Tb

APt

~ I_irnit State of Region 2.

F i g . 8. R e g i o n

2.

46

Ben Kato

where PR ( = 8mEst lotS2) represents a reduced modulus load. Substituting for P1 from eqn (14), one obtains [ Ap =

( 1 + S i - Y ) A6' ] y(1 + 6~)(1 + 6~+ A6') Py

(20)

where

"Y ey/ eR -m-

=

T \"~st ]

The axial thrust can now be expressed as P = P~ + AP =

y(1 + ~ + A~')

PY

(21)

Using eqn (17), the force increment in element t becomes 1

APt = AP - AEstAe c = A P - ~ PRA&

Substituting for AP from eqn (20) results in the relation Py[ (1-- 2Y -- 812)-- (l + 6]) A8' ] APt = ~ (-i7~)O+-6-~--A--~ A6'

(22)

From the form of eqn (22), three sub-regions can be defined as follows: (1) Sub-region 2 - A . It should be noted that when the relation 1 - 2y

-

812 ~ 0

(23)

is satisfied, AP t as given by eqn (22) is negative for positive values of A6'. This condition means that the strain reversal in element t continues with increasing axial deformation to the point of complete failure of the column. The entire post-yield behaviour of the column is therefore expressed by eqn (21). It should be noted that the terminal strength of the column as A6' approaches infinity is P R (2) Sub-region 2-B. If the term (1 - 2y - 6~2) is positive, AP t as given by eqn (22) will also be positive when A6' falls in the range

0 < A 6 ' < (1--2y--6~ 2) (1 + 8~)

Deformation capacity of steel structures

47

The peak value of AP t within this range corresponds to that value given by eqn (22) when A6' satisfies the maximum condition d(APt)

_ 0

d(A6')

thus, one obtains (APt)m = ~-~R{[x/2(1 + 61( 5 "/) ; ) ]-2 (1 ) ~+g ~ -

(24)

(aa')m

(25)

=

X/2(l+61-y)-(l+61)

If (APt) m given by eqn (24) is smaller than the decrease of compressive load undergone at the terminal state of Region 1, [V'2(1 + 6 1 - 3') - (1 + al)]

<

1 + al

1 + a~

which reduces to (1 - 2 , / -

81) 2 -

8y6] <

0

(26)

then the stress in element t never reaches the yield point, and the post-yield behaviour of the column is again defined by eqn (21).

(3) Sub-region 2-C. If on the other hand, the condition (1 - 2y - 61)2 - 8~,al -> 0

(27)

is satisfied, the stress in element t can recover the decrease given by eqn (13); thus, it will return to the yield level. As eqns (20)-(22) are valid only when the strain in element t remains in the elastic region, the terminal equilibrium state of Region 2 is obtained by equating the value of AP 1 given by eqn (22) to the negative value of AP1 given by eqn (13). Therefore, one obtains the terminal condition

Py[(1-2y-6~2)-(l+6[)A6']

T4L

(1+6[)(l+a~+A6')

( a[ J Aa'=

']

\l+6[/Py

which reduces to AS~2 -- (1 -- 23' -- 6~) A ~ + 23,8~ = 0

(28)

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Ben Kato

where A ~ is the non-dimensional deflection increment at the terminal state of Region 2. The corresponding strain increment in element c is AEc.2 = (~2)A32

(29)

Replacing AS' in eqn (21) by AS~ and making use of eqn (28), the axial thrust at the terminal state becomes P2

~ 2y82 / Py

(30)

where 8~ is the total non-dimensional deflection at the terminal state of Region 2, i.e. 6~ = 8i + A8~

(31)

3.2.3 Region 3--plastic f l o w takes place in element t while elastic strain reversal takes place in element c

If plastic flow takes place in element t while maintaining the yield stress, equilibrium is possible only when strain reversal occurs in element c. This situation is the reverse of the condition defined by Region 1. Referring to Fig. 9, the performance of the column in this region can be described by the relations OIS2 AE t 4

--

(32)

A t ' = APc

(33)

AMi = APc b = APb

(34)

AMe = PzA8 + AP62 + ApA6

(35)

A S ' --

Equating the increments of internal and external bending moments gives P2AS' + Ap6~ + ApAS' = Ap or

Ap = Apc --

P2A6' 1 - ~-

(36)

a~'

Thus e = Pa+AP =

1--~-;---~6,

P:

(37)

Deformation capacity of steel structures ~y

49

Est Ap~-~

~__

~-AEt ~

\

Limit State of Region 3.

Fig. 9. Region 3.

and 3' = 3~ + A6'

(38)

It should be noted that A6' has a negative sign in the above equations; also, if 61 were to exceed b, the stress in the convex-side element would become tension as b is the core radius of the section. This condition can never develop, however, in the present problem. Therefore, 6~ = 61/b must always be less than unity. Considering these two facts, it is clear that AP c in eqn (36) must be negative. Equations (36) and (37) are, of course, valid only until the strain in element t reaches the strain-hardening point. As seen in Fig. 9, the total increment of absolute deflection at the terminal point of Region 3 is equal to 61, i.e. A6~ = -- 6~

(39)

Substituting this value of A6~ into eqn (37) for A6', the terminal axial thrust for Region 3 is found to be P3 =

1 - 3~ ) 1 - 6 ~ - 8 ~ P2

Substituting for P2 from eqn (30) and making use of eqn (28), this equation reduces to the form

P3 = ( 1 ._-1-~i~2,)Py

(40)

The terminal deflection in Region 3 is

~;

= ~ + a~; = ~i-~

= A~

(41)

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Ben Kato

Making use of eqns (36) and (39), the decrease of axial thrust and thus the decrease of stress in element c is found to be AP3 = APc.3 =

1 - 6~ + 6 ]

P2 =

- 1 -

A6~

P2

Substituting for P2 from eqn (30), one obtains

a~A~ AP3 = -

2T6~(1- A~) Py

(42)

3.2.4 Region 4--compressive strain in element t increases into the strainhardening region while the compressive strain in element c increases again elastically The incremental changes which take place in Region 4 are shown in Fig. 10. One finds in this case that OrS2

A6' = - ~

Aet

(43)

Ap = AP~ + AEstAet

(44)

AM~ = ( A P c - A E ~ t A E t ) b = ( A P - 2 A E s t A e t ) b

(45)

A M e = P3A6 + Ap63 + A p A 6

(46)

E q u a t i n g t h e i n c r e m e n t s o f i n t e r n a l a n d e x t e r n a l b e n d i n g m o m e n t s gives Ap-

2AEstAet = p3A6' + ApA6~ + ApA6'

Solving for Ap, one obtains the relation Ap =

- ( P R - - P3) A6' ] -

6~ -

(47)

A6' ~y - -

Ep - -

~st

Y

b

b --'~AEtP~ \

Limit S t a t e of Region 4.

Fig. 10. Region 4.

Deformation capacity o f steel structures

51

The axial thrust and deflection are found to be P = P3 + A P =

Py- PRA8'

1 - 8~ - AS'

(48)

8' = 8;+A8'

(49)

It should be noted that Ate' in this region is negative. The increment of stress in element c is given by 1 APe = A P - A E s t A e t = AP + ~ P R A 8 '

PR[ (1-- 23,-- t~32) + (1-- 8;) AS' ] -

2

(-i"7~_6;)-~ "2 ~ - - A"~

A6'

(50)

If the term (1 - 23, - ~ 2 ) is positive, Ap~ as given by eqn (50) will also be positive when AS' falls in the range - (1 - 23,- 8:~2)

(1-8;)


Using eqns (28) and (41), the term (1 - 23' - a~2)can be written in the form (1 - 23,- 8~2) = (1 - 23,)(1 - A&5)+ 8~(A6-5+ 23,)

(51)

It is clear that the right-hand side of eqn (51) must be positive. Therefore it is shown that the compressive stress in element c, which decreased elastically in Region 3, is now increasing elastically. Obviously, AP e must have a peak value at some point in the above range for A~'. This peak value and its corresponding value of A~' are found using the maximum condition d(APc)/d(AS') = 0, which results in the relations

(AS')m

=

(1 - 8:]) - "V'2(1 - 8~- 3')

PR (APc)m = - ' ~

[ ( 1 -- 6~) -- X / 2 ( 1 -- 8~ -- 3')12 1 -- 8~

(52) (53)

If (APc)m is larger than the decrease of stress which occurred from the limit state of Region 2 to the limit state of Region 3 as given by eqn (42), the full strain reversal undergone in Region 3 will be recovered and the terminal point of Region 4 will be reached when the stress in both elements c and t

Ben Kato

52

arrives at the starting point of the virgin strain-hardening region. This condition can be checked by calculating the value

D

P.f

[(1- 6~) - %/2(1- 6~- 3,)]2

= ( A P c ) m -- A p 3 = L-g~-{ •

z

t

1 -

6.~

Making use of eqns (28) and (41), this relation simplifies to

D -

PR [1 -- %/2(1 -- 6~- 3')12 2 1 - 6~

(54)

From this form, it is obvious that D is positive; hence, it is shown that (APc) m > AP 3. The condition that the strain reversal in element c which took place in Region 3 is just recovered, thus defining the terminal point of Region 4, is A P 3 + APe = 0 or PR[ 61A~ 2 6~(1~-A6~) +

[(1--23,--6~2)+(1--6~)A6']A6' } ~-~----6~ZS-~

=0

(55)

Making use of eqns (28) and (41), A6' to be used in eqn (55) becomes equal to A~

=

- a~

(56)

= - ~

Introducing this relation into eqns (48) and (49), one obtains P4 = Py + PR~ = Py + PRA6~ = Py[1 + (A6~/3,)]

(57)

a~ = ~ + A ~

(58)

= ~ - ~

= 0

From eqn (58), it is seen that at the terminal point of Region 4, the column has returned to its original straight configuration.

3.2.5 Region 5--pure axial compression up to final failure As the column is now straight, it will be subjected to pure compression in Region 5. The tangent modulus load in this state is given by substituting Est for E in eqn (8); thus, PT

-

4A E~t -o~s2

=

1

~ PR

(59)

As soon as this critical load is reached, buckling will again take place, accompanied by strain reversal in one flange element. The post-buckling

Deformation capacity of steel structures

53

behaviour can be analysed using procedures similar to those previously presented, in which case one obtains P R ( 1 + 26'~

P=

(60)

2 \1+6'/

From this relation it is seen that as 8' approaches infinity, the column strength approaches PR- Actually, the column strength can never exceed the value Pu = 2A~ru, where O-uis the maximum strength of the material as controlled by local buckling. If Pa- is larger than Pu, the P vs 8' relation during the collapse state can be expressed in the form

PU P - ~1 + 8

(61)

In this case, the strain reversal which takes place in the convex-side element is elastic.

3.3 Examples The post-buckling behaviours of several columns with different slenderness, as predicted by the foregoing analysis, are illustrated in Fig. 11. In these examples, typical mild steel was assumed for the material and the mechanical properties were specified as follows: ey = 1"2 X 10 -3,

ep = 10ey,

o"u = l'8try,

Est = 0"03E

For actual columns, the ratio of cell length to column length represents the ratio of the longitudinal dimension of the inelastic zone to the column length. This ratio was assumed, rather arbitrarily, to be equal to 0.4 in the examples. From the results shown in Fig. 11, it is seen that columns with slenderness smaller than some critical value can return to their original straight position after having undergone limited amounts of buckling. The critical slenderness is given by eqn (27), which yields a numerical value of l l . 4 in this steel. In the present study, it was assumed that Est remained constant throughout the post-yield performance, even though it is obvious that it decreases with increasing inelastic strain. Ratio a = a/l was also assumed to remain constant in the post-yield range, even though the inelastic zone expands considerably with increasing column load, as will be shown in Section 4.

Ben Kato

54

P

3.0

2.0 ~

~.~,,,..,~..

~'~-

R.5)

"---.....~q.61

(~'¢)

1.0

~

'

X=~

7

~

-,,~ = 17 (~,T)

0.5

--k

1

I

I

I

I

5

10

15

20

25

= 20 I

3 0 x 10-~'

Fig. 11. Load-deflection curves. (1) ct = 0.2, Est = 0.03E, ey = 1.2 × 10 -3, ep = 10ey, tr, = 1-8try. (2) For A > 11.4, reversal of the lateral deflection does not occur. (3) (R.1), ( R . 2 ) . . . are the abbreviations of Region 1, Region 2 . . . respectively.

3.4 Test results

Tests were carried out on H-section steel columns as shown in Fig. 12. All test specimens were designed to buckle in the w e b direction. The average mechanical properties over the cross-section were d e t e r m i n e d from stub-column compressive tests, which gave the following results: O-y = 262 MPa,

o-u = 412 MPa = 1.57Ory,

est = 1270 x 10 -5 = 10ey,

ey = 1270 x 10 - 6 ,

Est = 7252 MPa = 0.0352E.

These properties are similar to those a d o p t e d in the calculations of the foregoing examples. T o d e m o n s t r a t e the validity of the basic theory p r e s e n t e d herein, the test results of two column specimens with different behaviours are shown in Fig. 13. In these tests, no local buckling of plate elements or lateral

Deformation capacity of steel structures

I

55

P

1152 I_ l - mm

No.1 600 No.2 900

~.

9.06 13.6

j,

IP Fig. 12. Test specimens.

torsional buckling was observed until the final collapse stage of the columns had been reached. Specimen No. 1 had a slight initial imperfection, which may have caused the small lateral deflection which occurred in the early stages of loading. This imperfection may also have had some influence on the post-buckling behaviour. The theoretically predicted behaviours for these test specimens are shown in Fig. 13 by dashed lines. The ratio o~was assumed to be equal to 0.2 in making these predictions. This value was selected because it gives an Euler load (eqn (8)) PE = 4AE/as2 = 20AE/s2, which is nearly equal to PE = 2~r2AE/s2 ~ 20AE/s2, which is the Euler load for the actual column. Although the theoretical predictions are based on an extreme simplification of the problem, the general post-buckling behaviours obtained show close resemblances to the actual column behaviours. The major differences found between theoretical predictions and actual column behaviours are thought to come from the neglect of web participation, the uncertainty of estimating equivalent cell length, and the assumption of a constant strain-hardening modulus, in making the theoretical predictions. It is believed that the participation of the web in actual columns makes the real post-buckling behaviour characteristics somewhat less pronounced than those predicted by theory. Obviously, the errors introduced by the assumption of a constant strain-hardening modulus become large with increasing strain.

56

Ben Kato

- -

I

0.5 0

l'Zxpe r i m e m a l

I

5

I

10

I~X 10-2

(a) P

i

0"50

10

20

t

3'0

40

i

50x10 -2

~'(crn)

(b) Fig. 13. Correlation between test and theory. (a) No.

1: l e n g t h = 6 0 0 m m . slenderness = 9-06. (b) No. 2: length = 900 ram, slenderness = 13.6.

Deformation capacity of steel structures

57

4 BEAM-COLUMN AND BEAM Basically, the argument put forward in Section 2 on the deformation capacity of a steel tension bar with a stress gradient along its length may be applicable to the assessment of the plastic deformation capacity of beams. To explain the situation, a cantilever H-section steel beam subject to a point load, P, at its free end as shown in Fig. 14 is taken as an example. The bending moment in this beam increases linearly from the loading point toward the fixed end as shown in Fig. 14(b). If the participation of the web plate is ignored for the sake of simplicity, the tension stress in the upper flange can be written as T = M/h, which means that the pattern of tensile stress distribution in the upper flange is the same as that of the moment distribution, as is shown in Fig. 14(b). Also, if Fig. 14(b) shows the bending moment distribution at the ultimate strength state, the ultimate bending capacity of the section, Mu = Tu" h, has been reached at the fixed end, where Tu = Af-o'u is the maximum tensile strength of the upper flange and ,'If is the sectional area of upper flange. The yield moment M y = A f " h . Cry, has been reached at section a-a, and the extension of plastic region is denoted by/31 (0-
try = Y _

Mu

tru

1(1-/3) _ 1-/3 l

then /3= 1 - Y

or /31= ( 1 - Y ) l

/I

(62)

or T MJh

Fig. 14. Bending of a cantilever beam.

58

Ben Kato

From eqn (62), it can be concluded that if the material is elasticperfectly plastic, namely Y = 1, no extension of the plastic region can be expected and fracture of the upper flange will take place as soon as the yield moment, My, is reached at the fixed end. In contrast, if the yield ratio, Y, is reasonably small, a substantial extension of the plastic region can be expected and thus a substantial hinge rotation of the beam will be obtained. In the short term, the tension behaviour of a tapered plate and the flange behaviour of an H-section beam with moment gradient are basically the same, and their plastic deformation capacity depends on the yield ratio of the base material. This deduction was verified by a test of high-strength steel beams in which H-section beams made of grade A514 steel (O-y = 764.4 MPa, O-u = 848.7 MPa, Y = 0.9) were tested, where the tension flanges were fractured and cracks had propagated into the webs immediately after the built-up of yield in the tension flanges. 9 The above description is suitable for an understanding of the basic characteristic of the problem; however, bending of the beam is accompanied by curvature of the flange, and its rotating behaviour is influenced by the moment vs curvature relationship of the member section; therefore a more accurate analysis is needed to assess the deformation capacity of a beam and a beam--column quantitatively. In the beam-column problem, an additional moment produced by the axial force and deflection must be taken into consideration. It is widely recognized that the direct solution of this type of inelastic beam--column problem is generally impossible to obtain because of the high non-linearity of the basic equation, and iterative numerical integration of the deflection is usually used to obtain the solution. However, the numerical analysis is thought to be a kind of numerical experiment and thus it is difficult to describe the whole structure of the problem; also, it is difficult to determine which parameters are most influential on the characteristic behaviour of the structure. Therefore, in this section, a closed-form mathematical solution is sought, using a simplified structural model. In this model, the stress-strain relationship is represented by the rigidplastic-flow-strain-hardening relation, and the H-shaped member section is reduced to a two-flange elements model similar to that used in Section 3. Using this model, a mathematical analysis of the inelastic behaviour of a cantilever beam--column (including a beam) is carried out first, and the effect of the yield ratio of the base material on the deformation capacity of beam is then assessed more quantitatively. 4.1 M o m e n t - c u r v a t u r e relation

The stress-strain relationship of structural steels is simplified by assuming the rigid-plastic flow-strain-hardening relation as shown in Fig. 15, where

Deformation capacity of steel structures

59

G

% =Ray m

,/I

"7

I

/m

I Ep

Eta

I i

_Ray Fig. 15. Stress-strain relation.

l-b+b4

T he ,

:2

Fig. 16. Equivalent section.

O'y is the yield point, o-u is the m a x i m u m strength of the material, R = tru/O'y = 1 / Y is the inverse of the yield ratio of the material, Ep = E s t - Ey is the plateau length, where est is the strain at the strainh a r d e n i n g point, eu is the strain at the m a x i m u m strength point in this m o d e l which is related to the real m a x i m u m strength point strain, ( % ) , as e u = (eu) r - ey, and Est is the strain-hardening modulus. T h e H-section is replaced by an equivalent two-flange m o d e l as shown in Fig. 16, where the equivalence can be maintained by equating the full plastic m o m e n t and the sectional area for both sections. T h e geometrical relations b e t w e e n these two sections which satisfy the assigned conditions are t h e n h 2 + 0.3(h/b) h---~ = 2 + O. 15(h/b) '

I le

[2 + 0.3(h/b)][2 + O.l(h/b)] [2 + 0" 15(h/b)] 2

(63)

60

Ben Kato

T h e n o r m a l stress is assumed to be c o n c e n t r a t e d at the centre Of gravity of each flange. Hereafter, h is used for the two-flange m o d e l instead of h e. T h e m e m b e r is subjected to bending m o m e n t , M, u n d e r the constant axial thrust, P. T h e stress due to axial thrust, cr0, is ~o = a t r y = P/2A = a P y / 2 A

where A is the area of either flange, Py = 2Acry is the axial yield load, and a = o'0/Cry is the axial stress ratio. T h e stress due to bending m o m e n t , M , is trf = M / h A

where h is the distance between the centres of gravity of the section. C a s e 1: a > ( R - 1 ) / 2

If a > (R - 1)/2, the tension flange does not yield until the stress in the compression flange reaches the m a x i m u m strength of the material, cru. T h e m o m e n t - c u r v a t u r e relationships (M-4, relationships) for this axial stress level are: (1) For 4' -< &p, the compression flange is in the plastic flow region, and the bending m o m e n t is M = (1 - a)Mp

(64)

where 4, = e / h is the curvature, 4,p = e p / h is the curvature at the strain-hardening point and M p = A h cry is the full plastic m o m e n t . (2) For 4,p < 4, -< 4,u, the compression flange is in the strain-hardening region, and the bending m o m e n t is M = (1

-

ot)Mp + D(~b - ~bp)

(65)

where 4,u = e u / h is the curvature at the m a x i m u m strength state, D = Dst = A M / A c k = A h A c r / ( A t / h ) = E s t A h 2 = 2Est" I is the flexural rigidity in the strain-hardening region and I = ~ A h 2 is the m o m e n t of inertia of the section. T h e u p p e r limit of the bending m o m e n t , Mu, is Mu = (t~u - oro)Ah = (R - a)Mp

(66)

By introducing Mu f r o m eqn (66) into M from eqn (65), the c o r r e s p o n d i n g critical curvature is ~bcr -

( R - 1)Mp t- ~bp D

(67)

Deformation capacity of steel structures

61

Case 2: a <- (R - 1)/2 If a -< (R - 1)/2, the tension flange yields after yielding of the compression flange. Until the tension flange yields, the M-~b relationships are exactly the same as for Case 1, so the foregoing descriptions of (1) and (2) are valid, except that the upper limit of the curvature for region (2) should be that at the yielding of the tension flange. The bending moment at the yielding of the tension flange is given as M = (1 + a ) g p and introducing this value into M of eqn (65) and solving for ~b, the limiting value of ~b is obtained as ~bp + 2aMp~D, so the equivalent expression for region (2) is written as follows: (2) For ~p ( t~ ~ (~p + (2aMp~D), the bending moment is

M = (1

-

ot)Mp + D(~b - ~bp)

(65)

The yield flow of the tension flange will take place instantaneously under the same moment, and the amount of this second flow of the curvature is ~bp, independent of a. Therefore, the upper bound curvature at M = ( l + a ) M p is Sbp = 2(~bp + (aMp~D)), and the M-~b relationship for the third region is given as: (3) For tkp + (2aMp~D) < q~--< 2[q~p + (aMp~D)], the tension flange is in the plastic flow region, and the bending moment is M = (1 + ~)Mp

(68)

For further loading, the tension flange is strained into the strain-hardening region, and the flexural rigidity in this region is , hA.Air 1 (A_~_~) Dst = ( ~ _ ~ ) = -~ Ah 2 = Est. I = D/2

The M-~b relationship for this fourth region is then given as: (4) For 2[~bp + (aMp~D)] < dp <- 49u

= Mp. + DI(dg2) - ~bp]

(69)

Equation (69) shows that the M-th relationship in this region is independent of the axial stress ratio, a, although the starting point of this region is a function of a. The upper limit of the bending moment is the same as given by eqn (66),

62

Ben Kato

and the critical curvature is obtained by introducing it into M of eqn (69), and solving for 4, as

~bu = 2 ~p +

(R -- ot -- 1 ) M p ]

D

/

(70)

C a s e 3: a = 0

If a = 0, both compression and tension flanges yield simultaneously, which represents the situation for the beam. (1) For 4 , <- 24,p, both flanges are in the plastic-flow region, and the bending m o m e n t is M = Mp

(71)

(2) For 24,p < 4,-< 4,u, both flanges are in the strain-hardening region, which is the same situation as that of (4) of Case 2, so the same M-4, relationship as eqn (69) is valid: M = Mp + D[(~b/2) -

¢~p]

(69)

The upper limit of the bending m o m e n t is Mu = RMp

(72)

and the critical curvature is

4~u= 2[ (~P+ (R- 1)Mp]

(73)

According to the equations derived above, an example of the M-4, relationships of an H-section steel is depicted in Fig. 17, in which the following mechanical properties of the steel are assumed: O-y = 262 MPa, o-u = 4 1 2 M P a , R = 1.57, e p = 1 . 2 7 x 1 0 -2 , e u = 3 . 3 4 x 1 0 -2 and Est = 7252 MPa.

4.2 Inelastic behaviour of cantilever beam--columns Cantilever beam--columns which are subjected to lateral shear force, Q, and a constant axial thrust, P, at their free end are considered. Most of the deformed configurations of rigid frames subjected to horizontal forces and gravity loads are thought to be assemblages of cantilever b e a m - c o l u m n s as shown in Fig. 18, so the study of the inelastic behaviour of such b e a m - c o l u m n s is important. The M-4, relationships were categorized according to the relation between the axial load ratio, a, and the inverse of the yield ratio of the material, R, as demonstrated in the previous section,

Deformation capacity of steel structures

63

1.5 a=O.l a=0.2

M

(x%)

1.0

a=O o.1

¢ I

a = -R--1 ~--

a=PIPy, Py=2Aay R = a./ay

0.5

bip m hA ay 0.7

I

i,

0.9 |.0 I

ID

I I

I

I

2 Cp

J

I

i

I 5

1

¢( X 10 -2 h-~)

Fig. 17. M - ¢ diagram.

Fig. 18. Deformation of a rigid frame.

II

i

8

64

Ben Kato

and the d e f o r m a t i o n analyses are carried out on the basis of that categorized M-4, relationship in the following. Case 1: a > (R - 1)/2 In this case, the tension flange does not yield until the compressive stress in the compression flange reaches the material's m a x i m u m stress, O-u, and the corresponding M-~b diagram is shown in Fig. 19, which was obtained in case 1 of Section 4.1. For the range of ~b- ~bp, the M-~b relation is given by eqn (65) as

M = ( 1 - o t ) M p + ( ~ b - ~bp)D = Dqb+m = D y " + m ~ J m = (1 - a ) g p -- Dq~p

(74)

Figure 20 is the deflected configuration of a cantilever b e a m - c o l u m n subjected to lateral shear force, Q, u n d e r a constant axial thrust, P, where l is the length of the cantilever b e a m - c o l u m n , AI is the length of the inelastic zone, 01 = 61/(1 - A)I and 6 = 60 + 61 is the total deflection• Using eqn (74), the equilibrium of the m o m e n t in the inelastic region, AI, of Fig. 20 is written as Dy" + m - a P y ( 6 - y) - Q ( l - x) = O,

O<-x<-Al

(71)

T h e solution of eqn (75) is • p

p

Q

-~y (l-x)-

Y = ASmlX+Bc°sTx+6+

m otPy

(72)

with the b o u n d a r y conditions of y=0

at x = 0,

y = 60 at x = ) t l

y' = 0 at x = O,

yr

=

61

O1 m

(1 - h ) /

at x = hl

where p2 __ 017r2 Py _ aPy

2

Pt

D 12'

Pt-

7r2Est I l--------~

T h e condition that the m e m b e r yields at x = Al is expressed as otPyt~1 + Q(1 - h)l = (1 - a ) M p

(75)

T h e four b o u n d a r y conditions together with eqn (75) are e n o u g h to

Deformation capacity of steel structures

65

M (R --Or) btp

.....

I

(1- O t (1 - - a ) Mp

jJJ / I

I Cp (R-1 Mp ~- Cp D

Xl

H Fig. 20. Configuration of beam-column, a > (R - 1)/2.

Fig. 19. M-~b diagram for a > (R - 1)/2.

determine the five unknowns A, B, 60, 61 and A in eqn (72). The final solution is

6 ~-

/1

tc°ssinincos ~--0-S_~ )

- ors (1 - psin pA - cos pA)

~)j

1

(76)

where K = E s t . ep/O'y = Est/E r is the material constant, Er = tyy/ep (see Fig. 15), s = l/i is the slenderness and i = h/2 is the radius of gyration. Equation (76) is the non-dimensional expression of the Q-6 relation taking A as a parameter. The ultimate state of this system is expressed as (77)

From eqns (76) and (77), eqn (78) is derived, which gives the value of A at the ultimate state of the beam-column: (R - a ) - ( 1 - a )

[

1+ p(1-A)

.1

J +K(1-c°spA)

= 0

(78)

Ql/Mp and 6/l at the ultimate state will be obtained by introducing h from eqn (78) into eqn (76).

66

Ben Kato

Case 2 : 0 < a <- ( R - 1)/2

(1) Until the bending m o m e n t at the fixed end reaches M = (1 + a) Mp, the tension flange does not yield, as was seen in the previous section, and the behaviour of the m e m b e r is identical to Case 1. Therefore, the Q - 6 relation is given by eqn (76). At this critical point, the equilibrium is given by Q l + P6 = (1 + a ) M p

(79)

T h e value of A at this critical point is obtained as the solution of eqn (80), which is derived from eqns (76) and (79): 2a

-

(1 - a)sinpA p(1 - A)

+ ,¢(1 - c o s p A )

= 0

(80)

Q l / M p and 6/! at this critical point will be obtained by introducing A from

eqn (80) into eqn (76). (2) For further loading, both flanges will yield at the vicinity of the fixed end, and the M-~b relationship in this region was given by eqn (69) as M m'

D m' = D ,, m' Mp + D (W2 - t~p) = T t~ + T y +

= =

mp

-

(81)

Dt~p

T h e M-~b diagram t h r o u g h o u t whole curvature range for Case 2 is depicted in Fig. 21, as was already shown in Fig. 17. Therefore, at this loading level, two types of plastic regions are distributed along the length of the m e m b e r , as shown in Fig. 22, i.e. both flanges yield in the region d e n o t e d by/xl, and the compression flange only yields in the region d e n o t e d by M. In the region d e n o t e d by p,l, the equilibrium is expressed as D

-}-y" + m'

-

o~Py(6

- y) - Q ( l - x) = O,

0 -< x -
(82)

T h e general solution of eqn (82) is y = A s •l n -V-~~p x

~ V/-2P + t~cos-----~ x +' 6 + - ~Qy

T h e b o u n d a r y conditions are y = 0 at x = 0,

y = 60 at x = pd

y' = 0 at x = 0,

y' = 01 at x =/xl

(l-x)-

- -m p

otPy

(83)

Deformation capacity of steel structures

(R-a) Mp (1 +a)Mp

67

M

(1 -a)Mp . --7/~ D ?n'

0 Fig. 21.

Cp

jA~ 2 ICe +1 ( R - . - 1D) % \

IH

I

)~ l

xlll A II

i I

| pl

amp ]

2 [¢p +-U-~

I" ~

M-~b diagram for a -< (R - 1)/2.

LI

Fig. 22. Configuration of beam-column, a -< (R - 1)/2.

The condition that both flanges should yield at x = / x l is expressed as otPyt~1 + Q(1 - ;z)l = (1 + a)Mp

(84)

Allowing 01 as known for the moment, the Q - 6 relation may be derived as follows: ,aPy 101 Ql (1 - V~p/z sin V~p/z - cos V2p/z) = - ~ cos V~plz ] Mp , V~pm' . . ~ -e ~ sm v zp/~

- o~t

(

l + a V~ " " ~--_ ~-~ V2p sm V2p/x

(85)

(a-cosV'~p~)+

m'

+--QQ ( 1 otPy \ "V'2p

sin V'2p/z

@P

_ 1 /1

sin ~/2P/z(1 + a)Mp aPy l

}

(86)

cosV o

As the second step, the equilibrium of the region denoted by AI should be considered. Taking point A in Fig. 22 as the origin, and rotating by 01, a local coordinate, ~-r/, is introduced as shown in Fig. 23. In this coordinate system, the equivalent shear force is approximated by Q + otPy 01 ~ Q, and the equivalent axial thrust is approximated by P - QO 1 ~. P. The equilibrium of the region denoted by Al is expressed as Dr/"+ m - aPy(62 + 83 - r/) - (Q + aPy01)[(1 - tt)l - ~:] = 0

(87)

68

Ben Kato P6

x ~Ql-p ~-~ ~/ Ol>~

-

_

~ ~) 5_

Fig. 23. Local coordinate. T h e g e n e r a l solution of eqn (87) is

~7 = A s i n / ~ + B c o s / ~ + (62 + 63)

+(~--~y+01) (1--tx)l--(a~y +Ol)'-- --aPy m

(88)

T h e b o u n d a r y c o n d i t i o n s are = 0 at ~ = 0,

~'=0at

7/

~=0,

=

at ~ = AI

62

7'=02

-

63 (1 - A - / x ) l

at ~: = Xl

T h e c o n d i t i o n t h a t the c o m p r e s s i o n flange s h o u l d yield at ~: = hi is e x p r e s s e d as

otPyt~3 + (Q + aPyO,)(1 - z - tx)l = (1 - a)Mp

(89)

F r o m e q n s (88) a n d (89), a n d the f o u r b o u n d a r y c o n d i t i o n s , the relation b e t w e e n the lateral force a n d the m e m b e r deflection is o b t a i n e d as

Q + otPyO1 =

(.,_

[cospA - (1 - A -/~)psinpZ] + tool sinpA (90)

m

(62 + 63) = - -

otPy

(1 - cospA) (91)

(1 - a)Mp [ +

~P-7 /c°sp* +

sin pA ] 0(1 - A - ~)

Deformation capacity of steel structures

69

The compatibility of the displacement at point B in Fig. 22 is 8, = (62 + 83) + (1 -/z)lO,

(92)

Now 01 and (~2 + 33) can be eliminated from eqns (84)-(86) and (90)-(92) as follows. The relationship between A and/z can be obtained from eqns (84), (91) and (92) as (1 - a)sinpX 0(1 - A -/Z)

2or - K(1 - cospA) = 0

(93)

When/~ = O, eqn (93) naturally takes the same expression as eqn (80). From eqns (85) and (90), Q is given as Ql

Mp _

(1 - X,/-2p/Zsin V2p/Z) =

(1 1A_a--/Z)cos PA "cos V~P/Z

\ / | a +/Z | V~p sin V~p/Z \ 1 -/Z,/ - x(V'2p sin V~p/Z + psinpA • cos V~p/Z)

(94)

From eqns (86), (90) and (94), ~ is given as as. 7 (1-V'2p/zsinV'2p/z) =

{[ 1+ X"2p(1-A-/Z) 1o cos pZ. sin V2p/z

+ a cos V'2p/Z] (1 - X,"2p/Zsin V'2p/Z) ( 1 -1A- - /az )

vzpsm cospA.cosV'2p/z+ a1+- / z . m . k/2p/z}

i[1 - (X/2p sin X/2p/z + p sin pA. cos X/2Nz) }

(95)

Equations (93)-(95) give the Q-~ relationship, taking A and /x as parameters. The ultimate state of this system is expressed as (96)

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70

The lateral force, Q, and the deflection, 6, at the ultimate state can be determined as follows. Substituting for Q I / M p from eqn (94), and for 6/l from eqn (95), eqn (96) can be written as X/2p(1 - A -/.~)

• cospA, sinX/2ptz + (K + a)COSX/2O~

K

X/2 sin pA- sin V2p/z =R+K-a-1

(97)

From eqns (93) and (97), eqn (98) is derived: (2a+K)cospA--K

sinX/2ptx+(K+et)COSX/2ptx = R + K - a - 1

sin ph

(98)

From eqns (93) and (98), A and/z are determined, and Q and 6 at the ultimate state will be obtained by introducing them into eqn (94) and eqn (95) respectively. Case 3: a = 0 ( b e a m )

As no axial thrust is acting, both flanges will yield simultaneously in this case. The M-~b diagram is depicted in Fig. 24, as was already shown in Fig. 17, and the M-(h relationship for the range of 24,p < (h -< (hu is given in eqn (69). Referring to Fig. 25, the equilibrium in the plastic region, pal, is given as

D

~ y" + m ' - Q(I - x) = 0

(99)

This linear equation is solved readily as follows. The slope of the beam at x = /xl, 01, is

2[

O, = --~

12

Ql2(tx - -~ I.t ) - m' txl

]

(100)

The deflection of the beam at x = / z l is (101)

Deformationcapacityof steelstructures

71

M RM M ,po"

I

¢

I

2 Cp

2[¢p +(R--l) M~

Fig. 24. M-¢ diagram for a = O.

Fig. 25. Configuration of beam (a = 0).

The deflection of the b e a m at x = l is

,2[(23)

8 = 5o + (1 - tx)t01 = -~ QI 2t~ - 2/.~2 + ~ be

- m'(2/z

(102)

The condition that both flanges yield at x = /zl is Q(1 - / x ) l = Mp, which directly gives the value of Q, taking p~ as the p a r a m e t e r , as

Ql Mp

1 1-

(103)

Introducing Ql from eqn (103) into eqns (100) and (102), 01 and 8 are written as 01 = s

eyE

tx2

2Est 1 - # k %Ix

)

SF O.y ( /./2 '~(1 1 ) --jt.~)] 7 = 2 k E s t \ l - / . t ] \ --~/x +ep,~(2

(104) (105)

Equations (103) and (105) are a parametric representation of the Q - 8 relationship. The ultimate strength of the b e a m is given by QI/Mp = R, and the u p p e r limit of ~ is obtained by introducing this value into eqn (103)

as /~u = 1 - -

1

R

(106)

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72

4.3 Examples

4.3.1 Q-6 relationships According to the predictions obtained in Section 4.2, the Q-6 relationships of cantilever beam-columns and beams are depicted in Fig. 26 for some variations of slenderness, s. The assumed mechanical properties of the member section are the same as those used in the M-~b diagram of Fig. 17, which are typical for mild steel. The Q-6 relationships for the case of R = 1, e.g. for steel with no strain hardening, are depicted by dashed lines, to demonstrate the significance of the yield ratio of the material. From this figure, it is observed that when the slenderness of a member and the axial force ratio exceed a certain limit (i) the slope of the Q-6 diagram becomes negative as soon as the member yields, and (ii) the resisting capacity at the ultimate state, Qu, becomes lower than the yield strength of the member.

QJ_

S INELASTIC ;LA S TIC

0

5 6

7

(%)

(a) Fig. 26. Q - 6 r e l a t i o n s h i p . (a) s = 8. (b) s = 10. (c) s = 15.

D e f o r m a t i o n capacity o f steel structures

73

F

1 . 5 1 I ; COMPRESSION FLANGE IS INELASTIC .II; BOTH FLANGESARE INELASTIC II

IIl; COLLAPSE DOMAIN

I

/

/

.

/

"

i

/

.

III

......

\

/

.

O. ~ ,

6 T

5 (~)

(b)

1.5 -

~

7

I ; COMPRESSION FLANGE IS INELASTIC II; BOTH FLANGES ARE

-I.~COLLAPSEDO'A'N

~

INELASTIC ~

-

/ II

~

-/'

~

--

1,0 ~ Mp

~

I

-- ~ "- ~ - - - - ~ _ . ~ - ' ~ " ~

k,

_

Ill

\

0.5

R'---\, I

I

5

\,

I 10 7

(c)

(~)

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74

4.3.2 Ultimate strength and deformation capacity The effects of the material's yield ratio and of slenderness on the ultimate strength and the deformation capacity of members are observed. The ultimate strength of a m e m b e r is defined as the lateral force, Q, at which the m e m b e r section is fully stressed to the material's maximum strength. Referring to Fig. 27, the deformation capacity of the m e m b e r can be defined either by using the deflection angle, ~b = 6/l, or by using the slope, 0, as 74 -

1Ou

70 -- 0y

1-

Ou-Oy 0y

(107) Oup

--

(108)

0y

~bu is the deflection angle and 0u is the slope of a m e m b e r at which it reaches the ultimate strength state or at which the lateral force, Q, comes down to Qy from its peak value, as illustrated in Fig. 28. 6u and 0up are the plastic parts of the deflection and the slope respectively, and they were obtained in Section 4.2 using the rigid-plasticstrain-hardening model. ~y is the elastic deflection and 0y the elastic rotation at the state when the m o m e n t of the critical section reaches the full plastic moment. The effects of the yield ratio of the material and of the slenderness on the ultimate strength and the deformation capacity of members are shown in Fig. 29 and Fig. 30 respectively. The mechanical properties of steels used for this parametric study are shown in Table 1. Typical structural steels are considered, and the yield ratio, Y, increases with increase of the yield

Q

qy P

0op,-OUp ~

0 Fig. 27. Deflection and slope.

i

l

I I

Oy

Ou

Ou

O

Fig. 28. Definition of deformation capacity.

Deformation capacity o f steel structures

75

Qut Mp

A,1.429 B,1.25 C,1.11

1.0

0.5

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0/

Fig. 29. Q-a relationship.

TABLE 1 Mechanical P r o p e r t i e s o f Steels

A-steel B-steel C-steel

Yield stress, Cry(mPa)

Tensile strength, cru (MPa)

Ey

265 353 441

378 441 490

1-3 x 10 -3 1.7 x 10 -3 2.1 x 10 -3

Est

1.3 x 10 -2 1.4 x 10 -2 1.3 x 10 -2

Yield ratio, Y

R = 1/Y

0.7 0.8 0.9

1.428 1.25 1-111

Est (MPa)

6880 5880 5880

76

Ben Kato

Y~q, 20-

AS'

15

I I

t

A18~ ~A12

A,13.73

6

I / / / / /

B 12i

///

"'~,,,~

,// 10

,.'//

, -

/

B12

/

_B18,,I B6

-,-,0.##

',\ ~

C18, C!2

__,y'/

,, , , /

. . . . . . . .

iI

i

//

~C,~c , , \ ~/ ,k! c° c,.

...+y ..-~Y"

fl°_;

231 _,,1~

om

0

t n . . . ~ , , , , ~ ' -

l ~ _ . J ~

0.1

0.2

,

i

i

i

,

0.3

0.4

0.5

0.6

0.7 Ot

Fig. 30. ~7,-a relationship.

stress. The values of slenderness of the members selected are 6, 12, and 18. A member made of A-steel with slenderness 12 is coded as A12, and so on. Figure 29 shows the relationship between the ultimate resisting capacity, Qu, and the axial force ratio, a. The normalized strength, Qul/Mp,is taken as the vertical axis and ot as the horizontal axis. It can be seen that the ultimate strength of members decreases with increase in slenderness and in yield ratio throughout the whole range of the axial force ratio. When the yield pattern changes, namely by the criterion o~~ (R - 1)/2, the curves take different shapes. The yield strength is defined by QI/Mp = 1 - a, and

Deformation capacity of steel structures

77

is depicted by the dash-and-dot line in Fig. 29. In the domain below this line, the ultimate strength is less than the yield strength. In example 4.3.1, it was observed that when the slenderness, s, and the axial force ratio, a, exceed a certain limit, the slope of the Q-6 diagram becomes negative as soon as the member yields. Using the first equation of eqn (76), this critical condition is written as O(QI/Mp) Oh

]x=o0

which results in o~~

2 2 + s2 ep

(109)

Critical points defined by eqn (109) are marked by large dots in Fig. 29. Using the definition of deformation capacity from eqn (107), the r / - a relations are depicted in Fig. 30. 6u in eqn (107) is obtained from the relevant equations of Section 4.2, and 6y is expressed according to the definition

6y =

(1 - a)Mpl 2 3EI

(110)

The difference in the deformation capacities among grades of steel is remarkable. The inferiority of the deformation capacity for the higher grades of steel is caused mainly by their higher yield ratio. The influence of other material parameters on the deformation capacity will be discussed in the next section. Deformation capacity increases slightly with increase of slenderness for all grades of steel, and it takes the minimum value at the critical axial force ratio, a = ( R - 1)/2. According to the definition of deformation capacity as illustrated in Fig. 28, the deformation capacity is cut at the point where the load-carrying capacity comes down to the yield load, O r = (1 - a)Mp/l. These points are marked by large dots in Fig. 30. In the domain of a < (R - 1)/2, the deformation capacity increases very sharply with decrease of a. In the theoretical predictions obtained in Section 4.2, the strain-hardening modulus, Est , w a s assumed to be constant; however, in the actual stress-strain relationship, Est decreases gradually with increase of strain, as was shown in Fig. 5. The value of Est is sensitive in this domain, and serious errors might be introduced by assuming a constant value of Est; therefore, the deformation capacity was cut, rather arbitrarily, at the level of that for the beam (a = 0).

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Ben Kato

4.4 Analysis of deformation capacity For the mechanical properties which influence the deformation capacity, an analysis is made for the case of a beam. The deformation capacity in terms of slope was defined by eqn (108) as Ou

1 --

~0 -- 0y

Ou--OY Oup 0y -- 0y

(108)

The slope, 01, as a function of/z was given in eqn (104) for the case of a beam as ey E

/x 2 ) + e p ~ 2E~t 1 - / x

01 = s

(104)

The value of/~ at the ultimate state was given in eqn (108) as /~u = 1

1 R

(106)

Replacing/x in eqn (104) by/z u from eqn (106),

sr ,

0up = -~--12 - ~ t (R - 1)2 + ep(R - 1)

]

0y is expressed according to the definition as _ Mpl _ Oy 2El

SOy 2E

Equation (108) is then written as _0y rio - 0up

R1 [ -Et~t ~ - (R - 1)e + 2 e P ( R Ey

1)]

(111)

Equation (111) can be rewritten in terms of the complementary energy of the stress-strain curve of the material as follows. The complementary energy of the material is defined as the area surrounded by the stress-strain curve and the vertical axis, as shown in Fig. 31. The

Deformation capacity of steel structures

0

79

I

~y

Est

Fig. 31. Complementary energy.

subdivided a r e a s C1, C 2 and C3 of the complementary energy in Fig. 31 are calculated as

C1 = O~y/2E=

C2 = ( R - 1)O'yest,

O-yey/2,

1

(R-

C3 = ~ (R - 1)Cry

1)o'y _

Est

1 (R - 1)2~

2

Est

It then follows that C3

--

C1

Est

E

( R - 1) 2

and

C2

-

2 ( R - 1)

C1

6st

ey

Introducing these expressions into eqn (111), the following relation is obtained:

y[.C2+_+_C3 2 ( R - 1 )

7/0 ----- L

C1

] [cp = Y ~

2(R-l)

]

(112)

where Cp = C2 + C3 is the complementary energy of the plastic part and Ce = Ca is the complementary energy of the elastic part. Equation (112) means that the deformation capacity of a beam can be expressed in terms of the complementary energy. Taking a steel with characteristic values of try---2.4 x 102MPa, Y = 0-6, E/Est = 50 and est/ey --- 5 as the reference material, the influences of the variation of these characteristic values on the deformation capacity are shown in Fig. 32. When Y, E/Est and est/ey a r e kept

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80

/'/(0.6)

t/(v) 1.0 . - - ~

+"°Io+,°++, 0 7/ ' / " "2 3

o+ , ,

4

5 0 y ( l ; / c m z)

(a) effect of O'y

L. ~.5 0.6 0.7 o.a 0.9 110 Y = O"y/O'u

(b) effect of Y

T/(50)

0.9 0.8

1.0

| ~bilinear

% ~' ' ' ~ ....

~

040 20 40 6'o +b ;~o ~o

~- = ep/•y

(c) effect of ep/ey

e = E/Est

(d) effect of E/Est

Fig. 32. Effects of characteristic values on rotation capacity.

TABLE 2

Mechanical Properties of Higher Grades of Steel

A-steel B-steel

O-y

O'u

(Mea)

(Mea)

661 483

716 656

Y

8st/Ey

E/Est

0-92 0-74

5.5 2.4

131-6 28.3

unchanged, variation of the yield stress, try, does not have any influence on the deformation capacity. The deformation capacity decreases gradually with increase of O-y,if Y, E/Est and est are kept unchanged, as shown in Fig. 32(a). Figure 32(b) shows the influence of the variation of Y (other characteristic values are kept unchanged), and it can be seen that the effect of Y is very strong. In a similar manner, the effects of e s t / e y and E/Est a r e shown in Figs 32(c) and (d) respectively. When a new high grade of steel is to be developed, this information may be useful in determining the target shape of the stress-strain curve. The stress-strain curves of the higher grades of steel with low yield ratio which are under trial production are shown in Fig. 33, and their mechanical properties are shown in Table 2.

Deformation capacity of steel structures

81

~ Prnax

P(KN)

Pmax

1000 800

o'(MPa)

A

VO-u

7 0"u

f

B

500

500

Idealize

il

100 .

I

5

.

.

,

10

,

,

J

e(%)

Fig. 33. Stress-strain curves of sample steels.

0

: 5

1 10

15

d(cm)

Fig. 34. Test results of beams.

Fig. 35. Local buckling of flange.

The experimental load--deflection curves of beams made of these steels are shown in Fig. 34, and the ratio of the deformation capacity of B-steel to that of A-steel is found to be 2-42. On the other hand, the ratio predicted by eqn (111) is 2.14, so eqn (111) seems to give a reasonable estimation on the conservative side.

5 L O C A L B U C K L I N G (WIDTH-TO-THICKNESS LIMITATION) The local buckling of flanges and webs will inevitably take place at the final stage of loading even if their width-to-thickness ratios are limited to a reasonably small value. The deformation capacity of a member is determined by the deformation capacity of the flange and web in their post-buckling stage. The equilibrium of a compressed flange in the post-buckling stage is similar to that of a centrally loaded compression member which was discussed in Section 3, as shown in Fig. 35, and therefore, it can be foreseen that the deformation capacity of a buckled flange and web is largely dependent on the yield ratio of their base material. However, the theoretical approach to this problem is extremely

82

Ben Kato

difficult because of its complex boundary conditions. Therefore, in this section, the maximum compressive strength of H-section members as determined by the local buckling of flanges and webs is evaluated by stub-column tests, and with the aid of these experimental findings, the deformation capacities of beam-columns and beams with H-sections are assessed using a simplified structural model. TABLE 3

Mechanical Properties of Grades of Steel

SM41 SM50 SM58L SM58H

Yield stress, O'y (MPa)

Tensile stress, tru (MPa)

Yield ratio, Y = O'y/Or u

299 377 460 525

458 521 635 599

0-65 0-72 0-73 0.88

ep/ey

E/ E~t

10.5 9-2 2.4 6.4

52 63 38 116

5.1 Buckling strength of H-section stub-column The local buckling strength of an H-section member is determined by the stiffness of the flanges and web, as the flange is restrained by the web, and vice versa. Therefore the local buckling strength of H-section members could be determined by using the slenderness parameter of the flange, ctf = E/ftry(tf/b) 2, and that of the web, aw = E/wtry(tw/d) 2, in which fOryis the yield stress of the flange and wtry is the yield stress of the web. The geometrical notations used are as shown in Fig. 16. A series of stub-column tests with different combinations of b/tf and d/tw are carried out for various grades of steel, the mechanical properties of which are shown in Table 3. In this table, SM58L is a new high-strength steel with low yield ratio which is under development to improve ductility, and SM58H is a conventional high-strength steel with high yield ratio. The plateau length ratio, e p / e y , and strain-hardening modulus ratio, E/Est, of SM58L are rather small compared with those of SM41 and SM50, although the yield ratio is comparable. The numbers of stub-columns tested are 25 for SM41, 20 for SM50, 21 for SM58L and 22 for SM58H. The normalized maximum strength, R b = trcr/O'y, obtained from stub-column tests is related to the slenderness parameter, off, and aw, by means of multiple regression analysis for each grade of steel as follows: for SM41 steel (8 -< b/tf <- 16, 40 -< d/tw <- 80),

1/Rb = 0.689 + 0.651(1/af) + 0-0553(1/aw) _+0-0303

(113)

Deformation capacity of steel structures

83

for SM50 steel (7 -< b/tr <- 15, 35 -< d/tw <- 65)

1/Rb = 0-689 + 0.586(1/af) + 0.0711(1/otw) + 0.0538

(114)

for SM58L steel (6 -< b/tf <- 14, 20 - d/tw <- 60)

1/Rb = 0.716 + 0-518(1/af) + 0.0389(1/otw) _+0.0325

(115)

for SM58H steel (8 -< b/tf <- 14, 30 -< d/tw <- 60)

1/Rb = 0.881 + 0.270(1/af) + 0"0365(1/aw)

(116)

where orcr = Pcr/At, Pcr is the maximum compressive load in a stubcolumn test, At is the sectional area of the stub-column, af = E/for, (tf/b) 2 is the slenderness parameter of the flange, and aw = E/wory(tw/d) ~ is the slenderness parameter of the web. The correlation between test points and regression points is shown in Fig. 36.

5.2 Deformation capacity (rotation capacity) of members The ultimate strengths of b e a m - c o l u m n s and beams are determined by the buckling strength of their most severely stressed region and thus their deformation capacity is also determined by this critical state. Associated with the concept of plastic hinge rotation, the term 'rotation capacity' is used in this section instead of deformation capacity. The relation between deformation and stress at the critical section of cantilever beam--columns

i

1.4

O SM41

1.3

• SM50

1.0

/ /

q, '

..~ 0.9

o.81

o.,t

o °~-

°10.6 v o.'7 ~.a 0'.9 1'.0 I'.T 1'.2 ?.3 1:4 Rb by Tests

Fig. 36. Correlation between test and regression points.

Ben Kato

84

and beams has already been m a d e clear in Section 4. In the present problem, it will be reasonable to assume that the critical state of m e m b e r s is w h e n the stress of the critical section reaches the m a x i m u m buckling stress obtained from the stub-column test in Section 5.1. H o w e v e r , in using the information from Section 5.1, it should be noted that webs in b e a m - c o l u m n s and beams have a stress gradient, whereas those of stub-columns are uniformly compressed. This difference is taken into account by introducing an effective width, de, as =

Aw

a d

(117)

where A t is the total area of the H-section and Aw is the sectional area of a web, i.e. a " = E/wtry(tw/de) 2 should be used instead of aw in eqns (113)-(116). The m a x i m u m stress-deformation relationships obtained in Section 4 are too complicated to combine with the buckling stresses given by eqns (113)-(116), and hence, neglecting the effect of additional m o m e n t produced by the deflection and axial thrust, simple linear relations are derived for cantilever b e a m - c o l u m n s and beams in the following.

5.2.1 Moment-curvature relation (M-49 relation) M-th relationships have been obtained in Section 4.1. The only modification necessary for the present analysis is to use the critical stress ratio Rb O'cr/O'y, as summarized below. =

Case 1: a > ( R (1) For (k -< (bp

b -

1)/2

M = (1 - a)Mp

(64)

(2) For ~bp < 4, <- ~bcr M = (1 - a)Mp + D ( 6 - 6p) Mcr = (Rb-- t~)Mp 6cr = 6p q- [ ( R b -- I)Mp/D]

(65) (118) (119)

Case 2 : 0 < a -< (Rb -- 1)/2 (1) For ~b< 6p M = (1 - a ) M o

(64)

(2) For thp < ~b <- @p + (2aMp/D) M = (1 - a)Mp + D ( 6 - 60)

(65)

Deformation capacity of steel structures

85

(3) For 4,p + 2aMp/D < 4, --- 214,p + (aMp/D)/ M = (1 +

a)Mp

(68)

(4) For 214,p + (aMp/D)l < 4, <- 4,or (69) ~bcr = 2{~bp+ [(Rb -- a -- 1) Mp/D]}

(120)

Mcr = (Rb-- ot)Mp

(118)

Case 3: a = 0 (1) For 4, - 24,p

M = Mp

(71)

(2) For 24,p < 4, --- 4,cr (69) Mcr ~-- RbMp

(121)

~bcr = 2{~bp+ [(Rb -- 1)Mp/DI}

(122)

5.2.2 Rotation capacity If the effect of additional m o m e n t produced by the deflection and axial thrust is ignored, the behaviours of cantilever b e a m - c o l u m n s and beams are expressed by the following simple linear relationships. Case 1: a > (Rb -- 1)/2 Figure 37 shows the ultimate state of a cantilever beam--column, in which yielding starts at point A and the critical m o m e n t is r e a c h e d at point B. F r o m the geometry, the length of the plastic region, M, is AI = ( R b - - l ~ l

(123)

\R b - af

The coordinates x and y are normalized by l as X = x/l and Y = y/l, respectively. The bending m o m e n t at X ( 0 - X -< it) is M = (Rb -- a)Mp - (Rb -- ot)mpX

(124)

Ben Kato

86

B

~

-~)Mp

Fig. 37. Configuration of beam--column, a > (Rb -- 1)/2.

Equating M from eqn (65) to that from eqn (124), the equilibrium equation is d2y

dX 2

_

Mpl (R b _ 1) + l ~ b p - Mpl (Ro - cOX

D

(125)

D

Integrating eqn (125), the slope, Ox, is dY Ox _ dX

_

( R b -- 1) + 14~p X -

~

( R b --

a)X e

(126)

The slope at the end of the plastic region, A, is dY) Ocr =

"~

Mpl (Rb-- 1)2 X=X-- "

2D

Rb- a

/Rb-- 1'~ + lg~p

\ R ~ - - ~ -a ]

(127)

The slope of the cantilever beam-column when the moment at fixed end B reaches (1 - a)Mp is Oy =

(1 -

a)

2El

(128)

Mpl

According to the definition of eqn (108), the rotation capacity in terms of the slope, r/0, is given as

(Rb--1) [-~st• (Rb-1)+2 (%)] Ty

cr ,7o __ 0o~-2(1;~-,~)

(129)

Deformation capacity of steel structures

87

Case 2 : 0 < a -< ( R b - 1)/2 In this case, the yielded region is further subdivided in two, i.e. yielding starts at point A in Fig. 38, and in region AB, only the compression flange yields, and the M - ¢ relationship is given by eqn (65). In region BC, both compression and tension flanges yield, and the M-~b relationship is given by eqn (69). From the geometry, the lengths of plastic region hl and/xl are given as, respectively, a =

1,

/z = \

)~b~-a-

1

(130)

c

;

l

', × 1__.~.

,x

Fig. 38. Configuration of b e a m - c o l u m n , a -< (R b - 1)/2.

The bending moment at X (0 -< X-
(124)

Equating M from eqn (69) to that from eqn (124), the equation of equilibrium is d2y [ - ~ dX2 =

(R b

-

a - 1 ) + 2/¢p]

-

Integrating eqn (131), the slope, 0x, is

2Mpl(Rb-a)X D

(131)

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Ben Kato

T h e slope at the end of plastic region/x (point B) is

1, T h e slope at the end of plastic region A (point A) is calculated by shifting the origin to point B. T h e bending m o m e n t at X (0 <- X <- A) is M = (1

+ ot)Mp - (R b -

ot)MpX

(134)

Equating M from eqn (65) to that from eqn (134), the equation of equilibrium is d2Y

dX 2

-[ 2Mpl

a + lq~p

-- - -

(135)

( R b -- a ) X

D

Integrating eqn (135), the slope, 0x, is Ox - - ~

-

a+14)p X -

-~(Rb--a)X2

(136)

+C

F r o m the b o u n d a r y condition that 0x at X = 0 should equal 0~, from eqn (133), C = 0/x is deduced. T h e slope at the end of plastic region A (point A) is then given by

1

dY

Rb -- a/--b--- [( b -- 1) 2 + 2a21

+ 2 1 ~ p ( R b - a --

1)} (137)

T h e rotation capacity, rio, is given as cr

"qO- 0y

(1

7 --

[(Rb -- 1) 2 + 2a 2] + 2 - - (Rb -- a -- 1) Ey

(138) Case 3: ot = 0 (beam) r/o is o b t a i n e d simply by introducing a = 0 into eqn (138) as no =

Rb

( R b - 1)

+22]

ey _1

(139)

Deformation capacity of steel structures

89

5.3 Width-to-thickness ratio limitations

In design specifications, width-to-thickness ratio limitations of the flange and web are prescribed independently according to the ductility d e m a n d for the structure. H o w e v e r , according to the present study, the rotation capacities are given by the interaction formulae of b/tf and d/tw, which seems to be very reasonable.

5.3.1 Beam--column As is seen in Fig. 36, the m a x i m u m stress ratio as d e t e r m i n e d by local buckling is at most 1.2; then the corresponding axial force ratio a is 0-1 according to the behavioural categorization of a X (Rb -- 1)/2, and most of the axial force ratios of b e a m - c o l u m n s are larger than 0.1. T h e r e f o r e , eqn (129) for case 1 can be used for the evaluation of the rotation capacity r/0. Solving eqn (129) for 1/Rb, 1 _ [e - e + (1 - a)rlo ] - "k/[e - e + (1 - a) r/o]2 - [e - 2e + 2a(1 - a)rlole = D e -- 2e + 2tl(1 - ct),lo

Rb

(140) w h e r e e = E/Est and e = ep/ey. O n the o t h e r hand, eqns (113)-(116) are expressed in general form as

--

'

Rb = A +

B

1

+C

1

(;w)

(141)

Substituting for 1/Rb from eqn (140), eqn (141) can be expressed as

.(.)

D-A

+ D-A

= 1

or

b)2 E (D-A)

,

tT:

( a 2 + E (D-A]

= 1

(142)

Equation (142) is the interaction formula for limiting the width-tothickness ratios of the flange, b/tf, and of the web, d/tw, for a given axial force ratio, a, and for a required rotation capacity, 7/o.

90

Ben Kato

To observe the practical features of the width-to-thickness ratio limitation for each grade of steel, a = 0.3 and a ductility demand of -% = 2 are assumed. As was seen in Fig. 30, the deformation capacity (rotation capacity) increases with the increase of axial force ratio, and a = 0.3 is adopted as a representative value for the design of beam-columns rather arbitrarily. Furthermore, the effective width, de, as given by eqn (117) is introduced to take the effect of the stress gradient of the web into consideration, where At/Aw = 2.5 is assumed for the standard H-section. Calculating A - D values for each grade of steel using mechanical properties as given in Table 3, the following formulae are obtained: for SM41 steel

(27 /2 +( 1065t2--1

(143)

\ v~--w-~w ~ /

for SM50 steel

2

( f)2

28212+ ( 92_~6 )2 -

1

(144)

~ wXwXwXwXwXwXwXwXwXw~7%/

for SM58L steel

(145)

21532+(895)2=1

for SM58H steel

159 )

+

( 49___._~3t2

= 1

where fOy and wO-yshould be given in MPa.

(146)

Deformation capacity of steel structures

91

5.3.2 B e a m

Equation (139) for case 3 is used for the evaluation of ~70.Solving eqn (139) for 1/Rb, 1

[(e - e + ('0o/2)] - V'[e - e + ('0o/2)]

R b

e -

2 -

2e

(e -

2e)e

(147)

where e = g/Est and e = ep/ey. By a similar calculation and postulation to those made in the foregoing section, the following formulae.are obtained for each grade of steel: for SM41 steel

271)2+(1860/2

= 1

(148)

= 1

(149)

= 1

(150)

=~

(15l)

for SM50 steel

283)2+(1624)

for SM58L steel

(2~__a_~t~+(,,9~/~ for SM58H steel

(~6_~_11~+ (, wV~yj ~/~ ~V~yj

where fO-yand w~ryshould be given in MPa.

Ben Kato

92 b

.

t (-:,fi~-,) 300

,oo

0

500

1000 d . ( + ~ ; )

(a)

20 15

-"SM50

SM41

10 5

50

d tw

(b)

Fig. 39, b/tf-d/tw interaction.

5.3.3 Example The width-to-thickness ratio limitations of H-section beam-columns with a = 0.3 and with the required rotation capacity, r/0 = 2, are prescribed by eqns (143)-(146) for various grades of steel. These equations are depicted in Fig. 39. In Fig. 39(a), the quantity on the vertical axis should be divided by VfO'y to read b/tf, and the quantity on the horizontal axis should be divided by ~/-~w ~y to read d/tw, i.e. the effect of the difference of yield stress is excluded in this figure; therefore, the differences observed in the width-to-thickness ratios among grades of steel are caused by differences in the yield ratio, ep/ey, and E/Est. For a specific observation, although the yield ratios of SM50 and SM58L are almost the s a m e , ep/ey and E/Est of SM58L are considerably smaller than those of SM50, and these differences seem to have resulted in the differences in the d/tf-d/tw interaction diagrams. This observation implies that the rotation capacity as controlled by local buckling should also be evaluated by the complementary energy of the stress-strain diagram of the base material. However, the effect of the yield ratio is dominant, as can be seen by comparing the diagrams for SM58L and SM58H. Even if the stress-strain diagrams of all grades of

Deformation capacity of steel structures

93

steel are similar, the width-to-thickness ratio limitation becomes severer with increase in yield stress, as is common for buckling problems. In Fig. 39(b), the width-to-thickness ratio limitation diagrams have b/tf on the vertical axis and d/tw on the horizontal axis. The differences between interaction diagrams for the various grades of steel are more pronounced. The correlation between these predictions and experimental results has been discussed elsewhere. 10,11

6 CONCLUSIONS The ultimate strength and plastic deformation capacity of the following steel structural members were analysed: (1) tension members with a stress gradient along their length; (2) centrally loaded short compression members in the post-buckling state; (3) beam--columns and beams which are not influenced by buckling; (4) beam-columns and beams whose strength and deformability are governed by local buckling of member sections. It was demonstrated that the strength and deformation capacity of all these members are strongly influenced by the yield ratio of the base material, i.e. the deformation capacity decreases substantially with increase of the yield ratio. The rotation capacity of beam--columns and beams can be evaluated more precisely by using the complementary energy of the stress-strain relationship of the steel, although the predominant part of the rotation capacity is that associated with the yield ratio. The rotation capacity of steel members is severely impaired by the occurrence of local buckling of plate elements of the constituent members, so limitations of the width-to-thickness ratios of the flange and web of H-section members are prescribed in the specifications of various countries. However, the theoretical or experimental background is not necessarily clear, and, furthermore, the width-to-thickness ratio limitations for the flange and for the web are prescribed independently of each other. Obviously, the flange is restrained by the web, and vice versa, and therefore an independent limitation is unreasonable. In Section 5, the inelastic rotation capacities of H-section steel members subject to bending with and without axial thrust as determined by local buckling were investigated. Also, the rotation capacities were predicted as functions of width-to-thickness ratios of the flange and web. In other words, the interaction formulae of the width-to-thickness ratio of flanges and that of

94

Ben Kato

the web for various loading conditions were given according to an assigned inelastic rotation capacity. F r o m the findings obtained in this study, it could be concluded that, to secure a reasonable and reliable deformation capacity of steel structural members, the upper limit of the yield stress and/or of the yield ratio should be specified for each grade of steel in the material standards as the basic requirement.

REFERENCES 1. Kato, B. & Aoki, H., Deformation capacity of steel plate elements. IABSE Publ., 30-I (1970), pp. 93-112. 2. Pope, G. G., The application of the matrix displacement method in plane elasto-plastic problems. Proc. Conference held at Wright-Patterson Air Force Base, OH, October 1965. 3. Kato, B. & Akiyama, H., The equilibrium of short strain-hardening steel columns. Int. J. Solids Struct., 11 (1975), 305-20. 4. Shanley, F. R., Inelastic column theory. J. Aerospace Sci., 14(5) (1947), 261-8. 5. Lee, G. C. & Galambos, T. V., Post-buckling strength of wide-flange beams. J. Engng Mech. Div., ASCE, 88(EM1) Proc. Paper 3059 (1962), 59-75. 6. Galambos, T. V. & Lay, M. G., Studies of the ductility of steel structures. J. Struct. Div., ASCE, 91(ST4) Proc. Paper 4444 (1965), 125-51. 7. Lay, M. G., Yielding of uniformly loaded steel members. J. Struct. Div., ASCE, 91(ST6) Proc. Paper 4580 (1965), 49-66. 8. Lay, M. G. & Galambos, T. V., Inelastic steel beams under uniform moment. J. Struct. Div., ASCE, 91(ST6) Proc. Paper 4566 (1965), 67-93. 9. McDermott, J. F., Plastic bending of A514 steel beams. J. Struct. Div., ASCE, 95(ST9) Proc. Paper 4566 (1969), 1851-71. 10. Oh, T. S. & Kato, B., Deformation capacity of H-shaped steel members governed by local buckling. J. Struct. Engng, 34B(AIJ) (1988), 161-8. 11. Kato, B., Rotation capacity of H-section members as determined by local buckling. J. Construct. Steel Res., 13 (1989), 95-109.