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Initiation of plastic folding mechanism in crushed box columns
Thin-Walled Structures 13 (1991) 115-143
!I Initiation of Plastic Folding Mechanism in Crushed Box Columns
Tomasz Wierzbicki & Joan Huang Department of Ocean Engineering. Massachusetts Institute of Technology, USA (Received 3 November 1990: accepted 25 March 1991)
ABSTRACT A simple computational model of a thin-walled prismatic column is developed, which describes a transition from the postbuckling to the pos~'ailure and crushing deformation phase. Energy methods are used to analyze the elastic post-buckling response of the column. Ultimate strength is calculated, using the first yield criterion. Limit analysis methods, properly generalized to large displacement and rotation problems, are employed in the postfailure range. The new model explains, with some realism, the process of strain localization and the formation of a plastic folding mechanism in the column. It also leads to a prediction of the entire load-deformation characteristics up to the internal touching and stiffening.
1 INTRODUCTION The response of a column to axial compressive loads consists of two distinct phases. In the prefailure phase the load is increasing up to the point of maximum strength; rotations are moderately large, but the strains remain small. Generally, each plane of the column can be considered as a plate. Except for very thin plates, there is little change in the buckled form of the plate, and the initially straight comer lines (intersections of the adjacent planes) are only slightly deformed. The first y i e l d o c c u r s n e a r the comers, but plastic deformations are contained within the plate and thus are relatively small. !i5 Thin-Walled Structures 0263-8231/91/$03.50© 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain
116
Tornasz Wierzbicki, Joan Huang
The postbuckling and ultimate strength response of plates and plate assemblages is well understood due to the fundamental work of von Karman, ~ Koiter 2 and Budiansky and Hutchinson) Since then, a number of authors have contributed to the problem of stiffness and strength of thin-walled structures, and clarified the effect of imperfections, aspect ratio, boundary conditions and plasticity9-6 The class of problems with diffused elastic-plastic deformations is relatively easy to solve numerically, and several general purpose nonlinear finite element codes such as ADINA, ABAQUS, etc., can be used to calculate the initial loaddeflection characteristics of box columns with great accuracy and speed. The subsequent phase of the column's response after reaching the peak load is entirely different. The deformations localize within narrow zones and an uncontained plastic flow takes place until first contact occurs. This process is accompanied by very large strains (up to the rupture strain of the material) and large rotations. Furthermore, the original cross-sectional shape of the column is subjected to a severe distortion and the initially straight corners open up and bend. Different analysis methods have been developed in the literature to tackle each of the above mentioned deformation stages separately. However, except for a preliminary analysis 7 and an unpublished report (Rhodes, J., pers. comm. 1989), no attempts have been made in the literature to study the process of localization of plastic deformation within the column. The objective of the present paper is to provide the missing link between these two distinct phases of column response, to explain the formation of plastic folding mechanism, and to present results for a few selected geometries of columns. A consistent and simple computational model of column deformation is developed, which describes the transition from postbuckling to the crushing deformation phase.
2 LOCALIZATION OF DEFORMATION AND PLASTIC F O L D I N G MECHANISM The buckled shape of a long thin-walled rectangular column is shown in Fig. 1. Except for very thin flanges, little change in the deformed shape of the column occurs until the point of maximum strength is reached. From this point on a dramatic change in failure mode takes place. A close examination of experimental results and computer simulation results of deformed shapes reveals the complexity of a transition phase. The transition phase continues until a clear plastic folding mechanism is formed (Fig. 2). The corner line then deforms from its initial vertical
Initiation of plastic folding mechanism in crushed box columns
I17
Fig. 1. A buckling mode of a rectangular column subjected to axial compression.
position. The right angle between the adjacent flanges opens up and two diagonal fold lines appear to grow on opposite flanges. The process of plastic strain localization is completed when the initial corner is completely flattened out. From this point on a crushing mechanism, described, for example, by Abramowicz and Wierzbicki, s takes over until first contact occurs. A simple two-degrees-of-freedom model of an imperfect column is developed, which captures the main features of the localization process. The model does not reveal any new features of the column's response in the postbuckling range which were not known before. However, the same model can be continued into the plastic range, thus providing an important insight into the complex physical problem. The model is explained in Section 3.
Fig. 2. Localization of plastic deformation along diagonal lines and opening up of the comer lines (courtesy of F. de Bruyne.)
Tomasz WierzbickL Joan Huang
! 18
3 BUCKLING AND POSTBUCKLING BEHAVIOR In the present model, the deformation of the column consists of in-plane compression and out-of-plane bending (Fig. 3). A uniform in-plane compression defines the prebuckling equilibrium path, P
u0
(1)
bh - E ~
where P is the compressive load, E is the Young's modulus, b is the column width, h is the column thickness, H is the half elastic wavelength which equals b/2, and u0 is the in-plane uniform displacement. Once buckling occurs, it is assumed that the deformed plates rotate with respect to each other and deform in the plate plane, as shown in Fig. 3(b). The out of plane displacement field w(x,y) takes the form of a 'prism' with four lines of slope discontinuities and a central amplitude, w0. The angle of slope discontinuities is defined by the distance (, which is considered to be an unknown parameter of the process (Fig. 4). An initial imperfection is introduced, which takes the same shape as in Fig. 3(b) and has a central amplitude w0. In the present analysis we follow the formulation presented by Wierzbicki and Recke 9 and Wierzbicki.7 The displacement field can be expressed by the following equations:
x
u° ( 1 - b ~ ) 2
u(x'Y) =
(x)1
l ( bw2- ~ )I1 - ~ ) ~x y w~
x)
0
and
v(x,y)
= 0
(2)
where v is the component of the displacement vector transverse to the direction of loading. (a)
[_
/1
_-
(b)
/
Fi~. 3. ~a) Uniform compression of a column, a n d (b) relieving tension due to out-of-
Initiation of plastic folding mechanism in crushed box columns
119
o
t
\
I 0/
Fig. 4. Illustration of the displacement field of a column under compression.
There are two contributions to the in-plane displacement u(x,y). One comes from the in-plane compression, which equals u0 at the upper and lower boundaries of the column and equals zero at the middle symmetry plane. The other contribution comes from the finite rotation of the plate element which actually relieves the in-plane compression. Additionally, it is assumed that the column is inextensional in the transverse directions, i.e, e v(X,y ) = O, and that no elastic work is done at the comers so that right angles are preserved during the deformation process. From the above assumptions, the strain field can be easily calculated using the infinitesimal strain definition, as u0 Sx
m
_
Ou
1
w~ y
0
- (b/2~-) + 2 (b/2) 2 (
Ox
u0
1 w02 - (b/2-----)+ 2 (b/2) 2
00
ey - Oy
b (
0
(3) ¢"
l(Ov
exv-~.
~+
•
O~y)
=
' 1 Wo 2 x 4(b/2)2(
t
0
0
The equilibrium path can be determined by using the principle of virtual work. Due to symmetry of the structures, only one quarter of the plate is considered here in the calculation, which gives, l"I(u0, w0) = Ub(U0, w0) --I- Urn(u0, w0) -
(')
~ u0
(4)
where II is the total potential energy, P is the compressive force applied at one comer of the column (the total force is 4P), and Ub and Um are the bending and membrane energies, respectively.
120
Tornasz Wierzbicki, Joan Huang
The calculations of Ub and U,, are carried out in Appendix A. It should be pointed out that the present model gives a good approximation of the membrane energy, but a rather poor approximation of the bending energy due to the discontinuities of slope. Because slope discontinuities are not admissible for elastic structures, we had to smooth-out the edges of the prism in the bending energy calculation. The arbitrariness in the smoothing procedure accounts for an inaccurate expression for Ub. The equilibrium condition 6II = 0 gives rise to the following system of nonlinear algebraic equations relating P, uo and w0: P-
( 1 - v 2) u0
~
I-
U o W o ( ( - b ) + w o ( w 2 - ~ 2)[1
(5)
4~'3 b+(1-v)12 ~]
+g-(Wo-~o) l+~+ y~
=o
(6)
where E and v denote, respectively, the Young's modulus and the Poisson's ratio. In the limiting case of a perfect plate, ~0 = 0, e q n s (5) and (6) define the primary and secondary equilibrium paths. Bifurcation occurs at 8 Eh 3 Per- 3 (1-v2)b '
4 h2 ucr- 3 b
(7)
The above result compares well with an exact buckling formula for a simply supported elastic plate where the coefficient 8/3 is replaced by n2/3. The postbuckling equilibrium path is given by the following equation: P -
AU°+B
ecr
(8)
Ucr
where the coefficients A(/~)
=
1
-
(l -
1
4
-g/~+
(1-~) B(fl)
=
[
1
4
t~) 2
(1 - v ) 1 12 fl
+~+~
(9)
Initiation of plastic folding mechanism in crushed box columns
121
are functions of the dimensionless parameter fl = (/b. Equation (8) is plotted for several values of(, as shown in Fig. 5. The lowest bifurcation load is obtained at ( = 0.5b, which corresponds to a roof mechanism with the slope discontinuity line inclined by 45 ° to the vertical direction. The onset of buckling leads to an immediate drop in axial stiffness to 0.440 of the prebuckling stiffness. This compares favorably with the exact values of 0-5 and 0-408 for plates with edges kept straight and edges free to wave, respectively, n° As the axial shortening increases, there is a further reduction in the column stiffness, and the load-deflection curve follows the envelope of a family of the straight lines with various values of(. Even more important than the reduction in the plate stiffness is the associated mode change. The present model predicts that the plate initially buckles in the triangular shape of w(y) in the cross-sectional plane. This shape then transforms gradually into trapezoidal shapes, shown by the labels in Fig. 5. It should be noted that in the classical approach to the postbuckling problem, the mode transition could only be described by retaining many 10
I
. . . .
~
. . . .
i
. . . .
i
. . . .
1
elastic / / / / / / / /
a.,
/ / / /
~.~.4
I
J
I
0
0
5
10
15
20
Uo/Ucr
Fig. 5. Load-deflection curve in the postbuckling range and mode transition.
Tomasz Wierzbicki, Joan Huang
122
terms in the modal expansion of the solution (cf. Koiter2). In the present formulation this result is obtained in a natural and elegant way.
4 FIRST YIELD AND ULTIMATE STRENGTH With increasing axial load there is a redistribution of stresses within the plate. Compressive stresses at the plate center are relieved while stresses near corner lines continue to increase. The plate material will then yield in the most stressed point. Assuming that the plate yields due to membrane stresses alone, the plane stress yield condition applies: 2 ? or,,2 - Crx~v + Cry + 3cr~v = or02
(10)
where cr0 is the yield strength of the material. At the commencement of yielding, eqn (10) can be expressed in terms of the components of the strain tensor given by eqn (3). In the case of a plate with ( = 0-5b and without initial imperfection, the corresponding equation takes the form: (l+v 2-v) 2 3 2 -(i e, + (1 + v) 2 e,:, =
(11)
A detailed analysis of the above equation reveals that the column will yield first at a point along the corner line, halfway between the supports (or nodal lines). Using the expressions of the axial and shear strains at the yielding point, the above equation becomes functions of u0 and w0: (l + v2-- v) 4u 2 (1 -- V2) 2
~b-
3
w4
q- ( l ~- v ) 2 b 4 =
(~)2 .--.
(12)
Equation (12), together with eqns (5) and (6). furnishes a system of three equations for three unknowns, P, (u0),, and (w0)u, from which the corresponding load at first yield can be obtained. Upon further compression there will be continued load shedding at the central region of the plate while the stresses near corner lines will increase only slightly due to the strain hardening of the material. The plate will exhaust its load carrying capacity soon after the first yield is reached. We thus identify the force corresponding to the first yield as the ultimate strength of the column, P,,. A plot of the normalized plate strength versus the slenderness parameter, hb / ~ ,
is shown in Fig. 6 for a few values of the
imperfection parameter. Also shown in this figure is the curve corresponding to the so-called squash load, which represents a full plastic capacity of the undeformed cross-section:
Initiation of plastic folding mechanism in crushed box columns
123
We/h =0.001 %/h=OA
2.0
"~
~,,'~,~
"
• Wo lh--0.2 , Wolh=0.5 ' Wolh=l.O
~',
~ ~
",, --... 1.2
R¢f. [151
~.
0.4 0
0.5
1.0
h
Fig. 6. Ultimate strength of the column versus the slenderness parameter for several values of the imperfection. Psq
=
bhao
(13)
The present solution overestimates by 10 to 15% the exact ultimate load calculated by means of the ABAQUS finite element code (circles in Fig. 6).
5 POSTFAILURE BEHAVIOR Once the ultimate strength is reached, the column will not bear any more load, and the compressive force decreases as the column is further deformed. Our simplified model can be extended into the postfailure range. The two-degrees-of-freedom model is valid up to the point of the first yield. Beyond the first yield the central part of each side plate of the column is subjected to further unloading, and plastic deformations spread into the edge zones. It is thus reasonable to assume that at an early stage of the crushing process the central trapezoidal sections of the column will be fully unloaded and will rotate as rigid bodied. From this point on, the in-plane and out-of-plane deformations are related through the geometry of the problem: Wo 2 = Uo(2H- Uo)
(14)
Tomasz Wierzbicki, Joan Huang
124
The above condition reduces the number of degrees of freedom of our model to just one. All kinematic quantities such as displacements ua, strains e~,t~and strain rates g,,t~become now unique functions of either u0 or w0 and their rates. In actual calculation it is more convenient to treat the rotation angle of the trapezoidal element, a, as an independent variable (see Fig. A.1 in the Appendix). Several interesting properties of the deformation process follow from the above assumptions. First, the initial right angle between the adjacent plates must now change as a increases (Fig. 7). Thus, the corner line opens up and flattens out and an inclined hinge line is gradually formed. As crushing progresses there is an overlap between the adjacent triangular elements in the lower and upper half of the column. This overlap is indicated by a shaded area in Fig. 7 and its maximum amplitude is denoted by u~. As a result, each of the deforming triangular elements is subjected to axial compression and shear. To simplify the calculation, it is assumed that the strain is distributed uniformly in the triangular elements. Additionally, the assumption e v = 0 is assumed to remain valid in the early postfailure stage. Under the above assumptions, the strain rate tensor is
H
2(
1 til
0
2(
(15)
The process of flattening out (phase I) is completed when a = a2 = arc sin ( ~ ) . F o r a > a2,a different failure mechanism takes over. Either, the inclined stationary hinge line splits up and starts to move so that further crushing is possible) ~or the inclined hinge line remains stationary and
Fig. 7. Plastic compression/tension in the triangular transition zones.
Initiation of plastic folding mechanism in crushed box columns
125
the side wall opens up, giving rise to the one-dimensional extension in the transverse direction (Fig. 8). The latter possibility will be explored here. When a > a2 (phase II), there is a relative open-up displacement between the adjacent triangular elements at the comer line connecting two plates. The maximum amplitude of the open-up in the second phase is denoted by un. Similarly, each of the deforming triangular elements is subjected to axial compression and shear. Since the strain in the transverse direction becomes dominant, it is reasonable to assume that ex = 0 while ey #: 0. Under these assumptions, the strain rate tensor in the second phase is
1 nil ] 0 2H
2 H
(16)
~"
The crushing process is completed when the two halves of the plate reach the point of touching (a = rt/2). The equilibrium of the column is expressed via the principle of virtual velocities
Pfto = h fs aaagaa dS + ~., MoliOi
(17)
i where the two terms at the right-hand side of eqn (17) denote, respectively, the rate of internal dissipation in the zones of continuous and discontinuous plastic deformations. The integration is performed over the area S of two rotating triangular elements. The tensorgao denotes the so-called strain velocity, calculated from the known velocity field in the current, deformed configuration. The components of the Cauchy stress tensor craa are calculated from the associated flow rule by using the
Fig. 8. Plastic collapse mode with opening up of the side walls in the transverse direction.
Tomasz Wierzbicki, Joan Huang
126
plane stress yield condition, eqn (10), as a plastic potential. The fully plastic bending moment is defined by M0 = aoh2/4, where tr0 is the yield strength, and li and 0; in eqn (17) denote, respectively, the length of the ith hinge line and the rate of rotation at the hinge. The summation is performed over all plastic hinges. In our one-degree-of-freedom model, all plastic hinge lines are fixed as the column deforms. Their lengths are thus constant and can be easily expressed in terms of b, H and (. The rates of relative rotations can be expressed in the form
0, = f ( b , H, ()gt
(18)
where f are length functions, which are calculated in Appendix C. The expression for the continuous energy dissipation is different depending on the range of a:
D = h
fsO,t,
dS =
~M0~-Iall
1+4
0
~3 Mof~la,, I
1+4
a2
The velocity vectors u~ and u. are calculated in Appendix B from the kinematics of our problem, which gives Ul
H S ( ( 2 - H 2 sin2a) - cos a v / ( 4 - H 4 / (-~¥~/~ sin2a - 2H 2
sin4a}
H sina& uw = 2H 2 _ (2 _ H 2 sin2a
2H(Ul + H)cosa
- ¢(4
_ H 4 sin4a
H4 _ -2-
¢¢4
sin 22a } Z~_/7sin4al
(20) and un = H sina (/-/2 sin2a - (2) (H 2 sin2a ~ (2) fin = a c o s a 6 [1 + 2 ( 2 ((nn 22 sin2 asin2a _+(2)-](2)2j
(21)
The displacements ul and un are plotted in Fig. 9(a) and (b) as functions of the rotation angle a. It is interesting to note that the velocity fir vanishes at a certain angle a~. Initially, the relative displacement u~ increases so that compressive stresses are developed in the triangular
Initiation of plastic folding mechanism in crushed box columns
127
0.20
;---o.s H 0.15
0. I0 0.6
0.05 0.4
0.2 0.00 t_ 0.0
1.0
0.5 (l
(a) 1.0 0.2
0,4
0.5
0.6
0,0
I
0.0
/ V
~=0.8 H
/ A/
J
0.5
1.0
1.5
2.0
a
Co) Fig. 9. The m a x i m u m amplitude of the relative displacement between the adjacent triangular elements as functions of the rotation angle a: (a) in phase I; (b) in phase II.
Tomasz Wierzbicki, Joan Huang
128
zones. As the maximum displacement is reached, u~ diminishes under further crushing and is brought back to zero at a = a2. This means that there is a strain reversal in the intermediate stage of the column deformation, where initial compression is followed by tension. This interesting effect was overlooked in all previous analyses of the crushing process based on the final configuration approach. ~2,~3
6 RESULTS AND DISCUSSION The results of the present calculations are shown in Figs l0 to 13 for typical material constants, cr0, E, and selected values of structure geometries, b/h. According to eqn (17), the postfailure characteristics of the column depends on, besides the structure geometries, the wavelength H and the parameter (. The wavelength H equals b/2 in the elastic range, and, in the plastic range, a family of curves with different values of H/h are plotted. The model parameter ( is an unknown of the crushing process. The solution presented in Section 8 reveals that ( is related to H. By
600
*
i
~
n
]
,
,
I
T
1
~
7000;¢:
n
J//~/
I
T
i
~
i
/
=,00 __~-------~---
squash
__
400
265
.....
"~'"9( first yield
200
0 0.00
I
I
;
I
1
0.05
I
I
I
I
I
t
0.10
t
k
l
]
0.15
I
I
I
t
0.20
uo/h Fig. 10. Load-deflection characteristics of the thinner column with normal yield strength. The c o m e r zones are yielded.
Initiation of plastic folding mechanism in crushed box columns 400
I
,
, ,
i
'
i
1
129
,
~o = 200 MPa
E = 200,0Pa 300
b -= h
50
~u~h
200
H --=20
100
]5I0
0
I
I
0.0
0.1
L
I
I
0.2
I
0.3
I
I
0.4
I
0.5
uo/h
Fig. 11. Load-deflection characteristics of the thicker column with normal yield strength. The full squash load is reached.
minimizing the energy absorbed in the collapsing column, (/H = 0.78 is obtained. Figures 10 and 11 show the normalized load-deformation curves for or0 = 200 MPa, E = 200 G P a and two aspect ratios ofb/h = 100 and 50. The shape of the load-deformation curve depends on the aspect ratio. At the very beginning, the column deforms elastically along the loaddeformation curve, then, for relatively thin columns (b/h = 100), as shown in Fig. 10, elastic buckling occurs, which leads to a reduction in the column stiffness. The axial load keeps increasing up to the point of first yield. Since the column will exhaust its load carrying capacity soon after the first yield is reached, the first yield stress is thus defined as the ultimate strength of the structure. The calculation shows that the normalized peak load corresponding to the first yield equals 265 for the column with b/h = 100. As the column is further deformed, the load will start to decrease along the postfailure branch with H/h ~ 30. Such a dramatic change from the elastic buckling wavelength to a much shorter plastic wavelength is always observed experimentally. For relatively thick columns with b/h = 50, Fig. 11, the squash load is reached at a load lower than the elastic buckling load. The column follows for a while the constant load (squash load) if the material
130
Tomasz Wierzbicki, Joan Huang 400
oo = 400 MPa E = 200 6 P a
_b=
300
h
50
squash
200 174 H --=20 h
i00
-|5---tO
0 0.0
0.1
I 0.3
0.2
~
I 0.4
0.5
uo/h Fig. 12. Load-deflection characteristics of the thinner column with high yield strength. 300
o 0
= 400 MPa
E = 200~t b -= h
30
200
squash H --=15_ h
100
- 10-
0
L._
0.00
0.05
0.10
0.15
0.20
uo/h Fig. 13. Load-deflection characteristics of the thicker column with high yield strength.
Initiation of plastic folding mechanism in crushed box columns
131
behaves ideally elastic-plastic, and then softens along the postfailure branch of the load-deformation curve. In most cases, however, the load can still increase after the squash load is reached due to the hardening of the materials. Eventually, the column will buckle in the plastic range. The plastic buckling of stocky columns with hardening behavior, as represented by the dotted line in Fig. 11, will be discussed in Section 7. In addition to the effect of the aspect ratio, the material constants of the column will also affect the structure behavior. Figures 12 and 13 show the normalized load-deformation curve for E = 200 GPa and a higher yield strength tr0 = 400 MPa. It is expected that the structures of higher yield strength are more likely to buckle elastically. The present model does predict this structure behavior. For example, with the same aspect ratio b/h = 50, the model predicts that an elastic buckling occurs for the highstrength column (Fig. 11). On the other hand, even with high strength, a very thick column ofb/h = 30 will still attain the squash load before the elastic buckling load is reached (Fig. 13). 7 PLASTIC BUCKLING OF STOCKY COLUMNS A theory of plastic buckling of columns was developed by Shanley and generalized by Stowell for rectangular plates. The formula for a critical plastic buckling stress of a simply supported rectangular plate, derived by Stowell, t4 is trcr _ rr2 ~ E s ( h ) 2 [2 + / 1
+ 3Et] EJ
(22)
where Et = dot/de is the tangent modulus, and Es = tr/e is the secant modulus. Because Et and Es are, in general, functions of the magnitude of stress (or strain), eqn (22) can be used only when the stress-strain diagram of the material is known. One of the typical stress-strain relations is the power type: cr = cr0
(;0)
(23)
where o'0 is the yield strength, e0 is the yield strain, and n is the hardening exponent. Having the stress-strain relation, one can easily calculate the critical plastic buckling stress and strain: ~cr
(eo)"
(2 + v / i + 3n)"
(24)
Tomasz Wierzbicki, Joan Huang
132
ecr
9
(2 + x/1 + 3n)
(25)
When the plate buckles in the plastic range, it cannot carry any additional load. The ultimate (peak) load of the plate is then obtained by multiplying ~ , by the cross-sectional area. In the case of our square column:
Pu
=
acrbh
(26)
where Pu is the force applied at one comer of the column. The results of the present calculation are plotted by the dotted lines in Figs 11 and 13 for the previous two examples, b/h = 50, o0/E = 0.1%, and b/h = 30, ao/E = 0.2%. The hardening exponent n is taken as 0-2, a typical value for mild steel. After the squash load is reached, the axial force will continue to increase along the material's hardening curve. The ultimate load is determined by the point of plastic buckling stress, which also determines the plastic wavelength. For b/h = 50, ao/E = 0.1%, the mode transits from HIh = 25 into HIh ~ 10, and for blh = 30, or0/ E = 0-2%, the mode transits from H/h = 15 into H/h ~ 5. It should be remembered that the real transition from elastic to plastic should be a smooth one. In view of this, the present model predictions for the plastic wavelength can only be considered as an estimation.
8 E N E R G Y ABSORPTION The energy absorbed by the collapsing column can be obtained by integrating both sides of eqn (17) with respect to time or the process parameter a.
f0',u0a, f0
f0
i
Of particular interest is the energy corresponding to a complete folding of the corner element up to the point of touching. This corresponds to a = n/2, for which all plastic hinges rotate by an angle rt/2, and the maximum value of horizontal opening is H 2 _ ~-2
(Ull)max = H ~
+ ~'2
(28)
The integration of the rate of dissipation, eqn (27), is straightforward once the deformation path is determined. The final result is
Initiation of plastic folding mechanism in crushed box columns em
4
M0- v/31/4+16 +2(b+
{4q"(if/()2
(f_/()2(",)max +h
l) + r t { l
133
(Ul_~max]
+ (~) 2
(29)
where Pm is the mean crushing force. The magnitude of(u,)max is obtained by optimizing numerically eqn (20) with respect to a (see Fig. 9(a)). It can be noted that the only unknown parameter in the right-hand side ofeqn (29) is ~'. Thus, the task is to find the optimal value of~"which minimizes the mean force Pro. According to eqn (29), the optimal value of~'will not depend on the aspect ratio b/h. On the other hand, calculation shows that for any value ofH/h, Pmattains a minimum value at ~/H = 0.78 (Fig. 14). Therefore, ~" can be taken as 0.78H in the postfailure range of the crushing process. This value, calculated from the theory, is very realistic for prismatic rectangular columns. It predicts that the diagonal plastic folding lines, as well as the plastic strains, are most likely to localize in the initial phase of postfailure response of columns. 150
,
,
,
b=50 h
~ 100
"---25
•
h
I
E Q.
50
0
I
0.0
I
i
I
I
0.5
i
I
I
0.779
1
1.0
¢/H Fig. 14. The mean crushing force versus the parameter ( for several values of the wavelength H.
Tomasz Wierzbicki, Joan Huang
134
9 COMPARISON WITH THE EXPERIMENTAL RESULTS The present model predictions are compared with two sets of experimental results for square columns made of mild steel. The experiments were carried out at the Impact Centre of the University of Liverpool, UK (Abramowicz, W., pers. comm). The material and geometric parameters of the specimens are listed in Table 1. Based on the information provided, the load-deformation curve can be obtained by using the present model, as shown in Figs 15 and 16. According to our model prediction, specimen 1 will buckle elastically and specimen 2 will buckle in the plastic range, which is exactly what has been observed in the experiments (Abramowicz, W., pers. comm.). In this case, the difference in the buckling mode is not due to the difference in geometry, but due to the substantial difference in the yield strength. The maximum load and the plastic wavelength predicted by our model are also compared with the experimental measured ones, in Table 2. It can be seen that in spite of the many assumptions built into the model, the model predictions agree reasonably well with the experimental results. 300 Test 1 a0 = 463
MPa
E = 2(:oePa
ram mm
b = 53.98 h = 1.42
200
yield
squash
H
---=15 h
100
0
It_.._
0.00
0.05
0.10
0,15
uo/h Fig. 15. L o a d - d e f l e c t i o n curve o f the c o l u m n in test 1.
0.20
135
Initiation of plastic folding mechanism in crushed box columns TABLE 1
Test
tro(MPa)
1 2
463 248
E(GPa)
b(mm)
h(mm)
b/h
53.98 37.32
!-42 1-16
38 32
200 200
TABLE 2
emax(kN)
H/h
151.24 127.61 18
10.0 11.0 8.9
52.73 47.0 12
7.0 9.8 28
Test 1 Predicted Measured Error (%)
Test 2 Predicted Measured Error (%)
400
Test 2 300
o 0 = 248 MPa
E
= 200@I~
b = 37.32 mm h = 1.16 mm o
200
......
\
~
~u~h H
--=15-h -10 -5
i00
0
L._
0.00
0.05
0.10
0.15
u0/h Fig. 16. Load-deflection curve of the column in test 2.
0.20
136
Tomasz Wierzbicki, Joan Huang
10 CONCLUSIONS The present approximate analysis of failure and postfailure response of box columns has revealed several interesting properties of the solution such as mode change, strain reversal and localization of plastic flow. The change from the elastic buckling mode to the plastic folding mode has been described with some realism by a simple two-degrees-of-freedom model with lines of slope discontinuities and deforming and/or rigid triangular or trapezoidal elements. As the column is compressed beyond buckling, the deflected form of the plates changes from triangular to trapezoidal shape. Additional flattening of the central portion of the plate takes place after the yield is reached and this process continues until the plastic collapse mechanism is fully developed. The next phase of the column's response is characterized by the opening up of the angle between the adjacent plates in the column and localization of plastic bending along inclined hinge lines. The initially compressive axial strain gives way to the tensile strain at the end of this phase. In the terminal phase of the collapse process, folding proceeds along diagonal hinge lines, but the direction of the additional in-plane deformations rotates by 90 ° with respect to the column axis. The extent of the edge zones (was shown to be related to the length of the folding wave H by a simple approximate expression ( = 0.78H. The entire load-deflection characteristic of the column was determined by combining the stiffening postbuckling equilibrium path and the softening postfailure path. An important conclusion is that the length of the plastic wave is decided early on in the crushing process and then stays constant for the continuation of the plastic collapse. In an alternative formulation, 7the length of the plastic folding wave was found from minimization of the mean crushing force. This means that the wavelength would depend on the entire loading history. We believe that the present formulation is closer to reality. The present energy method is being extended to the multicornered box column and other complex sheet metal joints, t~ The generalization of the failure analysis to the case of an impact loading is a subject of on-going research. 9
ACKNOWLEDGEMENTS The author is grateful to Mr Wolfgang Geiger and Mr OlafWasowski for their assistance in performing preliminary calculations which led to the development of the present theory. Special thanks are due to Mr Frank de Bruyne of the Computational Department at AUDI AG for the permission to publish Fig. 2 of this paper. The financial support of the
Initiation of plasticfolding mechanism in crushed box columns
137
Alexander von Humboldt Foundation and the M I T - - I n d u s t r y Crashworthiness Consortium during the course of this research is gratefully acknowledged. Also, thanks are due to the Department of Advance Calculations of BMW in Munich for providing one of us (T. W.) with excellent working conditions during a sabbatical leave in Germany. REFERENCES 1. von Karman, T., Sechler, E.E. & Donnel, L. H., Strength of thin plates in compression. Trans. ASME, 53 (1932) 53-7. 2. Koiter, W. T., On the concept of stability of equilibrium for continuous bodies. Proc. Kon. Ned. Akad. Wetensch, !i66 (1963) 173-7. 3. Budiansky, B. & Hutchinson, J. W., Dynamic buckling of imperfectionsensitive structures. Proc. Int. Conf. App. Mech., XI (1964) 636-51. 4. Grave Smith, T. R., Thin-walled Shell Structures, University College, Swansea, UIC 1967. 5. Walker, A. C., The post-buckling behavior of simply supported square plates, Aeronautical Quarterly, 20 (1969) 203-22. 6. Rhodes, J., On approximate prediction ofelastoplastic plate behavior. Proc. Inst. Ovil Engng, 71 (1981) 165-83. 7. Wierzbicki, T., Failure and post-failure behavior of box columns. Inelastic Solids and Structure, Pineridge Press, Swansea, UK 1990, pp. 385-403. 8. Abramowicz, W. & Wierzbicki, T., Axial crushing of multicorner sheet metal columns. J. Appl. Mech., 56 (1989) 113-120. 9. Wierzbicki, T. & Recke, L., Ultimate strength of box columns subjected to impact loading. Proc. 3rd Int. Conf. Mechanical Properties of Materials at high Rates of Strain, Oxford, UK, 19-23 March 1989, pp. 527-34. 10. Rhodes, J., Ultimate strength of thin-walled components. Manual of Crashworthiness Engineering, Vol. 14, Feb. 1989, Technical Report, Department of Ocean Engineering, M. I. T. 11. Wierzbicki, T. & Abramowicz, W., On the crushing mechanics of thinwalled structures. J. Appl. Mech., 50 (1983) 727-39. 12. Johnson, W., Soden, P. D. & AI-Hassani, S. T. S., Inextensional collapse of thin-walled tubes under axial compression,J. Strain Anal., 12 (1977) 317-30. 13. Pugsley, A. and Macaulay, M., The large scale crumpling of thin cylindrical column. Quart. J. Mech. Appl. Math., 13(1) (1960) 1-9. 14. Stowell, E. Z., A Unified Theory of the Plastic Buckling of Columns and Plates. National Advisory Committee for Aeronautics (NACA) TR898, 1948. 15. Geiger, W., Post-buckling Failure of Box Columns. MS. Thesis. Technical University, Munich, June 1989 (in German). A P P E N D I X A: CALCULATION OF THE B E N D I N G A N D MEMBRANE E N E R G Y The bending energy Ub is
Eh 3
fbl2 ~bl2(K x +
Ub - 2 4 ( 1 - v 2)J0
J0
Ky)2dxdy
(A.1)
138
Tomasz WierzbickL Joan Huang
where Kx and Ky are the bending curvatures, defined as
Kx-
O2w Ox 2
O2w K v - Oy 2
(A.2)
To smooth out the discontinuous edges, we use the average curvatures in the calculation: -
w0
f w0
-
Kx - (b/2)2
<~" ~"< y < b/2
~.~
0
0
Ky =
(A.3)
Substituting eqn (A.3) into eqn (A.1) gives
Eh 3
W2o
b2~2
Uh - 24(1_v2) b 3 [ ( 4 + ~ j
(A.4)
~+8(b-~')]
For the column with imperfection: eh 3
Ub - 2 4 ( 1 - v
2)
( W o - ~o) 2
-b-3
b:'~ ~
(A.5)
[ ( 4 + ~-2j ~ + 8 ( b - ~ ' ) ]
The membrane energy Um is U m - 2(1EZhv2) Jofb/2Jofb/2[e~. + e~ + 2vexev + 2(1 - v)e~vldxdy
(A.6)
Using eqn (3) in the text for strains, we have
eh [2¢b (ug w4 Um - 2(1 - v 2),_ k ~ + 3b 4 (w0~
+ 2 b ~ b-72
UoWg~
~
(1 - v) w 4
/I +
12
b~"
~)~ (~ - ~')]
(A.7)
For a column with imperfection, w02 in the above equation should be replaced by w02 - ~02. APPENDIX B: CALCULATION OF ul AND ull The deformed column in phase I(0 < a < a~) is illustrated in Fig. A.1. The maximum amplitude of the overlap is just the length QF. With the coordinates established as in the same figure, we have xB =
0
Xc = 0
YB =
Hsina
Yc = -~"
-( Zc = - H s i n a
z~ =
Let us denote the length DF by 17, then,
(B.1)
Initiation of plastic folding mechanism in crushed box columns
139
A
F
×
A'
A'
Fig. A.I. Geometry of the deformed c o l u m n in phase I.
17 = v / ( 2 - (BD)2 _ v ~2 v/(2 _ H 2 sin2ct
(B.2)
It can be seen from Fig. A. 1 that AF = H,
AO = H cosa,
1 17 - v / ~ v / ( 2 - H 2sin2a,
Q F = u, (B.3)
F r o m triangle D Q F and AQO, two equations can be derived from the geometric analysis; they are t a n 0 = uH, 17
(B.4)
(H-
(B.5)
ul)cos0 = H cosa
Solving the above two eqns (B.4) a n d (B.5) for two u n k n o w n s ul and 0 results in
140 Ul
Tomasz Wierzbicki, Joan Huang
-
{((2 __ H2 (~~t/-~sin2a) + - s~)C°sa~¢/~4~2H2_H 4 sin4a }
H
(B.6)
and sina~ H
{ 2cosa ( ~ + 1 ) - v ~ ( f _ / ( ) 4 - sin4a ]--12 v/(~)4 a s i n_2 si2n4a
} (B.7)
The illustration of the deformed column in phase II (a2 < a < n / 2 ) is shown in Fig. A.2. The shaded area represents the opening up of the adjacent triangular elements. Although the exact value of the maximum amplitude of the opening up is the length CC' along the conical surface, it is assumed that the conical surface can be replaced by a straight one. It can be shown that the maximum error introduced by approximating the conical surface by a straight one is less than 4.5%. Using the above approximation, the maximum amplitude u, is just the coordinate of point C in the y direction. The coordinates of the points A, B and C are XA = H cosa
YA = 0
ZA = 0
Xs : 0
YS = H sina
zs = --(
Xc
Yc
zc
= 0
-
uH
= unknown
G
Fig. A.2. Geometry of the deformed column in phase II.
(B.8)
Initiation of plastic folding mechanism in crushed box columns
141
The lengths of AC and BC remain unchanged during the deformation, i.e. AC = H:
H 2 cos2a + y~ + z 2 = H 2
BC = (:
(Yc - H s i n a ) 2 + (Zc + 0 2 = (2
(B.9) (B.10)
Solving eqns (B.9) and (B.10) gives Ull
H
sina (H2 sin2a - (2) (H 2 sin2a ¥ (2)
(B.11)
and
(2)1
ti. _ cosa& [1 + 2( 2 (H2 sin=a (H 2 sin=a + (2)1 H
(B.12)
A P P E N D I X C: CALCULATION OF T H E RATE OF ROTATION AT T H E H I N G E S Consider a quarter of the column due to symmetry. There are four hinge lines formed in phase I (0 < a < a2), as illustrated in Fig. A.1. It should be pointed out that the boundary of overlap will not be accounted for as a hinge line since the energy related to the overlapping has already been considered in the zones of continuous plastic deformation. The calculations of the rate of rotation and the length of hinge lines are as follows:
For Ùt 01 = a,
0t = h
and
Ii = b - 2 ~ "
(C.1)
12 = H
(C.2)
For 02 02 = Z C F B - , / 2 02 =
H 2 sin2a& V / ( 4 -- H 4 s i n 4 a
and
For 03 13 = v / - H ~ + ( 2 The calculation of 03 is more complicated. Here, we will only present the basic idea of the calculation. First, we calculate the unit vector n~, which is perpendicular to the plane ABG, as nl = s i n a i + cosaj
(C.3)
Tomasz Wierzbicki, Joan Huang
142
then we calculate the vectors AB a n d AQ: AB
=
-Hcosai
+ H sinaj - (k
-Hcosai-
AQ=
Acosa((-
H s i n a ) j - A c o s a ( ( + H s i n a ) k (C.4)
where A=
H 2 { x'/(¢2 -
H2sin2a)/(¢2+ H2 s i n 2 a ) - c o s a }
From the cross product of A B and A Q we can obtain the unit vector n2, which is perpendicular to the plane ,~Q, and the dot product ofn~ and a2 gives cos03. The rate of 03 can be obtained numerically.
For 04 04 = 03
and
14 = 13
(C.5)
Similarly, there are four hinge lines formed in phase II (a2 < a < ~/2) (see Fig. A.2).
For O, O, = h,
I, = b - 2 (
(C.6)
12 = (
(C.7)
14=¢
(c.8)
For 02 02 = a,
For 04
04=a, For 03
13 = v / a 2 +
¢2
To calculate 03, we first calculate the vectors AB a n d AC. F r o m the coordinates: XA = H c o s a
YA
xs = 0
Ys = H s i n a
zs = - (
Xc = 0
H 2 sinZa _ ¢2 Yc - H sina H2 sin2 a + (2
_ 2 ( H 2 sin2a Zc = (H 2 sin2a + ( ~
=
0
ZA =
0
(C.9)
Initiation of plastic folding mechanism in crushed box columns
143
we can obtain A B
=
-H cosai + H sinaj - (k .
AC = - H c o s a i
.
+ nsma
H 2 sin2a _ (2. 2 ( H 2 sin2a H2 sin2 a + (2J - H 2 sin2a + (2 k
(C.10)
F r o m the cross o f AB a n d AC we can obtain the unit vector n2, which is n o r m a l to the plane ABC: H 2 sin2a _ ( 2 . (H sin2a n2 = - s i n a i - cosa H2 sin2 a + (2J + H 2 sin2a + (2 k
(C.11)
We also k n o w the unit vector hi, which is n o r m a l to the plane ABG: nl = s i n a i + c o s a j
(C.12)
The dot o f nl a n d n2 gives cos03, which is 2 ( 2 cos2a cos03 = - 1 + H2 sin2 a + (2
(C.13)
The rate o f 03 is 0 _ -
1 sin0 3
2(2sin2a&{
1 _[ Hcosa ]2} H 2 sin2a + (2 [_H2~ 7 _ t - (2
(C.14)
!I Initiation of Plastic Folding Mechanism in Crushed Box Columns
Tomasz Wierzbicki & Joan Huang Department of Ocean Engineering. Massachusetts Institute of Technology, USA (Received 3 November 1990: accepted 25 March 1991)
ABSTRACT A simple computational model of a thin-walled prismatic column is developed, which describes a transition from the postbuckling to the pos~'ailure and crushing deformation phase. Energy methods are used to analyze the elastic post-buckling response of the column. Ultimate strength is calculated, using the first yield criterion. Limit analysis methods, properly generalized to large displacement and rotation problems, are employed in the postfailure range. The new model explains, with some realism, the process of strain localization and the formation of a plastic folding mechanism in the column. It also leads to a prediction of the entire load-deformation characteristics up to the internal touching and stiffening.
1 INTRODUCTION The response of a column to axial compressive loads consists of two distinct phases. In the prefailure phase the load is increasing up to the point of maximum strength; rotations are moderately large, but the strains remain small. Generally, each plane of the column can be considered as a plate. Except for very thin plates, there is little change in the buckled form of the plate, and the initially straight comer lines (intersections of the adjacent planes) are only slightly deformed. The first y i e l d o c c u r s n e a r the comers, but plastic deformations are contained within the plate and thus are relatively small. !i5 Thin-Walled Structures 0263-8231/91/$03.50© 1991 Elsevier Science Publishers Ltd, England. Printed in Great Britain
116
Tornasz Wierzbicki, Joan Huang
The postbuckling and ultimate strength response of plates and plate assemblages is well understood due to the fundamental work of von Karman, ~ Koiter 2 and Budiansky and Hutchinson) Since then, a number of authors have contributed to the problem of stiffness and strength of thin-walled structures, and clarified the effect of imperfections, aspect ratio, boundary conditions and plasticity9-6 The class of problems with diffused elastic-plastic deformations is relatively easy to solve numerically, and several general purpose nonlinear finite element codes such as ADINA, ABAQUS, etc., can be used to calculate the initial loaddeflection characteristics of box columns with great accuracy and speed. The subsequent phase of the column's response after reaching the peak load is entirely different. The deformations localize within narrow zones and an uncontained plastic flow takes place until first contact occurs. This process is accompanied by very large strains (up to the rupture strain of the material) and large rotations. Furthermore, the original cross-sectional shape of the column is subjected to a severe distortion and the initially straight corners open up and bend. Different analysis methods have been developed in the literature to tackle each of the above mentioned deformation stages separately. However, except for a preliminary analysis 7 and an unpublished report (Rhodes, J., pers. comm. 1989), no attempts have been made in the literature to study the process of localization of plastic deformation within the column. The objective of the present paper is to provide the missing link between these two distinct phases of column response, to explain the formation of plastic folding mechanism, and to present results for a few selected geometries of columns. A consistent and simple computational model of column deformation is developed, which describes the transition from postbuckling to the crushing deformation phase.
2 LOCALIZATION OF DEFORMATION AND PLASTIC F O L D I N G MECHANISM The buckled shape of a long thin-walled rectangular column is shown in Fig. 1. Except for very thin flanges, little change in the deformed shape of the column occurs until the point of maximum strength is reached. From this point on a dramatic change in failure mode takes place. A close examination of experimental results and computer simulation results of deformed shapes reveals the complexity of a transition phase. The transition phase continues until a clear plastic folding mechanism is formed (Fig. 2). The corner line then deforms from its initial vertical
Initiation of plastic folding mechanism in crushed box columns
I17
Fig. 1. A buckling mode of a rectangular column subjected to axial compression.
position. The right angle between the adjacent flanges opens up and two diagonal fold lines appear to grow on opposite flanges. The process of plastic strain localization is completed when the initial corner is completely flattened out. From this point on a crushing mechanism, described, for example, by Abramowicz and Wierzbicki, s takes over until first contact occurs. A simple two-degrees-of-freedom model of an imperfect column is developed, which captures the main features of the localization process. The model does not reveal any new features of the column's response in the postbuckling range which were not known before. However, the same model can be continued into the plastic range, thus providing an important insight into the complex physical problem. The model is explained in Section 3.
Fig. 2. Localization of plastic deformation along diagonal lines and opening up of the comer lines (courtesy of F. de Bruyne.)
Tomasz WierzbickL Joan Huang
! 18
3 BUCKLING AND POSTBUCKLING BEHAVIOR In the present model, the deformation of the column consists of in-plane compression and out-of-plane bending (Fig. 3). A uniform in-plane compression defines the prebuckling equilibrium path, P
u0
(1)
bh - E ~
where P is the compressive load, E is the Young's modulus, b is the column width, h is the column thickness, H is the half elastic wavelength which equals b/2, and u0 is the in-plane uniform displacement. Once buckling occurs, it is assumed that the deformed plates rotate with respect to each other and deform in the plate plane, as shown in Fig. 3(b). The out of plane displacement field w(x,y) takes the form of a 'prism' with four lines of slope discontinuities and a central amplitude, w0. The angle of slope discontinuities is defined by the distance (, which is considered to be an unknown parameter of the process (Fig. 4). An initial imperfection is introduced, which takes the same shape as in Fig. 3(b) and has a central amplitude w0. In the present analysis we follow the formulation presented by Wierzbicki and Recke 9 and Wierzbicki.7 The displacement field can be expressed by the following equations:
x
u° ( 1 - b ~ ) 2
u(x'Y) =
(x)1
l ( bw2- ~ )I1 - ~ ) ~x y w~
x)
0
and
v(x,y)
= 0
(2)
where v is the component of the displacement vector transverse to the direction of loading. (a)
[_
/1
_-
(b)
/
Fi~. 3. ~a) Uniform compression of a column, a n d (b) relieving tension due to out-of-
Initiation of plastic folding mechanism in crushed box columns
119
o
t
\
I 0/
Fig. 4. Illustration of the displacement field of a column under compression.
There are two contributions to the in-plane displacement u(x,y). One comes from the in-plane compression, which equals u0 at the upper and lower boundaries of the column and equals zero at the middle symmetry plane. The other contribution comes from the finite rotation of the plate element which actually relieves the in-plane compression. Additionally, it is assumed that the column is inextensional in the transverse directions, i.e, e v(X,y ) = O, and that no elastic work is done at the comers so that right angles are preserved during the deformation process. From the above assumptions, the strain field can be easily calculated using the infinitesimal strain definition, as u0 Sx
m
_
Ou
1
w~ y
0
- (b/2~-) + 2 (b/2) 2 (
Ox
u0
1 w02 - (b/2-----)+ 2 (b/2) 2
00
ey - Oy
b (
0
(3) ¢"
l(Ov
exv-~.
~+
•
O~y)
=
' 1 Wo 2 x 4(b/2)2(
t
0
0
The equilibrium path can be determined by using the principle of virtual work. Due to symmetry of the structures, only one quarter of the plate is considered here in the calculation, which gives, l"I(u0, w0) = Ub(U0, w0) --I- Urn(u0, w0) -
(')
~ u0
(4)
where II is the total potential energy, P is the compressive force applied at one comer of the column (the total force is 4P), and Ub and Um are the bending and membrane energies, respectively.
120
Tornasz Wierzbicki, Joan Huang
The calculations of Ub and U,, are carried out in Appendix A. It should be pointed out that the present model gives a good approximation of the membrane energy, but a rather poor approximation of the bending energy due to the discontinuities of slope. Because slope discontinuities are not admissible for elastic structures, we had to smooth-out the edges of the prism in the bending energy calculation. The arbitrariness in the smoothing procedure accounts for an inaccurate expression for Ub. The equilibrium condition 6II = 0 gives rise to the following system of nonlinear algebraic equations relating P, uo and w0: P-
( 1 - v 2) u0
~
I-
U o W o ( ( - b ) + w o ( w 2 - ~ 2)[1
(5)
4~'3 b+(1-v)12 ~]
+g-(Wo-~o) l+~+ y~
=o
(6)
where E and v denote, respectively, the Young's modulus and the Poisson's ratio. In the limiting case of a perfect plate, ~0 = 0, e q n s (5) and (6) define the primary and secondary equilibrium paths. Bifurcation occurs at 8 Eh 3 Per- 3 (1-v2)b '
4 h2 ucr- 3 b
(7)
The above result compares well with an exact buckling formula for a simply supported elastic plate where the coefficient 8/3 is replaced by n2/3. The postbuckling equilibrium path is given by the following equation: P -
AU°+B
ecr
(8)
Ucr
where the coefficients A(/~)
=
1
-
(l -
1
4
-g/~+
(1-~) B(fl)
=
[
1
4
t~) 2
(1 - v ) 1 12 fl
+~+~
(9)
Initiation of plastic folding mechanism in crushed box columns
121
are functions of the dimensionless parameter fl = (/b. Equation (8) is plotted for several values of(, as shown in Fig. 5. The lowest bifurcation load is obtained at ( = 0.5b, which corresponds to a roof mechanism with the slope discontinuity line inclined by 45 ° to the vertical direction. The onset of buckling leads to an immediate drop in axial stiffness to 0.440 of the prebuckling stiffness. This compares favorably with the exact values of 0-5 and 0-408 for plates with edges kept straight and edges free to wave, respectively, n° As the axial shortening increases, there is a further reduction in the column stiffness, and the load-deflection curve follows the envelope of a family of the straight lines with various values of(. Even more important than the reduction in the plate stiffness is the associated mode change. The present model predicts that the plate initially buckles in the triangular shape of w(y) in the cross-sectional plane. This shape then transforms gradually into trapezoidal shapes, shown by the labels in Fig. 5. It should be noted that in the classical approach to the postbuckling problem, the mode transition could only be described by retaining many 10
I
. . . .
~
. . . .
i
. . . .
i
. . . .
1
elastic / / / / / / / /
a.,
/ / / /
~.~.4
I
J
I
0
0
5
10
15
20
Uo/Ucr
Fig. 5. Load-deflection curve in the postbuckling range and mode transition.
Tomasz Wierzbicki, Joan Huang
122
terms in the modal expansion of the solution (cf. Koiter2). In the present formulation this result is obtained in a natural and elegant way.
4 FIRST YIELD AND ULTIMATE STRENGTH With increasing axial load there is a redistribution of stresses within the plate. Compressive stresses at the plate center are relieved while stresses near corner lines continue to increase. The plate material will then yield in the most stressed point. Assuming that the plate yields due to membrane stresses alone, the plane stress yield condition applies: 2 ? or,,2 - Crx~v + Cry + 3cr~v = or02
(10)
where cr0 is the yield strength of the material. At the commencement of yielding, eqn (10) can be expressed in terms of the components of the strain tensor given by eqn (3). In the case of a plate with ( = 0-5b and without initial imperfection, the corresponding equation takes the form: (l+v 2-v) 2 3 2 -(i e, + (1 + v) 2 e,:, =
(11)
A detailed analysis of the above equation reveals that the column will yield first at a point along the corner line, halfway between the supports (or nodal lines). Using the expressions of the axial and shear strains at the yielding point, the above equation becomes functions of u0 and w0: (l + v2-- v) 4u 2 (1 -- V2) 2
~b-
3
w4
q- ( l ~- v ) 2 b 4 =
(~)2 .--.
(12)
Equation (12), together with eqns (5) and (6). furnishes a system of three equations for three unknowns, P, (u0),, and (w0)u, from which the corresponding load at first yield can be obtained. Upon further compression there will be continued load shedding at the central region of the plate while the stresses near corner lines will increase only slightly due to the strain hardening of the material. The plate will exhaust its load carrying capacity soon after the first yield is reached. We thus identify the force corresponding to the first yield as the ultimate strength of the column, P,,. A plot of the normalized plate strength versus the slenderness parameter, hb / ~ ,
is shown in Fig. 6 for a few values of the
imperfection parameter. Also shown in this figure is the curve corresponding to the so-called squash load, which represents a full plastic capacity of the undeformed cross-section:
Initiation of plastic folding mechanism in crushed box columns
123
We/h =0.001 %/h=OA
2.0
"~
~,,'~,~
"
• Wo lh--0.2 , Wolh=0.5 ' Wolh=l.O
~',
~ ~
",, --... 1.2
R¢f. [151
~.
0.4 0
0.5
1.0
h
Fig. 6. Ultimate strength of the column versus the slenderness parameter for several values of the imperfection. Psq
=
bhao
(13)
The present solution overestimates by 10 to 15% the exact ultimate load calculated by means of the ABAQUS finite element code (circles in Fig. 6).
5 POSTFAILURE BEHAVIOR Once the ultimate strength is reached, the column will not bear any more load, and the compressive force decreases as the column is further deformed. Our simplified model can be extended into the postfailure range. The two-degrees-of-freedom model is valid up to the point of the first yield. Beyond the first yield the central part of each side plate of the column is subjected to further unloading, and plastic deformations spread into the edge zones. It is thus reasonable to assume that at an early stage of the crushing process the central trapezoidal sections of the column will be fully unloaded and will rotate as rigid bodied. From this point on, the in-plane and out-of-plane deformations are related through the geometry of the problem: Wo 2 = Uo(2H- Uo)
(14)
Tomasz Wierzbicki, Joan Huang
124
The above condition reduces the number of degrees of freedom of our model to just one. All kinematic quantities such as displacements ua, strains e~,t~and strain rates g,,t~become now unique functions of either u0 or w0 and their rates. In actual calculation it is more convenient to treat the rotation angle of the trapezoidal element, a, as an independent variable (see Fig. A.1 in the Appendix). Several interesting properties of the deformation process follow from the above assumptions. First, the initial right angle between the adjacent plates must now change as a increases (Fig. 7). Thus, the corner line opens up and flattens out and an inclined hinge line is gradually formed. As crushing progresses there is an overlap between the adjacent triangular elements in the lower and upper half of the column. This overlap is indicated by a shaded area in Fig. 7 and its maximum amplitude is denoted by u~. As a result, each of the deforming triangular elements is subjected to axial compression and shear. To simplify the calculation, it is assumed that the strain is distributed uniformly in the triangular elements. Additionally, the assumption e v = 0 is assumed to remain valid in the early postfailure stage. Under the above assumptions, the strain rate tensor is
H
2(
1 til
0
2(
(15)
The process of flattening out (phase I) is completed when a = a2 = arc sin ( ~ ) . F o r a > a2,a different failure mechanism takes over. Either, the inclined stationary hinge line splits up and starts to move so that further crushing is possible) ~or the inclined hinge line remains stationary and
Fig. 7. Plastic compression/tension in the triangular transition zones.
Initiation of plastic folding mechanism in crushed box columns
125
the side wall opens up, giving rise to the one-dimensional extension in the transverse direction (Fig. 8). The latter possibility will be explored here. When a > a2 (phase II), there is a relative open-up displacement between the adjacent triangular elements at the comer line connecting two plates. The maximum amplitude of the open-up in the second phase is denoted by un. Similarly, each of the deforming triangular elements is subjected to axial compression and shear. Since the strain in the transverse direction becomes dominant, it is reasonable to assume that ex = 0 while ey #: 0. Under these assumptions, the strain rate tensor in the second phase is
1 nil ] 0 2H
2 H
(16)
~"
The crushing process is completed when the two halves of the plate reach the point of touching (a = rt/2). The equilibrium of the column is expressed via the principle of virtual velocities
Pfto = h fs aaagaa dS + ~., MoliOi
(17)
i where the two terms at the right-hand side of eqn (17) denote, respectively, the rate of internal dissipation in the zones of continuous and discontinuous plastic deformations. The integration is performed over the area S of two rotating triangular elements. The tensorgao denotes the so-called strain velocity, calculated from the known velocity field in the current, deformed configuration. The components of the Cauchy stress tensor craa are calculated from the associated flow rule by using the
Fig. 8. Plastic collapse mode with opening up of the side walls in the transverse direction.
Tomasz Wierzbicki, Joan Huang
126
plane stress yield condition, eqn (10), as a plastic potential. The fully plastic bending moment is defined by M0 = aoh2/4, where tr0 is the yield strength, and li and 0; in eqn (17) denote, respectively, the length of the ith hinge line and the rate of rotation at the hinge. The summation is performed over all plastic hinges. In our one-degree-of-freedom model, all plastic hinge lines are fixed as the column deforms. Their lengths are thus constant and can be easily expressed in terms of b, H and (. The rates of relative rotations can be expressed in the form
0, = f ( b , H, ()gt
(18)
where f are length functions, which are calculated in Appendix C. The expression for the continuous energy dissipation is different depending on the range of a:
D = h
fsO,t,
dS =
~M0~-Iall
1+4
0
~3 Mof~la,, I
1+4
a2
The velocity vectors u~ and u. are calculated in Appendix B from the kinematics of our problem, which gives Ul
H S ( ( 2 - H 2 sin2a) - cos a v / ( 4 - H 4 / (-~¥~/~ sin2a - 2H 2
sin4a}
H sina& uw = 2H 2 _ (2 _ H 2 sin2a
2H(Ul + H)cosa
- ¢(4
_ H 4 sin4a
H4 _ -2-
¢¢4
sin 22a } Z~_/7sin4al
(20) and un = H sina (/-/2 sin2a - (2) (H 2 sin2a ~ (2) fin = a c o s a 6 [1 + 2 ( 2 ((nn 22 sin2 asin2a _+(2)-](2)2j
(21)
The displacements ul and un are plotted in Fig. 9(a) and (b) as functions of the rotation angle a. It is interesting to note that the velocity fir vanishes at a certain angle a~. Initially, the relative displacement u~ increases so that compressive stresses are developed in the triangular
Initiation of plastic folding mechanism in crushed box columns
127
0.20
;---o.s H 0.15
0. I0 0.6
0.05 0.4
0.2 0.00 t_ 0.0
1.0
0.5 (l
(a) 1.0 0.2
0,4
0.5
0.6
0,0
I
0.0
/ V
~=0.8 H
/ A/
J
0.5
1.0
1.5
2.0
a
Co) Fig. 9. The m a x i m u m amplitude of the relative displacement between the adjacent triangular elements as functions of the rotation angle a: (a) in phase I; (b) in phase II.
Tomasz Wierzbicki, Joan Huang
128
zones. As the maximum displacement is reached, u~ diminishes under further crushing and is brought back to zero at a = a2. This means that there is a strain reversal in the intermediate stage of the column deformation, where initial compression is followed by tension. This interesting effect was overlooked in all previous analyses of the crushing process based on the final configuration approach. ~2,~3
6 RESULTS AND DISCUSSION The results of the present calculations are shown in Figs l0 to 13 for typical material constants, cr0, E, and selected values of structure geometries, b/h. According to eqn (17), the postfailure characteristics of the column depends on, besides the structure geometries, the wavelength H and the parameter (. The wavelength H equals b/2 in the elastic range, and, in the plastic range, a family of curves with different values of H/h are plotted. The model parameter ( is an unknown of the crushing process. The solution presented in Section 8 reveals that ( is related to H. By
600
*
i
~
n
]
,
,
I
T
1
~
7000;¢:
n
J//~/
I
T
i
~
i
/
=,00 __~-------~---
squash
__
400
265
.....
"~'"9( first yield
200
0 0.00
I
I
;
I
1
0.05
I
I
I
I
I
t
0.10
t
k
l
]
0.15
I
I
I
t
0.20
uo/h Fig. 10. Load-deflection characteristics of the thinner column with normal yield strength. The c o m e r zones are yielded.
Initiation of plastic folding mechanism in crushed box columns 400
I
,
, ,
i
'
i
1
129
,
~o = 200 MPa
E = 200,0Pa 300
b -= h
50
~u~h
200
H --=20
100
]5I0
0
I
I
0.0
0.1
L
I
I
0.2
I
0.3
I
I
0.4
I
0.5
uo/h
Fig. 11. Load-deflection characteristics of the thicker column with normal yield strength. The full squash load is reached.
minimizing the energy absorbed in the collapsing column, (/H = 0.78 is obtained. Figures 10 and 11 show the normalized load-deformation curves for or0 = 200 MPa, E = 200 G P a and two aspect ratios ofb/h = 100 and 50. The shape of the load-deformation curve depends on the aspect ratio. At the very beginning, the column deforms elastically along the loaddeformation curve, then, for relatively thin columns (b/h = 100), as shown in Fig. 10, elastic buckling occurs, which leads to a reduction in the column stiffness. The axial load keeps increasing up to the point of first yield. Since the column will exhaust its load carrying capacity soon after the first yield is reached, the first yield stress is thus defined as the ultimate strength of the structure. The calculation shows that the normalized peak load corresponding to the first yield equals 265 for the column with b/h = 100. As the column is further deformed, the load will start to decrease along the postfailure branch with H/h ~ 30. Such a dramatic change from the elastic buckling wavelength to a much shorter plastic wavelength is always observed experimentally. For relatively thick columns with b/h = 50, Fig. 11, the squash load is reached at a load lower than the elastic buckling load. The column follows for a while the constant load (squash load) if the material
130
Tomasz Wierzbicki, Joan Huang 400
oo = 400 MPa E = 200 6 P a
_b=
300
h
50
squash
200 174 H --=20 h
i00
-|5---tO
0 0.0
0.1
I 0.3
0.2
~
I 0.4
0.5
uo/h Fig. 12. Load-deflection characteristics of the thinner column with high yield strength. 300
o 0
= 400 MPa
E = 200~t b -= h
30
200
squash H --=15_ h
100
- 10-
0
L._
0.00
0.05
0.10
0.15
0.20
uo/h Fig. 13. Load-deflection characteristics of the thicker column with high yield strength.
Initiation of plastic folding mechanism in crushed box columns
131
behaves ideally elastic-plastic, and then softens along the postfailure branch of the load-deformation curve. In most cases, however, the load can still increase after the squash load is reached due to the hardening of the materials. Eventually, the column will buckle in the plastic range. The plastic buckling of stocky columns with hardening behavior, as represented by the dotted line in Fig. 11, will be discussed in Section 7. In addition to the effect of the aspect ratio, the material constants of the column will also affect the structure behavior. Figures 12 and 13 show the normalized load-deformation curve for E = 200 GPa and a higher yield strength tr0 = 400 MPa. It is expected that the structures of higher yield strength are more likely to buckle elastically. The present model does predict this structure behavior. For example, with the same aspect ratio b/h = 50, the model predicts that an elastic buckling occurs for the highstrength column (Fig. 11). On the other hand, even with high strength, a very thick column ofb/h = 30 will still attain the squash load before the elastic buckling load is reached (Fig. 13). 7 PLASTIC BUCKLING OF STOCKY COLUMNS A theory of plastic buckling of columns was developed by Shanley and generalized by Stowell for rectangular plates. The formula for a critical plastic buckling stress of a simply supported rectangular plate, derived by Stowell, t4 is trcr _ rr2 ~ E s ( h ) 2 [2 + / 1
+ 3Et] EJ
(22)
where Et = dot/de is the tangent modulus, and Es = tr/e is the secant modulus. Because Et and Es are, in general, functions of the magnitude of stress (or strain), eqn (22) can be used only when the stress-strain diagram of the material is known. One of the typical stress-strain relations is the power type: cr = cr0
(;0)
(23)
where o'0 is the yield strength, e0 is the yield strain, and n is the hardening exponent. Having the stress-strain relation, one can easily calculate the critical plastic buckling stress and strain: ~cr
(eo)"
(2 + v / i + 3n)"
(24)
Tomasz Wierzbicki, Joan Huang
132
ecr
9
(2 + x/1 + 3n)
(25)
When the plate buckles in the plastic range, it cannot carry any additional load. The ultimate (peak) load of the plate is then obtained by multiplying ~ , by the cross-sectional area. In the case of our square column:
Pu
=
acrbh
(26)
where Pu is the force applied at one comer of the column. The results of the present calculation are plotted by the dotted lines in Figs 11 and 13 for the previous two examples, b/h = 50, o0/E = 0.1%, and b/h = 30, ao/E = 0.2%. The hardening exponent n is taken as 0-2, a typical value for mild steel. After the squash load is reached, the axial force will continue to increase along the material's hardening curve. The ultimate load is determined by the point of plastic buckling stress, which also determines the plastic wavelength. For b/h = 50, ao/E = 0.1%, the mode transits from HIh = 25 into HIh ~ 10, and for blh = 30, or0/ E = 0-2%, the mode transits from H/h = 15 into H/h ~ 5. It should be remembered that the real transition from elastic to plastic should be a smooth one. In view of this, the present model predictions for the plastic wavelength can only be considered as an estimation.
8 E N E R G Y ABSORPTION The energy absorbed by the collapsing column can be obtained by integrating both sides of eqn (17) with respect to time or the process parameter a.
f0',u0a, f0
f0
i
Of particular interest is the energy corresponding to a complete folding of the corner element up to the point of touching. This corresponds to a = n/2, for which all plastic hinges rotate by an angle rt/2, and the maximum value of horizontal opening is H 2 _ ~-2
(Ull)max = H ~
+ ~'2
(28)
The integration of the rate of dissipation, eqn (27), is straightforward once the deformation path is determined. The final result is
Initiation of plastic folding mechanism in crushed box columns em
4
M0- v/31/4+16 +2(b+
{4q"(if/()2
(f_/()2(",)max +h
l) + r t { l
133
(Ul_~max]
+ (~) 2
(29)
where Pm is the mean crushing force. The magnitude of(u,)max is obtained by optimizing numerically eqn (20) with respect to a (see Fig. 9(a)). It can be noted that the only unknown parameter in the right-hand side ofeqn (29) is ~'. Thus, the task is to find the optimal value of~"which minimizes the mean force Pro. According to eqn (29), the optimal value of~'will not depend on the aspect ratio b/h. On the other hand, calculation shows that for any value ofH/h, Pmattains a minimum value at ~/H = 0.78 (Fig. 14). Therefore, ~" can be taken as 0.78H in the postfailure range of the crushing process. This value, calculated from the theory, is very realistic for prismatic rectangular columns. It predicts that the diagonal plastic folding lines, as well as the plastic strains, are most likely to localize in the initial phase of postfailure response of columns. 150
,
,
,
b=50 h
~ 100
"---25
•
h
I
E Q.
50
0
I
0.0
I
i
I
I
0.5
i
I
I
0.779
1
1.0
¢/H Fig. 14. The mean crushing force versus the parameter ( for several values of the wavelength H.
Tomasz Wierzbicki, Joan Huang
134
9 COMPARISON WITH THE EXPERIMENTAL RESULTS The present model predictions are compared with two sets of experimental results for square columns made of mild steel. The experiments were carried out at the Impact Centre of the University of Liverpool, UK (Abramowicz, W., pers. comm). The material and geometric parameters of the specimens are listed in Table 1. Based on the information provided, the load-deformation curve can be obtained by using the present model, as shown in Figs 15 and 16. According to our model prediction, specimen 1 will buckle elastically and specimen 2 will buckle in the plastic range, which is exactly what has been observed in the experiments (Abramowicz, W., pers. comm.). In this case, the difference in the buckling mode is not due to the difference in geometry, but due to the substantial difference in the yield strength. The maximum load and the plastic wavelength predicted by our model are also compared with the experimental measured ones, in Table 2. It can be seen that in spite of the many assumptions built into the model, the model predictions agree reasonably well with the experimental results. 300 Test 1 a0 = 463
MPa
E = 2(:oePa
ram mm
b = 53.98 h = 1.42
200
yield
squash
H
---=15 h
100
0
It_.._
0.00
0.05
0.10
0,15
uo/h Fig. 15. L o a d - d e f l e c t i o n curve o f the c o l u m n in test 1.
0.20
135
Initiation of plastic folding mechanism in crushed box columns TABLE 1
Test
tro(MPa)
1 2
463 248
E(GPa)
b(mm)
h(mm)
b/h
53.98 37.32
!-42 1-16
38 32
200 200
TABLE 2
emax(kN)
H/h
151.24 127.61 18
10.0 11.0 8.9
52.73 47.0 12
7.0 9.8 28
Test 1 Predicted Measured Error (%)
Test 2 Predicted Measured Error (%)
400
Test 2 300
o 0 = 248 MPa
E
= 200@I~
b = 37.32 mm h = 1.16 mm o
200
......
\
~
~u~h H
--=15-h -10 -5
i00
0
L._
0.00
0.05
0.10
0.15
u0/h Fig. 16. Load-deflection curve of the column in test 2.
0.20
136
Tomasz Wierzbicki, Joan Huang
10 CONCLUSIONS The present approximate analysis of failure and postfailure response of box columns has revealed several interesting properties of the solution such as mode change, strain reversal and localization of plastic flow. The change from the elastic buckling mode to the plastic folding mode has been described with some realism by a simple two-degrees-of-freedom model with lines of slope discontinuities and deforming and/or rigid triangular or trapezoidal elements. As the column is compressed beyond buckling, the deflected form of the plates changes from triangular to trapezoidal shape. Additional flattening of the central portion of the plate takes place after the yield is reached and this process continues until the plastic collapse mechanism is fully developed. The next phase of the column's response is characterized by the opening up of the angle between the adjacent plates in the column and localization of plastic bending along inclined hinge lines. The initially compressive axial strain gives way to the tensile strain at the end of this phase. In the terminal phase of the collapse process, folding proceeds along diagonal hinge lines, but the direction of the additional in-plane deformations rotates by 90 ° with respect to the column axis. The extent of the edge zones (was shown to be related to the length of the folding wave H by a simple approximate expression ( = 0.78H. The entire load-deflection characteristic of the column was determined by combining the stiffening postbuckling equilibrium path and the softening postfailure path. An important conclusion is that the length of the plastic wave is decided early on in the crushing process and then stays constant for the continuation of the plastic collapse. In an alternative formulation, 7the length of the plastic folding wave was found from minimization of the mean crushing force. This means that the wavelength would depend on the entire loading history. We believe that the present formulation is closer to reality. The present energy method is being extended to the multicornered box column and other complex sheet metal joints, t~ The generalization of the failure analysis to the case of an impact loading is a subject of on-going research. 9
ACKNOWLEDGEMENTS The author is grateful to Mr Wolfgang Geiger and Mr OlafWasowski for their assistance in performing preliminary calculations which led to the development of the present theory. Special thanks are due to Mr Frank de Bruyne of the Computational Department at AUDI AG for the permission to publish Fig. 2 of this paper. The financial support of the
Initiation of plasticfolding mechanism in crushed box columns
137
Alexander von Humboldt Foundation and the M I T - - I n d u s t r y Crashworthiness Consortium during the course of this research is gratefully acknowledged. Also, thanks are due to the Department of Advance Calculations of BMW in Munich for providing one of us (T. W.) with excellent working conditions during a sabbatical leave in Germany. REFERENCES 1. von Karman, T., Sechler, E.E. & Donnel, L. H., Strength of thin plates in compression. Trans. ASME, 53 (1932) 53-7. 2. Koiter, W. T., On the concept of stability of equilibrium for continuous bodies. Proc. Kon. Ned. Akad. Wetensch, !i66 (1963) 173-7. 3. Budiansky, B. & Hutchinson, J. W., Dynamic buckling of imperfectionsensitive structures. Proc. Int. Conf. App. Mech., XI (1964) 636-51. 4. Grave Smith, T. R., Thin-walled Shell Structures, University College, Swansea, UIC 1967. 5. Walker, A. C., The post-buckling behavior of simply supported square plates, Aeronautical Quarterly, 20 (1969) 203-22. 6. Rhodes, J., On approximate prediction ofelastoplastic plate behavior. Proc. Inst. Ovil Engng, 71 (1981) 165-83. 7. Wierzbicki, T., Failure and post-failure behavior of box columns. Inelastic Solids and Structure, Pineridge Press, Swansea, UK 1990, pp. 385-403. 8. Abramowicz, W. & Wierzbicki, T., Axial crushing of multicorner sheet metal columns. J. Appl. Mech., 56 (1989) 113-120. 9. Wierzbicki, T. & Recke, L., Ultimate strength of box columns subjected to impact loading. Proc. 3rd Int. Conf. Mechanical Properties of Materials at high Rates of Strain, Oxford, UK, 19-23 March 1989, pp. 527-34. 10. Rhodes, J., Ultimate strength of thin-walled components. Manual of Crashworthiness Engineering, Vol. 14, Feb. 1989, Technical Report, Department of Ocean Engineering, M. I. T. 11. Wierzbicki, T. & Abramowicz, W., On the crushing mechanics of thinwalled structures. J. Appl. Mech., 50 (1983) 727-39. 12. Johnson, W., Soden, P. D. & AI-Hassani, S. T. S., Inextensional collapse of thin-walled tubes under axial compression,J. Strain Anal., 12 (1977) 317-30. 13. Pugsley, A. and Macaulay, M., The large scale crumpling of thin cylindrical column. Quart. J. Mech. Appl. Math., 13(1) (1960) 1-9. 14. Stowell, E. Z., A Unified Theory of the Plastic Buckling of Columns and Plates. National Advisory Committee for Aeronautics (NACA) TR898, 1948. 15. Geiger, W., Post-buckling Failure of Box Columns. MS. Thesis. Technical University, Munich, June 1989 (in German). A P P E N D I X A: CALCULATION OF THE B E N D I N G A N D MEMBRANE E N E R G Y The bending energy Ub is
Eh 3
fbl2 ~bl2(K x +
Ub - 2 4 ( 1 - v 2)J0
J0
Ky)2dxdy
(A.1)
138
Tomasz WierzbickL Joan Huang
where Kx and Ky are the bending curvatures, defined as
Kx-
O2w Ox 2
O2w K v - Oy 2
(A.2)
To smooth out the discontinuous edges, we use the average curvatures in the calculation: -
w0
f w0
-
Kx - (b/2)2
<~" ~"< y < b/2
~.~
0
0
Ky =
(A.3)
Substituting eqn (A.3) into eqn (A.1) gives
Eh 3
W2o
b2~2
Uh - 24(1_v2) b 3 [ ( 4 + ~ j
(A.4)
~+8(b-~')]
For the column with imperfection: eh 3
Ub - 2 4 ( 1 - v
2)
( W o - ~o) 2
-b-3
b:'~ ~
(A.5)
[ ( 4 + ~-2j ~ + 8 ( b - ~ ' ) ]
The membrane energy Um is U m - 2(1EZhv2) Jofb/2Jofb/2[e~. + e~ + 2vexev + 2(1 - v)e~vldxdy
(A.6)
Using eqn (3) in the text for strains, we have
eh [2¢b (ug w4 Um - 2(1 - v 2),_ k ~ + 3b 4 (w0~
+ 2 b ~ b-72
UoWg~
~
(1 - v) w 4
/I +
12
b~"
~)~ (~ - ~')]
(A.7)
For a column with imperfection, w02 in the above equation should be replaced by w02 - ~02. APPENDIX B: CALCULATION OF ul AND ull The deformed column in phase I(0 < a < a~) is illustrated in Fig. A.1. The maximum amplitude of the overlap is just the length QF. With the coordinates established as in the same figure, we have xB =
0
Xc = 0
YB =
Hsina
Yc = -~"
-( Zc = - H s i n a
z~ =
Let us denote the length DF by 17, then,
(B.1)
Initiation of plastic folding mechanism in crushed box columns
139
A
F
×
A'
A'
Fig. A.I. Geometry of the deformed c o l u m n in phase I.
17 = v / ( 2 - (BD)2 _ v ~2 v/(2 _ H 2 sin2ct
(B.2)
It can be seen from Fig. A. 1 that AF = H,
AO = H cosa,
1 17 - v / ~ v / ( 2 - H 2sin2a,
Q F = u, (B.3)
F r o m triangle D Q F and AQO, two equations can be derived from the geometric analysis; they are t a n 0 = uH, 17
(B.4)
(H-
(B.5)
ul)cos0 = H cosa
Solving the above two eqns (B.4) a n d (B.5) for two u n k n o w n s ul and 0 results in
140 Ul
Tomasz Wierzbicki, Joan Huang
-
{((2 __ H2 (~~t/-~sin2a) + - s~)C°sa~¢/~4~2H2_H 4 sin4a }
H
(B.6)
and sina~ H
{ 2cosa ( ~ + 1 ) - v ~ ( f _ / ( ) 4 - sin4a ]--12 v/(~)4 a s i n_2 si2n4a
} (B.7)
The illustration of the deformed column in phase II (a2 < a < n / 2 ) is shown in Fig. A.2. The shaded area represents the opening up of the adjacent triangular elements. Although the exact value of the maximum amplitude of the opening up is the length CC' along the conical surface, it is assumed that the conical surface can be replaced by a straight one. It can be shown that the maximum error introduced by approximating the conical surface by a straight one is less than 4.5%. Using the above approximation, the maximum amplitude u, is just the coordinate of point C in the y direction. The coordinates of the points A, B and C are XA = H cosa
YA = 0
ZA = 0
Xs : 0
YS = H sina
zs = --(
Xc
Yc
zc
= 0
-
uH
= unknown
G
Fig. A.2. Geometry of the deformed column in phase II.
(B.8)
Initiation of plastic folding mechanism in crushed box columns
141
The lengths of AC and BC remain unchanged during the deformation, i.e. AC = H:
H 2 cos2a + y~ + z 2 = H 2
BC = (:
(Yc - H s i n a ) 2 + (Zc + 0 2 = (2
(B.9) (B.10)
Solving eqns (B.9) and (B.10) gives Ull
H
sina (H2 sin2a - (2) (H 2 sin2a ¥ (2)
(B.11)
and
(2)1
ti. _ cosa& [1 + 2( 2 (H2 sin=a (H 2 sin=a + (2)1 H
(B.12)
A P P E N D I X C: CALCULATION OF T H E RATE OF ROTATION AT T H E H I N G E S Consider a quarter of the column due to symmetry. There are four hinge lines formed in phase I (0 < a < a2), as illustrated in Fig. A.1. It should be pointed out that the boundary of overlap will not be accounted for as a hinge line since the energy related to the overlapping has already been considered in the zones of continuous plastic deformation. The calculations of the rate of rotation and the length of hinge lines are as follows:
For Ùt 01 = a,
0t = h
and
Ii = b - 2 ~ "
(C.1)
12 = H
(C.2)
For 02 02 = Z C F B - , / 2 02 =
H 2 sin2a& V / ( 4 -- H 4 s i n 4 a
and
For 03 13 = v / - H ~ + ( 2 The calculation of 03 is more complicated. Here, we will only present the basic idea of the calculation. First, we calculate the unit vector n~, which is perpendicular to the plane ABG, as nl = s i n a i + cosaj
(C.3)
Tomasz Wierzbicki, Joan Huang
142
then we calculate the vectors AB a n d AQ: AB
=
-Hcosai
+ H sinaj - (k
-Hcosai-
AQ=
Acosa((-
H s i n a ) j - A c o s a ( ( + H s i n a ) k (C.4)
where A=
H 2 { x'/(¢2 -
H2sin2a)/(¢2+ H2 s i n 2 a ) - c o s a }
From the cross product of A B and A Q we can obtain the unit vector n2, which is perpendicular to the plane ,~Q, and the dot product ofn~ and a2 gives cos03. The rate of 03 can be obtained numerically.
For 04 04 = 03
and
14 = 13
(C.5)
Similarly, there are four hinge lines formed in phase II (a2 < a < ~/2) (see Fig. A.2).
For O, O, = h,
I, = b - 2 (
(C.6)
12 = (
(C.7)
14=¢
(c.8)
For 02 02 = a,
For 04
04=a, For 03
13 = v / a 2 +
¢2
To calculate 03, we first calculate the vectors AB a n d AC. F r o m the coordinates: XA = H c o s a
YA
xs = 0
Ys = H s i n a
zs = - (
Xc = 0
H 2 sinZa _ ¢2 Yc - H sina H2 sin2 a + (2
_ 2 ( H 2 sin2a Zc = (H 2 sin2a + ( ~
=
0
ZA =
0
(C.9)
Initiation of plastic folding mechanism in crushed box columns
143
we can obtain A B
=
-H cosai + H sinaj - (k .
AC = - H c o s a i
.
+ nsma
H 2 sin2a _ (2. 2 ( H 2 sin2a H2 sin2 a + (2J - H 2 sin2a + (2 k
(C.10)
F r o m the cross o f AB a n d AC we can obtain the unit vector n2, which is n o r m a l to the plane ABC: H 2 sin2a _ ( 2 . (H sin2a n2 = - s i n a i - cosa H2 sin2 a + (2J + H 2 sin2a + (2 k
(C.11)
We also k n o w the unit vector hi, which is n o r m a l to the plane ABG: nl = s i n a i + c o s a j
(C.12)
The dot o f nl a n d n2 gives cos03, which is 2 ( 2 cos2a cos03 = - 1 + H2 sin2 a + (2
(C.13)
The rate o f 03 is 0 _ -
1 sin0 3
2(2sin2a&{
1 _[ Hcosa ]2} H 2 sin2a + (2 [_H2~ 7 _ t - (2
(C.14)