Embed Size (px)
Citation preview
Interactive buckling in long thin-walled rectangular hollow section
struts
Jiajia Shen, M. Ahmer Wadee∗, Adam J. Sadowski
Department of Civil and Environmental Engineering, Imperial College London,
South Kensington Campus, London SW7 2AZ, UK
Abstract
An analytical model describing the nonlinear interaction between global and local buck-ling modes in long thin-walled rectangular hollow section struts under pure compressionfounded on variational principles is presented. A system of nonlinear differential and in-tegral equations subject to boundary conditions is formulated and solved using numericalcontinuation techniques. For the first time, the equilibrium behaviour of such struts withdifferent cross-section joint rigidities is highlighted with characteristically unstable inter-active buckling paths and a progressive change in the local buckling wavelength. Withincreasing joint rigidity within the cross-section, the severity of the unstable post-bucklingbehaviour is shown to be mollified. The results from the analytical model are validatedusing a nonlinear finite element model developed within the commercial package Abaqus
and show excellent comparisons. A simplified method to calculate the local buckling loadof the more compressed web undergoing global buckling and the corresponding global modeamplitude at the secondary bifurcation is also developed. Parametric studies on the effectof varying the length and cross-section aspect ratio are also presented that demonstratethe effectiveness of the currently developed models.
Keywords: Mode interaction; Global and local buckling; Variational principles;Thin-walled structures;
1. Introduction
Thin-walled plated structures are widely used in current structural engineering practiceowing to their mass efficiency and relative ease of manufacture. Buckling instabilitiesare practically always the governing failure mode of such structures [1, 2, 3]; moreover,compression members made from slender plate elements are vulnerable to a variety ofdifferent buckling phenomena [4, 5, 6, 7, 8]. The interaction between individual modes canlead to a profound change in the post-buckling behaviour, even though these modes may
∗Corresponding authorEmail addresses: [email protected] (Jiajia Shen), [email protected] (M. Ahmer
Wadee), [email protected] (Adam J. Sadowski)
Preprint submitted to International Journal of Non-Linear Mechanics November 17, 2016
be stable when triggered in isolation. In particular, such systems can exhibit a violentdestabilization after the peak load is reached [9, 10, 11] and, in certain parametric ranges,have been demonstrated to be highly sensitive to initial geometric imperfections [12, 13,14, 15, 16, 17, 18, 19].
Early work on the interactive buckling of columns was conducted by van der Neut[20]. Using Koiter’s theory [21], van der Neut developed a relatively simplified model thatcomprised two load-carrying flanges and a pair of rigid webs with no longitudinal stiffness.Moreover, the webs provided simply-supported boundary conditions to the flanges. Usingthis model, the initial post-buckling behaviour and imperfection sensitivity were investi-gated. The study identified the potentially dangerous consequences of columns exhibitinginteractive buckling. However, owing to increased technical complexity, the presentedmodel assumed no interaction between the individual plate elements that make up thecross-section.
Recently, Wadee and his collaborators have developed a series of analytical models usingvariational principles to investigate the interactive buckling of open-section compressionmembers, i.e. I-section struts [10, 22] and stiffened plates [11]. These analytical modelsshow very good quantitative comparisons with FE results and existing experimental results[9]. Moreover, snap-backs in the response, showing sequential destabilization and restabi-lization and a progressive spreading of the local buckling mode from mid-span, known ascellular buckling [23], are captured well by these models, which have also been observed inphysical experiments on thin-walled struts [9] and beams [24].
A number of experimental studies [7, 25] also exist on rectangular hollow sectioncolumns, in which an interaction between the local and the global buckling modes wasobserved. However, these studies primarily focused upon the ultimate capacity and thebehaviour in the inelastic range; only very limited details on the actual mechanism of themode interaction were presented.
In order to investigate how the interaction between individual plate elements affectsthe nonlinear interactive buckling behaviour, the aforementioned analytical models forstiffened plates and I-section struts were extended to include rotational springs at thejunctions within the cross-sections [26, 27]. In both cases, a rapid erosion of the snap-backs in the equilibrium path was observed with the increase of the rotational rigidity atthe junction and there was generally an increase in the residual post-buckling load carryingcapacity.
The present work aims to investigate the global–local mode interaction in rectangularhollow section struts. In comparison with open-section members, the interaction betweenindividual plates in a closed-section strut tends to be more significant. The principal focuscurrently is on how this affects the nonlinear behaviour, thus leading to a better under-standing of the underlying mechanics. An analytical model describing the behaviour of athin-walled rectangular hollow section strut with semi-rigid cross-section joints under axialcompression is developed founded on variational principles. The primary aim is to analysethe interaction of global and local buckling modes for the case where global buckling iscritical. A relationship describing how the cross-section joint rigidity affects the proper-ties of the system is obtained explicitly from the developed equilibrium equations. These
2
equations are solved using numerical continuation techniques through the well-known soft-ware Auto-07p [28]. The resulting equilibrium paths are presented for various differentcases and potentially dangerous unstable interactive buckling is found. The analyticalresults show excellent comparisons throughout the post-buckling range with numericalresults obtained using a nonlinear finite element (FE) model developed within the com-mercial package Abaqus [29]. A simplified method to predict the local buckling load ofthe more compressed web undergoing global buckling and the corresponding amplitude atthe secondary bifurcation is developed based on the validated analytical model. A cou-ple of parametric studies concerning the geometric properties are also presented that alsosuccessfully validate the simplified methodology.
The current work facilitates a better understanding of how the interaction between theindividual plates affects the post-buckling response of rectangular hollow section struts andin future would allow for imperfection sensitivity studies to be conducted such that morerobust design guidance can be established.
2. Development of the analytical model
A thin-walled simply supported rectangular hollow section strut of length L, loaded byan axial force P at the centroid of the cross-section is considered, as shown in Figure 1.The web depth and thickness are d and tw respectively; the flange width and thickness
Figure 1: (a) Plan view of the rectangular hollow section strut of length L under the concentric axial loadP . The lateral and longitudinal coordinates are x and z respectively. (b) Cross-section geometry of thestrut with semi-rigid joints including definitions of the rotational stiffness at junctions; the vertical axiscoordinate is y.
are b and tf respectively. The joints between the webs and the flanges are assumed to besemi-rigid and connected by a rotational spring with stiffness cθ. It should be stressedthat when cθ → ∞, it converges to a rigid joint; when cθ → 0, it converges to a pinnedjoint. The strut material is assumed to be linearly elastic, homogeneous and isotropic withYoung’s modulus E, Poisson’s ratio ν and shear modulus G = E/ [2(1 + ν)]. It is assumedthat global buckling occurs about the weak axis of bending.
3
2.1. Modal descriptions
The formulation begins with the definition of both the global and the local modaldisplacements based on a recent purely numerical study [30]. Since previous studies [31,32, 10, 11, 26, 27] have clearly demonstrated that it is essential to include the shearstrain contributions into the total potential energy formulation to model the interactivebuckling behaviour, Timoshenko beam theory is assumed currently. The global mode isdecomposed into two components: a pure lateral displacement W and a pure rotation ofthe plane sections θ, see Figure 2, known as the ‘sway’ and ‘tilt’ modes [33, 31] respectively.
Figure 2: (a) Sway and tilt components of the global buckling mode bending about the weak axis y. (b)Out-of-plane local mode in the flanges wf(x, z) and in the more compressed web wwc(y, z). Also shownare the in-plane local mode in the flanges uf(x, z) and in the more compressed web uwc(y, z).
The global buckling lateral displacement W and the corresponding rotation θ are definedby the following expressions:
W (z) = −qsL sin(πz
L
)
, θ(z) = −qtπ cos(πz
L
)
, (1)
4
where qs and qt are the generalized coordinates for the sway and tilt modes respectively.The shear strain in the flanges from global buckling is given by the following expression:
γxz =dW
dz− θ = − (qs − qt) π cos
(πz
L
)
. (2)
In the current study, the focus is on the cases where global buckling occurs first and so thelocal displacement in the less compressed web is assumed to be zero. The local bucklingmode, including out-of-plane and in-plane displacement components, shown in Figure 2(b),is defined with the following variables:
wf(x, z) = ff(x)w(z), wwc(y, z) = fwc(y)w(z), (3)
uf(x, z) = gf(x)u(z), uwc(y, z) = gwc(y)u(z), (4)
where f and g are the cross-section components for the out-of-plane and in-plane dis-placement components respectively; w(z) and u(z) are the longitudinal out-of-plane andin-plane displacement components respectively.
From an earlier numerical study [30], it was determined that the cross-section shapefunctions for the out-of-plane and the in-plane components, f and g, are approximatelythe same. Therefore, currently, these components are in fact assumed to be the same,i.e. gf(x) = ff(x) and gwc(y) = fwc(y). The cross-section components, ff(x) and fwc(y), asshown in Figure 3(a), are estimated by applying appropriate kinematic and static boundaryconditions for each plate in conjunction with the Rayleigh–Ritz method. It is assumed thatfwc has the functional form that is derived from the conditions of a simply-supported strut(a cosine wave) and a beam under pure bending (a parabola) such that the cases for a fullypinned joint or a joint that rotates as a rigid body can be modelled, thus:
fwc = B0 cos(πy
d
)
+ (1 − B0)
(
1 − 4y2
d2
)
. (5)
For ff , the functional form is derived from a beam with one end clamped and the other endsimply-supported with an end moment arising from the transfer of moment at a non-pinnedjoint. This naturally leads to a cubic polynomial form:
ff = A0
(
x +b
2
)
+ A1
(
x +b
2
)2
+ A2
(
x +b
2
)3
. (6)
The coefficients B0 in fwc and A0, A1 and A2 in ff are determined by applying ap-propriate boundary conditions at the junctions. The form of fwc automatically satisfiesthe natural boundary conditions for the web displacement function, i.e. fwc(±d/2) = 0.Since global buckling occurs first and the resulting less compressed web is assumed to havezero out-of-plane displacement, the flanges near the less compressed side also have zeroout-of-plane displacement. Therefore, the junction between the less compressed web andthe flanges is assumed to be rigid, as shown in Figure 3(a). At the junction between the
5
Figure 3: Semi-rigid joints with corresponding kinematic and static boundary conditions at the web–flangejunctions. (a) Cross-section component of the local mode in the flanges ff(x) and the more compressedweb fwc(y); the stiffness of the rotational spring at the joints is cθ. (b) Kinematic boundary conditionat the junction; θf and θw are the rotations of the flange and the more compressed web at the junctionrespectively. (c) Equilibrium condition at the junction; Mf and Mw are the bending moments in the flangeand the more compressed web at the junction respectively. (d) Equivalent rotational springs with stiffnesscθf attached to the more compressed web.
less compressed web and flanges, x = −b/2 and y = ±d/2, the boundary condition for theflanges are:
ff (−b/2) = f ′
f (−b/2) = 0, (7)
where the prime denotes differentiation with respect to x.Another boundary condition can be obtained by considering moment continuity at the
junction between the flanges and the more compressed web given that there is a rotationalspring of stiffness cθ present, as shown in Figure 3(b–c). Hence, the following boundaryconditions need to be satisfied:
Mf (x = b/2) + Mwc (y = −d/2) = cθ(θw − θf), (8)
where:
Mf (x = b/2) =
[
Df
(
∂2wf
∂x2+ ν
∂2wf
∂z2
)]
x=b/2
, (9)
Mwc (y = −d/2) =
[
Dw
(
∂2wwc
∂y2+ ν
∂2wwc
∂z2
)]
y=−d/2
, (10)
θw =dfwc
dy
∣
∣
∣
∣
y=−d/2
, θf =dff
dx
∣
∣
∣
∣
x=b/2
, (11)
with Df = Et3f /[12(1 − ν2)] and Dw = Et3w/[12(1 − ν2)] being the flexural rigidities of theindividual flanges and webs respectively.
6
As for the purely pinned or rigid joint, one more boundary condition at the morecompressed web and flange junction can be obtained. For the purely pinned joint case, theflanges do not buckle, hence:
θf = f ′
f (x = b/2) = 0. (12)
For the case where the joint rotates as a rigid body, the rotation of the more compressedweb and the less compressed web are the same, hence:
θf = f ′
f (x = b/2) = θw = f ′
wc(y = −d/2). (13)
These four equations above, i.e. Eqs. (7–8), (12) and (13), can resolve the four undeterminedcoefficients in fwc and ff for the pinned and rigid joint cases respectively.
However, for the semi-rigid joint case, the fourth boundary condition cannot be ob-tained directly as for the pinned and rigid joint cases above. When the more compressedweb buckles, both the flanges and the joint rotational springs provide the web with re-straints. Therefore, by isolating the more compressed web plate, the total rotational stiff-ness provided by the flanges together with the rotational spring can be replaced by anequivalent rotational spring cθf , as shown in Figure 3(d). Moreover, since the flanges andthe rotational springs are effectively in series, the following standard relationship can beused:
1
cθf
=1
cθ+
1
cf
, (14)
where cf is the equivalent rotational stiffness accounting for the rotational restraint providedby the flanges, as shown in Figure 4(a). In the rigid joint case, where cθ → ∞ and cθf = cf ,
Figure 4: (a) Equivalent rotational springs with stiffness cf on the more compressed web provided bythe connecting flange. (b) Cross-section component of the local mode in the flanges ff(x) and the morecompressed web fwc(y) for the rigid joint case due to the rotation of the flange and the more compressedweb at the junction, θw. (c) Free-bodies of the more compressed web–flange junctions; Mfw is the bendingmoment within the flange and the more compressed web at the junction.
the rotational stiffness cf can be determined by considering the continuity of moment and
7
rotation at the junctions, as shown in Figure 4(b–c), hence:
Mfw = cf θw =
[
Dw
(
∂2wwc
∂y2+ ν
∂2wwc
∂z2
)]
y=−d/2
, (15)
where:
θw =∂wwc
∂y
∣
∣
∣
∣
y=−d/2
= w [f ′
wc (y = −d/2)] , (16)
∂2wwc
∂y2
∣
∣
∣
∣
y=−d/2
= w [f ′′
wc (y = −d/2)] , (17)
∂2wwc
∂z2
∣
∣
∣
∣
y=−d/2
= w [fwc (y = −d/2)] = 0. (18)
Substituting the displacement function for the more compressed web in the rigid joint case,the rotational stiffness cf can be obtained:
cf =4Df
b(19)
and an expression for the equivalent rotational spring stiffness cθf can be expressed thus:
cθf =cθ
cθ/cf + 1. (20)
Therefore, the final boundary condition to determine the undetermined parameters isgiven by the relationship:
cθfθw =
[
Dw
(
∂2wwc
∂y2+ ν
∂2wwc
∂z2
)]
y=−d/2
. (21)
Based on these conditions, the coefficients Ai, where i = {0, 1, 2} and B0 can be determined:
A0 = 0, (22)
A1 = − 2πcθ (cθ + 2)
b2φc (cθ + 1) (πφcφ3t cθ − 4φcφ3
t cθ − 2cθ − 2), (23)
A2 =2πcθ (cθ + 2)
b3φc (cθ + 1) (πφcφ3t cθ − 4φcφ3
t cθ − 2cθ − 2), (24)
B0 = − 2 (2φcφ3t cθ + cθ + 1)
πφcφ3t cθ − 4φcφ
3t cθ − 2cθ − 2
, (25)
where cθ = cθ/cf , φt = tf/tw and φc = d/b. It should be stressed that when cθ → ∞, ff andfwc both converge to the rigid joint case, i.e. θf = f ′
f (x = b/2) = θw = f ′
wc(y = −d/2); whencθ → 0, ff and fwc both converge to the pinned joint case, i.e. ff = 0 and fwc = cos (πy/d).
8
2.2. Total potential energy
The total potential energy V comprises the contributions from the strain energy Ustored from the global bending of the strut, axial and shear stresses in the whole cross-section, the local bending of the flanges and the more compressed web, the rotationalsprings and the work done by the external load PE , where E is the total end-shortening.
The only contribution to the global bending strain energy Ub,o is from the webs throughthe sway mode, since the membrane strain energy contributions from the flanges and websaccount for the effect of global bending through the tilt mode, as shown in Figure 2(a).Therefore, the global bending strain energy, Ub,o, can be expressed thus:
Ub,o = 2
∫ L
0
EIw
2W 2 dz = EIw
∫ L
0
q2s
π4
L2sin2 πz
Ldz, (26)
where EIw = Edt3w/12 is the flexural rigidity about the local weak neutral axis of the weband dots represent differentiation with respect to z. The factor of 2 is included to accountfor both webs.
The local bending strain energy stored in both flanges and the more compressed webcan be determined with the following standard expressions:
Ub,fl = Df
∫ L
0
∫ b/2
−b/2
{
(
∂2wf
∂z2+
∂2wf
∂x2
)2
− 2(1 − ν)
[
∂2wf
∂z2
∂2wf
∂x2−(
∂2wf
∂z∂x
)2 ]}
dx dz,
(27)
Ub,wcl =Dw
2
∫ L
0
∫ d/2
−d/2
{
(
∂2wwc
∂z2+
∂2wwc
∂y2
)2
(28)
− 2(1 − ν)
[
∂2wwc
∂z2
∂2wwc
∂y2−(
∂2wwc
∂z∂y
)2 ]}
dy dz.
Since it is assumed that there is no buckling displacement in the less compressed web, itnaturally follows that there is zero local bending strain energy in that element.
The membrane strain energy in the flanges Um,f is derived from considering the directstrains (ε) and the shear strains (γ). The complete direct strain expression for the flangescan be written as:
εz,f =∂ut
∂z+
∂uf
∂z+
1
2
(
∂wf
∂z
)2
− ∆, (29)
where the first term is from the global mode and ut = xθ(z), being the ‘tilt’ in-planedisplacement; the second and third terms are the local components obtained based onvon Karman plate theory; the final term is the purely in-plane compressive strain. Thecorresponding shear strain component can be written thus:
γxz,f =∂uf
∂x+
∂W
∂z− θ +
∂wf
∂x
∂wf
∂z. (30)
9
From the previous numerical study [30], the transverse strain component was shown to bevery small when compared with the other two components, a finding that also coincideswith earlier analytical work [34], hence it is not included presently. The current nonlinearformulation is deemed sufficient since it is well-known that it accounts for local deflectionsthat are of the same order of magnitude as the plate thickness. The complete expressionfor the membrane strain energy stored in both flanges can be written thus:
Um,f = 2
∫ L
0
∫ tf/2
−tf/2
∫ b/2
−b/2
1
2
(
Eε2z,f + Gγ2
xz,f
)
dx dy dz. (31)
The membrane strain energy stored in the more compressed web also comprises direct andshear strain energy contributions. As for the less compressed web, the expression is morestraightforward since it is assumed that there are no local buckling related terms. Thecomplete expressions for the direct strain in the more and less compressed webs are:
εz,wc = εz,wco − ∆ +∂uwc
∂z+
1
2
(
∂wwc
∂z
)2
, (32)
εz,wt = εz,wto − ∆, (33)
where the direct strains from the global mode, i.e. εz,wco and εz,wto, can be written thus:
εz,wco = − b
2θ = −qt
bπ2
2Lsin
πz
L, (34)
εz,wto = +b
2θ = qt
bπ2
2Lsin
πz
L. (35)
Unlike the flanges, the shear strain within the webs only contain terms from the localmode. The shear strain in the less compressed web is zero and the shear strain in the morecompressed web can be written thus:
γyz,wc =∂uwc
∂y+
∂wwc
∂y
∂wwc
∂z. (36)
The transverse strain is once again neglected in the current formulation for the same reasonsas outlined above, thus the membrane strain energy contributions from the webs Um,wc andUm,wt can be given respectively thus:
Um,wc =
∫ L
0
∫ tw/2
−tw/2
∫ d/2
−d/2
1
2
(
Eε2z,wc + Gγ2
yz,wc
)
dy dx dz, (37)
Um,wt =
∫ L
0
∫ tw/2
−tw/2
∫ d/2
−d/2
1
2Eε2
z,wt dy dx dz. (38)
The strain energy stored in the rotational springs, Usp, accounting for the web–flange jointsin the side of the more compressed web, is given by the following expression:
Usp = 2
∫ L
0
1
2cθ
(
∂wf
∂x
∣
∣
∣
∣
x=b/2
− ∂wwc
∂y
∣
∣
∣
∣
y=−d/2
)2
dz. (39)
10
The factor of 2 is included to account for the rotation of both corners, as shown in Fig-ure 3(a).
The total end-shortening E comprises components from pure squash, the global swaymode and the local in-plane displacement. Hence, the work done by the external load Pis given by the expression:
PE = P
∫ L
0
(
q2s
π2
2cos2 πz
L+ ∆ − ∆m
)
dz, (40)
with:
∆m =
[
2 {gf}x + {gwc}y
]
u
2 (b + d), (41)
where {gf}x and {gwc}y are definite integrals with respect to their corresponding subscriptx or y, thus:
{gf}x =
∫ b/2
−b/2
gf dx, {gwc}y =
∫ d/2
−d/2
gwc dy. (42)
In summary, the total potential energy V can be expressed by the summation of all thestrain energy terms minus the work done by the external load:
V = Ub,o + Um,f + Um,wc + Um,wt + Ub,fl + Ub,wcl + Usp − PE . (43)
2.3. Governing equations
By performing the calculus of variations on the total potential energy V , the governingequations of equilibrium can be obtained. The integrand of the total potential energy Vcan be written as a Lagrangian (L) of the form thus:
V =
∫ L
0
L (w, w, w, u, u, z) dz. (44)
The equilibrium states of the system can be obtained by invoking the condition that V isstationary by setting the first variation of V , i.e. δV , to zero, where:
δV =
∫ L
0
(
∂L∂w
δw +∂L∂w
δw +∂L∂w
δw +∂L∂u
δu +∂L∂u
δu
)
dz. (45)
Since δw = d(δw)/ dz, δw = d(δw)/ dz and δu = d(δu)/ dz, integration by parts allowsthe development of the Euler–Lagrange equations for w and u, resulting in a fourth ordernonlinear ordinary differential equation (ODE) in w and a second order nonlinear ODE in
11
u:
Dw
[
2φ3t
{
f 2f
}
x+{
f 2wc
}
y
]
....w − 3Etw
2
[
2φt
{
f 4f
}
x+{
f 4wc
}
y
]
w2w
− Gtw
[
2φt
{
f ′
f
2f 2
f
}
x+{
f ′2wcf
2wc
}
y
]
(
ww2 + w2w)
− Etw
[
2φt
{
gff2f
}
x+{
gwcf2wc
}
y
]
(uw + uw)
+π2Etwbqt
2L
[
4φt
b
{
xf 2f
}
x+{
f 2wc
}
y
]
(
sinπz
Lw +
π
Lcos
πz
Lw)
+
{
2Dwν
[
2φ3t {fff
′′
f }x + {fwcf′′
wc}y
]
− 2Dw(1 − ν)
[
2φ3t
{
f ′
f2}
x+{
f ′
wc2}
y
]
+ Etw∆[
2φt
{
f 2f
}
x+{
f 2wc
}
y
]
}
w − Gtw
[
2φt {g′
ff′
fff}x + {g′
wcf′
wcfwc}y
]
uw
− 2π2GtfL
{f ′
fff}x (qs − qt) sinπz
Lw + Kw = 0,
(46)
Etw
[
2φt
{
g2f
}
x+{
g2wc
}
y
]
u + Etw
[
2φt
{
gff2f
}
x+{
gwcf2wc
}
y
]
ww
− Gtw
[
2φt {g′
ff′
fff}x + {g′
wcf′
wcfwc}y
]
ww + 2Gtf {g′
f}x (qs − qt)π cosπz
L
− π3Etwbqt
2L2
[
4φt
b{xgf}x + {gwc}y
]
cosπz
L− Gtw
[
2φt
{
g′
f
2}
x+{
g′2wc
}
y
]
u = 0,
(47)
where K is the coefficient of the linear term w, sometimes also referred to as the foundationterm, which is well known to affect the local buckling load [31]. Currently, it comprises aplate-related term Kp and a spring-related term Ks, i.e. K = Kp + Ks, where:
Kp = Dw
[
2φ3t
{
f ′′
f
2}
x+{
f ′′
wc2}
y
]
, Ks = 2cθ
(
f ′
wc|y=d/2 − f ′
f |x=b/2
)2. (48)
Moreover, equilibrium also requires the minimization of V with respect to the generalizedcoordinates qs, qt and ∆, leading to three integral equations:
∂V
∂qs= π2GtfbL (qs − qt) +
π4EIwqs
L− P
π2Lqs
2(49)
− 2πGtf
∫ L
0
[
{g′
f}x u + {f ′
fff}x ww
]
cosπz
Ldz = 0,
∂V
∂qt=
π4Etwb2dqt
4L
[
1 +φt
3φc
]
− π2GtfbL (qs − qt) (50)
+ 2πGtf
∫ L
0
[
{g′
f}x u + {f ′
fff}x ww
]
cosπz
Ldz
− π2Ebtw2L
∫ L
0
{[
{gwc}y +4φt
b{xgf}x
]
u
12
+
[
1
2
{
f 2wc
}
y+
2φt
b
{
xf 2f
}
x
]
w2
}
sinπz
Ldz = 0,
∂V
∂∆= −Etw
∫ L
0
{
[
{gwc}y + 2φt {gf}x
]
u +
[
1
2
{
f 2wc
}
y+ φt
{
f 2f
}
x
]
w2
}
dz (51)
+ 2EtwdL∆
(
1 +φt
φc
)
− PL = 0.
Since the strut is an integral member, Eq. (50) provides a relationship between qs and qt
before the local mode is triggered, i.e. when u = w = w = 0:
qs = (1 + s) qt, (52)
where:
s =π2Eb2
4GL2
(
1
3+
φc
φt
)
. (53)
The boundary conditions for w and u and their derivatives are for a simple-support atz = 0 and for symmetry at z = L/2:
w (0) = w (0) = w (L/2) =...w (L/2) = u (L/2) = 0. (54)
A further boundary condition can be obtained from minimizing V and it is a conditionthat relates to matching the in-plane strain at the ends:
u(0)[
{
g2wc
}
y+ 2φt
{
g2f
}
x
]
+w2(0)
2
[
{
gwcf2wc
}
y+ 2φt
{
gff2f
}
x
]
−∆[
{gwc}y + 2φt {gf}x
]
+P[
{gwc}y + 2φt{gf}x
]
2Etwd (1 + φt/φc)= 0. (55)
Linear eigenvalue analysis for the perfect column is conducted to determine the critical loadfor global buckling PC
o . This is achieved by considering the condition where the Hessianmatrix Vij is singular when qs = qt = w = u = 0, where:
Vij =
[
∂2V∂q2
s
∂2V∂qs∂qt
∂2V∂qt∂qs
∂2V∂q2
t
]
, (56)
which produces the following expression:
PCo =
2π2EIw
L2+
π2Etfb3
2 (1 + s)L2
(
1
3+
φc
φt
)
. (57)
Note that if Euler–Bernoulli bending theory had been assumed, the shear modulus G → ∞,which implies that s → 0, and PC
o would reduce to the classical Euler load, as would beexpected.
13
Table 1: Geometric properties of the rectangular hollow section strut in the numerical example, selectedto ensure global bucking is critical.
Length Flange width Web depth Flange thickness Web thicknessL b d tf tw
5250 mm 60 mm 120 mm 1 mm 1 mm
3. Numerical results
In this section, representative numerical examples from the analytical model with avarying rotational stiffness cθ are presented. The geometric properties of the example strutare presented in Table 1. The Young’s Modulus E and Poisson’s ratio ν of the material arechosen to be 210 kN/mm2 and 0.3 respectively. For the case where cθ = 0, the theoreticalbuckling stresses and critical mode are presented in Table 2. The global buckling stress
Table 2: Theoretical values of the global and local critical buckling stresses for the pinned cross-sectioncase.
σCo (N/mm2) σC
l,w (N/mm2) σCl,f(N/mm2) Critical mode
52.70 52.72 210.88 Global
is calculated using Eq. (57) where σCo = PC
o /A and A is the total area of the cross-section. The local buckling stress is estimated by using the classical plate buckling stressσC
l,w = kpDwπ2/(d2tw) for the webs and σCl,f = kpDfπ
2/(b2tf) for the flanges [35]. Since it isinitially assumed that cθ = 0, all the plates have effectively pinned edges and the lengthof the strut is much larger than the width of the plate, kp = 4 is adopted for the platebuckling coefficient to obtain a lower bound. With the increase of the rotational stiffness cθ,the local buckling load PC
l would also increase whereas the global buckling load PCo would
remain the same. Therefore, the selected geometric dimensions and material propertiesensure that global buckling is always critical for any positive value of cθ in the examplespresented.
The system of nonlinear ordinary differential equations is solved numerically using thecontinuation and bifurcation software Auto [28]. The software is not only capable ofsolving the nonlinear ordinary differential equations numerically, but it also maintains theintrinsic bifurcational structure of the solutions. Moreover, importantly, the software canswitch between, as well as trace, different equilibrium paths, allowing the behaviouralevolution of the geometrically perfect cases to be studied.
The solution strategy for using Auto is shown diagrammatically in Figure 5. Thecritical buckling load PC
o is obtained explicitly from Eq. (57). Using the continuationmethod, the generalized global sway mode amplitude qs is first varied, while P = PC
o ,to obtain the secondary bifurcation points Si, where the first one (S1 ≡ S) pinpoints thelocation where interactive buckling is practically triggered. Subsequently, the second runis started at the secondary bifurcation point S using the branch switching facility withinthe software and P is varied to compute the interactive buckling path. With the increase
14
Figure 5: Numerical continuation procedure for determining the interactive buckling equilibrium path forperfect struts where global buckling is critical. The thicker solid line shows the actual solution path. Circlesmarked C and Si represent the critical and secondary bifurcation points respectively. The generalizedcoordinate of the sway mode at the secondary bifurcation point is defined as qS
s .
of cθ, the value of qs at the secondary bifurcation point qSs increases due to the higher local
buckling stress for the more compressed web relative to the constant global buckling load.However, before conducting numerical continuation in Auto, the effects of the rota-
tional springs on the nonlinear ODEs are investigated. The explicit spring related term inthe ODEs is the coefficient of linear term w, defined as K in Eq. (46). Figure 6(a) showsthe relationship between the individual components of K, i.e. Kp and Ks defined in Eq.(48), while varying the normalized joint rigidity cθ. It can be seen that the plate-relatedterm Kp rises with the increase of the rotational spring stiffness. Moreover, it reaches aplateau with the value being the same as the rigid joint case when the normalized jointstiffness cθ is close to 7. As for the spring-related term Ks, it reaches a peak and thendecays to zero as cθ is increased further. This can be attributed to the fact that the rota-tion of the flange and the more compressed web at the joint are approximately the sameand thus f ′
wc|y=d/2 = f ′
wt|x=b/2. Based on this analysis, struts with the cross-section jointstiffness cθ listed in Table 3 are used in the subsequent numerical study.
Table 3: Rotational stiffness cθ and the corresponding normalized stiffness cθ values used in the numericalstudies.
cθ (Nm/m) 0 160.26 641.03 2564.10 ∞cθ 0 0.125 0.5 2 ∞
Unstable post-buckling behaviours arising from the triggering of interactive bucklingare observed in all example struts presented in Figure 7. Figure 7(a) shows that the severesnap-back phenomenon in the normalized load p = P/PC
o versus the normalized end-shortening E/L relationship is mollified with the increasing joint rigidity cθ. Moreover,stiffer joints within the cross-section also lead to a higher residual post-buckling capacity.A gradual transition from highly unstable behaviour to less unstable behaviour can be
15
cθ = cθb/(4Df)0 2 4 6 8
Coeffi
cients
ofthelinearterm
w
0
0.2
0.4
0.6
0.8
1
1.2
1.4(a)
Kp
Ks
Kp(cθ → ∞)
cθ = cθb/(4Df)0 2 4 6 8
Kan
dqS s
0
0.2
0.4
0.6
0.8
1(b)
KqSs
Figure 6: (a) The influence of the rotational spring stiffness on the coefficient of the linear term w in Eq.(46) for the example strut with the properties listed in Table 1. The quantities Kp and Ks are the plateand spring related terms respectively, given in Eq. (48). (b) The normalized values of the coefficient of thelinear term K and the generalized coordinate of global sway mode qS
s versus the normalized joint rigiditycθ.
observed. At the same load level in the post-buckling range, a higher joint rigidity casecorresponds to larger global and local mode amplitudes, as shown in Figure 7(b, c). Thegeneralized global mode amplitude to trigger the mode interaction qS
s increases significantlyespecially between cθ = 0 and cθ = 0.5. The rate of increase in qS
s , however, begins to reducesignificantly as cθ is increased further; for example, the equilibrium path of the case withcθ = 2 is very similar to the rigid case. By normalizing K and qS
s between the pinned andrigid cases, thus:
K =K − K0
KR − K0
, qSs =
qSs − qS
s0
qSsR − qS
s0
, (58)
where K0 is the coefficient of the linear term for the pinned case (cθ = 0); KR is thecoefficient of the linear term for the rigid case (cθ → ∞); qS
s0 is the value of qSs for the
pinned case (cθ = 0); and qSsR is the value of qS
s for the rigid case (cθ → ∞). It can beobserved in Figure 6(b) that K and qS
s are practically identically distributed with cθ, whichis entirely logical since K is known to control the local buckling load [31] and that in turncontrols when the secondary bifurcation occurs.
From the solutions of the out-of-plane components of the local mode w, a wavelengthvariation is observed with the progress of interactive buckling, as shown in Figure 8. Theinitially localized buckling mode spreads outwards from the mid-span of the column, de-veloping with more peaks and troughs alongside a clear reduction in wavelength as themodal amplitude becomes larger and the load drops in the post-buckling range. Since the
16
E/L ×10-40 2 4 6 8
p
0
0.2
0.4
0.6
0.8
1
(a)
Pinned (cθ = 0)cθ = 0.125cθ = 0.5cθ = 2Rigid (cθ → ∞)
qs ×10-30 5 10 15
p0
0.2
0.4
0.6
0.8
1
(b)
Pinned (cθ = 0)cθ = 0.125cθ = 0.5cθ = 2Rigid (cθ → ∞)
wwc,max/tw-2 -1.5 -1 -0.5 0
p
0
0.2
0.4
0.6
0.8
1
(c)
Pinned (cθ = 0)cθ = 0.125cθ = 0.5cθ = 2Rigid (cθ → ∞)
qs0 0.005 0.01 0.015
wwc,max/t
w
-2
-1.5
-1
-0.5
0
(d)
Pinned (cθ = 0)cθ = 0.125cθ = 0.5cθ = 2Rigid (cθ → ∞)
cθ increasing cθ increasing
cθ increasing
cθ increasing
Figure 7: Numerical equilibrium paths for struts with varying joint rigidity cθ. Graphs of the normalizedload p = P/PC
o versus (a) the normalized end-shortening E/L, (b) the generalized coordinate of thesway mode qs and (c) the normalized peak amplitude of local deformation in the more compressed webwwc,max/tw; (d) wwc,max/tw versus qs.
global buckling mode amplitude at the secondary bifurcation is relatively larger for strutswith a higher joint rigidity, the local buckling profile is initially more localized at mid-spanat p = 0.995. Moreover, the higher joint rigidity also leads to a smaller wavelength at thesame load level. This is in accord with results in previous work on I-section struts [27] andon struts on elastic softening–hardening foundations [36].
Three dimensional representations of the numerical solutions with cθ = 0, cθ = 0.5 andthe rigid case (cθ → ∞) at load levels p = 0.995 and p = 0.790 in the post-buckling rangeare shown in Figure 9. It should be emphasized that there is no buckle in the flanges forthe cθ = 0 case, as shown in Figure 9(a), owing to the lack of interaction between adjacentplates within the cross-section.
17
Figure 8: Numerical solutions of the normalized local out-of-plane displacement in the more compressedweb wwc/tw for the cases where: (a) cθ = 0 (pinned), (b) cθ = 0.5 and (c) cθ → ∞ (rigid). The leftand right columns correspond to the normalized load in the post-buckling range p = 0.995 and p = 0.790respectively. Note that the longitudinal coordinates are normalized with respect to half the strut lengthz = 2z/L and that the buckling wavelengths are reduced at the lower load.
4. Finite element model
The commercial finite element (FE) package Abaqus [29] has been demonstrated to bean excellent tool to conduct nonlinear post-buckling analysis, especially with its implemen-tation of the Riks arc-length method [38]. In previous works, it has been used successfullyfor mode interaction problems on stainless steel I-section columns [39], stainless steel boxsection columns [25], linear elastic stiffened plates [26] and linear elastic I-section columns[27]. Therefore, it is implemented currently to establish a purely numerical model for thestrut with semi-rigid joints with the same structural properties as the analytical modelshown in Figure 1(a) together with the properties presented in Table 1. The purpose ofthe FE model is that it is used as a benchmark to validate the analytical model.
4.1. Element type and meshing scheme
The struts are modelled using the four-node, reduced-integration S4R shell elements[29]. Previous research studies [40, 10, 26, 25] have demonstrated that this element canmodel plate buckling problems with very good accuracy. Since the wavelength of the localbuckling mode is much smaller than that of the global one, a meshing scheme suitable forcapturing the local buckling mode naturally would be sufficiently good to capture globalbuckling. Moreover, to increase computational accuracy, the shape of the element shouldbe made to be as square as possible [41]. A sensitivity analysis was conducted to find anacceptable meshing scheme that not only yields accurate results but is also computationallyefficient. As demonstrated in [30], the meshing scheme of 20 elements per wavelength yieldsa more than satisfactory result.
18
Figure 9: 3D visualization of the numerical solutions from the analytical model, plotted using Matlab [37].The results are shown for the post-buckling equilibrium states for the cases where: (a) cθ = 0 (pinned), (b)cθ = 0.5 and (c) cθ → ∞ (rigid) from the top to the bottom row respectively. The left and right columnscorrespond to the normalized load in the post-buckling range p = 0.995 and p = 0.790 respectively. Notethat the deformations shown have been amplified by a factor of 5 and the longitudinal coordinate (z) hasbeen scaled by a factor of 0.25, both to aid visualization. All dimensions are in millimetres.
19
4.2. Strut modelling
To capture the local deformation of each individual plate and the effects of the rotationalstiffness at the junction, each plate is modelled separately. The nodes at the junctions ofthe webs and the flanges are defined and labelled separately but share the same coordinates.The translational degrees of freedom (DOFs) of these nodes are then tied together but therotational DOFs are not, thus approximating a pinned joint. A rotational spring element,‘Spring2’ in the Abaqus element library, is then introduced to connect the end nodesof each flange and web, as shown in Figure 10(a). As in the analytical model, with the
Figure 10: Semi-rigid joints in the FE model. (a) Node-to-node rotational springs in the cross-section; (b)Springs along the length of the strut; ml represents the number of elements along the length of the strut;kθ is the stiffness of an individual rotational spring.
varying stiffness of the spring element, cross-section joint properties ranging from pinnedto rigid can be modelled.
Unlike the rotational springs in the analytical model, however, the rotational springsare discretely distributed in the FE model, as shown in Figure 10(b). To make the springsequivalent in these two cases, the following relationship is applied:
kθ =cθL
ml + 1, (59)
where cθ and L were defined earlier in the analytical formulation; kθ is the rigidity ofan individual rotational spring in the FE model with its units being Nmm and ml is thenumber of elements along the length of strut. As for the two limiting joint cases, i.e.the pinned and rigid cases, special treatments are adopted. In the FE model, if kθ is setto be zero, the strut would in fact be a mechanism and hence would not satisfy staticequilibrium. Therefore, kθ is set to be a very small nominal value of 10−6 Nmm such thatpotential kinematic mechanisms can be avoided and the pinned case is essentially satisfiednumerically. As for the rigid joint case, the rotational DOFs at the junctions of the websand flanges are tied together in the same way as the translational DOFs.
In terms of modelling the simply-supported boundary condition, a reference point isintroduced at the cross-section centroid of the loaded section and linked to the loadededges via rigid body kinematic coupling. This ensures that all the boundary and loading
20
conditions, defined at the reference point, are uniformly transmitted to the whole sectionand the pinned–roller support assumption is therefore satisfied. The translational DOFs inthe x and y directions at the reference point are restrained. Moreover, symmetry boundaryconditions are adopted at mid-span to reduce computational effort. The compressive axialload is applied to the column by specifying a concentrated force to the reference point inthe z-direction.
4.3. Analysis procedure
It is not possible to analyse perfect post-buckling behaviour in Abaqus directly owingto the discontinuous pitchfork bifurcation response at the initial instability. Therefore, itis necessary to transform the bifurcation problem into a continuous one by introducing aninitial perturbation in the geometry [42], the analysis procedure comprising two stages. Inthe first stage, a linear eigenvalue analysis is performed to obtain the buckling loads andmodes, some of which are introduced as a perturbation in the second stage of analysis.In the second stage, the Riks arc-length method [38] is used to conduct the post-bucklinganalysis with initial geometric perturbations. The scale factors for the global and localbuckling modes are set to 10−6L and 10−3tw respectively. These values are sufficientlysmall to ensure that the response essentially mimics the perfect case as far as possible,without encountering the pitchfork bifurcations that would lead to convergence problems.A study of the sensitivity to more realistic imperfection sizes is left for future work.
5. Validation and discussion
A linear eigenvalue analysis is first conducted in Abaqus to obtain the global bucklingload of the FE model with the same geometric properties as the analytical model. Theglobal critical load is found to be 18.90 kN, approximately 0.21% smaller than the analyticalsolution for PC
o using Eq. (57). The insignificant error is postulated to be derived from theglobal mode displacement field assumption in Eq. (1).
In terms of the nonlinear behaviour, equilibrium paths obtained from both the ana-lytical and FE models show excellent agreement for all rotational stiffness cases listed inTable 3, with the analytical model exhibiting a very slightly stiffer response in the ad-vanced post-buckling range. Three typical cases, i.e. cθ = 0 (pinned), cθ = 0.5 and cθ → ∞(rigid), are presented and discussed currently, as shown in Figure 11. From both the p–qs
and wwc,max–qs equilibrium diagrams, it can be observed that the values of qSs for all cases
are extremely close between the two models. As for the local–global mode relationship,the present model matches better with FE results, when compared to previous studieson I-section struts [27, 22] using the same methodology. Apart from the fact that thecross-section functions ff , fwc, gf and gwc have assumed forms, an additional source for thestiffer response of the analytical model is derived from the underlying assumption that theneutral axis location remains unchanged. In fact, the neutral axis would move to the lesscompressed web side when local buckling occurs in the more compressed web and flanges;however, it is worth emphasising that presently the errors are fundamentally small.
21
×10-40 5
p
0
0.5
1
×10-30 5 10
0
0.5
1
-1.5 -1 -0.5 00
0.5
1
0 0.005 0.01
wwc,max/t w
-2
-1.5
-1
-0.5
0
×10-40 5
p
0
0.5
1
×10-30 5 10
0
0.5
1
-1.5 -1 -0.5 00
0.5
1
0 0.005 0.01
wwc,max/t
w
-2
-1.5
-1
-0.5
0
E/L ×10-40 5
p
0
0.5
1
qs ×10-30 5 10
0
0.5
1
wwc,max/tw-1.5 -1 -0.5 0
0
0.5
1
qs0 0.005 0.01
wwc,max/t
w
-2
-1.5
-1
-0.5
0
cθ = 0
cθ = 0.5
cθ → ∞
(a)
(b)
(c)
Figure 11: Comparison of the post-buckling equilibrium paths for the cases where: (a) cθ = 0 (pinned),(b) cθ = 0.5, and (c) cθ → ∞ (rigid) from the analytical (solid line) and FE (dashed line) models. Graphsof the normalized load ratio p = P/PC
o versus the normalized end-shortening E/L in the first column, thegeneralized coordinate of the sway mode qs in the second column, and the normalized maximum amplitudeof local deflection in the more compressed web wwc,max/tw in the third column; the fourth column showswwc,max/tw versus qs.
Figure 12 shows the evolution of the cross-section deformation at mid-span for thecases where cθ = 0 (pinned), cθ = 0.5, and cθ → ∞ (rigid). The excellent comparisonsthroughout validate the effectiveness of the assumed cross-sectional shape functions. Atcθ = 0, there is no out-of-plane displacement in the flanges. With the increase of the jointrigidity cθ, the flanges bulge increasingly and finish with the rotation at the joint being equalto the more compressed web, i.e. θf(x = b/2) = θwc(y = d/2), as shown in Figure 12(c). Asmall difference in the more compressed web deflection can be observed between the FE andthe analytical results in the advanced post-buckling range; the difference increases as cθ isincreased, as shown in the fourth column of Figure 12(b, c). Moreover, the discrepancy inthe less compressed web is caused by the large amount of bending in that web [30], whichis currently not included as an extra local displacement function in the analytical model.
22
-30 -10 10 30
y (
mm
)
0
20
40
60
p =0.950
-30 -10 10 300
20
40
60
p =0.900
-30 -10 10 300
20
40
60
p =0.850
-30 -10 10 300
20
40
60
p =0.790
-30 -10 10 30
y (
mm
)
0
20
40
60
-30 -10 10 300
20
40
60
-30 -10 10 300
20
40
60
-30 -10 10 300
20
40
60
x (mm)
-30 -10 10 30
y (
mm
)
0
20
40
60
x (mm)
-30 -10 10 300
20
40
60
x (mm)
-30 -10 10 300
20
40
60
x (mm)
-30 -10 10 300
20
40
60
cθ = 0
cθ = 0.5
cθ → ∞
(a)
(b)
(c)
Figure 12: Local deformation of the cross-section for the example struts at mid-span at different load levelsfor the cases where: (a) cθ = 0 (pinned), (b) cθ = 0.5, and (c) cθ → ∞ (rigid) from the analytical (solidline) and FE (dashed line) models. Note that the displacements shown have been amplified by a factor of20 to aid visualization.
23
Figure 13 shows the comparison for the normalized solutions of the out-of-plane dis-placement in the more compressed web wwc/tw for the cases where cθ = 0 (pinned), cθ = 0.5,and cθ → ∞ (rigid), at p = 0.950 and p = 0.790 respectively. An excellent comparison is
Figure 13: Comparisons of the numerical solutions for the normalized local out-of-plane displacementwwc/tw in the more compressed web from the analytical (solid line) and FE (dashed line) models forcθ = 0 (pinned), cθ = 0.5 and cθ → ∞ (rigid) cases respectively. Note that the longitudinal coordinate isnormalized with respect to half of the strut length z = 2z/L.
observed in all the cases, especially for the case where cθ = 0. However, the slight errorincreases with the commensurate increase of the joint rigidity and the progression of inter-active buckling, as described for Figure 12. In the analytical model, the profile of the localmode is assumed to be the same in the whole strut and the amplitude is allowed to vary,see Eqs. (3) and (4). This assumption holds true for the cθ = 0 case, where the profile isalways the sine function, since the flanges provide no rotational restraints to the buckledweb. For the cases where cθ > 0, the cross-section profile depends on the bending momentat the web–flange junctions, which varies along the length. Figure 12 demonstrates thatthe currently assumed shape function matches well with the cross-section deformation atmid-span. Therefore, the errors, although very small, become increasingly larger towardsthe ends, where the profile is slightly different from that at mid-span.
Since the FE package Abaqus can output the strain energy in individual plates andthe work done by load as standard, it provides an additional perspective for validatingthe analytical model. Figure 14 presents the comparisons between the components of thepotential energy during the loading for the cases where cθ = 0 (pinned), cθ = 0.5 andcθ → ∞ (rigid). As for the energy in the analytical model, the strain energy in the flangesUf comprises the local bending energy Ub,fl, given in Eq. (27), and the membrane strainenergy Um,f , given in Eq. (31). The strain energy in the more compressed web Uwc compriseshalf of the total global bending energy Ub,o/2, given in Eq. (26), the local bending energyUb,wcl, given in Eq. (28), and the membrane strain energy Um,wc, given in Eq. (37). Thestrain energy in the less compressed web Uwt comprises half of the total global bending
24
×10-30 5 10
Uf
0
5000
10000
×10-30 5 10
Uwc
×104
0
1
2
3
×10-30 5 10
Uwt
0
2000
4000
6000
×10-30 5 10
PE
×104
0
1
2
3
4
×10-30 5 10
Uf
0
5000
10000
×10-30 5 10
Uwc
×104
0
1
2
3
×10-30 5 10
Uwt
0
2000
4000
6000
×10-30 5 10
PE
×104
0
1
2
3
4
qs ×10-30 5 10
Uf
0
5000
10000
qs ×10-30 5 10
Uwc
×104
0
1
2
3
qs ×10-30 5 10
Uwt
0
2000
4000
6000
qs ×10-30 5 10
PE
×104
0
1
2
3
4
cθ = 0
cθ = 0.5
cθ → ∞
(a)
(b)
(c)
Figure 14: Comparisons of the potential energy components from the example struts with joint rigiditiescθ = 0 (pinned), cθ = 0.5 and cθ → ∞ (rigid) from the analytical (solid line) and FE (dashed line) models.Graphs of the total strain energy stored in the flanges Uf ; the total strain energy stored in the morecompressed web Uwc; the total strain energy stored in the less compressed web Uwt; the work done by theload PE , all versus the generalized coordinate of the sway mode qs.
energy Ub,o/2, given in Eq. (26), and the membrane strain energy Um,wt, given in Eq. (38).The work done by load term PE is given in Eq. (40).
There are three individual stages that may be observed in the energy relationshipsversus the generalized coordinate of the sway mode qs. The first stage corresponds tothe purely axial deformation of the struts under compression before the buckling load isreached. The second stage is where pure global buckling is triggered and the third stageis where interactive buckling progresses with the simultaneous increase of the global andlocal modes. Except for the strain energy in the more compressed web, an energy reductioncan be observed at the initial stage of interactive buckling. For the strain energy in theflanges and the work done by the load, the reduction corresponds to the ‘snap-back’ thatfeatures in the load–end-shortening relationship shown in Figure 7(a) for small values ofcθ. With the increase of the joint rigidity, the reduction diminishes; for the rigid joint case,
25
the energy reduction is essentially negligible, which perhaps explains why no snap-back isobserved for that case.
The comparisons of the strain energy terms and the work done by the load betweenthe FE and the analytical models are generally excellent for all values of cθ. The mainsource of discrepancy resides in the strain energy stored in the less compressed web Uwt,with the relative errors being 40%, 33% and 28% at qs = 0.01 for cθ = 0 (pinned), cθ = 0.5and cθ → ∞ (rigid) cases respectively. However, the proportion of the strain energy storedin the less compressed web compared to the total strain energy stored in the rectangularhollow section strut, Uwt/U , is relatively small; for the rigid joint case, Uwt/U ≈ 0.07 whenqs = 0.01. Therefore, the errors for the entire system due to errors from the less compressedweb are in fact below 3%. Referring to Figure 7(d), at the same value of qs, the local modeamplitude is higher for smaller values of cθ. Therefore, any neutral axis movement wouldbe larger for smaller values of cθ. This contributes to the reason why the error in thestrain energy stored in the less compressed web is the largest in the effective pinned jointcase at the same value of qs. The neutral axis movement due to the plate buckling and theassumed cross-section shape functions (see Figure 12) are the two principal factors that arepostulated to be responsible for the small overall discrepancy in the strain energy of themore compressed and less compressed webs. All of these factors taken together lead to avery marginally stiffer response in the analytical model, but it is not particularly large andis only really significant in the far-field post-buckling range. Hence, it may be concludedthat the developed analytical model has been validated and may now be exploited further.
6. Simplified approach to predicting the location of secondary bifurcation
From the numerical results, as presented in Figure 7, unstable post-buckling equilib-rium paths were observed after the secondary bifurcation point and the severely unstablebehaviour is somewhat mollified with the increase of cθ, which in turn shows an increase inthe generalized coordinate of the sway mode at the secondary bifurcation point qS
s . Earlierstudies [20, 13, 4, 6, 17] on thin-walled structures susceptible to mode interaction alsoshow that the imperfection sensitivity decreases with the relative increase of qS
s since theproximity of the critical and secondary bifurcation points is reduced. A simplified approachto predicting qS
s based on the validated analytical model is presented currently since thisquantity provides a valuable indication of the potential sensitivity to imperfections.
When the strut buckles in a purely global mode, the direct strain εz,wc can be writtenas Eq. (32) by assuming the local buckling components are zero:
εz,wc = εz,wco − ∆, (60)
where εz,wco is obtained from Eq. (34); ∆ is also obtained by assuming the local bucklingcomponents in Eq. (51) are zero. Since the transverse strain is neglected, the compressivestress in the more compressed web σz,wc can be written thus:
σz,wc = Eεz,wc = −π2Ebqt
2Lsin
πz
L− PC
o
2twd (1 + φt/φc). (61)
26
From the numerical results in Figure 8, the local mode is initially localized. Insteadof analysing the whole web with the entire strut length, a plate element at mid-spanwith length le is isolated to compute the approximate local buckling coefficient kp, asshown in Figure 15. It is assumed that within this plate element, the axial stress is
Figure 15: Deformed shape of the isolated plate element from the more compressed web under the criticalbuckling stress σC
wcl. The effective length of the element is le with depth d and thickness tw. The equivalentrotational stiffness provided by the flanges and the rotational springs at the web–flange joints is cθf .
constant along the length with the value of the direct stress at mid-span. Therefore,when the direct stress in the more compressed web σz,wc reaches the local buckling stress,σC
wcl = kpπ2E/[12(1− ν2)(d/tw)2], it may be assumed that interactive buckling will also be
triggered.Since the cross-section shape function for the more compressed web has already been
obtained, with reference to Eq. (5) and Figure 3(d), the local buckling coefficient kp may becalculated by applying minimum potential energy principles on the isolated plate elementof the more compressed web. The buckled displacement field is thus assumed to be:
wwc(y, z) = Qfwc(y) sinπz
le, (62)
where Q is a new generalized coordinate representing the amplitude of the local bucklingmode within the plate element shown in Figure 15.
The strain energy U in the plate element comprises two components: the strain energystored from local buckling Ub,wcl and the strain energy stored in the equivalent rotationalsprings Usp,θf :
Ub,wcl =Dw
2
∫ le
0
∫ d/2
−d/2
{
(
∂2wwc
∂z2+
∂2wwc
∂y2
)2
(63)
− 2(1 − ν)
[
∂2wwc
∂z2
∂2wwc
∂y2−(
∂2wwc
∂z∂y
)2 ]}
dy dz,
27
Usp,θf = 2
∫ le
0
1
2cθf
(
∂wwc
∂y
∣
∣
∣
∣
y=−d/2
)2
dz, (64)
where the expression for cθf was presented in Eq. (20). The work done by load term isgiven by the following standard expression:
P∆ =σC
wcltw2
∫ le
0
∫ d/2
−d/2
(
∂wwc
∂z
)2
dy dz. (65)
The total potential energy can thus be written as:
V = Ub,wcl + Usp,θf − P∆, (66)
and by setting ∂V/∂Q = 0 for equilibrium, the following expression for kp is obtained:
kp = a0 + a1φ2l +
1
φ2l
, (67)
where φl = le/d with a0 and a1 being constants that are functions of φc, φt and cθ, thus:
a0 =10{
4φcφ3t cθ [φcφ
3t cθ (5π2 − 48) + 3 (π2 − 8) (cθ + 1)] + 3π2 (cθ + 1)2
}
4φcφ3t cθ [φcφ
3t cθ (π4 + 15π2 − 240) + 15 (π2 − 8) (cθ + 1)] + 15π2 (cθ + 1)2
, (68)
a1 =15{
4φcφ3t cθ [φcφ
3t cθ (π2 − 8) + (π2 − 4) (cθ + 1)] + π2 (cθ + 1)2
}
4φcφ3t cθ [φcφ3
t cθ (π4 + 15π2 − 240) + 15 (π2 − 8) (cθ + 1)] + 15π2 (cθ + 1)2. (69)
Defining φl = (a1)−1/4, an expression for the minimum value of kp is found:
kp = a0 + 2√
a1. (70)
By referring to the relationship between qs and qt given in Eq. (52), an explicit expressionfor the secondary bifurcation point qS
s is duly obtained:
qSs =
2(
σCwcl − σC
o
)
(1 + s) L
π2Eb. (71)
7. Parametric studies
To validate the simplified approach, a couple of parametric studies are presented wherethe length and the cross-section aspect ratios are varied. The results from the simplifiedapproach are compared to the full analytical model solved using numerical continuation inAuto.
28
Length of strut L (mm)5000 5500 6000 6500 7000 7500
qS s
×10-3
0
0.5
1
1.5
2
2.5
3
3.5
(a) cθ = 0
Length of strut L (mm)4500 5000 5500 6000 6500 7000 7500
qS s
×10-3
0
1
2
3
4
(b) cθ = 0.5
Length of strut L (mm)4500 5000 5500 6000 6500 7000 7500
qS s
×10-3
0
1
2
3
4
5
(c) cθ = 1
Length of strut L (mm)4500 5000 5500 6000 6500 7000 7500
qS s
×10-3
0
1
2
3
4
5
6
(d) cθ → ∞
AUTOSimplified method
Figure 16: Comparison of the generalized coordinate of the sway mode at the secondary bifurcation pointqSs using numerical continuation with Auto for the full analytical model alongside the presented simplified
method.
7.1. Length variation
The strut geometries have the same cross-section properties as shown in Table 1 andthe joint rigidity values are cθ = {0, 0.5, 1, ∞}. The length of the struts is varied from thecase where global buckling is marginally critical to L = 7200 mm. The comparison of thegeneralized coordinate of the sway mode at the secondary bifurcation point qS
s between thefull analytical model using numerical continuation and the simplified approach using Eq.(71) is shown in Figure 16. With the increase of the length L, qS
s increases; the simplifiedapproach predicts qS
s with good accuracy and is always on the safe side. The source ofdifference in the prediction is derived from the fact that the simplified model assumes thatthe stress is constant along the length of the plate element. However, the stress distributionis effectively a combination of the uniform stress from the axial load and the superpositionof the sine function from global buckling, as given in Eq. (61).
29
7.2. Cross-section aspect ratio variation
For the cross-section aspect ratio parametric study, the geometric properties of thestruts are shown in Table 4. The cross-section aspect ratio φc ranges from 1 to 2.5; the
Table 4: Geometric properties of the rectangular hollow section struts in the parametric study, selectedto ensure global bucking is critical. The flange width b = 60 mm and the wall thickness tf = tw = 1 mmthroughout.
Cross-section aspect ratio Web depth d Length Lφc (mm) (mm)1 60 2430
1.25 75 31201.5 90 38301.75 105 45402 120 5250
2.5 150 6700
width of the flange b is fixed and the wall thickness is fixed and uniform throughoutthe cross-section. The length of each strut is selected to ensure that global buckling ismarginally critical (PC
o /PCl ≈ 0.995) for the pinned joint case. Since qS
s is related to thestrut length, the current focus is on the local buckling coefficient kp at the secondarybifurcation point.
Table 5 shows the comparisons between the evaluation of the local buckling coefficient
Table 5: Comparison of the local buckling coefficient kp for the more compressed web at the secondarybifurcation point from the full analytical model, kp,AUTO, solved using numerical continuation with Auto
and the simplified method, kp,EQ, using Eq. (70) from the pinned case (cθ = 0) to rigid case (cθ → ∞) fordifferent cross-section aspect ratios.
φcRanges kp,EQ/kp,AUTO
kp,AUTO kp,EQ/kp,AUTO Mean COV1 4.01 → 5.03 0.968 → 0.998 0.980 1.16%
1.25 4.01 → 5.32 0.941 → 0.998 0.959 2.22%1.5 4.01 → 5.47 0.936 → 0.997 0.956 2.38%1.75 4.01 → 5.56 0.938 → 0.998 0.957 2.31%2 4.01 → 5.63 0.941 → 1.000 0.960 2.28%
2.5 4.01 → 5.76 0.947 → 0.998 0.964 1.93%
kp at the secondary bifurcation point from the full analytical model solved by numericalcontinuation and the approximation presented in Eq. (70). In the same way, as shown inthe length parameter study results, the simplified method is demonstrated to predict kp
with very good accuracy yet being always on the safe side for the cases studied. Definingkp,EQ as the prediction of kp from the simplified method using Eq. (70) and kp,AUTO asthe value of kp from the full analytical model, for each cross-section case, the mean value
30
of kp,EQ/kp,AUTO ranges between 0.956 and 0.980 and the maximum COV (coefficient ofvariation) is 2.38%. With the increase of the aspect ratio φc, an increase in kp is observeddue to the rotational restraint provided by the relatively narrower flanges. Therefore, alarger cross-section aspect ratio would lead to a relatively higher post-buckling strength.
Since cases with rigid joints (cθ → ∞) and uniform thickness (φt = 1) are most commonin practice, a power series approximation for Eq. (70) can be derived to order φ2
c for suchcases:
kp = 4.33 + 0.76φc − 0.10φ2c. (72)
The comparison between this function and Eq. (70) is shown in Figure 17 and can be seento be practically perfect for the range shown.
φc
1 1.5 2 2.5
kp
4.9
5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
Simplified method
Power series approximation
Figure 17: The relationship between the local buckling coefficient kp and the cross-section aspect ratio φc
for the rigid joint case from the simplified method using Eq. (70) and the curve fit function given in Eq.(72).
8. Concluding remarks
A nonlinear analytical model describing the interactive buckling of a thin-walled rect-angular hollow section strut with varying rigidities of the web–flange joints under purecompression has been developed using variational principles. Numerical examples, focus-ing on cases where global buckling is critical, have been presented and validated usingthe FE package Abaqus. Unstable post-buckling behaviour due to mode interaction wasobserved. A progressive change in the local buckling mode is identified in terms of boththe wavelength and the amplitude. As far as the authors are aware, it is the first timethat this has been demonstrated in rectangular hollow section struts. With the increase ofthe cross-section joint rigidity, a transition from highly unstable to more mildly unstable
31
post-buckling behaviour is observed. The excellent comparisons between the analyticaland FE results validate the effectiveness of the presented methodology.
A simplified method to predict the local buckling coefficient in the more compressedweb and the global buckling amplitude at the secondary bifurcation point is proposed basedon the validated analytical model; it is demonstrated to be simple, yet safe and accurate forthe cases studied. Work is currently being conducted to extend the model to describe thescenarios where local buckling is critical and for struts with initial geometric imperfections.The ultimate aim of this work is to provide guidance to engineers for designing thin-walledrectangular hollow section struts with geometries that are susceptible to modal interactions.
Acknowledgement
Financial support for Jiajia Shen was provided by the Imperial PhD scholarship scheme.
References
[1] F. Bleich, Buckling strength of metal structures, McGraw-Hill, 1952.
[2] S. Timoshenko, J. M. Gere, Theory of elastic stability, McGraw-Hill, 1961.
[3] Z. P. Bazant, L. Cedolin, Stability of structures: Elastic, inelastic, fracture and damagetheories, World Scientific, 2010.
[4] J. M. T. Thompson, G. W. Hunt, A general theory of elastic stability, Wiley, London,1973.
[5] J. Loughlan, Mode interaction in lipped channel columns under concentric or eccentricloading, Ph.D. thesis, University of Strathclyde (1979).
[6] J. M. T. Thompson, G. W. Hunt, Elastic instability phenomena, Wiley, London, 1984.
[7] L. Gardner, D. A. Nethercot, Experiments on stainless steel hollow sections – Part 2:Member behaviour of columns and beams, Journal of Constructional Steel Research60 (9) (2004) 1319–1332.
[8] H. Degee, A. Detzel, U. Kuhlmann, Interaction of global and local buckling in weldedRHS compression members, Journal of Constructional Steel Research 64 (7) (2008)755–765.
[9] J. Becque, K. J. R. Rasmussen, Experimental investigation of the interaction of localand overall buckling of stainless steel I-columns, ASCE Journal of Structural Engi-neering 135 (11) (2009) 1340–1348.
[10] M. A. Wadee, L. Bai, Cellular buckling in I-section struts, Thin-Walled Structures 81(2014) 89–100.
32
[11] M. A. Wadee, M. Farsi, Cellular buckling in stiffened plates, Proceedings of the RoyalSociety A 470 (2168) (2014) 20140094.
[12] V. Tvergaard, Imperfection-sensitivity of a wide integrally stiffened panel under com-pression, International Journal of Solids and Structures 9 (1) (1973) 177–192.
[13] A. van der Neut, The sensitivity of thin-walled compression members to column axisimperfection, International Journal of Solids and Structures 9 (8) (1973) 999–1011.
[14] J. Loughlan, The ultimate load sensitivity of lipped channel columns to column axisimperfection, Thin-Walled Structures 1 (1) (1983) 75–96.
[15] L. Bai, M. A. Wadee, Imperfection sensitivity of thin-walled I-section struts susceptibleto cellular buckling, International Journal of Mechanical Sciences 104 (2015) 162–173.
[16] E. L. Liu, M. A. Wadee, Mode interaction in perfect and imperfect thin-walled I-section struts susceptible to global buckling about the strong axis, Thin-Walled Struc-tures 106 (2016) 228–243.
[17] M. A. Wadee, M. Farsi, Imperfection sensitivity and geometric effects in stiffenedplates susceptible to cellular buckling, Structures 3 (2015) 172–186.
[18] L. Bai, M. A. Wadee, Slenderness effects in thin-walled I-section struts susceptible tolocal–global mode interaction, Engineering Structures 124 (2016) 128–141.
[19] E. L. Liu, M. A. Wadee, Geometric factors affecting i-section struts experiencing localand strong-axis global buckling mode interaction, Thin-Walled Structures 109 (2016)319–331.
[20] A. van der Neut, The interaction of local buckling and column failure of thin-walledcompression members, in: Proceedings of the 12th International Congress on AppliedMechanics, Springer, 1969, pp. 389–399.
[21] W. T. Koiter, The stability of elastic equilibrium, Ph.D. thesis, Delft University ofTechnology (1945).
[22] E. L. Liu, M. A. Wadee, Interactively induced localization in thin-walled I-sectionstruts buckling about the strong axis, Structures 4 (2015) 13–26.
[23] G. W. Hunt, M. A. Peletier, A. R. Champneys, P. D. Woods, M. A. Wadee, C. J.Budd, G. J. Lord, Cellular buckling in long structures, Nonlinear Dynamics 21 (1)(2000) 3–29.
[24] M. A. Wadee, L. Gardner, Cellular buckling from mode interaction in I-beams underuniform bending, Proceedings of the Royal Society A 468 (2137) (2012) 245–268.
33
[25] H. X. Yuan, Y. Q. Wang, L. Gardner, Y. J. Shi, Local-overall interactive bucklingof welded stainless steel box section compression members, Engineering Structures 67(2014) 62–76.
[26] M. A. Wadee, M. Farsi, Local–global mode interaction in stringer-stiffened plates,Thin-Walled Structures 85 (2014) 419–430.
[27] L. Bai, M. A. Wadee, Mode interaction in thin-walled I-section struts with semi-rigidflange-web joints, International Journal of Non-Linear Mechanics 69 (2015) 71–83.
[28] E. J. Doedel, B. E. Oldeman, AUTO-07p: Continuation and bifurcation softwarefor ordinary differential equations, available from http://indy.cs.concordia.ca/auto/(2009).
[29] ABAQUS, Version 6.14, Dassault Systemes, Providence RI, USA, 2014.
[30] J. Shen, M. A. Wadee, A. J. Sadowski, Numerical study of interactive buckling inthin-walled section box columns under pure compression, in: D. Camotim, P. B. Dinis,S. L. Chan, C. M. Wang, R. Goncalves, N. Silvestre, C. Basaglia, A. Landesmann,R. Bebiano (Eds.), Proceedings of the 8th International Conference on Advances inSteel Structures (ICASS 2015), 2015, paper number: 44.
[31] G. W. Hunt, M. A. Wadee, Localization and mode interaction in sandwich structures,Proceedings of the Royal Society A 454 (1972) (1998) 1197–1216.
[32] M. A. Wadee, S. Yiatros, M. Theofanous, Comparative studies of localized bucklingin sandwich struts with different core bending models, International Journal of Non-Linear Mechanics 45 (2) (2010) 111–120.
[33] G. W. Hunt, L. S. Da Silva, G. M. E. Manzocchi, Interactive buckling in sandwichstructures, Proceedings of the Royal Society A 417 (1852) (1988) 155–177.
[34] W. T. Koiter, M. Pignataro, An alternative approach to the interaction betweenlocal and overall buckling in stiffened panels, in: B. Budiansky (Ed.), Buckling ofStructures, International Union of Theoretical and Applied Mechanics, Springer BerlinHeidelberg, 1976, pp. 133–148.
[35] P. S. Bulson, The stability of flat plates, Chatto and Windus, London, UK, 1970.
[36] C. J. Budd, G. W. Hunt, R. Kuske, Asymptotics of cellular buckling close to theMaxwell load, Proceedings of the Royal Society A 457 (2016) (2001) 2935–2964.
[37] MATLAB, version 7.10.0 (R2012a), The MathWorks Inc., Natick, Massachusetts,2010.
[38] E. Riks, An incremental approach to the solution of snapping and buckling problems,International Journal of Solids and Structures 15 (7) (1979) 529–551.
34
[39] J. Becque, K. J. R. Rasmussen, Numerical investigation of the interaction of local andoverall buckling of stainless steel I-columns, ASCE Journal of Structural Engineering135 (11) (2009) 1349–1356.
[40] A. J. Sadowski, J. M. Rotter, On the relationship between mesh and stress fieldorientations in linear stability analyses of thin plates and shells, Finite Elements inAnalysis and Design 73 (2013) 42–54.
[41] R. D. Cook, S. M. David, E. P. Michael, Concepts and Applications of Finite ElementAnalysis, Wiley, New York, 2007.
[42] T. Belytschko, W. K. Liu, B. Moran, K. Elkhodary, Nonlinear finite elements forcontinua and structures, Wiley, New York, 2000.
35
