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Thermoelastic buckling of steel columns with load-dependent supports
International Journal of Non-Linear Mechanics 47 (2012) 8–15
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Thermoelastic buckling of steel columns with load-dependent supports Jianguo Cai, Jian Feng n, Jin Zhang Key Lab. of C and PC Structures of Ministry of Education, Southeast University, Nanjing 210096, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 13 May 2010 Received in revised form 24 February 2012 Accepted 7 March 2012 Available online 16 March 2012
The in-plane elastic buckling of a steel column with load-dependent supports under thermal loading is investigated. Two elastic rotational springs at the column ends are used to model the restraints which are provided by adjacent structural members or elastic foundations. The temperature is assumed to be linearly distributed across the section. Based on a nonlinear strain–displacement relationship, both the equilibrium and buckling equations are obtained by using the energy method. Then the limits for different buckling modes and the critical temperature of columns with different cases are studied. The results show that the proposed analytical solution can be used to predict the critical temperature for elastic buckling. The effect of thermal loading on the buckling of steel columns is significant. Furthermore, the thermal gradient plays a positive role in improving the stability of columns, and the effect of thermal gradients decreases while decreasing the modified slenderness ratios of columns. It can also be found that rotational restraints can significantly affect the column elastic buckling loads. Increasing the initial stiffness coefficient a or the stiffening rate b of thermal restraints will increase the critical temperature. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Buckling Steel column Thermal restraint Analytical analysis Elasticity
1. Introduction As a structural form, columns are the main load-carrying members governing the collapse of structures. The stability of columns at ambient temperature is a classical problem, and the elastic buckling loads can be obtained by the classical Euler equation [1,2]. In some situations, a column may be subjected to thermal loading produced by high temperatures caused by fire. Since the behavior of columns in fire is vastly different from those under ambient temperature, the subject of steel columns subjected to fire conditions has received increasing attention in recent years. General background information about the behavior of steel structures at elevated temperatures can be referred to books by Buchanan [3] and Wang [4]. The fire resistance of steel columns has been investigated analytically, experimentally and theoretically by many researchers. The stability of wide-flanged steel columns subject to elevated temperature using a finite difference approach was studied by Culver [5]. He then extended this research to steel columns subjected to thermal gradients [6]. Also, the effects of axial and rotational restraints on the fire resistance of steel column have been investigated based on a finite element method [7,8]. Poh and Bennetts [9,10] developed a numerical model to calculate the critical temperature of steel columns based on the experimental
n
Corresponding author. Tel.: þ86 25 83793150; fax: þ86 25 83793150. E-mail address: [email protected] (J. Feng).
0020-7462/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijnonlinmec.2012.03.004
results for steel columns. Franssen et al. [11–13] predicted the critical temperature of axially loaded members using a nonlinear computer code. Yang et al. [14] loaded a series of steel H columns to their limit states at specified temperature levels to examine the structural behavior of steel columns in fire. Based on Rankine’s principle, a simple analytical formula was derived to determine the compressive resistance of steel columns subjected to fire conditions [15]. Then a revised approach was presented based on test results [16]. Furthermore, they extended Rankine’s formula to predict the critical temperature of a rotational–restrained steel column, and the results agreed well with experimental and finite element predictions [17]. Huang et al. [18] also presented a series of numerical studies on thermally restrained steel columns taking into account of the effect of creep. Ali and O’Connor [19] conducted a series of tests on rotationally restrained steel columns subjected to quasi-standard fire. Bradford [20] presented a generic elastic analysis of a straight member with translational and rotational elastic supports at its ends by using the principle of virtual work. Wang et al. [21] carried out numerical analyses and conducted a comprehensive parametric study to investigate the effect of end restraints on steel frame behavior in fire. Hozjan et al. [22] gave an analytical procedure for determining the buckling load and temperature of a translationally and rotationally restrained steel column exposed to fire. In some of the previous work, the thermal loading is assumed uniform across the depth of the cross section. However, in many practical cases, the temperature on both sides of the column is not the same, such as a column on the perimeter of a compartment or
J. Cai et al. / International Journal of Non-Linear Mechanics 47 (2012) 8–15
Nomenclature A1, B1, C1, A2, B2 coefficients. E(y) temperature-dependent elastic modulus of steel at the coordinate y. (EA)eq, (EIz)eq temperature-dependent cross-section properties. G1, G2, G3, G4 coefficients. iz the modified radius of gyration of the cross section about the z axis. N axial compression in the column. T0 the temperature increment relative to its ambient value. T0,cr critical temperature. T1,T2 temperature change respectively along the depth of steel columns. k the stiffness of rotational elastic springs at the end of the column. h the depth of steel columns. m ratio of critical temperature with thermal gradient to that without thermal gradient. n ratio of critical temperature for rotationally restrained columns to that for pin-ended columns. u axial displacement. v transverse displacement. e strain.
a building during a fire [23–27]. Then the non-uniform thermal loading across the cross-section is considered in this paper. It is very important to produce an accurate and reliable theoretical modeling for the behavior of steel members at elevated temperature because of the high cost of the full-scale experiments. The structural response under thermal loading is very complicated and often counter-intuitive, and the finite element method has been preferred to more generic or analytical solutions [20]. Because of this, the significance of some parameters that greatly influence the behavior may be lost. In all aforementioned works the supports with constant stiffness were considered. However, the stiffness of structural supports may change as they are subjected to increasing loads. When a steel column is welded to a base plate, and the base plate is connected to a concrete footing with anchor bolts, the rotational stiffness of the base connection varies with the compressive loading. The characteristic of this connection for steel columns has been identified by Picard et al. in a series of experiments [28,29]. The influence of such supports on the stability of compressible columns was carried by Plaut [30] and Guran [31]. Plaut [32] extended his study to the elastic instability of a shallow sinusoidal arch by the classic method. No study appears to have been reported which has considered the in-plane elastic buckling of a steel column under thermal loading with load-dependent supports. The objective of this paper is to formulate an analytical model for the elastically in-plane buckling of a steel column with loaddependent supports under thermal loading, and to investigate in particular the effect of rotational restraints on the buckling behavior of these columns. Based on the nonlinear strain– displacement relationship, the principle of virtual work is used to establish both the nonlinear equilibrium and buckling equations. The formulas for the critical temperature are also given. Then the limits for different buckling modes (antisymmetric and symmetric) and the effects of the rotational restraints on the critical temperature are discussed.
ee em eb et eT eTc rT m Z Zc
9
mechanical elastic strain. mechanical axial strain. mechanical bending strain. thermal strain. thermal axial strain. critical thermal axial strain. thermal bending curvature. q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi N=ðEIz Þeq . mL/2, axial force coefficient. axial force parameter for the antisymmetric buckling of the column. axial force parameter for the symmetric buckling of the column. the coefficient of thermal expansion. the initial stiffness coefficient. the stiffening rate. coefficients. rTL/20eT, thermal gradient parameter. the modified slenderness ratio of the column. virtual axial displacement. virtual transverse displacement. buckling axial displacement. buckling transverse displacement. mechanical axial strain during column buckling.
Zs c A
b
g, g0 z l du dv Du Dv Dem
2. Formulation Fig. 1 shows a rotational restrained column under thermal loading. Two rotational springs attached to the column ends provide flexural resistance of the elastic support. In this investigation, the following assumptions are used: 1. The columns are slender; i.e., the dimensions of the cross section are much smaller than the height of the column. Hence, deformations of the column are assumed to satisfy the Euler–Bernoulli hypothesis. 2. The cross section of the column is symmetric about the axis. 3. The derivative of the temperature with respect to time vanishes. The axis system is selected as being centroidal, with the origin at mid-span. When temperature changes, the strain at the axial line and bending curvature induced by an assumed linear temperature gradient are respectively
eT ¼ cT 0 ¼ c
T 1 þT 2 , 2
rT ¼ c
DT h
¼c
T 1 T 2 , h
ð1Þ
where T0 is the temperature increment relative to its ambient value, T1 and T2 the temperature changes along the depth h of steel columns, and c the coefficient of thermal expansion that is set as 1.4 10 5/1C in this study. The nonlinear strain–displacement relationship can be written as [26] 1 2
e ¼ u0 þ ðv0 Þ2 yv00 ,
ð2Þ
where u0 ¼du/dx, v0 ¼dv/dx, v00 ¼d2v/dx2, and u and v the axial and transverse displacements of the column. On the other hand, the total strain of the beam can be described by
e ¼ ee þ et ,
ð3Þ
where ee denotes the mechanical elastic strain, and et the thermally induced strain, which includes the axial thermal strain
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J. Cai et al. / International Journal of Non-Linear Mechanics 47 (2012) 8–15
For equilibrium in the transverse direction, integrating Eq. (5) by parts and combining Eq. (7), the differential equilibrium equation can be given as
x k
ðEIz Þeq viv þ Nv00 ¼ 0:
ð8Þ
For simplicity, the following new parameters are introduced:
m2 ¼ T2 o
L
h
T1
viv þ m2 v00 ¼ 0:
y
v00 þ rT þ
k v0 9 ¼ 0, ðEIz Þeq x ¼ L=2
ð11Þ
v00 þ rT
k v0 9 ¼ 0: ðEIz Þeq x ¼ L=2
ð12Þ
k
In addition, the kinematic boundary conditions given as u ¼ 0, and v ¼ 0 at x ¼ 7 L=2
Fig. 1. Geometry model of rotationally restrained columns.
and the bending thermal strain. All the strains are defined as positive in tension. The mechanical strains ee can be divided into two components, the axial strains em and bending strains eb. From Eqs. (1)–(3), we can obtain the mechanical axial strain and bending strain as 1 2
em ¼ u0 þ ðv0 Þ2 cT 0 , eb ¼ yðv00 þ rT Þ:
ð4Þ
The differential equations of equilibrium for a steel column with elastic supports under thermal actions can be derived using the principle of virtual work that requires Z L=2 X ½ðEAÞeq ðdu0 þv0 dv0 Þem þðEIz Þeq ðv00 þ rT Þdv00 dx þ kv0i dv0i ¼ 0 i ¼ 7 L=2
ð5Þ for all sets of kinematically admissible virtual displacements du and dv, k is the stiffness of rotational elastic springs at the end of the column, and (EA)eq and (EIz)eq are temperature-dependent cross-section properties defined by Z ðEAÞeq ¼ EðyÞdA, ð5aÞ
ðEIz Þeq ¼
EðyÞy2 dA,
ð5bÞ
A
in which y is referred to an axis that is defined by Z EðyÞydA ¼ 0,
v¼
ð5cÞ
where E(y) is the temperature-dependent elastic modulus at the coordinate y. The integration by parts of Eq. (5) leads to the differential equilibrium equation for the axial direction as
e ¼ 0:
ð6Þ
From Eq. (6), the axial strain em is constant and can be written as
em ¼
N , ðEAÞeq
where N is the axial compression in the column.
rT L2 cosðmxÞcos Z , 2Z g
ð14Þ
where Z ¼ mL=2 is the axial force coefficient, and a, b are the initial stiffness coefficient and the stiffening rate of the rotational stiffness of structural supports which changes with the variation of axial force as k¼
ða þ bZ2 ÞðEIz Þeq , L
ð14aÞ
and coefficient g can be written as g ¼ 2Z cos Z þ ða þ bZ2 Þsin Z: The nonlinear relationship between the thermal loading and the axial compression N can be established by considering the constant axial strain given by Eq. (7), which should be equal to the axial stain averaged mathematically over the column length L calculated form Eq. (4). Therefore, Z N 1 L=2 0 1 02 em ¼ ¼ ðu þ v cT 0 Þdx: ð15Þ ðEAÞeq L L=2 2 Since the axial displacement at both ends are fully prevented, 1 L
Z
L=2
u0 dx ¼ 0,
ð16Þ
L=2
and rewriting the left-hand side of Eq. (15) as
A
ðEAÞeq 0m
ð13Þ
need to be satisfied. The transverse displacement can be obtained by solving Eq. (10) with the boundary conditions Eqs. (11)–(13), which yields
A
Z
ð10Þ
The boundary equations for rotationally restrained columns can also be obtained by integrating Eq. (5) by parts
T1
L=2
ð9Þ
and then the differential equilibrium equation for the transverse direction can be written as
y T2
N , ðEIz Þeq
ð7Þ
em ¼
N N ðEIz Þeq ¼ ¼ m2 i2z , ðEAÞeq ðEIz Þeq ðEAÞeq
ð17Þ
where iz is the modified radius of gyration of the cross section qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðEIz Þeq =ðEAÞeq .
about the z axis given by iz ¼
Then substituting Eq. (14) into Eq. (15) leads to the nonlinear equilibrium condition for rotationally restrained columns as
r2T L2 4Z2 ðZsin Z cos ZÞeT þ 2 ¼ 0, 4Zg2 l
ð18Þ
where l ¼ L=iz is the modified slenderness ratio of the column. For the neutrally stable configuration, it is required that the second variation of the energy is equal to zero. The condition of this neutral equilibrium is equivalent to the variation of the
J. Cai et al. / International Journal of Non-Linear Mechanics 47 (2012) 8–15
virtual work Eq. (5) as Z L=2 ½ðEAÞeq ðDu0 þv0 Dv0 ÞDem þðEAÞeq Dv02 þ ðEIz Þeq Dv002 dx L=2 X þ kDv02 i ¼ 0,
ð19Þ
i ¼ 7 L=2
where Du and Dv are the buckling displacements that are taken from the prebuckling equilibrium position u and v to the adjacent buckling equilibrium position uþ Du and v þ Dv, and Dem the axial strain during buckling given by 0
0
0
Dem ¼ Du þ v Dv :
ð19aÞ
ð20Þ
so the axial strain during buckling Dem is constant, and the differential equation for the transverse direction is
Dviv þ m2 Dv00 ¼
Dem i2z
v00 :
A column may buckle in an antisymmetric mode, where the prebuckling displacement v is symmetric while the buckling displacement Dv is antisymmetric, so the prebuckling slope v0 is antisymmetric and Dv0 is symmetric. Therefore the term v0 Dv0 in Eq. (26) is antisymmetric and so is the odd function of the axial coordinate x. The axial strain during buckling is obtained as Z 1 L=2 Dem ¼ ðDu0 þ v0 Dv0 Þdx ¼ 0: ð29Þ L L=2 Substituting Eq. (29) into Eq. (21) gives the differential equation for antisymmetric buckling as
Dviv þ m2 Dv00 ¼ 0:
Integrating Eq. (19) by parts leads to EADe0m ¼ 0,
ð21Þ
The corresponding boundary conditions for the buckled configuration can be obtained by integrating Eq. (19) and combining Eqs. (11)-(13), which gives
Du9x ¼ 7 L=2 ¼ 0,
ð22Þ
Dv9x ¼ 7 L=2 ¼ 0,
ð23Þ
ððEIz Þeq Dv00 þ kDv0 Þx ¼ L=2 ¼ 0, and ððEIz Þeq Dv00 kDv0 Þx ¼ L=2 ¼ 0:
ð30Þ
The general solution of Eq. (30) can be written as
Dv ¼ G1 þG2 x þG3 sinðmxÞ þG4 cosðmxÞ:
When the first factor of Eq. (32) vanishes, the axial force parameter Z ¼ Zc. Then the coefficients of Eq. (31) yield G1 ¼ G4 ¼ 0, and G2 ¼ G3
The solution of Eq. (21) that satisfies the boundary conditions of Eqs. (23) and (24) is
Dem rT L2 8m2 i2z Zg2
A1 cosðmxÞ þ B1 sinðmxÞmx þ C 1 ,
ð25Þ
ð31Þ
Using the boundary conditions expressed as Eqs. (23) and (24) leads to four linear homogeneous algebraic equations with respect to G1–G4. The requirement for the existence of non-trivial solution for G1–G4 is the vanishing of determinant of the four linear algebraic equations’ coefficient matrix, which yields ða þ bZ2 Þsin Z ða þ bZ2 Þcos Z ða þ bZ2 Þsin Z þ Z2 ðsin Z Þ þ cos Z ¼ 0: 2 2Z 2Z ð32Þ
ð24Þ
Dv ¼
11
sin Z
Z
,
ð33Þ
and substituting Eq. (33) to Eq. (31) leads to the antisymmetric buckling shape given by mx cosðmxÞ Dv ¼ G3 sinðmxÞ : ð34Þ
Z
Substituting the parameter Zc into Eq. (18) gives the equation for the antisymmetric buckling load. For a fixed column, the parameter Zc ¼1.4303p, and for a pin-ended column, the parameter Zc ¼ p, so the parameter Zc is in the range [p, 1.4303p]. When the second factor of Eq. (32) is equal to zero, the axial force parameter Z ¼ Zs, and the coefficients of Eq. (31) yield
where the coefficients A1, B1, C1 are expressed as A1 ¼ 2Z cos Z½4 þða þ bZ2 Þþ 2sin Z½ða þ bZ2 Þ2Z2 , B1 ¼ 4Z cos Z þ 2ða þ bZ2 Þsin Z, C 1 ¼ 2Z½2 þ ða þ bZ2 Þ þ 2 cosð2ZÞða þ bZ2 Þsinð2ZÞ: The axial strain during buckling can also be calculated by Z 1 L=2 Dem ¼ ðDu0 þ v0 Dv0 Þdx: ð26Þ L L=2
G2 ¼ G3 ¼ 0, and G1 ¼ G4 cos Z,
Because the axial buckling displacement is Du¼ 0 at both ends of the column (x ¼ 7L/2), we have Z 1 L=2 0 u dx ¼ 0: ð27Þ L L=2
It can be seen form Eq. (36) that the buckling shape is symmetric, which will not induce antisymmetric buckling. For a fixed column, the parameter Zs ¼ p, and for a pin-ended column, the parameter Zs ¼ p/2, so the parameter Zc is in the range of [p/2, p].
Substituting Eqs. (14), (25), (27) into Eq. (26), Eqs. (23) and (24) produce the equation for buckling with the corresponding boundary conditions as
3. Discussion
A2 r2T L2 þ B2 ¼ 0,
ð28Þ
where the coefficient A2, B2 are given as 2
A2 ¼
2Zg sin ZðZsin Z cos ZÞðg þ2Zg Þ 8Z , B2 ¼ 2 , 4Z2 g3 l 0
and the coefficient g0 can be written as g0 ¼ ð2 þ a þ bZ2 Þcos Z 2Zð1bÞsin Z. For a given temperature loading, the critical modified slenderness ratio of the column and the corresponding axial force coefficient can be obtained by solving Eqs. (18) and (28) simultaneously.
ð35Þ
then substituting Eq. (35) to Eq. (31) leads to
Dv ¼ G4 ½cosðmxÞcos Z:
ð36Þ
3.1. Limits for different buckling modes The solution of Eq. (28) for axial force parameter Z as a function of the parameter rTLl is shown in Fig. 2. It should be noted that the axial force coefficient Z Z p/2 for in-plane buckling of pin-ended columns (a ¼ b ¼0). For the column with rotational restraints, it will buckle when the axial force coefficient is higher than Zs. The initial stiffness coefficient a denoted as 0.5, 1.0 and 5.0 shown in Fig. 2(a) corresponds to the parameter Zs of 1.716, 1.837 and 2.381 respectively. Also, the stiffening rate b denoted as 0.5, 1.0, 2.0 and 5.0 shown in Fig. 2(b) corresponds to the parameter Zs of 2.043, 2.459, 2.798 and 3.010 respectively.
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J. Cai et al. / International Journal of Non-Linear Mechanics 47 (2012) 8–15
Fig. 3. Comparisons with finite element results.
Fig. 2. Axial force coefficients at elastic buckling. (a) The stiffening rate b ¼0 and (b) the initial stiffness coefficient a ¼ 0. Fig. 4. Critical temperature against modified slenderness ratios of columns with different thermal gradients when a ¼ b ¼ 0.
The column with load-dependent supports will buckle in an antisymmetric mode when the axial force coefficient Z ¼ Zc. And the lower limit axial force coefficient for column buckling is Zs, so the column will have a bifurcation symmetric buckling when the axial force coefficient Zs r Z o Zc. Moreover, it can also be seen from Fig. 2 that the axial force coefficient for column buckling increases with the thermal gradient rT. It can be concluded that the thermal gradient plays an important role in improving the stability of columns.
It can be seen from Fig. 3 that the critical temperature for columns with different thermal gradients agrees well with the finite element predictions. Thus, comparisons with finite element predictions have shown that the analytical solutions given in this paper are accurate and they validate the assumptions made in the derivations. 3.3. Critical temperature for the pin-ended column
3.2. Comparison with finite element results The solution for the critical temperature T0,cr for pin-ended columns is compared with finite element predictions in Fig. 3. The critical temperature T0,cr is defined as the average temperature increment over the whole cross-section relative to its ambient value when the column buckles under thermal loading. Fig. 3 shows the variation of the critical temperature with different thermal gradient for a column. The finite element package ANSYS was used in the numerical analysis where an I-section was used. The dimensions of the I-section are: overall depth D ¼0.3 m, flange width B¼0.3 m, flange thickness tf ¼0.014 m, and web thickness tw ¼ 0.008 m.
Fig. 4 shows the relation between critical temperature T0,cr and modified slenderness ratios of columns l with different thermal gradients. The elasticity of rotational supports is not considered in this figure (a ¼ b ¼0). Assuming that the thermal gradient parameter z as
z¼
rT L DTL ¼ , 20eT 20hT 0
ð37Þ
the thermal regimes shown in Fig. 5 correspond to the values of z of 0.2, 1.0, and 5.0 respectively. The critical temperature of columns with no thermal gradient (z ¼0) is also given for comparison. It can be seen that within the range of parameters considered in Fig. 4, the critical temperature T0,cr decreases with the increase of
J. Cai et al. / International Journal of Non-Linear Mechanics 47 (2012) 8–15
13
Fig. 5. Critical temperature ratio m against thermal gradient parameter z. (a) The stiffening rate b ¼ 0 and (b) the initial stiffness coefficient a ¼ 0.
Fig. 7. Critical temperature ratio n against stiffness of elastic supports. (a) The stiffening rate b ¼ 0 and (b) the initial stiffness coefficient a ¼0.
Fig. 6. Critical temperature against slenderness ratios of columns with different rotational stiffness when g ¼1.0. (a) The stiffening rate b ¼ 0 and (b) the initial stiffness coefficient a ¼ 0.
as the thermal gradient parameter increases. This is because the axial force coefficient for column buckling increases with the thermal gradient as illustrated in Fig. 2. For a pin-ended column with no thermal gradient (z ¼0), the critical temperature is below 100 1C when the slenderness ratios are greater than 100. Normally, the limit for slenderness ratio of compression members is 150 or 200. The corresponding critical temperature is 31.33 1C and 17.62 1C respectively. It can be concluded that the effect of temperature loads on the buckling of steel columns is significant. For the columns with larger slenderness ratios, the column will buckle under smaller temperature loads. Fig. 5 shows the positive role of thermal gradient for the columns with different slenderness ratios. The ratio of critical temperature for columns with thermal gradient to that with no thermal gradient is denoted as m. It can be seen from Fig. 5 that m increases with the increasing of thermal gradient parameter. It is also noted that the effect of thermal gradient becomes more significant when the slenderness ratios of columns decreases. 3.4. Effects of elastic supports
modified slenderness ratios of columns. Furthermore, the critical temperature of columns with thermal gradient is higher than those with no thermal gradient. Also, the critical temperature increases
Fig. 6 shows the influence of the elasticity of rotational restraints on the critical temperature with the relation between
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J. Cai et al. / International Journal of Non-Linear Mechanics 47 (2012) 8–15
the critical temperature T0,cr and the modified slenderness ratios of columns l. The critical temperature of pin-ended columns is also shown in Fig. 6 for comparison. It can be seen that the effects of elastic supports on critical temperature with different rotational stiffness are similar to the pin-ended columns. However, the critical temperature of elastically restrained columns is higher than pin-ended columns. Moreover, the critical temperature increases with an increase of the initial stiffness coefficient a and the stiffening rate b. The effects of the elastic supports on critical loads when the thermal gradient parameter is equal to 1.0 are shown in Fig. 7, which plots the variations of the critical temperature ratio n with the stiffness of elastic supports. Ratio n is defined as the ratio of critical temperature for rotationally restrained columns to that for pin-ended columns. It can be found that the critical temperature ratio n increases with the increase of the initial stiffness coefficient a and the stiffening rate b of rotational restraints, but the trend becomes slow when the initial stiffness coefficient a or the stiffening rate b is larger. Furthermore, the influence of rotational stiffness is more notable when the modified slenderness ratio becomes higher.
4. Conclusions The in-plane stability of steel columns with load-dependent supports at its ends under thermal loading was studied in this paper. In the model, two elastic springs connected to the ends of the member were modeled to simulate the rotational restraints. The temperature distribution over the thickness of columns was assumed to be non-uniform. Based on a nonlinear strain– displacement relationship, the principal of virtual work was used to establish the differential equations of equilibrium of the column in the axial and transverse directions, as well as the static boundary conditions. It is very important to derive the static boundary conditions in this way for columns under thermal loading, because their mechanism is not always intuitive as that of external loading. The in-plane equilibrium of a column in closed form was obtained by solving the differential equations with the boundary conditions. Then the buckling equation for a column under thermal loading was produced by using the energy method. The limits for different buckling modes and the critical temperature for columns were studied. The column with loaddependent supports will buckle in an antisymmetric mode when the axial force coefficient Z ¼ Zc, and it will have a bifurcation symmetric buckling when the axial force coefficient Zs r Z o Zc. The results show that the effect of thermal loading on the buckling of steel columns is significant. For columns with larger modified slenderness ratios, the column will buckle at a lower temperature. It can also be found that the thermal gradient plays a positive role in improving the stability of columns. Comparing these predictions with uniform temperature distribution over cross-section, it can be shown that the buckling load is greatly underestimated. The degree of conservatism increases when decreasing of the modified slenderness ratios of columns. The results also show that thermal restraints can significantly affect the column elastic buckling loads. The critical temperature increases with the initial stiffness coefficient a and the stiffening rate b of thermal restraints. It can also be found that the trend becomes slow when the rotational stiffness is higher. Furthermore, the effect of rotational stiffness increases as the modified slenderness ratio of columns increases. For the future work, the effects of initial imperfection and plasticity will be investigated.
Acknowledgments This research work was supported in part by the National Natural Science Foundation of China (50478075), the Priority Academic Program Development of Jiangsu Higher Education Institutions and also by the Scientific Research Foundation of Graduate School of Southeast University (YBJJ0817). The authors gratefully acknowledge both their supports. The first author would also like to thank the China Scholarship Council to support his research at California Institute of Technology and the help of Prof. Pellegrino S., Mr. Deng X., and Mr. Kwok K. Fruitful interactions leading to the improvement of the manuscript, which were brought by the review process, are also gratefully acknowledged.
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[27] Y.L. Pi, M.A. Bradford, Nonlinear thermoelastic buckling of pin-ended shallow arches under temperature gradient, Journal of Engineering Mechanics 136 (8) (2010) 960–968. [28] A. Picard, D. Beaulieu, Behaviour of a simple column base connection, Canadian Journal of Civil Engineering 12 (1985) 126–136. [29] A. Picard, D. Beaulieu, B. Crusse, Rotational restraint of a simple column base connection, Canadian Journal of Civil Engineering 14 (1987) 49–57.
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[30] R.H. Plaut, Column buckling when support stiffens under compression, Journal of Applied Mechanics 56 (2) (1989) 484. [31] A. Guran, Influence of various types of load-dependent supports on the stability of a compressible column model, Acta Mechanica 97 (1993) 91–100. [32] R.H. Plaut, Buckling of shallow arches with supports that stiffen when compressed, Journal of Engineering Mechanics 116 (4) (1990) 973–976.
Contents lists available at SciVerse ScienceDirect
International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm
Thermoelastic buckling of steel columns with load-dependent supports Jianguo Cai, Jian Feng n, Jin Zhang Key Lab. of C and PC Structures of Ministry of Education, Southeast University, Nanjing 210096, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 13 May 2010 Received in revised form 24 February 2012 Accepted 7 March 2012 Available online 16 March 2012
The in-plane elastic buckling of a steel column with load-dependent supports under thermal loading is investigated. Two elastic rotational springs at the column ends are used to model the restraints which are provided by adjacent structural members or elastic foundations. The temperature is assumed to be linearly distributed across the section. Based on a nonlinear strain–displacement relationship, both the equilibrium and buckling equations are obtained by using the energy method. Then the limits for different buckling modes and the critical temperature of columns with different cases are studied. The results show that the proposed analytical solution can be used to predict the critical temperature for elastic buckling. The effect of thermal loading on the buckling of steel columns is significant. Furthermore, the thermal gradient plays a positive role in improving the stability of columns, and the effect of thermal gradients decreases while decreasing the modified slenderness ratios of columns. It can also be found that rotational restraints can significantly affect the column elastic buckling loads. Increasing the initial stiffness coefficient a or the stiffening rate b of thermal restraints will increase the critical temperature. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Buckling Steel column Thermal restraint Analytical analysis Elasticity
1. Introduction As a structural form, columns are the main load-carrying members governing the collapse of structures. The stability of columns at ambient temperature is a classical problem, and the elastic buckling loads can be obtained by the classical Euler equation [1,2]. In some situations, a column may be subjected to thermal loading produced by high temperatures caused by fire. Since the behavior of columns in fire is vastly different from those under ambient temperature, the subject of steel columns subjected to fire conditions has received increasing attention in recent years. General background information about the behavior of steel structures at elevated temperatures can be referred to books by Buchanan [3] and Wang [4]. The fire resistance of steel columns has been investigated analytically, experimentally and theoretically by many researchers. The stability of wide-flanged steel columns subject to elevated temperature using a finite difference approach was studied by Culver [5]. He then extended this research to steel columns subjected to thermal gradients [6]. Also, the effects of axial and rotational restraints on the fire resistance of steel column have been investigated based on a finite element method [7,8]. Poh and Bennetts [9,10] developed a numerical model to calculate the critical temperature of steel columns based on the experimental
n
Corresponding author. Tel.: þ86 25 83793150; fax: þ86 25 83793150. E-mail address: [email protected] (J. Feng).
0020-7462/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijnonlinmec.2012.03.004
results for steel columns. Franssen et al. [11–13] predicted the critical temperature of axially loaded members using a nonlinear computer code. Yang et al. [14] loaded a series of steel H columns to their limit states at specified temperature levels to examine the structural behavior of steel columns in fire. Based on Rankine’s principle, a simple analytical formula was derived to determine the compressive resistance of steel columns subjected to fire conditions [15]. Then a revised approach was presented based on test results [16]. Furthermore, they extended Rankine’s formula to predict the critical temperature of a rotational–restrained steel column, and the results agreed well with experimental and finite element predictions [17]. Huang et al. [18] also presented a series of numerical studies on thermally restrained steel columns taking into account of the effect of creep. Ali and O’Connor [19] conducted a series of tests on rotationally restrained steel columns subjected to quasi-standard fire. Bradford [20] presented a generic elastic analysis of a straight member with translational and rotational elastic supports at its ends by using the principle of virtual work. Wang et al. [21] carried out numerical analyses and conducted a comprehensive parametric study to investigate the effect of end restraints on steel frame behavior in fire. Hozjan et al. [22] gave an analytical procedure for determining the buckling load and temperature of a translationally and rotationally restrained steel column exposed to fire. In some of the previous work, the thermal loading is assumed uniform across the depth of the cross section. However, in many practical cases, the temperature on both sides of the column is not the same, such as a column on the perimeter of a compartment or
J. Cai et al. / International Journal of Non-Linear Mechanics 47 (2012) 8–15
Nomenclature A1, B1, C1, A2, B2 coefficients. E(y) temperature-dependent elastic modulus of steel at the coordinate y. (EA)eq, (EIz)eq temperature-dependent cross-section properties. G1, G2, G3, G4 coefficients. iz the modified radius of gyration of the cross section about the z axis. N axial compression in the column. T0 the temperature increment relative to its ambient value. T0,cr critical temperature. T1,T2 temperature change respectively along the depth of steel columns. k the stiffness of rotational elastic springs at the end of the column. h the depth of steel columns. m ratio of critical temperature with thermal gradient to that without thermal gradient. n ratio of critical temperature for rotationally restrained columns to that for pin-ended columns. u axial displacement. v transverse displacement. e strain.
a building during a fire [23–27]. Then the non-uniform thermal loading across the cross-section is considered in this paper. It is very important to produce an accurate and reliable theoretical modeling for the behavior of steel members at elevated temperature because of the high cost of the full-scale experiments. The structural response under thermal loading is very complicated and often counter-intuitive, and the finite element method has been preferred to more generic or analytical solutions [20]. Because of this, the significance of some parameters that greatly influence the behavior may be lost. In all aforementioned works the supports with constant stiffness were considered. However, the stiffness of structural supports may change as they are subjected to increasing loads. When a steel column is welded to a base plate, and the base plate is connected to a concrete footing with anchor bolts, the rotational stiffness of the base connection varies with the compressive loading. The characteristic of this connection for steel columns has been identified by Picard et al. in a series of experiments [28,29]. The influence of such supports on the stability of compressible columns was carried by Plaut [30] and Guran [31]. Plaut [32] extended his study to the elastic instability of a shallow sinusoidal arch by the classic method. No study appears to have been reported which has considered the in-plane elastic buckling of a steel column under thermal loading with load-dependent supports. The objective of this paper is to formulate an analytical model for the elastically in-plane buckling of a steel column with loaddependent supports under thermal loading, and to investigate in particular the effect of rotational restraints on the buckling behavior of these columns. Based on the nonlinear strain– displacement relationship, the principle of virtual work is used to establish both the nonlinear equilibrium and buckling equations. The formulas for the critical temperature are also given. Then the limits for different buckling modes (antisymmetric and symmetric) and the effects of the rotational restraints on the critical temperature are discussed.
ee em eb et eT eTc rT m Z Zc
9
mechanical elastic strain. mechanical axial strain. mechanical bending strain. thermal strain. thermal axial strain. critical thermal axial strain. thermal bending curvature. q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi N=ðEIz Þeq . mL/2, axial force coefficient. axial force parameter for the antisymmetric buckling of the column. axial force parameter for the symmetric buckling of the column. the coefficient of thermal expansion. the initial stiffness coefficient. the stiffening rate. coefficients. rTL/20eT, thermal gradient parameter. the modified slenderness ratio of the column. virtual axial displacement. virtual transverse displacement. buckling axial displacement. buckling transverse displacement. mechanical axial strain during column buckling.
Zs c A
b
g, g0 z l du dv Du Dv Dem
2. Formulation Fig. 1 shows a rotational restrained column under thermal loading. Two rotational springs attached to the column ends provide flexural resistance of the elastic support. In this investigation, the following assumptions are used: 1. The columns are slender; i.e., the dimensions of the cross section are much smaller than the height of the column. Hence, deformations of the column are assumed to satisfy the Euler–Bernoulli hypothesis. 2. The cross section of the column is symmetric about the axis. 3. The derivative of the temperature with respect to time vanishes. The axis system is selected as being centroidal, with the origin at mid-span. When temperature changes, the strain at the axial line and bending curvature induced by an assumed linear temperature gradient are respectively
eT ¼ cT 0 ¼ c
T 1 þT 2 , 2
rT ¼ c
DT h
¼c
T 1 T 2 , h
ð1Þ
where T0 is the temperature increment relative to its ambient value, T1 and T2 the temperature changes along the depth h of steel columns, and c the coefficient of thermal expansion that is set as 1.4 10 5/1C in this study. The nonlinear strain–displacement relationship can be written as [26] 1 2
e ¼ u0 þ ðv0 Þ2 yv00 ,
ð2Þ
where u0 ¼du/dx, v0 ¼dv/dx, v00 ¼d2v/dx2, and u and v the axial and transverse displacements of the column. On the other hand, the total strain of the beam can be described by
e ¼ ee þ et ,
ð3Þ
where ee denotes the mechanical elastic strain, and et the thermally induced strain, which includes the axial thermal strain
10
J. Cai et al. / International Journal of Non-Linear Mechanics 47 (2012) 8–15
For equilibrium in the transverse direction, integrating Eq. (5) by parts and combining Eq. (7), the differential equilibrium equation can be given as
x k
ðEIz Þeq viv þ Nv00 ¼ 0:
ð8Þ
For simplicity, the following new parameters are introduced:
m2 ¼ T2 o
L
h
T1
viv þ m2 v00 ¼ 0:
y
v00 þ rT þ
k v0 9 ¼ 0, ðEIz Þeq x ¼ L=2
ð11Þ
v00 þ rT
k v0 9 ¼ 0: ðEIz Þeq x ¼ L=2
ð12Þ
k
In addition, the kinematic boundary conditions given as u ¼ 0, and v ¼ 0 at x ¼ 7 L=2
Fig. 1. Geometry model of rotationally restrained columns.
and the bending thermal strain. All the strains are defined as positive in tension. The mechanical strains ee can be divided into two components, the axial strains em and bending strains eb. From Eqs. (1)–(3), we can obtain the mechanical axial strain and bending strain as 1 2
em ¼ u0 þ ðv0 Þ2 cT 0 , eb ¼ yðv00 þ rT Þ:
ð4Þ
The differential equations of equilibrium for a steel column with elastic supports under thermal actions can be derived using the principle of virtual work that requires Z L=2 X ½ðEAÞeq ðdu0 þv0 dv0 Þem þðEIz Þeq ðv00 þ rT Þdv00 dx þ kv0i dv0i ¼ 0 i ¼ 7 L=2
ð5Þ for all sets of kinematically admissible virtual displacements du and dv, k is the stiffness of rotational elastic springs at the end of the column, and (EA)eq and (EIz)eq are temperature-dependent cross-section properties defined by Z ðEAÞeq ¼ EðyÞdA, ð5aÞ
ðEIz Þeq ¼
EðyÞy2 dA,
ð5bÞ
A
in which y is referred to an axis that is defined by Z EðyÞydA ¼ 0,
v¼
ð5cÞ
where E(y) is the temperature-dependent elastic modulus at the coordinate y. The integration by parts of Eq. (5) leads to the differential equilibrium equation for the axial direction as
e ¼ 0:
ð6Þ
From Eq. (6), the axial strain em is constant and can be written as
em ¼
N , ðEAÞeq
where N is the axial compression in the column.
rT L2 cosðmxÞcos Z , 2Z g
ð14Þ
where Z ¼ mL=2 is the axial force coefficient, and a, b are the initial stiffness coefficient and the stiffening rate of the rotational stiffness of structural supports which changes with the variation of axial force as k¼
ða þ bZ2 ÞðEIz Þeq , L
ð14aÞ
and coefficient g can be written as g ¼ 2Z cos Z þ ða þ bZ2 Þsin Z: The nonlinear relationship between the thermal loading and the axial compression N can be established by considering the constant axial strain given by Eq. (7), which should be equal to the axial stain averaged mathematically over the column length L calculated form Eq. (4). Therefore, Z N 1 L=2 0 1 02 em ¼ ¼ ðu þ v cT 0 Þdx: ð15Þ ðEAÞeq L L=2 2 Since the axial displacement at both ends are fully prevented, 1 L
Z
L=2
u0 dx ¼ 0,
ð16Þ
L=2
and rewriting the left-hand side of Eq. (15) as
A
ðEAÞeq 0m
ð13Þ
need to be satisfied. The transverse displacement can be obtained by solving Eq. (10) with the boundary conditions Eqs. (11)–(13), which yields
A
Z
ð10Þ
The boundary equations for rotationally restrained columns can also be obtained by integrating Eq. (5) by parts
T1
L=2
ð9Þ
and then the differential equilibrium equation for the transverse direction can be written as
y T2
N , ðEIz Þeq
ð7Þ
em ¼
N N ðEIz Þeq ¼ ¼ m2 i2z , ðEAÞeq ðEIz Þeq ðEAÞeq
ð17Þ
where iz is the modified radius of gyration of the cross section qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðEIz Þeq =ðEAÞeq .
about the z axis given by iz ¼
Then substituting Eq. (14) into Eq. (15) leads to the nonlinear equilibrium condition for rotationally restrained columns as
r2T L2 4Z2 ðZsin Z cos ZÞeT þ 2 ¼ 0, 4Zg2 l
ð18Þ
where l ¼ L=iz is the modified slenderness ratio of the column. For the neutrally stable configuration, it is required that the second variation of the energy is equal to zero. The condition of this neutral equilibrium is equivalent to the variation of the
J. Cai et al. / International Journal of Non-Linear Mechanics 47 (2012) 8–15
virtual work Eq. (5) as Z L=2 ½ðEAÞeq ðDu0 þv0 Dv0 ÞDem þðEAÞeq Dv02 þ ðEIz Þeq Dv002 dx L=2 X þ kDv02 i ¼ 0,
ð19Þ
i ¼ 7 L=2
where Du and Dv are the buckling displacements that are taken from the prebuckling equilibrium position u and v to the adjacent buckling equilibrium position uþ Du and v þ Dv, and Dem the axial strain during buckling given by 0
0
0
Dem ¼ Du þ v Dv :
ð19aÞ
ð20Þ
so the axial strain during buckling Dem is constant, and the differential equation for the transverse direction is
Dviv þ m2 Dv00 ¼
Dem i2z
v00 :
A column may buckle in an antisymmetric mode, where the prebuckling displacement v is symmetric while the buckling displacement Dv is antisymmetric, so the prebuckling slope v0 is antisymmetric and Dv0 is symmetric. Therefore the term v0 Dv0 in Eq. (26) is antisymmetric and so is the odd function of the axial coordinate x. The axial strain during buckling is obtained as Z 1 L=2 Dem ¼ ðDu0 þ v0 Dv0 Þdx ¼ 0: ð29Þ L L=2 Substituting Eq. (29) into Eq. (21) gives the differential equation for antisymmetric buckling as
Dviv þ m2 Dv00 ¼ 0:
Integrating Eq. (19) by parts leads to EADe0m ¼ 0,
ð21Þ
The corresponding boundary conditions for the buckled configuration can be obtained by integrating Eq. (19) and combining Eqs. (11)-(13), which gives
Du9x ¼ 7 L=2 ¼ 0,
ð22Þ
Dv9x ¼ 7 L=2 ¼ 0,
ð23Þ
ððEIz Þeq Dv00 þ kDv0 Þx ¼ L=2 ¼ 0, and ððEIz Þeq Dv00 kDv0 Þx ¼ L=2 ¼ 0:
ð30Þ
The general solution of Eq. (30) can be written as
Dv ¼ G1 þG2 x þG3 sinðmxÞ þG4 cosðmxÞ:
When the first factor of Eq. (32) vanishes, the axial force parameter Z ¼ Zc. Then the coefficients of Eq. (31) yield G1 ¼ G4 ¼ 0, and G2 ¼ G3
The solution of Eq. (21) that satisfies the boundary conditions of Eqs. (23) and (24) is
Dem rT L2 8m2 i2z Zg2
A1 cosðmxÞ þ B1 sinðmxÞmx þ C 1 ,
ð25Þ
ð31Þ
Using the boundary conditions expressed as Eqs. (23) and (24) leads to four linear homogeneous algebraic equations with respect to G1–G4. The requirement for the existence of non-trivial solution for G1–G4 is the vanishing of determinant of the four linear algebraic equations’ coefficient matrix, which yields ða þ bZ2 Þsin Z ða þ bZ2 Þcos Z ða þ bZ2 Þsin Z þ Z2 ðsin Z Þ þ cos Z ¼ 0: 2 2Z 2Z ð32Þ
ð24Þ
Dv ¼
11
sin Z
Z
,
ð33Þ
and substituting Eq. (33) to Eq. (31) leads to the antisymmetric buckling shape given by mx cosðmxÞ Dv ¼ G3 sinðmxÞ : ð34Þ
Z
Substituting the parameter Zc into Eq. (18) gives the equation for the antisymmetric buckling load. For a fixed column, the parameter Zc ¼1.4303p, and for a pin-ended column, the parameter Zc ¼ p, so the parameter Zc is in the range [p, 1.4303p]. When the second factor of Eq. (32) is equal to zero, the axial force parameter Z ¼ Zs, and the coefficients of Eq. (31) yield
where the coefficients A1, B1, C1 are expressed as A1 ¼ 2Z cos Z½4 þða þ bZ2 Þþ 2sin Z½ða þ bZ2 Þ2Z2 , B1 ¼ 4Z cos Z þ 2ða þ bZ2 Þsin Z, C 1 ¼ 2Z½2 þ ða þ bZ2 Þ þ 2 cosð2ZÞða þ bZ2 Þsinð2ZÞ: The axial strain during buckling can also be calculated by Z 1 L=2 Dem ¼ ðDu0 þ v0 Dv0 Þdx: ð26Þ L L=2
G2 ¼ G3 ¼ 0, and G1 ¼ G4 cos Z,
Because the axial buckling displacement is Du¼ 0 at both ends of the column (x ¼ 7L/2), we have Z 1 L=2 0 u dx ¼ 0: ð27Þ L L=2
It can be seen form Eq. (36) that the buckling shape is symmetric, which will not induce antisymmetric buckling. For a fixed column, the parameter Zs ¼ p, and for a pin-ended column, the parameter Zs ¼ p/2, so the parameter Zc is in the range of [p/2, p].
Substituting Eqs. (14), (25), (27) into Eq. (26), Eqs. (23) and (24) produce the equation for buckling with the corresponding boundary conditions as
3. Discussion
A2 r2T L2 þ B2 ¼ 0,
ð28Þ
where the coefficient A2, B2 are given as 2
A2 ¼
2Zg sin ZðZsin Z cos ZÞðg þ2Zg Þ 8Z , B2 ¼ 2 , 4Z2 g3 l 0
and the coefficient g0 can be written as g0 ¼ ð2 þ a þ bZ2 Þcos Z 2Zð1bÞsin Z. For a given temperature loading, the critical modified slenderness ratio of the column and the corresponding axial force coefficient can be obtained by solving Eqs. (18) and (28) simultaneously.
ð35Þ
then substituting Eq. (35) to Eq. (31) leads to
Dv ¼ G4 ½cosðmxÞcos Z:
ð36Þ
3.1. Limits for different buckling modes The solution of Eq. (28) for axial force parameter Z as a function of the parameter rTLl is shown in Fig. 2. It should be noted that the axial force coefficient Z Z p/2 for in-plane buckling of pin-ended columns (a ¼ b ¼0). For the column with rotational restraints, it will buckle when the axial force coefficient is higher than Zs. The initial stiffness coefficient a denoted as 0.5, 1.0 and 5.0 shown in Fig. 2(a) corresponds to the parameter Zs of 1.716, 1.837 and 2.381 respectively. Also, the stiffening rate b denoted as 0.5, 1.0, 2.0 and 5.0 shown in Fig. 2(b) corresponds to the parameter Zs of 2.043, 2.459, 2.798 and 3.010 respectively.
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J. Cai et al. / International Journal of Non-Linear Mechanics 47 (2012) 8–15
Fig. 3. Comparisons with finite element results.
Fig. 2. Axial force coefficients at elastic buckling. (a) The stiffening rate b ¼0 and (b) the initial stiffness coefficient a ¼ 0. Fig. 4. Critical temperature against modified slenderness ratios of columns with different thermal gradients when a ¼ b ¼ 0.
The column with load-dependent supports will buckle in an antisymmetric mode when the axial force coefficient Z ¼ Zc. And the lower limit axial force coefficient for column buckling is Zs, so the column will have a bifurcation symmetric buckling when the axial force coefficient Zs r Z o Zc. Moreover, it can also be seen from Fig. 2 that the axial force coefficient for column buckling increases with the thermal gradient rT. It can be concluded that the thermal gradient plays an important role in improving the stability of columns.
It can be seen from Fig. 3 that the critical temperature for columns with different thermal gradients agrees well with the finite element predictions. Thus, comparisons with finite element predictions have shown that the analytical solutions given in this paper are accurate and they validate the assumptions made in the derivations. 3.3. Critical temperature for the pin-ended column
3.2. Comparison with finite element results The solution for the critical temperature T0,cr for pin-ended columns is compared with finite element predictions in Fig. 3. The critical temperature T0,cr is defined as the average temperature increment over the whole cross-section relative to its ambient value when the column buckles under thermal loading. Fig. 3 shows the variation of the critical temperature with different thermal gradient for a column. The finite element package ANSYS was used in the numerical analysis where an I-section was used. The dimensions of the I-section are: overall depth D ¼0.3 m, flange width B¼0.3 m, flange thickness tf ¼0.014 m, and web thickness tw ¼ 0.008 m.
Fig. 4 shows the relation between critical temperature T0,cr and modified slenderness ratios of columns l with different thermal gradients. The elasticity of rotational supports is not considered in this figure (a ¼ b ¼0). Assuming that the thermal gradient parameter z as
z¼
rT L DTL ¼ , 20eT 20hT 0
ð37Þ
the thermal regimes shown in Fig. 5 correspond to the values of z of 0.2, 1.0, and 5.0 respectively. The critical temperature of columns with no thermal gradient (z ¼0) is also given for comparison. It can be seen that within the range of parameters considered in Fig. 4, the critical temperature T0,cr decreases with the increase of
J. Cai et al. / International Journal of Non-Linear Mechanics 47 (2012) 8–15
13
Fig. 5. Critical temperature ratio m against thermal gradient parameter z. (a) The stiffening rate b ¼ 0 and (b) the initial stiffness coefficient a ¼ 0.
Fig. 7. Critical temperature ratio n against stiffness of elastic supports. (a) The stiffening rate b ¼ 0 and (b) the initial stiffness coefficient a ¼0.
Fig. 6. Critical temperature against slenderness ratios of columns with different rotational stiffness when g ¼1.0. (a) The stiffening rate b ¼ 0 and (b) the initial stiffness coefficient a ¼ 0.
as the thermal gradient parameter increases. This is because the axial force coefficient for column buckling increases with the thermal gradient as illustrated in Fig. 2. For a pin-ended column with no thermal gradient (z ¼0), the critical temperature is below 100 1C when the slenderness ratios are greater than 100. Normally, the limit for slenderness ratio of compression members is 150 or 200. The corresponding critical temperature is 31.33 1C and 17.62 1C respectively. It can be concluded that the effect of temperature loads on the buckling of steel columns is significant. For the columns with larger slenderness ratios, the column will buckle under smaller temperature loads. Fig. 5 shows the positive role of thermal gradient for the columns with different slenderness ratios. The ratio of critical temperature for columns with thermal gradient to that with no thermal gradient is denoted as m. It can be seen from Fig. 5 that m increases with the increasing of thermal gradient parameter. It is also noted that the effect of thermal gradient becomes more significant when the slenderness ratios of columns decreases. 3.4. Effects of elastic supports
modified slenderness ratios of columns. Furthermore, the critical temperature of columns with thermal gradient is higher than those with no thermal gradient. Also, the critical temperature increases
Fig. 6 shows the influence of the elasticity of rotational restraints on the critical temperature with the relation between
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the critical temperature T0,cr and the modified slenderness ratios of columns l. The critical temperature of pin-ended columns is also shown in Fig. 6 for comparison. It can be seen that the effects of elastic supports on critical temperature with different rotational stiffness are similar to the pin-ended columns. However, the critical temperature of elastically restrained columns is higher than pin-ended columns. Moreover, the critical temperature increases with an increase of the initial stiffness coefficient a and the stiffening rate b. The effects of the elastic supports on critical loads when the thermal gradient parameter is equal to 1.0 are shown in Fig. 7, which plots the variations of the critical temperature ratio n with the stiffness of elastic supports. Ratio n is defined as the ratio of critical temperature for rotationally restrained columns to that for pin-ended columns. It can be found that the critical temperature ratio n increases with the increase of the initial stiffness coefficient a and the stiffening rate b of rotational restraints, but the trend becomes slow when the initial stiffness coefficient a or the stiffening rate b is larger. Furthermore, the influence of rotational stiffness is more notable when the modified slenderness ratio becomes higher.
4. Conclusions The in-plane stability of steel columns with load-dependent supports at its ends under thermal loading was studied in this paper. In the model, two elastic springs connected to the ends of the member were modeled to simulate the rotational restraints. The temperature distribution over the thickness of columns was assumed to be non-uniform. Based on a nonlinear strain– displacement relationship, the principal of virtual work was used to establish the differential equations of equilibrium of the column in the axial and transverse directions, as well as the static boundary conditions. It is very important to derive the static boundary conditions in this way for columns under thermal loading, because their mechanism is not always intuitive as that of external loading. The in-plane equilibrium of a column in closed form was obtained by solving the differential equations with the boundary conditions. Then the buckling equation for a column under thermal loading was produced by using the energy method. The limits for different buckling modes and the critical temperature for columns were studied. The column with loaddependent supports will buckle in an antisymmetric mode when the axial force coefficient Z ¼ Zc, and it will have a bifurcation symmetric buckling when the axial force coefficient Zs r Z o Zc. The results show that the effect of thermal loading on the buckling of steel columns is significant. For columns with larger modified slenderness ratios, the column will buckle at a lower temperature. It can also be found that the thermal gradient plays a positive role in improving the stability of columns. Comparing these predictions with uniform temperature distribution over cross-section, it can be shown that the buckling load is greatly underestimated. The degree of conservatism increases when decreasing of the modified slenderness ratios of columns. The results also show that thermal restraints can significantly affect the column elastic buckling loads. The critical temperature increases with the initial stiffness coefficient a and the stiffening rate b of thermal restraints. It can also be found that the trend becomes slow when the rotational stiffness is higher. Furthermore, the effect of rotational stiffness increases as the modified slenderness ratio of columns increases. For the future work, the effects of initial imperfection and plasticity will be investigated.
Acknowledgments This research work was supported in part by the National Natural Science Foundation of China (50478075), the Priority Academic Program Development of Jiangsu Higher Education Institutions and also by the Scientific Research Foundation of Graduate School of Southeast University (YBJJ0817). The authors gratefully acknowledge both their supports. The first author would also like to thank the China Scholarship Council to support his research at California Institute of Technology and the help of Prof. Pellegrino S., Mr. Deng X., and Mr. Kwok K. Fruitful interactions leading to the improvement of the manuscript, which were brought by the review process, are also gratefully acknowledged.
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