research article lateral buckling analysis of the steel...
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Research ArticleLateral Buckling Analysis of the Steel-ConcreteComposite Beams in Negative Moment Region
Fengqi Guo Shun Zhou and Lizhong Jiang
School of Civil Engineering Central South University Changsha 410075 China
Correspondence should be addressed to Fengqi Guo fengqiguocsueducn
Received 30 April 2015 Revised 7 July 2015 Accepted 7 July 2015
Academic Editor Joao M P Q Delgado
Copyright copy 2015 Fengqi Guo et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Distortional buckling is one of the important buckling models of steel-concrete composite beam in negative moment regionRotation restraining rigidity and lateral restraining rigidity which steel beam web to bottom plate of steel-concrete compositeare the key factors to influence the distortional buckling behavior A comprehensive and intensive study on rotation restrainingrigidity and lateral restraining rigidity which steel beamweb to bottom plate of I-shaped steel-concrete composite beam in negativemoment region is conducted in this paper Energy variation principle is adopted to deduce the analytical expressions to calculate therotation restraining rigidity and lateral restraining rigidity Combined with the buckling theory of axial compression thin-walledbars in elastic medium the buckling moment is obtained Theoretical analysis shows that the rotation restraining rigidity andlateral restraining rigidity of steel beam web appear to have a linear relationship with the external loads and could also be negativeCompared with other methods the results calculated by the proposed expressions agree well with the numerical results by ANSYSThe proposed expressions are more concise and suitable than the existing formulas for the engineering application
1 Introduction
The steel-concrete composite beam is composed of profiledsteel or welded I-shaped beam and concrete slab throughshear connectors which can resist external loads togetherBecause this composite structure combines the tensile resis-tance of steel with the compressive resistance of concrete ithas the advantages of higher bearing capacity better plasticityand ductility constructing conveniently and lower costwhich makes it widely used in long-span bridges and high-rise buildings In practical engineering it is unnecessary toverify the lateral buckling of composite beams in positivebending moment region because of the enough bendingrigidity and torsional rigidity of concrete slab However withbigger variable loads andunfavorable loads the bottomflangeof steel beam in the negative moment region would yieldlateral buckling associated with web transverse deformationThe distortion buckling is then likely to occur
In recent years several scholars used the energy varia-tional methods to analyze global stability of composite beamSome authors computed the critical buckling load [1ndash3] and
others compiled the corresponding specifications [4] Thesespecifications only consider global bending instability of thesteel beam web but fail to take into account the distortionalbuckling Besides the critical load formulas by thesemethodsare a little tedious for engineering calculations Based onthe elastic foundation compressive bar method Williamsand Jemah [5] Svensson [6] Goltermann and Svensson [7]and Ronagh [8] successively study the stability of compositebeam under constant axial force increasing the contributionof torsional rigidity of the concrete slabs and participatingarea of the steel girder web plate In order to consider theeffect of bending moment gradient compressive bar variablethe axial force is introduced Diansheng and Xiaomin [9]presented a model for analyzing the local buckling propertyof cold-formed thin-wall steel-concrete composite beamTheelastic buckling stresses at steel beam web and flange wallare calculated by the energy method Jiang [10 11] researchedthe local stability in the negative moment region for steelbeam web of continuous composite beams in attempt toestablish the simplified calculation model of local stabilityunder various loads and propose the critical local buckling
Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2015 Article ID 763634 8 pageshttpdxdoiorg1011552015763634
2 Advances in Materials Science and Engineering
MxMx
xy y
z0
1 bcbf
hwyc
0 x
tf hctw
Figure 1 Cross section dimensions of steel-concrete composite beams and axes
stress formula under a variety of stress states Ye and Chen[12] improved Svensson compressive bar model appropri-ately Considering the steel beam web effective participationpart two variable axial forces distortional buckling stabilitycritical load calculation formulas were deduced based onthe improved model Making use of the finite element theaccuracy of the abovemethodwas analyzed by calculating thecomposite beam constraint distortional buckling load Zhouet al [13 14] used energy variation principle to deduce thecalculationmethod of rotation restraining rigidity and lateralrestraining rigidity
In this paper a comprehensive and intensive study onrotation restraining rigidity and lateral restraining rigiditywhich steel beam web to bottom plate of steel-concretecomposite beam in negative moment region is conductedThe energy method is used to deduce the theoretical formulafor rotation restraining rigidity and lateral restraining rigiditywhich the steel beam web provides for bottom plate Energyvariation principle is adopted to derive the steel beamcritical stress of positive symmetry bending buckling anddissymmetry bending-torsion buckling in order to calculatethe buckling moment In the end of the paper the proposedformulas are discussed and analyzed
2 Basic Assumptions
The schematic diagram of steel-concrete composite beamis shown in Figure 1 The lateral buckling model of steel I-shaped beam in composite beam is different from the freesimply supported steel beam (unconstrained steel beam)The top flange of steel beam constituted by concrete slabhas big stiffness so the lateral deformation and torsionaldeformation are restricted to a certain degree The bottomflange of steel beam is under compression Although it canyield lateral displacement and torsion angle the bottomflange constrained by the web is not perfectly free Thereforethe lateral buckling of composite beam can be regarded asthe distortional buckling in company with lateral bendingdeformation of steel beam web
The right handed coordinate system 119909119910119911 is fixed to thecentroid of steel beam bottom flange As shown in Figure 1the monosymmetrical composite beam bears a bendingmoment 119872
119909in the 119910119911 plane which shows big stiffness In
order to analyze the rotation restraining rigidity 119896120593of steel
y
bf
tf
x
k1205930kx
ky = infin
Figure 2 Simplified calculation model of steel-concrete compositebeams
beam bottom flange to web and the buckling moment someassumptions are made as follows
(1) The materials are isotropic and perfectly elastic body(2) The element is constant section beam and there were
no initial imperfections(3) The cross-sectional shape of steel beam bottom flange
does not change during distortional buckling yield-ing
(4) The lateral deformation and torsional deformation ofsteel beam top flange could not happen because ofenough stiffness of concrete slab
(5) Due to the negative moment most of concrete incomposite beam has been cracked when bucklingyields Therefore the bending capacity of concrete isignored which means that only the bending capacityof the steel reinforcements in concrete slab is consid-ered
(6) The vertical restraining rigidity whichwebs to bottomflange 119896
119910= infin
Based on above assumptions the problem to be analyzedcan be simplified as a thin-walled constraint distortionproblem which is restricted by spring restraint and verticalrigid constraint in horizontal and distortion direction Thesimplified model is plotted in Figure 2
3 Web Constraint Factor 119896120601
and 119896119909
31 Rotation Constraint Rigidity 119896120601 Figure 3 presents a half-
wave length of web section under consideration The width
Advances in Materials Science and Engineering 3
1205901 1205901
yc ycz
y
1205902 1205902
120582 tw
h
m(z)
The longitudinal edges of the web
The longitudinal edges of the web
Figure 3 Rectangular plate subjected to compression andmoments
and thickness of web section are ℎ119908and 119905119908 respectively 120582
refers to the half-wave length of web caused by distortionbuckling in longitudinal direction (called as the half-wavelength hereafter) Two transversal opposite sides are simplysupported The side connected to top flange is fixed and theother side connected to bottom flange is simply supportedThe two simply supported sides bear the longitudinal lineardistributed stress 120590 in 119885 direction (compressive stress ispositive and tension stress is negative)The side connected tobottom flange bears the equivalent spring constraint moment119898(119911) which bottom flange exerted on web The coordinate ofgravity centre of steel beam total cross section is representedby minus119910
119888 and the moment of inertia is 119868 According to the
assumptions mentioned above when the negative moment119872119909acts on the reinforcement in concrete slab the axial
compressive stress at bottom edge of web is 1205901 = 119872119909119910119888119868
and the axial compressive stress at top edge of web is 1205902 =
1205901(119910119888minusℎ119908)119910119888Therefore the axial compressive stress at otherpoints of web is 120590 = 1205901(119910119888 + 119910)119910119888
Assuming 119863 = 1198641199053
11990812(1 minus 120583
2) 120583 is Poissonrsquos ratio
of steel 119864 is the elastic modulus of steel 119906 denotes thedeformation function of web The boundary conditions of 119906can be expressed as
[119906]119911=0120582 = 0
[119906]119910=0minusℎ
119908
= 0
[
120597119906
120597119910
]
119910=minusℎ119908
= 0
[minus119863(
1205972119906
1205971199112 +120583
1205972119906
1205971199102)]119911=0120582
= 0
(1)
With (1) the displacement functions are written as
119906 = 119888 [
119910
ℎ119908
+ 2(119910
ℎ119908
)
2+(
119910
ℎ119908
)
3] sin120587119911
120582
(2)
The strain energy of half-wave length web in the case ofsmall deformations is [16ndash18]
1198801 =119863
2int
120582
0int
0
minusℎ119908
[(
1205972119906
1205971199102)
2
+(
1205972119906
1205971199112)
2
+ 21205831205972119906
12059711991021205972119906
1205971199112
+ 2 (1minus120583)( 1205972119906
120597119910120597119911
)
2
]119889119910119889119911
(3)
Substituting (2) into (3) leads to the fact that
1198801 =120582119863
2[
21198882
ℎ3
119908
+
21198882
15ℎ119908
(
120587
120582
)
2+
1198882ℎ119908
210(
120587
120582
)
4] (4)
The elastic potential energy caused by half-wave lengthlongitudinal side spring constraint is
1198802 =119896120593
2int
120582
0(
120597119906
120597119910
)
2
119910=0119889119911 (5)
Substituting (2) into (5)
1198802 =1205821198961205931198882
4ℎ119908
2 (6)
The external force work of half-wave length web can becomputed by [16ndash18]
119882 =
119905119908
2int
120582
0int
0
minusℎ119908
120590(
120597119906
120597119911
)
2119889119910119889119911
=
119905119908
2int
120582
0int
0
minusℎ119908
1205901 (119910119888 + 119910)
119910119888
(
120597119906
120597119911
)
2119889119910119889119911
(7)
Substituting (5) into (8)
119882 =
12058211988821205901119905119908ℎ119908420
(
120587
120582
)
2minus
12058211988821205901119905119908ℎ
2
119908
1120119910119888
(
120587
120582
)
2 (8)
The total potential energy of half-wave length web is
Π = 1198801 +1198802 minus119882 (9)
Substitution of (4) (6) and (8) into (9) results in
Π =
120582119863
2[
21198882
ℎ3
119908
+
21198882
15ℎ119908
(
120587
120582
)
2+
1198882ℎ119908
210(
120587
120582
)
4]
+
1205821198961205931198882
4ℎ2119908
minus
12058211988821205901119905119908ℎ119908420
(
120587
120582
)
2
+
12058211988821205901119905119908ℎ
2
119908
1120119910119888
(
120587
120582
)
2
(10)
Based on principle of resident potential energy weobtained the following
119863
2[
2ℎ3
119908
+
215ℎ119908
(
120587
120582
)
2+
ℎ119908
210(
120587
120582
)
4]+
119896120593
4ℎ2119908
minus
1205901119905119908ℎ119908420
(
120587
120582
)
2+
1205901119905119908ℎ2
119908
1120119910119888
(
120587
120582
)
2= 0
(11)
By solving (11) one can obtain
119896120593= (
119905119908ℎ3
119908
105minus
119905119908ℎ4
119908
280119910119888
)(
120587
120582
)
21205901
minus119863[
4ℎ119908
+
4ℎ119908
15(
120587
120582
)
2+
ℎ3
119908
105(
120587
120582
)
4]
(12)
4 Advances in Materials Science and Engineering
1205901 1205901
yc ycz
y
1205902 1205902
120582 tw
h
f(z)
The longitudinal edges of the web
The longitudinal edges of the web
Figure 4 Rectangular plate subjected to compression and lateralstress
32 Lateral Constraint Rigidity 119896119909 The half-wave length of
web section is shown in Figure 4 Two transversal oppositesides are simply supported The side connected to top flangeis fixed and the other side connected to bottom flangecan move laterally The two simply supported sides bearthe longitudinal linear distributed stress 120590 in 119885 direction(similarly compressive stress is taken as positive and tensionstress is negative) The side connected to bottom flange bearsthe equivalent spring constraint distributed force 119891(119911) whichbottom flange exerted on web
Based on above analysis the boundary conditions of119906 canbe expressed as
[119906]119911=0120582 = 0
[119906]119910=minusℎ
119908
= 0
[
120597119906
120597119910
]
119910=0minusℎ119908
= 0
[minus119863(
1205972119906
1205971199112 +120583
1205972119906
1205971199102)]119911=0120582
= 0
(13)
According to above boundary conditions the displace-ment functions can be written as
119906 = [119888 minus 3119888 (119910
ℎ119908
)
2minus 2119888 (
119910
ℎ119908
)
3] sin120587119911
120582
(14)
Substituting (14) into (3) the strain energy of half-wavelength web in the case of small deformations is then obtainedas follows
1198801 =120582119863
2[
61198882
ℎ3
119908
+
131198882ℎ119908
70(
120587
120582
)
4+
61198882
5ℎ119908
(
120587
120582
)
2] (15)
The elastic potential energy caused by half-wave lengthlongitudinal side spring constraint is
1198802 =119896119909
2int
120582
0[119906]
2119910=0 119889119911 (16)
Substituting (14) into (16) leads to the fact that
1198802 =1205821198961199091198882
4 (17)
Substituting (14) into (7) the external force work of half-wave length web can be obtained as follows
119882 =
1312058211988821205901119905119908ℎ119908140
(
120587
120582
)
2minus
312058211988821205901119905119908ℎ2
119908
140119910119888
(
120587
120582
)
2 (18)
Substituting (15) (17) and (18) into (9) the total potentialenergy of half-wave length web is
Π =
120582119863
2[
61198882
ℎ3
119908
+
131198882ℎ119908
70(
120587
120582
)
4+
61198882
5ℎ119908
(
120587
120582
)
2]
+
1205821198961199091198882
4minus
1312058211988821205901119905119908ℎ119908140
(
120587
120582
)
2
+
312058211988821205901119905119908ℎ2
119908
140119910119888
(
120587
120582
)
2
(19)
Based on principle of resident potential energy one canhave
119863
2[
6ℎ3
119908
+
65ℎ119908
(
120587
120582
)
2+
13ℎ119908
70(
120587
120582
)
4]+
119896119909
4
minus
131205901119905119908ℎ119908140
(
120587
120582
)
2+
31205901119905119908ℎ2
119908
140119910119888
(
120587
120582
)
2= 0
(20)
By solving (20) we obtained the following
119896119909= (
13119905119908ℎ119908
35minus
3119905119908ℎ2
119908
35119910119888
)(
120587
120582
)
21205901
minus119863[
12ℎ119908
3 +125ℎ119908
(
120587
120582
)
2+
13ℎ119908
35(
120587
120582
)
4]
(21)
33 Discussion about 119896119909and 119896120601
(1) Equations (12) and (21) indicated that both 119896120601and
119896119909show a linear relationship with the longitudinal
compressive stress 1205901 Generally ℎ119908119910119888 is less than2 so the coefficient before 1205901 is positive for mostsituations The bigger 1205901 is the higher 119896120601 and 119896119909 areAt the same time it is of interest to note that both119896120601and 119896119909which steel beam bottom flange to web are
determined by the compressive stress 1205901 but not bycomposite beam section properties
(2) Since the polynomials on right-hand side of (12) and(21) have negative terms 119896
120601and and 119896
119909could be
negative This is not consistent with regular positivedefinite rigidity and rigidity matrix If the rotationconstraint rigidity and lateral constraint rigidity arenegative the rotation and lateral displacement of steelbeam bottomflange will be restricted by web Namelythe steel beam web will restrict bottom flange tobuckle but the steel beam bottom flange will inducethe web to buckle According to [16] the lateralconstraint rigidity 119896 = 119864119905
3
119908(4ℎ3
119908) is obtained by
using strip method in the elastic constraint compres-sion member buckling model However the restraintaction of two adjacent strips is not considered inthis method Therefore the lateral constraint rigiditywhich has nothing to do with external forces isalways positive But this does not agree with theactual situation Furthermore the neglected rotationconstraint rigidity will lead to certain errors whencalculating buckling load of composite beam
Advances in Materials Science and Engineering 5
(3) The ratios of the first term second term and thirdterm on the right side of (21) and (12) are
(13120590119905119908ℎ11990835) (120587120582)2
(120590119905119908ℎ3
119908105) (120587120582)2
=
39ℎ2
119908
(31205901119905119908ℎ2
11990835119910119888) (120587120582)
2
(1205901119905119908ℎ4
119908280119910
119888) (120587120582)
2=
24
ℎ2
119908
119863 [12ℎ3
119908+ (125ℎ
119908) (120587120582)
2
+ (13ℎ11990835) (120587120582)
4
]
119863 [4ℎ119908+ (4ℎ11990815) (120587120582)
2
+ (ℎ3
119908105) (120587120582)
4
]
asymp
3
ℎ2
119908
(22)
From (22) ℎ2119908119896119909119896120593is not an infinitesimal value so the
lateral constraint rigidity of bottom flange to web cannot bedisregarded Namely in the calculation the equation 119896
119909= 0
is not available Therefore the lateral constraint rigidity ofbottom flange to web cannot be approximated by zero Thisis different from [16] in which the lateral constraint rigiditywhich the cold-formed thin-walled lipped channel steel websto the top and bottom flange is taken as zero
4 Theoretical Derivation of Critical Moment
41 Derivation of Critical Moment The lateral constraintrigidity and rotation constraint rigidity which steel beamwebto bottom flange can be simplified respectively as
119896119909= 1205721120573120590+119863120572
2 (23)
119896120593= 1205723120573120590+119863120572
4 (24)
120573 = (
120587
120582
)
2
(25)
1205721= 119905119908ℎ119908(
0086ℎ119908
119910119888
minus 037) (26a)
1205722=
12
ℎ3
119908
+
24120573
ℎ119908
+ 037ℎ1199081205732
(26b)
1205723= 119905119908ℎ3
119908(
00036ℎ119908
119910119888
minus 00095) (26c)
1205724=
4
ℎ119908
+ 027ℎ119908120573+ 00095ℎ
3
1199081205732
(26d)
120590 =
119872119909119910119888
119868
=
119875
119860119891
(27)
Therein119863 = 1198641199053
119908[12(1minus120583
2
)] 120583 is Poissonrsquos ratio of steel120590 represents the compressive stress of bottom flange 119875 refersto the compressive force of bottom flange minus119910
119888is the gravity
centre coordinate of steel beam total cross section 119868 is theinertia moment of steel beam 120582 = 119897119899 119899 is the buckling half-wave number 119860
119891is the area of bottom flange
As shown in Figure 2 the thin-walled member is doublysymmetric along 119909-axis and 119910-axis and the centre of origin119874 coincides with the flexural center The displacements oforigin 119874 in 119909 direction and 119910 direction are denoted as 119906 andV respectively Because the rigidity in 119910 direction is infinityV is equal to zero The equivalent distributed forces causedby elastic medium as a result of displacements of thin-walledmember can be written as
119903119909= 119896119909119906
119903119910= 119896119910V
(28)
where 119896119909and 119896119910are the lateral and vertical constraint rigidity
which web to bottom flange respectivelyThe torsional angle which the member rotates around the
bending center is assumed to be 120593The equivalent distributedmoment of torsional thin-walled member induced by equiv-alent spring is
119898 = 119896120593120593 (29)
Neutral balance differential equation of thin-walledmember can be expressed as [13 14]
119864119868119910119906119868119881
+119875 (11990610158401015840
+11991011988612059310158401015840
) + 119896119909[119906 minus (119910
119889minus119910119886) 120593] = 0
119864119868119909V119868119881 +119875 (V10158401015840 minus119909
11988612059310158401015840
) + 119896119910[V+ (119909
119889minus119909119886) 120593] = 0
119864119868119908120593119868119881
+ (11990320119875minus119866119869) 120593
10158401015840
minus119875 (119909119886V10158401015840 minus119910
11988611990610158401015840
)
minus 119896119909[119906 minus (119910
119889minus119910119886) 120593] (119910
119889minus119910119886)
+ 119896119910[V+ (119909
119889minus119909119886) 120593] (119909
119889minus119909119886) + 119896120593120593 = 0
(30)
Therein 119868119910= 119905119891119887311989112 119868
119909= 119887119891119905311989112 119869 = 119887
11989111990531198913 11990320 =
1199092119886+119910
2119886+(119868119909+119868119910)119860119904 119860119904being the area of steel beam 119909
119886is the
horizontal coordinate of the bottom flange section bendingcentre 119909
119886= 0 119910
119886is the vertical coordinate of the bottom
flange section bending centre 119910119886= 0 119909
119889is the horizontal
coordinate of the bottom flange section rotation axis 119909119889=
0 119910119889is the vertical coordinate of the bottom flange section
rotation axis 119910119889= 0 119868119908is the fan-shaped inertia moment of
bottom flange section 119868119908= 0 119864 is the tensile elastic modulus
of steel 119866 is the shear elastic modulus of steelWith substitution of 119910
119886= 0 119910
119889= 0 119909
119886= 0 V = 0 119868
119908= 0
119896119909= 0 119909
119886= 0 119910
119886= 0 and 119910
119889= 0 into (30) one can obtain
119864119868119910119868119906119868119881
+11991011988811986011989111987211990911990610158401015840
+ (1205721120573119910119888119872119909 + 1198681198631205722) 119906 = 0 (31)
119896119910[V+ (119909
119889minus119909119886) 120593] = 0 (32)
(11990320119910119888119860119891119872119909 minus119866119869119868) 120593
10158401015840
+ (1205723120573119910119888119872119909 + 1198681198631205724) 120593 = 0 (33)
When the steel-concrete composite beam in negativemoment region bears lateral bending buckling its neutralbalance equation is shown in (31) and the correspondingboundary conditions are
[119906]119911=0119897 = 0
[11990610158401015840
]119911=0119897 = 0
(34)
6 Advances in Materials Science and Engineering
Assuming 119906 = 119860 sin(radic120573119911) which satisfies the boundaryconditions according to Galerkinrsquos method
1205732minus
119872119909119910119888119860119891
119864119868119910119868
120573 +
(1205721120573119872119909119910119888119868 + 119863120572
2)
119864119868119910
= 0 (35)
By solving (35)
1198721198881199031 =
119864119868119910120573 + 1198631205722120573
(119860119891minus 1205721) 119910119888
119868 (36)
Due to 1198891198721198881199031119889120573 = 0 we obtained the following
1205731198881199031 =
346
radic119864119868119910ℎ3119908119863 + 037ℎ4
119908
1198991198881199031 =
119897radic1205731198881199031120587
(37)
If 119897radic1205731198881199031120587 is an integer substitution of 120573
1198881199031into (36) leads
to the lateral bending critical moment If 119897radic1205731198881199031120587 is not an
integer the two values of 1205731198881199031
which makes 119897radic1205731198881199031120587 be two
integers most near to 119897radic1205731198881199031120587 are then substituted into (36)
and the smaller value is chosen to be the lateral bendingcritical moment
When the steel beam bottom flange of steel-concretecomposite beam in negativemoment region suffers rotationalbuckling the steel-concrete composite beam will yield lateralbending and torsional bucklingTheneutral balance equationis seen in (33) and the boundary conditions are
[120593]119911=0119897 = 0
[12059310158401015840
]119911=0119897 = 0
(38)
Assume 1205852 = (1205723120573119872119909119910119888119868 + 119863120572
4)(1199032
0119872119909119910119888119860119891119868 minus 119866119869)
Solving (33) we obtained the following
120593 = 119860 sin 120585119911 +119861 cos 120585119911 (39)
Substituting (39) into (38) and with 120593 = 0 one can have
sin 120585119897 = 0 (40)
120585 =
119899120587
119897
= radic120573 (41)
(1205723120573119872119909119910119888119868 + 1198631205724)
(11990320119872119909119910119888119860119891119868 minus 119866119869)
= 120573 (42)
Solving (42)
1198721198881199032 =
1198631205724120573 + 119866119869
(11990320119860119891 minus 1205723) 119910119888
119868 (43)
Because of the fact that 1198891198721198881199032119889120573 = 0 then
1205731198881199032
=
205
ℎ2
119908
1198991198881199032
=
119897radic1205731198881199032
120587
(44)
Table 1 Geometric dimension of examples
Number of example ℎ119908mm 119905
119908mm 119905
119891mm 119887
119891mm
1 1886 7 114 1002 1886 7 104 1003 1886 7 94 1004 1886 8 94 1005 1886 9 94 1006 150 8 94 1007 200 8 94 1008 200 8 94 809 200 8 94 120
When 119897radic1205731198881199032120587 is an integer the substitution of 120573
1198881199032into
(43) gets the lateral bending critical momentWhen 119897radic1205731198881199032120587
is not an integer a similar calculation mentioned before istaken to obtain the lateral bending critical moment
After getting the lateral bending buckling criticalmoment1198721198881199031
and the bending and torsional buckling critical moment1198721198881199032
the smaller one of these two is taken as the buckling loadof composite beam in negativemoment regionThe analyticalresults indicate that the composite beam in negative momentregion yields as a result of lateral bending and torsionalbuckling acting together Therefore in [11 12] the methodin which only one case is considered is questionable Thepresented work is an improvement to them
To sum up the calculation formula of buckling momentcan be expressed as
119872119888119903
= min1198641198681199101205731198881199031 + 11986312057221205731198881199031
(119860119891minus 1205721) 119910119888
119868
11986312057241205731198881199032 + 119866119869
(11990320119860119891 minus 1205723) 119910119888
119868
1205731198881199031
=
346
radic119864119868119910ℎ3
119908119863 + 037ℎ
4
119908
1205731198881199032
=
205
ℎ2
119908
1205721= 119905119908ℎ119908(
0086ℎ119908
119910119888
minus 037)
1205722=
12
ℎ3
119908
+
241205731198881199031
ℎ119908
+ 037ℎ1199081205732
1198881199031
1205723= 119905119908ℎ3
119908(
00036ℎ119908
119910119888
minus 00095)
1205724=
4
ℎ119908
+ 027ℎ1199081205731198881199032+ 00095ℎ
3
1199081205732
1198881199032
(45)
When 119897radic1205731198881199031120587 is not an integer substituting 120573
1198881199031that
corresponds to two integers of the left and right side of119897radic1205731198881199031120587 into (36) the smaller resulting value of 120573
1198881199031is
the desired value And when 119897radic1205731198881199032120587 is not an integer
substituting 1205731198881199032
that corresponds to two integers of the left
Advances in Materials Science and Engineering 7
Table 2 Calculation results
Number of example ANSYSMPa Literature method [4]MPa Literature method [15]MPa The proposed methodMPa1 581 times 102 221 times 102 838 times 102 583 times 102
2 546 times 102 205 times 102 832 times 102 551 times 102
3 513 times 102 187 times 102 827 times 102 518 times 102
4 613 times 102 213 times 102 122 times 102 628 times 102
5 703 times 102 246 times 102 173 times 102 769 times 102
6 632 times 102 256 times 102 192 times 102 629 times 102
7 608 times 102 206 times 102 109 times 102 609 times 102
8 480 times 102 138 times 102 108 times 102 489 times 102
9 789 times 102 292 times 102 104 times 102 824 times 102
and right side of 119897radic1205731198881199032120587 into (43) the smaller resulting
value of 1205731198881199032
is the desired value
42 Practical Example Analysis Nine cases of I-shape steel-concrete composite beam are shown in Table 1 The length ofspecimen in negative moment region is equal to 4000mmFinite element method is adopted to calculate these nineexamples by using ANSYS in which the concrete part ofcomposite beam is substituted by lateral restraint and the steelI-beam is simulated by SHELL 43 The method proposed byBritish Steel Construction Institution [4] and the bucklingmodel method [15] are also used to calculate the bucklingloads The obtained results are shown in Table 2
As shown in Table 2 the results obtained by bucklingmodel method [15] tend to be excessively unsafe The resultsby using the method proposed by British Steel ConstructionInstitution [4] are too conservative However the results ofthe proposed method are in good agreement with thoseof ANSYS which means that the proposed method is areasonable and effective method
5 Conclusions
Based on energy method a comprehensive and intensivestudy on rotation restraining rigidity 119896
120601and lateral restrain-
ing rigidity 119896119909which steel beam web to bottom plate of
steel-concrete composite beam in negative moment region isperformed in this paper The expressions of lateral bendingbuckling stress lateral bending and torsional buckling stressand buckling moments of steel-concrete composite beam innegative moment region are deduced Some conclusions aredrawn as follows
(1) Both the rotation constraint rigidity 119896120601and the lateral
constraint rigidity 119896119909show a linear relationship with
longitudinal compressive stress 1205901 at bottom flange
(2) The rotation constraint rigidity 119896120601and the lateral
constraint rigidity 119896119909could be negative When the
rotation constraint rigidity and the lateral constraintrigidity are negative that means the bottom flange ofsteel beam can be restrained by rotation or lateralconstraint when steel beam was in negative momentregion buckling
(3) Since ℎ2119908119896119909119896120593is not infinitesimal the lateral con-
straint rigidity of bottom flange to web cannot beneglected In other words in the calculation theequation 119896
119909= 0 cannot be used Therefore it is
proved theoretically that the lateral constraint rigidityof bottom flange to web cannot be approximated to bezero This point is different with the steel structure
(4) The results obtained by literatures [4 15] showsome deviations with the ANSYS results But theresults by the presented expressions agree well withthe ANSYS results The reason is that the proposedexpressions consider the lateral bending buckling andlateral bending-torsion buckling simultaneously Theproposed method is more reasonable and clearer inphysical concepts in comparison with those methodswhich consider only one buckling mode Besides theproposed expressions are more concise and suitablefor the engineering application
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge financial supportfrom Projects (51078355) supported by National NaturalScience Foundation of China Projects (IRT1296) supportedby the Program for Changjiang Scholars and InnovativeResearch Team in University (PCSIRT) and Project (12K104)supported by Scientific Research Fund of Hunan ProvincialEducation Department China Dr Jing-Jing Qi and DrWang-Bao Zhou have tried their best to help the authorsTheauthors express their sincere gratitude to Dr Jing-Jing Qi andDr Wang-Bao Zhou
References
[1] M Ma and O Hughes ldquoLateral distortional buckling ofmonosymmetric I-beams under distributed vertical loadrdquoThin-Walled Structures vol 26 no 2 pp 123ndash143 1996
[2] M A Bradford ldquoLateral-distortional buckling of continuouslyrestrained columnsrdquo Journal of Constructional Steel Researchvol 42 no 2 pp 121ndash139 1997
8 Advances in Materials Science and Engineering
[3] M L-H Ng and H R Ronagh ldquoAn analytical solution forthe elastic lateral-distortional buckling of I-section beamsrdquoAdvances in Structural Engineering vol 7 no 2 pp 189ndash2002004
[4] R M Lawson and J W RaekhamDesign of Haunched Compos-ite Beams in Buildings Steel Construction Institute 1989
[5] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987
[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985
[7] P Goltermann and S E Svensson ldquoLateral distortional buck-ling predicting elastic critical stressrdquo Journal of StructuralEngineering vol 114 no 7 pp 1606ndash1625 1988
[8] H R Ronagh ldquoProgress in the methods of analysis of restricteddistortional buckling of composite bridge girdersrdquo Progress inStructural Engineering and Materials vol 3 no 2 pp 141ndash1482001
[9] Z Diansheng and Y Xiaomin ldquoStudy on beam local buckling ofcold-formed thin-wall steel and concrete compositerdquo IndustrialArchitecture vol 38 supplement pp 539ndash542 2008
[10] L-Z Jiang L-J Zeng and L-L Sun ldquoSteel-concrete compositecontinuous beam web local buckling analysisrdquo Journal of CivilEngineering and Architecture vol 26 no 3 pp 27ndash31 2009
[11] L Jiang and L Sun ldquoThe lateral buckling of steel-concrete com-posite box-beamsrdquo Journal of Huazhong University of Scienceand Technology (Urban Science Edition) vol 25 no 3 pp 5ndash92008
[12] J Ye and W Chen ldquoBeam with elastic constraints of I-shapedsteelmdashconcrete composite distortional bucklingrdquo Journal ofBuilding Structures vol 32 no 6 pp 82ndash91 2011
[13] W-B Zhou L-Z Jiang and Z-W Yu ldquoAnalysis of beamnegative moment zone of the elastic distortional buckling ofsteel-concrete compositerdquo Journal of Central South University(Natural Science Edition) vol 43 no 6 pp 2316ndash2323 2012
[14] W-B Zhou L-Z Jiang and Z-W Yu ldquoThe calculation formulaof steel concrete composite beams under negative bendingzonedistortional buckling momentrdquo Chinese Journal of Compu-tational Mechanics vol 29 no 3 pp 446ndash451 2012
[15] P Zhu Steel-Concrete Composite Beam Design Theory ChinaArchitecture amp Building Press Beijing China 1989
[16] J Yao and J-G Teng ldquoWeb rotational restraint in elasticdistortional buckling of cold-formed lipped channel sectionsrdquoEngineering Mechanics vol 25 no 4 pp 65ndash69 2008
[17] W-B Zhou L-Z Jiang and Z-W Yu ldquoClosed-form solutionfor shear lag effects of steel-concrete composite box beamsconsidering shear deformation and sliprdquo Journal of CentralSouth University vol 19 no 10 pp 2976ndash2982 2012
[18] W-B Zhou L-Z Jiang and Z-W Yu ldquoClosed-form solutionto thin-walled box girders considering the effects of sheardeformation and shear lagrdquo Journal of Central South Universityvol 19 no 9 pp 2650ndash2655 2012
Submit your manuscripts athttpwwwhindawicom
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BioMed Research International
MaterialsJournal of
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Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
2 Advances in Materials Science and Engineering
MxMx
xy y
z0
1 bcbf
hwyc
0 x
tf hctw
Figure 1 Cross section dimensions of steel-concrete composite beams and axes
stress formula under a variety of stress states Ye and Chen[12] improved Svensson compressive bar model appropri-ately Considering the steel beam web effective participationpart two variable axial forces distortional buckling stabilitycritical load calculation formulas were deduced based onthe improved model Making use of the finite element theaccuracy of the abovemethodwas analyzed by calculating thecomposite beam constraint distortional buckling load Zhouet al [13 14] used energy variation principle to deduce thecalculationmethod of rotation restraining rigidity and lateralrestraining rigidity
In this paper a comprehensive and intensive study onrotation restraining rigidity and lateral restraining rigiditywhich steel beam web to bottom plate of steel-concretecomposite beam in negative moment region is conductedThe energy method is used to deduce the theoretical formulafor rotation restraining rigidity and lateral restraining rigiditywhich the steel beam web provides for bottom plate Energyvariation principle is adopted to derive the steel beamcritical stress of positive symmetry bending buckling anddissymmetry bending-torsion buckling in order to calculatethe buckling moment In the end of the paper the proposedformulas are discussed and analyzed
2 Basic Assumptions
The schematic diagram of steel-concrete composite beamis shown in Figure 1 The lateral buckling model of steel I-shaped beam in composite beam is different from the freesimply supported steel beam (unconstrained steel beam)The top flange of steel beam constituted by concrete slabhas big stiffness so the lateral deformation and torsionaldeformation are restricted to a certain degree The bottomflange of steel beam is under compression Although it canyield lateral displacement and torsion angle the bottomflange constrained by the web is not perfectly free Thereforethe lateral buckling of composite beam can be regarded asthe distortional buckling in company with lateral bendingdeformation of steel beam web
The right handed coordinate system 119909119910119911 is fixed to thecentroid of steel beam bottom flange As shown in Figure 1the monosymmetrical composite beam bears a bendingmoment 119872
119909in the 119910119911 plane which shows big stiffness In
order to analyze the rotation restraining rigidity 119896120593of steel
y
bf
tf
x
k1205930kx
ky = infin
Figure 2 Simplified calculation model of steel-concrete compositebeams
beam bottom flange to web and the buckling moment someassumptions are made as follows
(1) The materials are isotropic and perfectly elastic body(2) The element is constant section beam and there were
no initial imperfections(3) The cross-sectional shape of steel beam bottom flange
does not change during distortional buckling yield-ing
(4) The lateral deformation and torsional deformation ofsteel beam top flange could not happen because ofenough stiffness of concrete slab
(5) Due to the negative moment most of concrete incomposite beam has been cracked when bucklingyields Therefore the bending capacity of concrete isignored which means that only the bending capacityof the steel reinforcements in concrete slab is consid-ered
(6) The vertical restraining rigidity whichwebs to bottomflange 119896
119910= infin
Based on above assumptions the problem to be analyzedcan be simplified as a thin-walled constraint distortionproblem which is restricted by spring restraint and verticalrigid constraint in horizontal and distortion direction Thesimplified model is plotted in Figure 2
3 Web Constraint Factor 119896120601
and 119896119909
31 Rotation Constraint Rigidity 119896120601 Figure 3 presents a half-
wave length of web section under consideration The width
Advances in Materials Science and Engineering 3
1205901 1205901
yc ycz
y
1205902 1205902
120582 tw
h
m(z)
The longitudinal edges of the web
The longitudinal edges of the web
Figure 3 Rectangular plate subjected to compression andmoments
and thickness of web section are ℎ119908and 119905119908 respectively 120582
refers to the half-wave length of web caused by distortionbuckling in longitudinal direction (called as the half-wavelength hereafter) Two transversal opposite sides are simplysupported The side connected to top flange is fixed and theother side connected to bottom flange is simply supportedThe two simply supported sides bear the longitudinal lineardistributed stress 120590 in 119885 direction (compressive stress ispositive and tension stress is negative)The side connected tobottom flange bears the equivalent spring constraint moment119898(119911) which bottom flange exerted on web The coordinate ofgravity centre of steel beam total cross section is representedby minus119910
119888 and the moment of inertia is 119868 According to the
assumptions mentioned above when the negative moment119872119909acts on the reinforcement in concrete slab the axial
compressive stress at bottom edge of web is 1205901 = 119872119909119910119888119868
and the axial compressive stress at top edge of web is 1205902 =
1205901(119910119888minusℎ119908)119910119888Therefore the axial compressive stress at otherpoints of web is 120590 = 1205901(119910119888 + 119910)119910119888
Assuming 119863 = 1198641199053
11990812(1 minus 120583
2) 120583 is Poissonrsquos ratio
of steel 119864 is the elastic modulus of steel 119906 denotes thedeformation function of web The boundary conditions of 119906can be expressed as
[119906]119911=0120582 = 0
[119906]119910=0minusℎ
119908
= 0
[
120597119906
120597119910
]
119910=minusℎ119908
= 0
[minus119863(
1205972119906
1205971199112 +120583
1205972119906
1205971199102)]119911=0120582
= 0
(1)
With (1) the displacement functions are written as
119906 = 119888 [
119910
ℎ119908
+ 2(119910
ℎ119908
)
2+(
119910
ℎ119908
)
3] sin120587119911
120582
(2)
The strain energy of half-wave length web in the case ofsmall deformations is [16ndash18]
1198801 =119863
2int
120582
0int
0
minusℎ119908
[(
1205972119906
1205971199102)
2
+(
1205972119906
1205971199112)
2
+ 21205831205972119906
12059711991021205972119906
1205971199112
+ 2 (1minus120583)( 1205972119906
120597119910120597119911
)
2
]119889119910119889119911
(3)
Substituting (2) into (3) leads to the fact that
1198801 =120582119863
2[
21198882
ℎ3
119908
+
21198882
15ℎ119908
(
120587
120582
)
2+
1198882ℎ119908
210(
120587
120582
)
4] (4)
The elastic potential energy caused by half-wave lengthlongitudinal side spring constraint is
1198802 =119896120593
2int
120582
0(
120597119906
120597119910
)
2
119910=0119889119911 (5)
Substituting (2) into (5)
1198802 =1205821198961205931198882
4ℎ119908
2 (6)
The external force work of half-wave length web can becomputed by [16ndash18]
119882 =
119905119908
2int
120582
0int
0
minusℎ119908
120590(
120597119906
120597119911
)
2119889119910119889119911
=
119905119908
2int
120582
0int
0
minusℎ119908
1205901 (119910119888 + 119910)
119910119888
(
120597119906
120597119911
)
2119889119910119889119911
(7)
Substituting (5) into (8)
119882 =
12058211988821205901119905119908ℎ119908420
(
120587
120582
)
2minus
12058211988821205901119905119908ℎ
2
119908
1120119910119888
(
120587
120582
)
2 (8)
The total potential energy of half-wave length web is
Π = 1198801 +1198802 minus119882 (9)
Substitution of (4) (6) and (8) into (9) results in
Π =
120582119863
2[
21198882
ℎ3
119908
+
21198882
15ℎ119908
(
120587
120582
)
2+
1198882ℎ119908
210(
120587
120582
)
4]
+
1205821198961205931198882
4ℎ2119908
minus
12058211988821205901119905119908ℎ119908420
(
120587
120582
)
2
+
12058211988821205901119905119908ℎ
2
119908
1120119910119888
(
120587
120582
)
2
(10)
Based on principle of resident potential energy weobtained the following
119863
2[
2ℎ3
119908
+
215ℎ119908
(
120587
120582
)
2+
ℎ119908
210(
120587
120582
)
4]+
119896120593
4ℎ2119908
minus
1205901119905119908ℎ119908420
(
120587
120582
)
2+
1205901119905119908ℎ2
119908
1120119910119888
(
120587
120582
)
2= 0
(11)
By solving (11) one can obtain
119896120593= (
119905119908ℎ3
119908
105minus
119905119908ℎ4
119908
280119910119888
)(
120587
120582
)
21205901
minus119863[
4ℎ119908
+
4ℎ119908
15(
120587
120582
)
2+
ℎ3
119908
105(
120587
120582
)
4]
(12)
4 Advances in Materials Science and Engineering
1205901 1205901
yc ycz
y
1205902 1205902
120582 tw
h
f(z)
The longitudinal edges of the web
The longitudinal edges of the web
Figure 4 Rectangular plate subjected to compression and lateralstress
32 Lateral Constraint Rigidity 119896119909 The half-wave length of
web section is shown in Figure 4 Two transversal oppositesides are simply supported The side connected to top flangeis fixed and the other side connected to bottom flangecan move laterally The two simply supported sides bearthe longitudinal linear distributed stress 120590 in 119885 direction(similarly compressive stress is taken as positive and tensionstress is negative) The side connected to bottom flange bearsthe equivalent spring constraint distributed force 119891(119911) whichbottom flange exerted on web
Based on above analysis the boundary conditions of119906 canbe expressed as
[119906]119911=0120582 = 0
[119906]119910=minusℎ
119908
= 0
[
120597119906
120597119910
]
119910=0minusℎ119908
= 0
[minus119863(
1205972119906
1205971199112 +120583
1205972119906
1205971199102)]119911=0120582
= 0
(13)
According to above boundary conditions the displace-ment functions can be written as
119906 = [119888 minus 3119888 (119910
ℎ119908
)
2minus 2119888 (
119910
ℎ119908
)
3] sin120587119911
120582
(14)
Substituting (14) into (3) the strain energy of half-wavelength web in the case of small deformations is then obtainedas follows
1198801 =120582119863
2[
61198882
ℎ3
119908
+
131198882ℎ119908
70(
120587
120582
)
4+
61198882
5ℎ119908
(
120587
120582
)
2] (15)
The elastic potential energy caused by half-wave lengthlongitudinal side spring constraint is
1198802 =119896119909
2int
120582
0[119906]
2119910=0 119889119911 (16)
Substituting (14) into (16) leads to the fact that
1198802 =1205821198961199091198882
4 (17)
Substituting (14) into (7) the external force work of half-wave length web can be obtained as follows
119882 =
1312058211988821205901119905119908ℎ119908140
(
120587
120582
)
2minus
312058211988821205901119905119908ℎ2
119908
140119910119888
(
120587
120582
)
2 (18)
Substituting (15) (17) and (18) into (9) the total potentialenergy of half-wave length web is
Π =
120582119863
2[
61198882
ℎ3
119908
+
131198882ℎ119908
70(
120587
120582
)
4+
61198882
5ℎ119908
(
120587
120582
)
2]
+
1205821198961199091198882
4minus
1312058211988821205901119905119908ℎ119908140
(
120587
120582
)
2
+
312058211988821205901119905119908ℎ2
119908
140119910119888
(
120587
120582
)
2
(19)
Based on principle of resident potential energy one canhave
119863
2[
6ℎ3
119908
+
65ℎ119908
(
120587
120582
)
2+
13ℎ119908
70(
120587
120582
)
4]+
119896119909
4
minus
131205901119905119908ℎ119908140
(
120587
120582
)
2+
31205901119905119908ℎ2
119908
140119910119888
(
120587
120582
)
2= 0
(20)
By solving (20) we obtained the following
119896119909= (
13119905119908ℎ119908
35minus
3119905119908ℎ2
119908
35119910119888
)(
120587
120582
)
21205901
minus119863[
12ℎ119908
3 +125ℎ119908
(
120587
120582
)
2+
13ℎ119908
35(
120587
120582
)
4]
(21)
33 Discussion about 119896119909and 119896120601
(1) Equations (12) and (21) indicated that both 119896120601and
119896119909show a linear relationship with the longitudinal
compressive stress 1205901 Generally ℎ119908119910119888 is less than2 so the coefficient before 1205901 is positive for mostsituations The bigger 1205901 is the higher 119896120601 and 119896119909 areAt the same time it is of interest to note that both119896120601and 119896119909which steel beam bottom flange to web are
determined by the compressive stress 1205901 but not bycomposite beam section properties
(2) Since the polynomials on right-hand side of (12) and(21) have negative terms 119896
120601and and 119896
119909could be
negative This is not consistent with regular positivedefinite rigidity and rigidity matrix If the rotationconstraint rigidity and lateral constraint rigidity arenegative the rotation and lateral displacement of steelbeam bottomflange will be restricted by web Namelythe steel beam web will restrict bottom flange tobuckle but the steel beam bottom flange will inducethe web to buckle According to [16] the lateralconstraint rigidity 119896 = 119864119905
3
119908(4ℎ3
119908) is obtained by
using strip method in the elastic constraint compres-sion member buckling model However the restraintaction of two adjacent strips is not considered inthis method Therefore the lateral constraint rigiditywhich has nothing to do with external forces isalways positive But this does not agree with theactual situation Furthermore the neglected rotationconstraint rigidity will lead to certain errors whencalculating buckling load of composite beam
Advances in Materials Science and Engineering 5
(3) The ratios of the first term second term and thirdterm on the right side of (21) and (12) are
(13120590119905119908ℎ11990835) (120587120582)2
(120590119905119908ℎ3
119908105) (120587120582)2
=
39ℎ2
119908
(31205901119905119908ℎ2
11990835119910119888) (120587120582)
2
(1205901119905119908ℎ4
119908280119910
119888) (120587120582)
2=
24
ℎ2
119908
119863 [12ℎ3
119908+ (125ℎ
119908) (120587120582)
2
+ (13ℎ11990835) (120587120582)
4
]
119863 [4ℎ119908+ (4ℎ11990815) (120587120582)
2
+ (ℎ3
119908105) (120587120582)
4
]
asymp
3
ℎ2
119908
(22)
From (22) ℎ2119908119896119909119896120593is not an infinitesimal value so the
lateral constraint rigidity of bottom flange to web cannot bedisregarded Namely in the calculation the equation 119896
119909= 0
is not available Therefore the lateral constraint rigidity ofbottom flange to web cannot be approximated by zero Thisis different from [16] in which the lateral constraint rigiditywhich the cold-formed thin-walled lipped channel steel websto the top and bottom flange is taken as zero
4 Theoretical Derivation of Critical Moment
41 Derivation of Critical Moment The lateral constraintrigidity and rotation constraint rigidity which steel beamwebto bottom flange can be simplified respectively as
119896119909= 1205721120573120590+119863120572
2 (23)
119896120593= 1205723120573120590+119863120572
4 (24)
120573 = (
120587
120582
)
2
(25)
1205721= 119905119908ℎ119908(
0086ℎ119908
119910119888
minus 037) (26a)
1205722=
12
ℎ3
119908
+
24120573
ℎ119908
+ 037ℎ1199081205732
(26b)
1205723= 119905119908ℎ3
119908(
00036ℎ119908
119910119888
minus 00095) (26c)
1205724=
4
ℎ119908
+ 027ℎ119908120573+ 00095ℎ
3
1199081205732
(26d)
120590 =
119872119909119910119888
119868
=
119875
119860119891
(27)
Therein119863 = 1198641199053
119908[12(1minus120583
2
)] 120583 is Poissonrsquos ratio of steel120590 represents the compressive stress of bottom flange 119875 refersto the compressive force of bottom flange minus119910
119888is the gravity
centre coordinate of steel beam total cross section 119868 is theinertia moment of steel beam 120582 = 119897119899 119899 is the buckling half-wave number 119860
119891is the area of bottom flange
As shown in Figure 2 the thin-walled member is doublysymmetric along 119909-axis and 119910-axis and the centre of origin119874 coincides with the flexural center The displacements oforigin 119874 in 119909 direction and 119910 direction are denoted as 119906 andV respectively Because the rigidity in 119910 direction is infinityV is equal to zero The equivalent distributed forces causedby elastic medium as a result of displacements of thin-walledmember can be written as
119903119909= 119896119909119906
119903119910= 119896119910V
(28)
where 119896119909and 119896119910are the lateral and vertical constraint rigidity
which web to bottom flange respectivelyThe torsional angle which the member rotates around the
bending center is assumed to be 120593The equivalent distributedmoment of torsional thin-walled member induced by equiv-alent spring is
119898 = 119896120593120593 (29)
Neutral balance differential equation of thin-walledmember can be expressed as [13 14]
119864119868119910119906119868119881
+119875 (11990610158401015840
+11991011988612059310158401015840
) + 119896119909[119906 minus (119910
119889minus119910119886) 120593] = 0
119864119868119909V119868119881 +119875 (V10158401015840 minus119909
11988612059310158401015840
) + 119896119910[V+ (119909
119889minus119909119886) 120593] = 0
119864119868119908120593119868119881
+ (11990320119875minus119866119869) 120593
10158401015840
minus119875 (119909119886V10158401015840 minus119910
11988611990610158401015840
)
minus 119896119909[119906 minus (119910
119889minus119910119886) 120593] (119910
119889minus119910119886)
+ 119896119910[V+ (119909
119889minus119909119886) 120593] (119909
119889minus119909119886) + 119896120593120593 = 0
(30)
Therein 119868119910= 119905119891119887311989112 119868
119909= 119887119891119905311989112 119869 = 119887
11989111990531198913 11990320 =
1199092119886+119910
2119886+(119868119909+119868119910)119860119904 119860119904being the area of steel beam 119909
119886is the
horizontal coordinate of the bottom flange section bendingcentre 119909
119886= 0 119910
119886is the vertical coordinate of the bottom
flange section bending centre 119910119886= 0 119909
119889is the horizontal
coordinate of the bottom flange section rotation axis 119909119889=
0 119910119889is the vertical coordinate of the bottom flange section
rotation axis 119910119889= 0 119868119908is the fan-shaped inertia moment of
bottom flange section 119868119908= 0 119864 is the tensile elastic modulus
of steel 119866 is the shear elastic modulus of steelWith substitution of 119910
119886= 0 119910
119889= 0 119909
119886= 0 V = 0 119868
119908= 0
119896119909= 0 119909
119886= 0 119910
119886= 0 and 119910
119889= 0 into (30) one can obtain
119864119868119910119868119906119868119881
+11991011988811986011989111987211990911990610158401015840
+ (1205721120573119910119888119872119909 + 1198681198631205722) 119906 = 0 (31)
119896119910[V+ (119909
119889minus119909119886) 120593] = 0 (32)
(11990320119910119888119860119891119872119909 minus119866119869119868) 120593
10158401015840
+ (1205723120573119910119888119872119909 + 1198681198631205724) 120593 = 0 (33)
When the steel-concrete composite beam in negativemoment region bears lateral bending buckling its neutralbalance equation is shown in (31) and the correspondingboundary conditions are
[119906]119911=0119897 = 0
[11990610158401015840
]119911=0119897 = 0
(34)
6 Advances in Materials Science and Engineering
Assuming 119906 = 119860 sin(radic120573119911) which satisfies the boundaryconditions according to Galerkinrsquos method
1205732minus
119872119909119910119888119860119891
119864119868119910119868
120573 +
(1205721120573119872119909119910119888119868 + 119863120572
2)
119864119868119910
= 0 (35)
By solving (35)
1198721198881199031 =
119864119868119910120573 + 1198631205722120573
(119860119891minus 1205721) 119910119888
119868 (36)
Due to 1198891198721198881199031119889120573 = 0 we obtained the following
1205731198881199031 =
346
radic119864119868119910ℎ3119908119863 + 037ℎ4
119908
1198991198881199031 =
119897radic1205731198881199031120587
(37)
If 119897radic1205731198881199031120587 is an integer substitution of 120573
1198881199031into (36) leads
to the lateral bending critical moment If 119897radic1205731198881199031120587 is not an
integer the two values of 1205731198881199031
which makes 119897radic1205731198881199031120587 be two
integers most near to 119897radic1205731198881199031120587 are then substituted into (36)
and the smaller value is chosen to be the lateral bendingcritical moment
When the steel beam bottom flange of steel-concretecomposite beam in negativemoment region suffers rotationalbuckling the steel-concrete composite beam will yield lateralbending and torsional bucklingTheneutral balance equationis seen in (33) and the boundary conditions are
[120593]119911=0119897 = 0
[12059310158401015840
]119911=0119897 = 0
(38)
Assume 1205852 = (1205723120573119872119909119910119888119868 + 119863120572
4)(1199032
0119872119909119910119888119860119891119868 minus 119866119869)
Solving (33) we obtained the following
120593 = 119860 sin 120585119911 +119861 cos 120585119911 (39)
Substituting (39) into (38) and with 120593 = 0 one can have
sin 120585119897 = 0 (40)
120585 =
119899120587
119897
= radic120573 (41)
(1205723120573119872119909119910119888119868 + 1198631205724)
(11990320119872119909119910119888119860119891119868 minus 119866119869)
= 120573 (42)
Solving (42)
1198721198881199032 =
1198631205724120573 + 119866119869
(11990320119860119891 minus 1205723) 119910119888
119868 (43)
Because of the fact that 1198891198721198881199032119889120573 = 0 then
1205731198881199032
=
205
ℎ2
119908
1198991198881199032
=
119897radic1205731198881199032
120587
(44)
Table 1 Geometric dimension of examples
Number of example ℎ119908mm 119905
119908mm 119905
119891mm 119887
119891mm
1 1886 7 114 1002 1886 7 104 1003 1886 7 94 1004 1886 8 94 1005 1886 9 94 1006 150 8 94 1007 200 8 94 1008 200 8 94 809 200 8 94 120
When 119897radic1205731198881199032120587 is an integer the substitution of 120573
1198881199032into
(43) gets the lateral bending critical momentWhen 119897radic1205731198881199032120587
is not an integer a similar calculation mentioned before istaken to obtain the lateral bending critical moment
After getting the lateral bending buckling criticalmoment1198721198881199031
and the bending and torsional buckling critical moment1198721198881199032
the smaller one of these two is taken as the buckling loadof composite beam in negativemoment regionThe analyticalresults indicate that the composite beam in negative momentregion yields as a result of lateral bending and torsionalbuckling acting together Therefore in [11 12] the methodin which only one case is considered is questionable Thepresented work is an improvement to them
To sum up the calculation formula of buckling momentcan be expressed as
119872119888119903
= min1198641198681199101205731198881199031 + 11986312057221205731198881199031
(119860119891minus 1205721) 119910119888
119868
11986312057241205731198881199032 + 119866119869
(11990320119860119891 minus 1205723) 119910119888
119868
1205731198881199031
=
346
radic119864119868119910ℎ3
119908119863 + 037ℎ
4
119908
1205731198881199032
=
205
ℎ2
119908
1205721= 119905119908ℎ119908(
0086ℎ119908
119910119888
minus 037)
1205722=
12
ℎ3
119908
+
241205731198881199031
ℎ119908
+ 037ℎ1199081205732
1198881199031
1205723= 119905119908ℎ3
119908(
00036ℎ119908
119910119888
minus 00095)
1205724=
4
ℎ119908
+ 027ℎ1199081205731198881199032+ 00095ℎ
3
1199081205732
1198881199032
(45)
When 119897radic1205731198881199031120587 is not an integer substituting 120573
1198881199031that
corresponds to two integers of the left and right side of119897radic1205731198881199031120587 into (36) the smaller resulting value of 120573
1198881199031is
the desired value And when 119897radic1205731198881199032120587 is not an integer
substituting 1205731198881199032
that corresponds to two integers of the left
Advances in Materials Science and Engineering 7
Table 2 Calculation results
Number of example ANSYSMPa Literature method [4]MPa Literature method [15]MPa The proposed methodMPa1 581 times 102 221 times 102 838 times 102 583 times 102
2 546 times 102 205 times 102 832 times 102 551 times 102
3 513 times 102 187 times 102 827 times 102 518 times 102
4 613 times 102 213 times 102 122 times 102 628 times 102
5 703 times 102 246 times 102 173 times 102 769 times 102
6 632 times 102 256 times 102 192 times 102 629 times 102
7 608 times 102 206 times 102 109 times 102 609 times 102
8 480 times 102 138 times 102 108 times 102 489 times 102
9 789 times 102 292 times 102 104 times 102 824 times 102
and right side of 119897radic1205731198881199032120587 into (43) the smaller resulting
value of 1205731198881199032
is the desired value
42 Practical Example Analysis Nine cases of I-shape steel-concrete composite beam are shown in Table 1 The length ofspecimen in negative moment region is equal to 4000mmFinite element method is adopted to calculate these nineexamples by using ANSYS in which the concrete part ofcomposite beam is substituted by lateral restraint and the steelI-beam is simulated by SHELL 43 The method proposed byBritish Steel Construction Institution [4] and the bucklingmodel method [15] are also used to calculate the bucklingloads The obtained results are shown in Table 2
As shown in Table 2 the results obtained by bucklingmodel method [15] tend to be excessively unsafe The resultsby using the method proposed by British Steel ConstructionInstitution [4] are too conservative However the results ofthe proposed method are in good agreement with thoseof ANSYS which means that the proposed method is areasonable and effective method
5 Conclusions
Based on energy method a comprehensive and intensivestudy on rotation restraining rigidity 119896
120601and lateral restrain-
ing rigidity 119896119909which steel beam web to bottom plate of
steel-concrete composite beam in negative moment region isperformed in this paper The expressions of lateral bendingbuckling stress lateral bending and torsional buckling stressand buckling moments of steel-concrete composite beam innegative moment region are deduced Some conclusions aredrawn as follows
(1) Both the rotation constraint rigidity 119896120601and the lateral
constraint rigidity 119896119909show a linear relationship with
longitudinal compressive stress 1205901 at bottom flange
(2) The rotation constraint rigidity 119896120601and the lateral
constraint rigidity 119896119909could be negative When the
rotation constraint rigidity and the lateral constraintrigidity are negative that means the bottom flange ofsteel beam can be restrained by rotation or lateralconstraint when steel beam was in negative momentregion buckling
(3) Since ℎ2119908119896119909119896120593is not infinitesimal the lateral con-
straint rigidity of bottom flange to web cannot beneglected In other words in the calculation theequation 119896
119909= 0 cannot be used Therefore it is
proved theoretically that the lateral constraint rigidityof bottom flange to web cannot be approximated to bezero This point is different with the steel structure
(4) The results obtained by literatures [4 15] showsome deviations with the ANSYS results But theresults by the presented expressions agree well withthe ANSYS results The reason is that the proposedexpressions consider the lateral bending buckling andlateral bending-torsion buckling simultaneously Theproposed method is more reasonable and clearer inphysical concepts in comparison with those methodswhich consider only one buckling mode Besides theproposed expressions are more concise and suitablefor the engineering application
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge financial supportfrom Projects (51078355) supported by National NaturalScience Foundation of China Projects (IRT1296) supportedby the Program for Changjiang Scholars and InnovativeResearch Team in University (PCSIRT) and Project (12K104)supported by Scientific Research Fund of Hunan ProvincialEducation Department China Dr Jing-Jing Qi and DrWang-Bao Zhou have tried their best to help the authorsTheauthors express their sincere gratitude to Dr Jing-Jing Qi andDr Wang-Bao Zhou
References
[1] M Ma and O Hughes ldquoLateral distortional buckling ofmonosymmetric I-beams under distributed vertical loadrdquoThin-Walled Structures vol 26 no 2 pp 123ndash143 1996
[2] M A Bradford ldquoLateral-distortional buckling of continuouslyrestrained columnsrdquo Journal of Constructional Steel Researchvol 42 no 2 pp 121ndash139 1997
8 Advances in Materials Science and Engineering
[3] M L-H Ng and H R Ronagh ldquoAn analytical solution forthe elastic lateral-distortional buckling of I-section beamsrdquoAdvances in Structural Engineering vol 7 no 2 pp 189ndash2002004
[4] R M Lawson and J W RaekhamDesign of Haunched Compos-ite Beams in Buildings Steel Construction Institute 1989
[5] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987
[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985
[7] P Goltermann and S E Svensson ldquoLateral distortional buck-ling predicting elastic critical stressrdquo Journal of StructuralEngineering vol 114 no 7 pp 1606ndash1625 1988
[8] H R Ronagh ldquoProgress in the methods of analysis of restricteddistortional buckling of composite bridge girdersrdquo Progress inStructural Engineering and Materials vol 3 no 2 pp 141ndash1482001
[9] Z Diansheng and Y Xiaomin ldquoStudy on beam local buckling ofcold-formed thin-wall steel and concrete compositerdquo IndustrialArchitecture vol 38 supplement pp 539ndash542 2008
[10] L-Z Jiang L-J Zeng and L-L Sun ldquoSteel-concrete compositecontinuous beam web local buckling analysisrdquo Journal of CivilEngineering and Architecture vol 26 no 3 pp 27ndash31 2009
[11] L Jiang and L Sun ldquoThe lateral buckling of steel-concrete com-posite box-beamsrdquo Journal of Huazhong University of Scienceand Technology (Urban Science Edition) vol 25 no 3 pp 5ndash92008
[12] J Ye and W Chen ldquoBeam with elastic constraints of I-shapedsteelmdashconcrete composite distortional bucklingrdquo Journal ofBuilding Structures vol 32 no 6 pp 82ndash91 2011
[13] W-B Zhou L-Z Jiang and Z-W Yu ldquoAnalysis of beamnegative moment zone of the elastic distortional buckling ofsteel-concrete compositerdquo Journal of Central South University(Natural Science Edition) vol 43 no 6 pp 2316ndash2323 2012
[14] W-B Zhou L-Z Jiang and Z-W Yu ldquoThe calculation formulaof steel concrete composite beams under negative bendingzonedistortional buckling momentrdquo Chinese Journal of Compu-tational Mechanics vol 29 no 3 pp 446ndash451 2012
[15] P Zhu Steel-Concrete Composite Beam Design Theory ChinaArchitecture amp Building Press Beijing China 1989
[16] J Yao and J-G Teng ldquoWeb rotational restraint in elasticdistortional buckling of cold-formed lipped channel sectionsrdquoEngineering Mechanics vol 25 no 4 pp 65ndash69 2008
[17] W-B Zhou L-Z Jiang and Z-W Yu ldquoClosed-form solutionfor shear lag effects of steel-concrete composite box beamsconsidering shear deformation and sliprdquo Journal of CentralSouth University vol 19 no 10 pp 2976ndash2982 2012
[18] W-B Zhou L-Z Jiang and Z-W Yu ldquoClosed-form solutionto thin-walled box girders considering the effects of sheardeformation and shear lagrdquo Journal of Central South Universityvol 19 no 9 pp 2650ndash2655 2012
Submit your manuscripts athttpwwwhindawicom
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MaterialsJournal of
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Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Advances in Materials Science and Engineering 3
1205901 1205901
yc ycz
y
1205902 1205902
120582 tw
h
m(z)
The longitudinal edges of the web
The longitudinal edges of the web
Figure 3 Rectangular plate subjected to compression andmoments
and thickness of web section are ℎ119908and 119905119908 respectively 120582
refers to the half-wave length of web caused by distortionbuckling in longitudinal direction (called as the half-wavelength hereafter) Two transversal opposite sides are simplysupported The side connected to top flange is fixed and theother side connected to bottom flange is simply supportedThe two simply supported sides bear the longitudinal lineardistributed stress 120590 in 119885 direction (compressive stress ispositive and tension stress is negative)The side connected tobottom flange bears the equivalent spring constraint moment119898(119911) which bottom flange exerted on web The coordinate ofgravity centre of steel beam total cross section is representedby minus119910
119888 and the moment of inertia is 119868 According to the
assumptions mentioned above when the negative moment119872119909acts on the reinforcement in concrete slab the axial
compressive stress at bottom edge of web is 1205901 = 119872119909119910119888119868
and the axial compressive stress at top edge of web is 1205902 =
1205901(119910119888minusℎ119908)119910119888Therefore the axial compressive stress at otherpoints of web is 120590 = 1205901(119910119888 + 119910)119910119888
Assuming 119863 = 1198641199053
11990812(1 minus 120583
2) 120583 is Poissonrsquos ratio
of steel 119864 is the elastic modulus of steel 119906 denotes thedeformation function of web The boundary conditions of 119906can be expressed as
[119906]119911=0120582 = 0
[119906]119910=0minusℎ
119908
= 0
[
120597119906
120597119910
]
119910=minusℎ119908
= 0
[minus119863(
1205972119906
1205971199112 +120583
1205972119906
1205971199102)]119911=0120582
= 0
(1)
With (1) the displacement functions are written as
119906 = 119888 [
119910
ℎ119908
+ 2(119910
ℎ119908
)
2+(
119910
ℎ119908
)
3] sin120587119911
120582
(2)
The strain energy of half-wave length web in the case ofsmall deformations is [16ndash18]
1198801 =119863
2int
120582
0int
0
minusℎ119908
[(
1205972119906
1205971199102)
2
+(
1205972119906
1205971199112)
2
+ 21205831205972119906
12059711991021205972119906
1205971199112
+ 2 (1minus120583)( 1205972119906
120597119910120597119911
)
2
]119889119910119889119911
(3)
Substituting (2) into (3) leads to the fact that
1198801 =120582119863
2[
21198882
ℎ3
119908
+
21198882
15ℎ119908
(
120587
120582
)
2+
1198882ℎ119908
210(
120587
120582
)
4] (4)
The elastic potential energy caused by half-wave lengthlongitudinal side spring constraint is
1198802 =119896120593
2int
120582
0(
120597119906
120597119910
)
2
119910=0119889119911 (5)
Substituting (2) into (5)
1198802 =1205821198961205931198882
4ℎ119908
2 (6)
The external force work of half-wave length web can becomputed by [16ndash18]
119882 =
119905119908
2int
120582
0int
0
minusℎ119908
120590(
120597119906
120597119911
)
2119889119910119889119911
=
119905119908
2int
120582
0int
0
minusℎ119908
1205901 (119910119888 + 119910)
119910119888
(
120597119906
120597119911
)
2119889119910119889119911
(7)
Substituting (5) into (8)
119882 =
12058211988821205901119905119908ℎ119908420
(
120587
120582
)
2minus
12058211988821205901119905119908ℎ
2
119908
1120119910119888
(
120587
120582
)
2 (8)
The total potential energy of half-wave length web is
Π = 1198801 +1198802 minus119882 (9)
Substitution of (4) (6) and (8) into (9) results in
Π =
120582119863
2[
21198882
ℎ3
119908
+
21198882
15ℎ119908
(
120587
120582
)
2+
1198882ℎ119908
210(
120587
120582
)
4]
+
1205821198961205931198882
4ℎ2119908
minus
12058211988821205901119905119908ℎ119908420
(
120587
120582
)
2
+
12058211988821205901119905119908ℎ
2
119908
1120119910119888
(
120587
120582
)
2
(10)
Based on principle of resident potential energy weobtained the following
119863
2[
2ℎ3
119908
+
215ℎ119908
(
120587
120582
)
2+
ℎ119908
210(
120587
120582
)
4]+
119896120593
4ℎ2119908
minus
1205901119905119908ℎ119908420
(
120587
120582
)
2+
1205901119905119908ℎ2
119908
1120119910119888
(
120587
120582
)
2= 0
(11)
By solving (11) one can obtain
119896120593= (
119905119908ℎ3
119908
105minus
119905119908ℎ4
119908
280119910119888
)(
120587
120582
)
21205901
minus119863[
4ℎ119908
+
4ℎ119908
15(
120587
120582
)
2+
ℎ3
119908
105(
120587
120582
)
4]
(12)
4 Advances in Materials Science and Engineering
1205901 1205901
yc ycz
y
1205902 1205902
120582 tw
h
f(z)
The longitudinal edges of the web
The longitudinal edges of the web
Figure 4 Rectangular plate subjected to compression and lateralstress
32 Lateral Constraint Rigidity 119896119909 The half-wave length of
web section is shown in Figure 4 Two transversal oppositesides are simply supported The side connected to top flangeis fixed and the other side connected to bottom flangecan move laterally The two simply supported sides bearthe longitudinal linear distributed stress 120590 in 119885 direction(similarly compressive stress is taken as positive and tensionstress is negative) The side connected to bottom flange bearsthe equivalent spring constraint distributed force 119891(119911) whichbottom flange exerted on web
Based on above analysis the boundary conditions of119906 canbe expressed as
[119906]119911=0120582 = 0
[119906]119910=minusℎ
119908
= 0
[
120597119906
120597119910
]
119910=0minusℎ119908
= 0
[minus119863(
1205972119906
1205971199112 +120583
1205972119906
1205971199102)]119911=0120582
= 0
(13)
According to above boundary conditions the displace-ment functions can be written as
119906 = [119888 minus 3119888 (119910
ℎ119908
)
2minus 2119888 (
119910
ℎ119908
)
3] sin120587119911
120582
(14)
Substituting (14) into (3) the strain energy of half-wavelength web in the case of small deformations is then obtainedas follows
1198801 =120582119863
2[
61198882
ℎ3
119908
+
131198882ℎ119908
70(
120587
120582
)
4+
61198882
5ℎ119908
(
120587
120582
)
2] (15)
The elastic potential energy caused by half-wave lengthlongitudinal side spring constraint is
1198802 =119896119909
2int
120582
0[119906]
2119910=0 119889119911 (16)
Substituting (14) into (16) leads to the fact that
1198802 =1205821198961199091198882
4 (17)
Substituting (14) into (7) the external force work of half-wave length web can be obtained as follows
119882 =
1312058211988821205901119905119908ℎ119908140
(
120587
120582
)
2minus
312058211988821205901119905119908ℎ2
119908
140119910119888
(
120587
120582
)
2 (18)
Substituting (15) (17) and (18) into (9) the total potentialenergy of half-wave length web is
Π =
120582119863
2[
61198882
ℎ3
119908
+
131198882ℎ119908
70(
120587
120582
)
4+
61198882
5ℎ119908
(
120587
120582
)
2]
+
1205821198961199091198882
4minus
1312058211988821205901119905119908ℎ119908140
(
120587
120582
)
2
+
312058211988821205901119905119908ℎ2
119908
140119910119888
(
120587
120582
)
2
(19)
Based on principle of resident potential energy one canhave
119863
2[
6ℎ3
119908
+
65ℎ119908
(
120587
120582
)
2+
13ℎ119908
70(
120587
120582
)
4]+
119896119909
4
minus
131205901119905119908ℎ119908140
(
120587
120582
)
2+
31205901119905119908ℎ2
119908
140119910119888
(
120587
120582
)
2= 0
(20)
By solving (20) we obtained the following
119896119909= (
13119905119908ℎ119908
35minus
3119905119908ℎ2
119908
35119910119888
)(
120587
120582
)
21205901
minus119863[
12ℎ119908
3 +125ℎ119908
(
120587
120582
)
2+
13ℎ119908
35(
120587
120582
)
4]
(21)
33 Discussion about 119896119909and 119896120601
(1) Equations (12) and (21) indicated that both 119896120601and
119896119909show a linear relationship with the longitudinal
compressive stress 1205901 Generally ℎ119908119910119888 is less than2 so the coefficient before 1205901 is positive for mostsituations The bigger 1205901 is the higher 119896120601 and 119896119909 areAt the same time it is of interest to note that both119896120601and 119896119909which steel beam bottom flange to web are
determined by the compressive stress 1205901 but not bycomposite beam section properties
(2) Since the polynomials on right-hand side of (12) and(21) have negative terms 119896
120601and and 119896
119909could be
negative This is not consistent with regular positivedefinite rigidity and rigidity matrix If the rotationconstraint rigidity and lateral constraint rigidity arenegative the rotation and lateral displacement of steelbeam bottomflange will be restricted by web Namelythe steel beam web will restrict bottom flange tobuckle but the steel beam bottom flange will inducethe web to buckle According to [16] the lateralconstraint rigidity 119896 = 119864119905
3
119908(4ℎ3
119908) is obtained by
using strip method in the elastic constraint compres-sion member buckling model However the restraintaction of two adjacent strips is not considered inthis method Therefore the lateral constraint rigiditywhich has nothing to do with external forces isalways positive But this does not agree with theactual situation Furthermore the neglected rotationconstraint rigidity will lead to certain errors whencalculating buckling load of composite beam
Advances in Materials Science and Engineering 5
(3) The ratios of the first term second term and thirdterm on the right side of (21) and (12) are
(13120590119905119908ℎ11990835) (120587120582)2
(120590119905119908ℎ3
119908105) (120587120582)2
=
39ℎ2
119908
(31205901119905119908ℎ2
11990835119910119888) (120587120582)
2
(1205901119905119908ℎ4
119908280119910
119888) (120587120582)
2=
24
ℎ2
119908
119863 [12ℎ3
119908+ (125ℎ
119908) (120587120582)
2
+ (13ℎ11990835) (120587120582)
4
]
119863 [4ℎ119908+ (4ℎ11990815) (120587120582)
2
+ (ℎ3
119908105) (120587120582)
4
]
asymp
3
ℎ2
119908
(22)
From (22) ℎ2119908119896119909119896120593is not an infinitesimal value so the
lateral constraint rigidity of bottom flange to web cannot bedisregarded Namely in the calculation the equation 119896
119909= 0
is not available Therefore the lateral constraint rigidity ofbottom flange to web cannot be approximated by zero Thisis different from [16] in which the lateral constraint rigiditywhich the cold-formed thin-walled lipped channel steel websto the top and bottom flange is taken as zero
4 Theoretical Derivation of Critical Moment
41 Derivation of Critical Moment The lateral constraintrigidity and rotation constraint rigidity which steel beamwebto bottom flange can be simplified respectively as
119896119909= 1205721120573120590+119863120572
2 (23)
119896120593= 1205723120573120590+119863120572
4 (24)
120573 = (
120587
120582
)
2
(25)
1205721= 119905119908ℎ119908(
0086ℎ119908
119910119888
minus 037) (26a)
1205722=
12
ℎ3
119908
+
24120573
ℎ119908
+ 037ℎ1199081205732
(26b)
1205723= 119905119908ℎ3
119908(
00036ℎ119908
119910119888
minus 00095) (26c)
1205724=
4
ℎ119908
+ 027ℎ119908120573+ 00095ℎ
3
1199081205732
(26d)
120590 =
119872119909119910119888
119868
=
119875
119860119891
(27)
Therein119863 = 1198641199053
119908[12(1minus120583
2
)] 120583 is Poissonrsquos ratio of steel120590 represents the compressive stress of bottom flange 119875 refersto the compressive force of bottom flange minus119910
119888is the gravity
centre coordinate of steel beam total cross section 119868 is theinertia moment of steel beam 120582 = 119897119899 119899 is the buckling half-wave number 119860
119891is the area of bottom flange
As shown in Figure 2 the thin-walled member is doublysymmetric along 119909-axis and 119910-axis and the centre of origin119874 coincides with the flexural center The displacements oforigin 119874 in 119909 direction and 119910 direction are denoted as 119906 andV respectively Because the rigidity in 119910 direction is infinityV is equal to zero The equivalent distributed forces causedby elastic medium as a result of displacements of thin-walledmember can be written as
119903119909= 119896119909119906
119903119910= 119896119910V
(28)
where 119896119909and 119896119910are the lateral and vertical constraint rigidity
which web to bottom flange respectivelyThe torsional angle which the member rotates around the
bending center is assumed to be 120593The equivalent distributedmoment of torsional thin-walled member induced by equiv-alent spring is
119898 = 119896120593120593 (29)
Neutral balance differential equation of thin-walledmember can be expressed as [13 14]
119864119868119910119906119868119881
+119875 (11990610158401015840
+11991011988612059310158401015840
) + 119896119909[119906 minus (119910
119889minus119910119886) 120593] = 0
119864119868119909V119868119881 +119875 (V10158401015840 minus119909
11988612059310158401015840
) + 119896119910[V+ (119909
119889minus119909119886) 120593] = 0
119864119868119908120593119868119881
+ (11990320119875minus119866119869) 120593
10158401015840
minus119875 (119909119886V10158401015840 minus119910
11988611990610158401015840
)
minus 119896119909[119906 minus (119910
119889minus119910119886) 120593] (119910
119889minus119910119886)
+ 119896119910[V+ (119909
119889minus119909119886) 120593] (119909
119889minus119909119886) + 119896120593120593 = 0
(30)
Therein 119868119910= 119905119891119887311989112 119868
119909= 119887119891119905311989112 119869 = 119887
11989111990531198913 11990320 =
1199092119886+119910
2119886+(119868119909+119868119910)119860119904 119860119904being the area of steel beam 119909
119886is the
horizontal coordinate of the bottom flange section bendingcentre 119909
119886= 0 119910
119886is the vertical coordinate of the bottom
flange section bending centre 119910119886= 0 119909
119889is the horizontal
coordinate of the bottom flange section rotation axis 119909119889=
0 119910119889is the vertical coordinate of the bottom flange section
rotation axis 119910119889= 0 119868119908is the fan-shaped inertia moment of
bottom flange section 119868119908= 0 119864 is the tensile elastic modulus
of steel 119866 is the shear elastic modulus of steelWith substitution of 119910
119886= 0 119910
119889= 0 119909
119886= 0 V = 0 119868
119908= 0
119896119909= 0 119909
119886= 0 119910
119886= 0 and 119910
119889= 0 into (30) one can obtain
119864119868119910119868119906119868119881
+11991011988811986011989111987211990911990610158401015840
+ (1205721120573119910119888119872119909 + 1198681198631205722) 119906 = 0 (31)
119896119910[V+ (119909
119889minus119909119886) 120593] = 0 (32)
(11990320119910119888119860119891119872119909 minus119866119869119868) 120593
10158401015840
+ (1205723120573119910119888119872119909 + 1198681198631205724) 120593 = 0 (33)
When the steel-concrete composite beam in negativemoment region bears lateral bending buckling its neutralbalance equation is shown in (31) and the correspondingboundary conditions are
[119906]119911=0119897 = 0
[11990610158401015840
]119911=0119897 = 0
(34)
6 Advances in Materials Science and Engineering
Assuming 119906 = 119860 sin(radic120573119911) which satisfies the boundaryconditions according to Galerkinrsquos method
1205732minus
119872119909119910119888119860119891
119864119868119910119868
120573 +
(1205721120573119872119909119910119888119868 + 119863120572
2)
119864119868119910
= 0 (35)
By solving (35)
1198721198881199031 =
119864119868119910120573 + 1198631205722120573
(119860119891minus 1205721) 119910119888
119868 (36)
Due to 1198891198721198881199031119889120573 = 0 we obtained the following
1205731198881199031 =
346
radic119864119868119910ℎ3119908119863 + 037ℎ4
119908
1198991198881199031 =
119897radic1205731198881199031120587
(37)
If 119897radic1205731198881199031120587 is an integer substitution of 120573
1198881199031into (36) leads
to the lateral bending critical moment If 119897radic1205731198881199031120587 is not an
integer the two values of 1205731198881199031
which makes 119897radic1205731198881199031120587 be two
integers most near to 119897radic1205731198881199031120587 are then substituted into (36)
and the smaller value is chosen to be the lateral bendingcritical moment
When the steel beam bottom flange of steel-concretecomposite beam in negativemoment region suffers rotationalbuckling the steel-concrete composite beam will yield lateralbending and torsional bucklingTheneutral balance equationis seen in (33) and the boundary conditions are
[120593]119911=0119897 = 0
[12059310158401015840
]119911=0119897 = 0
(38)
Assume 1205852 = (1205723120573119872119909119910119888119868 + 119863120572
4)(1199032
0119872119909119910119888119860119891119868 minus 119866119869)
Solving (33) we obtained the following
120593 = 119860 sin 120585119911 +119861 cos 120585119911 (39)
Substituting (39) into (38) and with 120593 = 0 one can have
sin 120585119897 = 0 (40)
120585 =
119899120587
119897
= radic120573 (41)
(1205723120573119872119909119910119888119868 + 1198631205724)
(11990320119872119909119910119888119860119891119868 minus 119866119869)
= 120573 (42)
Solving (42)
1198721198881199032 =
1198631205724120573 + 119866119869
(11990320119860119891 minus 1205723) 119910119888
119868 (43)
Because of the fact that 1198891198721198881199032119889120573 = 0 then
1205731198881199032
=
205
ℎ2
119908
1198991198881199032
=
119897radic1205731198881199032
120587
(44)
Table 1 Geometric dimension of examples
Number of example ℎ119908mm 119905
119908mm 119905
119891mm 119887
119891mm
1 1886 7 114 1002 1886 7 104 1003 1886 7 94 1004 1886 8 94 1005 1886 9 94 1006 150 8 94 1007 200 8 94 1008 200 8 94 809 200 8 94 120
When 119897radic1205731198881199032120587 is an integer the substitution of 120573
1198881199032into
(43) gets the lateral bending critical momentWhen 119897radic1205731198881199032120587
is not an integer a similar calculation mentioned before istaken to obtain the lateral bending critical moment
After getting the lateral bending buckling criticalmoment1198721198881199031
and the bending and torsional buckling critical moment1198721198881199032
the smaller one of these two is taken as the buckling loadof composite beam in negativemoment regionThe analyticalresults indicate that the composite beam in negative momentregion yields as a result of lateral bending and torsionalbuckling acting together Therefore in [11 12] the methodin which only one case is considered is questionable Thepresented work is an improvement to them
To sum up the calculation formula of buckling momentcan be expressed as
119872119888119903
= min1198641198681199101205731198881199031 + 11986312057221205731198881199031
(119860119891minus 1205721) 119910119888
119868
11986312057241205731198881199032 + 119866119869
(11990320119860119891 minus 1205723) 119910119888
119868
1205731198881199031
=
346
radic119864119868119910ℎ3
119908119863 + 037ℎ
4
119908
1205731198881199032
=
205
ℎ2
119908
1205721= 119905119908ℎ119908(
0086ℎ119908
119910119888
minus 037)
1205722=
12
ℎ3
119908
+
241205731198881199031
ℎ119908
+ 037ℎ1199081205732
1198881199031
1205723= 119905119908ℎ3
119908(
00036ℎ119908
119910119888
minus 00095)
1205724=
4
ℎ119908
+ 027ℎ1199081205731198881199032+ 00095ℎ
3
1199081205732
1198881199032
(45)
When 119897radic1205731198881199031120587 is not an integer substituting 120573
1198881199031that
corresponds to two integers of the left and right side of119897radic1205731198881199031120587 into (36) the smaller resulting value of 120573
1198881199031is
the desired value And when 119897radic1205731198881199032120587 is not an integer
substituting 1205731198881199032
that corresponds to two integers of the left
Advances in Materials Science and Engineering 7
Table 2 Calculation results
Number of example ANSYSMPa Literature method [4]MPa Literature method [15]MPa The proposed methodMPa1 581 times 102 221 times 102 838 times 102 583 times 102
2 546 times 102 205 times 102 832 times 102 551 times 102
3 513 times 102 187 times 102 827 times 102 518 times 102
4 613 times 102 213 times 102 122 times 102 628 times 102
5 703 times 102 246 times 102 173 times 102 769 times 102
6 632 times 102 256 times 102 192 times 102 629 times 102
7 608 times 102 206 times 102 109 times 102 609 times 102
8 480 times 102 138 times 102 108 times 102 489 times 102
9 789 times 102 292 times 102 104 times 102 824 times 102
and right side of 119897radic1205731198881199032120587 into (43) the smaller resulting
value of 1205731198881199032
is the desired value
42 Practical Example Analysis Nine cases of I-shape steel-concrete composite beam are shown in Table 1 The length ofspecimen in negative moment region is equal to 4000mmFinite element method is adopted to calculate these nineexamples by using ANSYS in which the concrete part ofcomposite beam is substituted by lateral restraint and the steelI-beam is simulated by SHELL 43 The method proposed byBritish Steel Construction Institution [4] and the bucklingmodel method [15] are also used to calculate the bucklingloads The obtained results are shown in Table 2
As shown in Table 2 the results obtained by bucklingmodel method [15] tend to be excessively unsafe The resultsby using the method proposed by British Steel ConstructionInstitution [4] are too conservative However the results ofthe proposed method are in good agreement with thoseof ANSYS which means that the proposed method is areasonable and effective method
5 Conclusions
Based on energy method a comprehensive and intensivestudy on rotation restraining rigidity 119896
120601and lateral restrain-
ing rigidity 119896119909which steel beam web to bottom plate of
steel-concrete composite beam in negative moment region isperformed in this paper The expressions of lateral bendingbuckling stress lateral bending and torsional buckling stressand buckling moments of steel-concrete composite beam innegative moment region are deduced Some conclusions aredrawn as follows
(1) Both the rotation constraint rigidity 119896120601and the lateral
constraint rigidity 119896119909show a linear relationship with
longitudinal compressive stress 1205901 at bottom flange
(2) The rotation constraint rigidity 119896120601and the lateral
constraint rigidity 119896119909could be negative When the
rotation constraint rigidity and the lateral constraintrigidity are negative that means the bottom flange ofsteel beam can be restrained by rotation or lateralconstraint when steel beam was in negative momentregion buckling
(3) Since ℎ2119908119896119909119896120593is not infinitesimal the lateral con-
straint rigidity of bottom flange to web cannot beneglected In other words in the calculation theequation 119896
119909= 0 cannot be used Therefore it is
proved theoretically that the lateral constraint rigidityof bottom flange to web cannot be approximated to bezero This point is different with the steel structure
(4) The results obtained by literatures [4 15] showsome deviations with the ANSYS results But theresults by the presented expressions agree well withthe ANSYS results The reason is that the proposedexpressions consider the lateral bending buckling andlateral bending-torsion buckling simultaneously Theproposed method is more reasonable and clearer inphysical concepts in comparison with those methodswhich consider only one buckling mode Besides theproposed expressions are more concise and suitablefor the engineering application
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge financial supportfrom Projects (51078355) supported by National NaturalScience Foundation of China Projects (IRT1296) supportedby the Program for Changjiang Scholars and InnovativeResearch Team in University (PCSIRT) and Project (12K104)supported by Scientific Research Fund of Hunan ProvincialEducation Department China Dr Jing-Jing Qi and DrWang-Bao Zhou have tried their best to help the authorsTheauthors express their sincere gratitude to Dr Jing-Jing Qi andDr Wang-Bao Zhou
References
[1] M Ma and O Hughes ldquoLateral distortional buckling ofmonosymmetric I-beams under distributed vertical loadrdquoThin-Walled Structures vol 26 no 2 pp 123ndash143 1996
[2] M A Bradford ldquoLateral-distortional buckling of continuouslyrestrained columnsrdquo Journal of Constructional Steel Researchvol 42 no 2 pp 121ndash139 1997
8 Advances in Materials Science and Engineering
[3] M L-H Ng and H R Ronagh ldquoAn analytical solution forthe elastic lateral-distortional buckling of I-section beamsrdquoAdvances in Structural Engineering vol 7 no 2 pp 189ndash2002004
[4] R M Lawson and J W RaekhamDesign of Haunched Compos-ite Beams in Buildings Steel Construction Institute 1989
[5] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987
[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985
[7] P Goltermann and S E Svensson ldquoLateral distortional buck-ling predicting elastic critical stressrdquo Journal of StructuralEngineering vol 114 no 7 pp 1606ndash1625 1988
[8] H R Ronagh ldquoProgress in the methods of analysis of restricteddistortional buckling of composite bridge girdersrdquo Progress inStructural Engineering and Materials vol 3 no 2 pp 141ndash1482001
[9] Z Diansheng and Y Xiaomin ldquoStudy on beam local buckling ofcold-formed thin-wall steel and concrete compositerdquo IndustrialArchitecture vol 38 supplement pp 539ndash542 2008
[10] L-Z Jiang L-J Zeng and L-L Sun ldquoSteel-concrete compositecontinuous beam web local buckling analysisrdquo Journal of CivilEngineering and Architecture vol 26 no 3 pp 27ndash31 2009
[11] L Jiang and L Sun ldquoThe lateral buckling of steel-concrete com-posite box-beamsrdquo Journal of Huazhong University of Scienceand Technology (Urban Science Edition) vol 25 no 3 pp 5ndash92008
[12] J Ye and W Chen ldquoBeam with elastic constraints of I-shapedsteelmdashconcrete composite distortional bucklingrdquo Journal ofBuilding Structures vol 32 no 6 pp 82ndash91 2011
[13] W-B Zhou L-Z Jiang and Z-W Yu ldquoAnalysis of beamnegative moment zone of the elastic distortional buckling ofsteel-concrete compositerdquo Journal of Central South University(Natural Science Edition) vol 43 no 6 pp 2316ndash2323 2012
[14] W-B Zhou L-Z Jiang and Z-W Yu ldquoThe calculation formulaof steel concrete composite beams under negative bendingzonedistortional buckling momentrdquo Chinese Journal of Compu-tational Mechanics vol 29 no 3 pp 446ndash451 2012
[15] P Zhu Steel-Concrete Composite Beam Design Theory ChinaArchitecture amp Building Press Beijing China 1989
[16] J Yao and J-G Teng ldquoWeb rotational restraint in elasticdistortional buckling of cold-formed lipped channel sectionsrdquoEngineering Mechanics vol 25 no 4 pp 65ndash69 2008
[17] W-B Zhou L-Z Jiang and Z-W Yu ldquoClosed-form solutionfor shear lag effects of steel-concrete composite box beamsconsidering shear deformation and sliprdquo Journal of CentralSouth University vol 19 no 10 pp 2976ndash2982 2012
[18] W-B Zhou L-Z Jiang and Z-W Yu ldquoClosed-form solutionto thin-walled box girders considering the effects of sheardeformation and shear lagrdquo Journal of Central South Universityvol 19 no 9 pp 2650ndash2655 2012
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Nano
materials
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Journal ofNanomaterials
4 Advances in Materials Science and Engineering
1205901 1205901
yc ycz
y
1205902 1205902
120582 tw
h
f(z)
The longitudinal edges of the web
The longitudinal edges of the web
Figure 4 Rectangular plate subjected to compression and lateralstress
32 Lateral Constraint Rigidity 119896119909 The half-wave length of
web section is shown in Figure 4 Two transversal oppositesides are simply supported The side connected to top flangeis fixed and the other side connected to bottom flangecan move laterally The two simply supported sides bearthe longitudinal linear distributed stress 120590 in 119885 direction(similarly compressive stress is taken as positive and tensionstress is negative) The side connected to bottom flange bearsthe equivalent spring constraint distributed force 119891(119911) whichbottom flange exerted on web
Based on above analysis the boundary conditions of119906 canbe expressed as
[119906]119911=0120582 = 0
[119906]119910=minusℎ
119908
= 0
[
120597119906
120597119910
]
119910=0minusℎ119908
= 0
[minus119863(
1205972119906
1205971199112 +120583
1205972119906
1205971199102)]119911=0120582
= 0
(13)
According to above boundary conditions the displace-ment functions can be written as
119906 = [119888 minus 3119888 (119910
ℎ119908
)
2minus 2119888 (
119910
ℎ119908
)
3] sin120587119911
120582
(14)
Substituting (14) into (3) the strain energy of half-wavelength web in the case of small deformations is then obtainedas follows
1198801 =120582119863
2[
61198882
ℎ3
119908
+
131198882ℎ119908
70(
120587
120582
)
4+
61198882
5ℎ119908
(
120587
120582
)
2] (15)
The elastic potential energy caused by half-wave lengthlongitudinal side spring constraint is
1198802 =119896119909
2int
120582
0[119906]
2119910=0 119889119911 (16)
Substituting (14) into (16) leads to the fact that
1198802 =1205821198961199091198882
4 (17)
Substituting (14) into (7) the external force work of half-wave length web can be obtained as follows
119882 =
1312058211988821205901119905119908ℎ119908140
(
120587
120582
)
2minus
312058211988821205901119905119908ℎ2
119908
140119910119888
(
120587
120582
)
2 (18)
Substituting (15) (17) and (18) into (9) the total potentialenergy of half-wave length web is
Π =
120582119863
2[
61198882
ℎ3
119908
+
131198882ℎ119908
70(
120587
120582
)
4+
61198882
5ℎ119908
(
120587
120582
)
2]
+
1205821198961199091198882
4minus
1312058211988821205901119905119908ℎ119908140
(
120587
120582
)
2
+
312058211988821205901119905119908ℎ2
119908
140119910119888
(
120587
120582
)
2
(19)
Based on principle of resident potential energy one canhave
119863
2[
6ℎ3
119908
+
65ℎ119908
(
120587
120582
)
2+
13ℎ119908
70(
120587
120582
)
4]+
119896119909
4
minus
131205901119905119908ℎ119908140
(
120587
120582
)
2+
31205901119905119908ℎ2
119908
140119910119888
(
120587
120582
)
2= 0
(20)
By solving (20) we obtained the following
119896119909= (
13119905119908ℎ119908
35minus
3119905119908ℎ2
119908
35119910119888
)(
120587
120582
)
21205901
minus119863[
12ℎ119908
3 +125ℎ119908
(
120587
120582
)
2+
13ℎ119908
35(
120587
120582
)
4]
(21)
33 Discussion about 119896119909and 119896120601
(1) Equations (12) and (21) indicated that both 119896120601and
119896119909show a linear relationship with the longitudinal
compressive stress 1205901 Generally ℎ119908119910119888 is less than2 so the coefficient before 1205901 is positive for mostsituations The bigger 1205901 is the higher 119896120601 and 119896119909 areAt the same time it is of interest to note that both119896120601and 119896119909which steel beam bottom flange to web are
determined by the compressive stress 1205901 but not bycomposite beam section properties
(2) Since the polynomials on right-hand side of (12) and(21) have negative terms 119896
120601and and 119896
119909could be
negative This is not consistent with regular positivedefinite rigidity and rigidity matrix If the rotationconstraint rigidity and lateral constraint rigidity arenegative the rotation and lateral displacement of steelbeam bottomflange will be restricted by web Namelythe steel beam web will restrict bottom flange tobuckle but the steel beam bottom flange will inducethe web to buckle According to [16] the lateralconstraint rigidity 119896 = 119864119905
3
119908(4ℎ3
119908) is obtained by
using strip method in the elastic constraint compres-sion member buckling model However the restraintaction of two adjacent strips is not considered inthis method Therefore the lateral constraint rigiditywhich has nothing to do with external forces isalways positive But this does not agree with theactual situation Furthermore the neglected rotationconstraint rigidity will lead to certain errors whencalculating buckling load of composite beam
Advances in Materials Science and Engineering 5
(3) The ratios of the first term second term and thirdterm on the right side of (21) and (12) are
(13120590119905119908ℎ11990835) (120587120582)2
(120590119905119908ℎ3
119908105) (120587120582)2
=
39ℎ2
119908
(31205901119905119908ℎ2
11990835119910119888) (120587120582)
2
(1205901119905119908ℎ4
119908280119910
119888) (120587120582)
2=
24
ℎ2
119908
119863 [12ℎ3
119908+ (125ℎ
119908) (120587120582)
2
+ (13ℎ11990835) (120587120582)
4
]
119863 [4ℎ119908+ (4ℎ11990815) (120587120582)
2
+ (ℎ3
119908105) (120587120582)
4
]
asymp
3
ℎ2
119908
(22)
From (22) ℎ2119908119896119909119896120593is not an infinitesimal value so the
lateral constraint rigidity of bottom flange to web cannot bedisregarded Namely in the calculation the equation 119896
119909= 0
is not available Therefore the lateral constraint rigidity ofbottom flange to web cannot be approximated by zero Thisis different from [16] in which the lateral constraint rigiditywhich the cold-formed thin-walled lipped channel steel websto the top and bottom flange is taken as zero
4 Theoretical Derivation of Critical Moment
41 Derivation of Critical Moment The lateral constraintrigidity and rotation constraint rigidity which steel beamwebto bottom flange can be simplified respectively as
119896119909= 1205721120573120590+119863120572
2 (23)
119896120593= 1205723120573120590+119863120572
4 (24)
120573 = (
120587
120582
)
2
(25)
1205721= 119905119908ℎ119908(
0086ℎ119908
119910119888
minus 037) (26a)
1205722=
12
ℎ3
119908
+
24120573
ℎ119908
+ 037ℎ1199081205732
(26b)
1205723= 119905119908ℎ3
119908(
00036ℎ119908
119910119888
minus 00095) (26c)
1205724=
4
ℎ119908
+ 027ℎ119908120573+ 00095ℎ
3
1199081205732
(26d)
120590 =
119872119909119910119888
119868
=
119875
119860119891
(27)
Therein119863 = 1198641199053
119908[12(1minus120583
2
)] 120583 is Poissonrsquos ratio of steel120590 represents the compressive stress of bottom flange 119875 refersto the compressive force of bottom flange minus119910
119888is the gravity
centre coordinate of steel beam total cross section 119868 is theinertia moment of steel beam 120582 = 119897119899 119899 is the buckling half-wave number 119860
119891is the area of bottom flange
As shown in Figure 2 the thin-walled member is doublysymmetric along 119909-axis and 119910-axis and the centre of origin119874 coincides with the flexural center The displacements oforigin 119874 in 119909 direction and 119910 direction are denoted as 119906 andV respectively Because the rigidity in 119910 direction is infinityV is equal to zero The equivalent distributed forces causedby elastic medium as a result of displacements of thin-walledmember can be written as
119903119909= 119896119909119906
119903119910= 119896119910V
(28)
where 119896119909and 119896119910are the lateral and vertical constraint rigidity
which web to bottom flange respectivelyThe torsional angle which the member rotates around the
bending center is assumed to be 120593The equivalent distributedmoment of torsional thin-walled member induced by equiv-alent spring is
119898 = 119896120593120593 (29)
Neutral balance differential equation of thin-walledmember can be expressed as [13 14]
119864119868119910119906119868119881
+119875 (11990610158401015840
+11991011988612059310158401015840
) + 119896119909[119906 minus (119910
119889minus119910119886) 120593] = 0
119864119868119909V119868119881 +119875 (V10158401015840 minus119909
11988612059310158401015840
) + 119896119910[V+ (119909
119889minus119909119886) 120593] = 0
119864119868119908120593119868119881
+ (11990320119875minus119866119869) 120593
10158401015840
minus119875 (119909119886V10158401015840 minus119910
11988611990610158401015840
)
minus 119896119909[119906 minus (119910
119889minus119910119886) 120593] (119910
119889minus119910119886)
+ 119896119910[V+ (119909
119889minus119909119886) 120593] (119909
119889minus119909119886) + 119896120593120593 = 0
(30)
Therein 119868119910= 119905119891119887311989112 119868
119909= 119887119891119905311989112 119869 = 119887
11989111990531198913 11990320 =
1199092119886+119910
2119886+(119868119909+119868119910)119860119904 119860119904being the area of steel beam 119909
119886is the
horizontal coordinate of the bottom flange section bendingcentre 119909
119886= 0 119910
119886is the vertical coordinate of the bottom
flange section bending centre 119910119886= 0 119909
119889is the horizontal
coordinate of the bottom flange section rotation axis 119909119889=
0 119910119889is the vertical coordinate of the bottom flange section
rotation axis 119910119889= 0 119868119908is the fan-shaped inertia moment of
bottom flange section 119868119908= 0 119864 is the tensile elastic modulus
of steel 119866 is the shear elastic modulus of steelWith substitution of 119910
119886= 0 119910
119889= 0 119909
119886= 0 V = 0 119868
119908= 0
119896119909= 0 119909
119886= 0 119910
119886= 0 and 119910
119889= 0 into (30) one can obtain
119864119868119910119868119906119868119881
+11991011988811986011989111987211990911990610158401015840
+ (1205721120573119910119888119872119909 + 1198681198631205722) 119906 = 0 (31)
119896119910[V+ (119909
119889minus119909119886) 120593] = 0 (32)
(11990320119910119888119860119891119872119909 minus119866119869119868) 120593
10158401015840
+ (1205723120573119910119888119872119909 + 1198681198631205724) 120593 = 0 (33)
When the steel-concrete composite beam in negativemoment region bears lateral bending buckling its neutralbalance equation is shown in (31) and the correspondingboundary conditions are
[119906]119911=0119897 = 0
[11990610158401015840
]119911=0119897 = 0
(34)
6 Advances in Materials Science and Engineering
Assuming 119906 = 119860 sin(radic120573119911) which satisfies the boundaryconditions according to Galerkinrsquos method
1205732minus
119872119909119910119888119860119891
119864119868119910119868
120573 +
(1205721120573119872119909119910119888119868 + 119863120572
2)
119864119868119910
= 0 (35)
By solving (35)
1198721198881199031 =
119864119868119910120573 + 1198631205722120573
(119860119891minus 1205721) 119910119888
119868 (36)
Due to 1198891198721198881199031119889120573 = 0 we obtained the following
1205731198881199031 =
346
radic119864119868119910ℎ3119908119863 + 037ℎ4
119908
1198991198881199031 =
119897radic1205731198881199031120587
(37)
If 119897radic1205731198881199031120587 is an integer substitution of 120573
1198881199031into (36) leads
to the lateral bending critical moment If 119897radic1205731198881199031120587 is not an
integer the two values of 1205731198881199031
which makes 119897radic1205731198881199031120587 be two
integers most near to 119897radic1205731198881199031120587 are then substituted into (36)
and the smaller value is chosen to be the lateral bendingcritical moment
When the steel beam bottom flange of steel-concretecomposite beam in negativemoment region suffers rotationalbuckling the steel-concrete composite beam will yield lateralbending and torsional bucklingTheneutral balance equationis seen in (33) and the boundary conditions are
[120593]119911=0119897 = 0
[12059310158401015840
]119911=0119897 = 0
(38)
Assume 1205852 = (1205723120573119872119909119910119888119868 + 119863120572
4)(1199032
0119872119909119910119888119860119891119868 minus 119866119869)
Solving (33) we obtained the following
120593 = 119860 sin 120585119911 +119861 cos 120585119911 (39)
Substituting (39) into (38) and with 120593 = 0 one can have
sin 120585119897 = 0 (40)
120585 =
119899120587
119897
= radic120573 (41)
(1205723120573119872119909119910119888119868 + 1198631205724)
(11990320119872119909119910119888119860119891119868 minus 119866119869)
= 120573 (42)
Solving (42)
1198721198881199032 =
1198631205724120573 + 119866119869
(11990320119860119891 minus 1205723) 119910119888
119868 (43)
Because of the fact that 1198891198721198881199032119889120573 = 0 then
1205731198881199032
=
205
ℎ2
119908
1198991198881199032
=
119897radic1205731198881199032
120587
(44)
Table 1 Geometric dimension of examples
Number of example ℎ119908mm 119905
119908mm 119905
119891mm 119887
119891mm
1 1886 7 114 1002 1886 7 104 1003 1886 7 94 1004 1886 8 94 1005 1886 9 94 1006 150 8 94 1007 200 8 94 1008 200 8 94 809 200 8 94 120
When 119897radic1205731198881199032120587 is an integer the substitution of 120573
1198881199032into
(43) gets the lateral bending critical momentWhen 119897radic1205731198881199032120587
is not an integer a similar calculation mentioned before istaken to obtain the lateral bending critical moment
After getting the lateral bending buckling criticalmoment1198721198881199031
and the bending and torsional buckling critical moment1198721198881199032
the smaller one of these two is taken as the buckling loadof composite beam in negativemoment regionThe analyticalresults indicate that the composite beam in negative momentregion yields as a result of lateral bending and torsionalbuckling acting together Therefore in [11 12] the methodin which only one case is considered is questionable Thepresented work is an improvement to them
To sum up the calculation formula of buckling momentcan be expressed as
119872119888119903
= min1198641198681199101205731198881199031 + 11986312057221205731198881199031
(119860119891minus 1205721) 119910119888
119868
11986312057241205731198881199032 + 119866119869
(11990320119860119891 minus 1205723) 119910119888
119868
1205731198881199031
=
346
radic119864119868119910ℎ3
119908119863 + 037ℎ
4
119908
1205731198881199032
=
205
ℎ2
119908
1205721= 119905119908ℎ119908(
0086ℎ119908
119910119888
minus 037)
1205722=
12
ℎ3
119908
+
241205731198881199031
ℎ119908
+ 037ℎ1199081205732
1198881199031
1205723= 119905119908ℎ3
119908(
00036ℎ119908
119910119888
minus 00095)
1205724=
4
ℎ119908
+ 027ℎ1199081205731198881199032+ 00095ℎ
3
1199081205732
1198881199032
(45)
When 119897radic1205731198881199031120587 is not an integer substituting 120573
1198881199031that
corresponds to two integers of the left and right side of119897radic1205731198881199031120587 into (36) the smaller resulting value of 120573
1198881199031is
the desired value And when 119897radic1205731198881199032120587 is not an integer
substituting 1205731198881199032
that corresponds to two integers of the left
Advances in Materials Science and Engineering 7
Table 2 Calculation results
Number of example ANSYSMPa Literature method [4]MPa Literature method [15]MPa The proposed methodMPa1 581 times 102 221 times 102 838 times 102 583 times 102
2 546 times 102 205 times 102 832 times 102 551 times 102
3 513 times 102 187 times 102 827 times 102 518 times 102
4 613 times 102 213 times 102 122 times 102 628 times 102
5 703 times 102 246 times 102 173 times 102 769 times 102
6 632 times 102 256 times 102 192 times 102 629 times 102
7 608 times 102 206 times 102 109 times 102 609 times 102
8 480 times 102 138 times 102 108 times 102 489 times 102
9 789 times 102 292 times 102 104 times 102 824 times 102
and right side of 119897radic1205731198881199032120587 into (43) the smaller resulting
value of 1205731198881199032
is the desired value
42 Practical Example Analysis Nine cases of I-shape steel-concrete composite beam are shown in Table 1 The length ofspecimen in negative moment region is equal to 4000mmFinite element method is adopted to calculate these nineexamples by using ANSYS in which the concrete part ofcomposite beam is substituted by lateral restraint and the steelI-beam is simulated by SHELL 43 The method proposed byBritish Steel Construction Institution [4] and the bucklingmodel method [15] are also used to calculate the bucklingloads The obtained results are shown in Table 2
As shown in Table 2 the results obtained by bucklingmodel method [15] tend to be excessively unsafe The resultsby using the method proposed by British Steel ConstructionInstitution [4] are too conservative However the results ofthe proposed method are in good agreement with thoseof ANSYS which means that the proposed method is areasonable and effective method
5 Conclusions
Based on energy method a comprehensive and intensivestudy on rotation restraining rigidity 119896
120601and lateral restrain-
ing rigidity 119896119909which steel beam web to bottom plate of
steel-concrete composite beam in negative moment region isperformed in this paper The expressions of lateral bendingbuckling stress lateral bending and torsional buckling stressand buckling moments of steel-concrete composite beam innegative moment region are deduced Some conclusions aredrawn as follows
(1) Both the rotation constraint rigidity 119896120601and the lateral
constraint rigidity 119896119909show a linear relationship with
longitudinal compressive stress 1205901 at bottom flange
(2) The rotation constraint rigidity 119896120601and the lateral
constraint rigidity 119896119909could be negative When the
rotation constraint rigidity and the lateral constraintrigidity are negative that means the bottom flange ofsteel beam can be restrained by rotation or lateralconstraint when steel beam was in negative momentregion buckling
(3) Since ℎ2119908119896119909119896120593is not infinitesimal the lateral con-
straint rigidity of bottom flange to web cannot beneglected In other words in the calculation theequation 119896
119909= 0 cannot be used Therefore it is
proved theoretically that the lateral constraint rigidityof bottom flange to web cannot be approximated to bezero This point is different with the steel structure
(4) The results obtained by literatures [4 15] showsome deviations with the ANSYS results But theresults by the presented expressions agree well withthe ANSYS results The reason is that the proposedexpressions consider the lateral bending buckling andlateral bending-torsion buckling simultaneously Theproposed method is more reasonable and clearer inphysical concepts in comparison with those methodswhich consider only one buckling mode Besides theproposed expressions are more concise and suitablefor the engineering application
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge financial supportfrom Projects (51078355) supported by National NaturalScience Foundation of China Projects (IRT1296) supportedby the Program for Changjiang Scholars and InnovativeResearch Team in University (PCSIRT) and Project (12K104)supported by Scientific Research Fund of Hunan ProvincialEducation Department China Dr Jing-Jing Qi and DrWang-Bao Zhou have tried their best to help the authorsTheauthors express their sincere gratitude to Dr Jing-Jing Qi andDr Wang-Bao Zhou
References
[1] M Ma and O Hughes ldquoLateral distortional buckling ofmonosymmetric I-beams under distributed vertical loadrdquoThin-Walled Structures vol 26 no 2 pp 123ndash143 1996
[2] M A Bradford ldquoLateral-distortional buckling of continuouslyrestrained columnsrdquo Journal of Constructional Steel Researchvol 42 no 2 pp 121ndash139 1997
8 Advances in Materials Science and Engineering
[3] M L-H Ng and H R Ronagh ldquoAn analytical solution forthe elastic lateral-distortional buckling of I-section beamsrdquoAdvances in Structural Engineering vol 7 no 2 pp 189ndash2002004
[4] R M Lawson and J W RaekhamDesign of Haunched Compos-ite Beams in Buildings Steel Construction Institute 1989
[5] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987
[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985
[7] P Goltermann and S E Svensson ldquoLateral distortional buck-ling predicting elastic critical stressrdquo Journal of StructuralEngineering vol 114 no 7 pp 1606ndash1625 1988
[8] H R Ronagh ldquoProgress in the methods of analysis of restricteddistortional buckling of composite bridge girdersrdquo Progress inStructural Engineering and Materials vol 3 no 2 pp 141ndash1482001
[9] Z Diansheng and Y Xiaomin ldquoStudy on beam local buckling ofcold-formed thin-wall steel and concrete compositerdquo IndustrialArchitecture vol 38 supplement pp 539ndash542 2008
[10] L-Z Jiang L-J Zeng and L-L Sun ldquoSteel-concrete compositecontinuous beam web local buckling analysisrdquo Journal of CivilEngineering and Architecture vol 26 no 3 pp 27ndash31 2009
[11] L Jiang and L Sun ldquoThe lateral buckling of steel-concrete com-posite box-beamsrdquo Journal of Huazhong University of Scienceand Technology (Urban Science Edition) vol 25 no 3 pp 5ndash92008
[12] J Ye and W Chen ldquoBeam with elastic constraints of I-shapedsteelmdashconcrete composite distortional bucklingrdquo Journal ofBuilding Structures vol 32 no 6 pp 82ndash91 2011
[13] W-B Zhou L-Z Jiang and Z-W Yu ldquoAnalysis of beamnegative moment zone of the elastic distortional buckling ofsteel-concrete compositerdquo Journal of Central South University(Natural Science Edition) vol 43 no 6 pp 2316ndash2323 2012
[14] W-B Zhou L-Z Jiang and Z-W Yu ldquoThe calculation formulaof steel concrete composite beams under negative bendingzonedistortional buckling momentrdquo Chinese Journal of Compu-tational Mechanics vol 29 no 3 pp 446ndash451 2012
[15] P Zhu Steel-Concrete Composite Beam Design Theory ChinaArchitecture amp Building Press Beijing China 1989
[16] J Yao and J-G Teng ldquoWeb rotational restraint in elasticdistortional buckling of cold-formed lipped channel sectionsrdquoEngineering Mechanics vol 25 no 4 pp 65ndash69 2008
[17] W-B Zhou L-Z Jiang and Z-W Yu ldquoClosed-form solutionfor shear lag effects of steel-concrete composite box beamsconsidering shear deformation and sliprdquo Journal of CentralSouth University vol 19 no 10 pp 2976ndash2982 2012
[18] W-B Zhou L-Z Jiang and Z-W Yu ldquoClosed-form solutionto thin-walled box girders considering the effects of sheardeformation and shear lagrdquo Journal of Central South Universityvol 19 no 9 pp 2650ndash2655 2012
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Nano
materials
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Journal ofNanomaterials
Advances in Materials Science and Engineering 5
(3) The ratios of the first term second term and thirdterm on the right side of (21) and (12) are
(13120590119905119908ℎ11990835) (120587120582)2
(120590119905119908ℎ3
119908105) (120587120582)2
=
39ℎ2
119908
(31205901119905119908ℎ2
11990835119910119888) (120587120582)
2
(1205901119905119908ℎ4
119908280119910
119888) (120587120582)
2=
24
ℎ2
119908
119863 [12ℎ3
119908+ (125ℎ
119908) (120587120582)
2
+ (13ℎ11990835) (120587120582)
4
]
119863 [4ℎ119908+ (4ℎ11990815) (120587120582)
2
+ (ℎ3
119908105) (120587120582)
4
]
asymp
3
ℎ2
119908
(22)
From (22) ℎ2119908119896119909119896120593is not an infinitesimal value so the
lateral constraint rigidity of bottom flange to web cannot bedisregarded Namely in the calculation the equation 119896
119909= 0
is not available Therefore the lateral constraint rigidity ofbottom flange to web cannot be approximated by zero Thisis different from [16] in which the lateral constraint rigiditywhich the cold-formed thin-walled lipped channel steel websto the top and bottom flange is taken as zero
4 Theoretical Derivation of Critical Moment
41 Derivation of Critical Moment The lateral constraintrigidity and rotation constraint rigidity which steel beamwebto bottom flange can be simplified respectively as
119896119909= 1205721120573120590+119863120572
2 (23)
119896120593= 1205723120573120590+119863120572
4 (24)
120573 = (
120587
120582
)
2
(25)
1205721= 119905119908ℎ119908(
0086ℎ119908
119910119888
minus 037) (26a)
1205722=
12
ℎ3
119908
+
24120573
ℎ119908
+ 037ℎ1199081205732
(26b)
1205723= 119905119908ℎ3
119908(
00036ℎ119908
119910119888
minus 00095) (26c)
1205724=
4
ℎ119908
+ 027ℎ119908120573+ 00095ℎ
3
1199081205732
(26d)
120590 =
119872119909119910119888
119868
=
119875
119860119891
(27)
Therein119863 = 1198641199053
119908[12(1minus120583
2
)] 120583 is Poissonrsquos ratio of steel120590 represents the compressive stress of bottom flange 119875 refersto the compressive force of bottom flange minus119910
119888is the gravity
centre coordinate of steel beam total cross section 119868 is theinertia moment of steel beam 120582 = 119897119899 119899 is the buckling half-wave number 119860
119891is the area of bottom flange
As shown in Figure 2 the thin-walled member is doublysymmetric along 119909-axis and 119910-axis and the centre of origin119874 coincides with the flexural center The displacements oforigin 119874 in 119909 direction and 119910 direction are denoted as 119906 andV respectively Because the rigidity in 119910 direction is infinityV is equal to zero The equivalent distributed forces causedby elastic medium as a result of displacements of thin-walledmember can be written as
119903119909= 119896119909119906
119903119910= 119896119910V
(28)
where 119896119909and 119896119910are the lateral and vertical constraint rigidity
which web to bottom flange respectivelyThe torsional angle which the member rotates around the
bending center is assumed to be 120593The equivalent distributedmoment of torsional thin-walled member induced by equiv-alent spring is
119898 = 119896120593120593 (29)
Neutral balance differential equation of thin-walledmember can be expressed as [13 14]
119864119868119910119906119868119881
+119875 (11990610158401015840
+11991011988612059310158401015840
) + 119896119909[119906 minus (119910
119889minus119910119886) 120593] = 0
119864119868119909V119868119881 +119875 (V10158401015840 minus119909
11988612059310158401015840
) + 119896119910[V+ (119909
119889minus119909119886) 120593] = 0
119864119868119908120593119868119881
+ (11990320119875minus119866119869) 120593
10158401015840
minus119875 (119909119886V10158401015840 minus119910
11988611990610158401015840
)
minus 119896119909[119906 minus (119910
119889minus119910119886) 120593] (119910
119889minus119910119886)
+ 119896119910[V+ (119909
119889minus119909119886) 120593] (119909
119889minus119909119886) + 119896120593120593 = 0
(30)
Therein 119868119910= 119905119891119887311989112 119868
119909= 119887119891119905311989112 119869 = 119887
11989111990531198913 11990320 =
1199092119886+119910
2119886+(119868119909+119868119910)119860119904 119860119904being the area of steel beam 119909
119886is the
horizontal coordinate of the bottom flange section bendingcentre 119909
119886= 0 119910
119886is the vertical coordinate of the bottom
flange section bending centre 119910119886= 0 119909
119889is the horizontal
coordinate of the bottom flange section rotation axis 119909119889=
0 119910119889is the vertical coordinate of the bottom flange section
rotation axis 119910119889= 0 119868119908is the fan-shaped inertia moment of
bottom flange section 119868119908= 0 119864 is the tensile elastic modulus
of steel 119866 is the shear elastic modulus of steelWith substitution of 119910
119886= 0 119910
119889= 0 119909
119886= 0 V = 0 119868
119908= 0
119896119909= 0 119909
119886= 0 119910
119886= 0 and 119910
119889= 0 into (30) one can obtain
119864119868119910119868119906119868119881
+11991011988811986011989111987211990911990610158401015840
+ (1205721120573119910119888119872119909 + 1198681198631205722) 119906 = 0 (31)
119896119910[V+ (119909
119889minus119909119886) 120593] = 0 (32)
(11990320119910119888119860119891119872119909 minus119866119869119868) 120593
10158401015840
+ (1205723120573119910119888119872119909 + 1198681198631205724) 120593 = 0 (33)
When the steel-concrete composite beam in negativemoment region bears lateral bending buckling its neutralbalance equation is shown in (31) and the correspondingboundary conditions are
[119906]119911=0119897 = 0
[11990610158401015840
]119911=0119897 = 0
(34)
6 Advances in Materials Science and Engineering
Assuming 119906 = 119860 sin(radic120573119911) which satisfies the boundaryconditions according to Galerkinrsquos method
1205732minus
119872119909119910119888119860119891
119864119868119910119868
120573 +
(1205721120573119872119909119910119888119868 + 119863120572
2)
119864119868119910
= 0 (35)
By solving (35)
1198721198881199031 =
119864119868119910120573 + 1198631205722120573
(119860119891minus 1205721) 119910119888
119868 (36)
Due to 1198891198721198881199031119889120573 = 0 we obtained the following
1205731198881199031 =
346
radic119864119868119910ℎ3119908119863 + 037ℎ4
119908
1198991198881199031 =
119897radic1205731198881199031120587
(37)
If 119897radic1205731198881199031120587 is an integer substitution of 120573
1198881199031into (36) leads
to the lateral bending critical moment If 119897radic1205731198881199031120587 is not an
integer the two values of 1205731198881199031
which makes 119897radic1205731198881199031120587 be two
integers most near to 119897radic1205731198881199031120587 are then substituted into (36)
and the smaller value is chosen to be the lateral bendingcritical moment
When the steel beam bottom flange of steel-concretecomposite beam in negativemoment region suffers rotationalbuckling the steel-concrete composite beam will yield lateralbending and torsional bucklingTheneutral balance equationis seen in (33) and the boundary conditions are
[120593]119911=0119897 = 0
[12059310158401015840
]119911=0119897 = 0
(38)
Assume 1205852 = (1205723120573119872119909119910119888119868 + 119863120572
4)(1199032
0119872119909119910119888119860119891119868 minus 119866119869)
Solving (33) we obtained the following
120593 = 119860 sin 120585119911 +119861 cos 120585119911 (39)
Substituting (39) into (38) and with 120593 = 0 one can have
sin 120585119897 = 0 (40)
120585 =
119899120587
119897
= radic120573 (41)
(1205723120573119872119909119910119888119868 + 1198631205724)
(11990320119872119909119910119888119860119891119868 minus 119866119869)
= 120573 (42)
Solving (42)
1198721198881199032 =
1198631205724120573 + 119866119869
(11990320119860119891 minus 1205723) 119910119888
119868 (43)
Because of the fact that 1198891198721198881199032119889120573 = 0 then
1205731198881199032
=
205
ℎ2
119908
1198991198881199032
=
119897radic1205731198881199032
120587
(44)
Table 1 Geometric dimension of examples
Number of example ℎ119908mm 119905
119908mm 119905
119891mm 119887
119891mm
1 1886 7 114 1002 1886 7 104 1003 1886 7 94 1004 1886 8 94 1005 1886 9 94 1006 150 8 94 1007 200 8 94 1008 200 8 94 809 200 8 94 120
When 119897radic1205731198881199032120587 is an integer the substitution of 120573
1198881199032into
(43) gets the lateral bending critical momentWhen 119897radic1205731198881199032120587
is not an integer a similar calculation mentioned before istaken to obtain the lateral bending critical moment
After getting the lateral bending buckling criticalmoment1198721198881199031
and the bending and torsional buckling critical moment1198721198881199032
the smaller one of these two is taken as the buckling loadof composite beam in negativemoment regionThe analyticalresults indicate that the composite beam in negative momentregion yields as a result of lateral bending and torsionalbuckling acting together Therefore in [11 12] the methodin which only one case is considered is questionable Thepresented work is an improvement to them
To sum up the calculation formula of buckling momentcan be expressed as
119872119888119903
= min1198641198681199101205731198881199031 + 11986312057221205731198881199031
(119860119891minus 1205721) 119910119888
119868
11986312057241205731198881199032 + 119866119869
(11990320119860119891 minus 1205723) 119910119888
119868
1205731198881199031
=
346
radic119864119868119910ℎ3
119908119863 + 037ℎ
4
119908
1205731198881199032
=
205
ℎ2
119908
1205721= 119905119908ℎ119908(
0086ℎ119908
119910119888
minus 037)
1205722=
12
ℎ3
119908
+
241205731198881199031
ℎ119908
+ 037ℎ1199081205732
1198881199031
1205723= 119905119908ℎ3
119908(
00036ℎ119908
119910119888
minus 00095)
1205724=
4
ℎ119908
+ 027ℎ1199081205731198881199032+ 00095ℎ
3
1199081205732
1198881199032
(45)
When 119897radic1205731198881199031120587 is not an integer substituting 120573
1198881199031that
corresponds to two integers of the left and right side of119897radic1205731198881199031120587 into (36) the smaller resulting value of 120573
1198881199031is
the desired value And when 119897radic1205731198881199032120587 is not an integer
substituting 1205731198881199032
that corresponds to two integers of the left
Advances in Materials Science and Engineering 7
Table 2 Calculation results
Number of example ANSYSMPa Literature method [4]MPa Literature method [15]MPa The proposed methodMPa1 581 times 102 221 times 102 838 times 102 583 times 102
2 546 times 102 205 times 102 832 times 102 551 times 102
3 513 times 102 187 times 102 827 times 102 518 times 102
4 613 times 102 213 times 102 122 times 102 628 times 102
5 703 times 102 246 times 102 173 times 102 769 times 102
6 632 times 102 256 times 102 192 times 102 629 times 102
7 608 times 102 206 times 102 109 times 102 609 times 102
8 480 times 102 138 times 102 108 times 102 489 times 102
9 789 times 102 292 times 102 104 times 102 824 times 102
and right side of 119897radic1205731198881199032120587 into (43) the smaller resulting
value of 1205731198881199032
is the desired value
42 Practical Example Analysis Nine cases of I-shape steel-concrete composite beam are shown in Table 1 The length ofspecimen in negative moment region is equal to 4000mmFinite element method is adopted to calculate these nineexamples by using ANSYS in which the concrete part ofcomposite beam is substituted by lateral restraint and the steelI-beam is simulated by SHELL 43 The method proposed byBritish Steel Construction Institution [4] and the bucklingmodel method [15] are also used to calculate the bucklingloads The obtained results are shown in Table 2
As shown in Table 2 the results obtained by bucklingmodel method [15] tend to be excessively unsafe The resultsby using the method proposed by British Steel ConstructionInstitution [4] are too conservative However the results ofthe proposed method are in good agreement with thoseof ANSYS which means that the proposed method is areasonable and effective method
5 Conclusions
Based on energy method a comprehensive and intensivestudy on rotation restraining rigidity 119896
120601and lateral restrain-
ing rigidity 119896119909which steel beam web to bottom plate of
steel-concrete composite beam in negative moment region isperformed in this paper The expressions of lateral bendingbuckling stress lateral bending and torsional buckling stressand buckling moments of steel-concrete composite beam innegative moment region are deduced Some conclusions aredrawn as follows
(1) Both the rotation constraint rigidity 119896120601and the lateral
constraint rigidity 119896119909show a linear relationship with
longitudinal compressive stress 1205901 at bottom flange
(2) The rotation constraint rigidity 119896120601and the lateral
constraint rigidity 119896119909could be negative When the
rotation constraint rigidity and the lateral constraintrigidity are negative that means the bottom flange ofsteel beam can be restrained by rotation or lateralconstraint when steel beam was in negative momentregion buckling
(3) Since ℎ2119908119896119909119896120593is not infinitesimal the lateral con-
straint rigidity of bottom flange to web cannot beneglected In other words in the calculation theequation 119896
119909= 0 cannot be used Therefore it is
proved theoretically that the lateral constraint rigidityof bottom flange to web cannot be approximated to bezero This point is different with the steel structure
(4) The results obtained by literatures [4 15] showsome deviations with the ANSYS results But theresults by the presented expressions agree well withthe ANSYS results The reason is that the proposedexpressions consider the lateral bending buckling andlateral bending-torsion buckling simultaneously Theproposed method is more reasonable and clearer inphysical concepts in comparison with those methodswhich consider only one buckling mode Besides theproposed expressions are more concise and suitablefor the engineering application
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge financial supportfrom Projects (51078355) supported by National NaturalScience Foundation of China Projects (IRT1296) supportedby the Program for Changjiang Scholars and InnovativeResearch Team in University (PCSIRT) and Project (12K104)supported by Scientific Research Fund of Hunan ProvincialEducation Department China Dr Jing-Jing Qi and DrWang-Bao Zhou have tried their best to help the authorsTheauthors express their sincere gratitude to Dr Jing-Jing Qi andDr Wang-Bao Zhou
References
[1] M Ma and O Hughes ldquoLateral distortional buckling ofmonosymmetric I-beams under distributed vertical loadrdquoThin-Walled Structures vol 26 no 2 pp 123ndash143 1996
[2] M A Bradford ldquoLateral-distortional buckling of continuouslyrestrained columnsrdquo Journal of Constructional Steel Researchvol 42 no 2 pp 121ndash139 1997
8 Advances in Materials Science and Engineering
[3] M L-H Ng and H R Ronagh ldquoAn analytical solution forthe elastic lateral-distortional buckling of I-section beamsrdquoAdvances in Structural Engineering vol 7 no 2 pp 189ndash2002004
[4] R M Lawson and J W RaekhamDesign of Haunched Compos-ite Beams in Buildings Steel Construction Institute 1989
[5] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987
[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985
[7] P Goltermann and S E Svensson ldquoLateral distortional buck-ling predicting elastic critical stressrdquo Journal of StructuralEngineering vol 114 no 7 pp 1606ndash1625 1988
[8] H R Ronagh ldquoProgress in the methods of analysis of restricteddistortional buckling of composite bridge girdersrdquo Progress inStructural Engineering and Materials vol 3 no 2 pp 141ndash1482001
[9] Z Diansheng and Y Xiaomin ldquoStudy on beam local buckling ofcold-formed thin-wall steel and concrete compositerdquo IndustrialArchitecture vol 38 supplement pp 539ndash542 2008
[10] L-Z Jiang L-J Zeng and L-L Sun ldquoSteel-concrete compositecontinuous beam web local buckling analysisrdquo Journal of CivilEngineering and Architecture vol 26 no 3 pp 27ndash31 2009
[11] L Jiang and L Sun ldquoThe lateral buckling of steel-concrete com-posite box-beamsrdquo Journal of Huazhong University of Scienceand Technology (Urban Science Edition) vol 25 no 3 pp 5ndash92008
[12] J Ye and W Chen ldquoBeam with elastic constraints of I-shapedsteelmdashconcrete composite distortional bucklingrdquo Journal ofBuilding Structures vol 32 no 6 pp 82ndash91 2011
[13] W-B Zhou L-Z Jiang and Z-W Yu ldquoAnalysis of beamnegative moment zone of the elastic distortional buckling ofsteel-concrete compositerdquo Journal of Central South University(Natural Science Edition) vol 43 no 6 pp 2316ndash2323 2012
[14] W-B Zhou L-Z Jiang and Z-W Yu ldquoThe calculation formulaof steel concrete composite beams under negative bendingzonedistortional buckling momentrdquo Chinese Journal of Compu-tational Mechanics vol 29 no 3 pp 446ndash451 2012
[15] P Zhu Steel-Concrete Composite Beam Design Theory ChinaArchitecture amp Building Press Beijing China 1989
[16] J Yao and J-G Teng ldquoWeb rotational restraint in elasticdistortional buckling of cold-formed lipped channel sectionsrdquoEngineering Mechanics vol 25 no 4 pp 65ndash69 2008
[17] W-B Zhou L-Z Jiang and Z-W Yu ldquoClosed-form solutionfor shear lag effects of steel-concrete composite box beamsconsidering shear deformation and sliprdquo Journal of CentralSouth University vol 19 no 10 pp 2976ndash2982 2012
[18] W-B Zhou L-Z Jiang and Z-W Yu ldquoClosed-form solutionto thin-walled box girders considering the effects of sheardeformation and shear lagrdquo Journal of Central South Universityvol 19 no 9 pp 2650ndash2655 2012
Submit your manuscripts athttpwwwhindawicom
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Advances in
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BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
6 Advances in Materials Science and Engineering
Assuming 119906 = 119860 sin(radic120573119911) which satisfies the boundaryconditions according to Galerkinrsquos method
1205732minus
119872119909119910119888119860119891
119864119868119910119868
120573 +
(1205721120573119872119909119910119888119868 + 119863120572
2)
119864119868119910
= 0 (35)
By solving (35)
1198721198881199031 =
119864119868119910120573 + 1198631205722120573
(119860119891minus 1205721) 119910119888
119868 (36)
Due to 1198891198721198881199031119889120573 = 0 we obtained the following
1205731198881199031 =
346
radic119864119868119910ℎ3119908119863 + 037ℎ4
119908
1198991198881199031 =
119897radic1205731198881199031120587
(37)
If 119897radic1205731198881199031120587 is an integer substitution of 120573
1198881199031into (36) leads
to the lateral bending critical moment If 119897radic1205731198881199031120587 is not an
integer the two values of 1205731198881199031
which makes 119897radic1205731198881199031120587 be two
integers most near to 119897radic1205731198881199031120587 are then substituted into (36)
and the smaller value is chosen to be the lateral bendingcritical moment
When the steel beam bottom flange of steel-concretecomposite beam in negativemoment region suffers rotationalbuckling the steel-concrete composite beam will yield lateralbending and torsional bucklingTheneutral balance equationis seen in (33) and the boundary conditions are
[120593]119911=0119897 = 0
[12059310158401015840
]119911=0119897 = 0
(38)
Assume 1205852 = (1205723120573119872119909119910119888119868 + 119863120572
4)(1199032
0119872119909119910119888119860119891119868 minus 119866119869)
Solving (33) we obtained the following
120593 = 119860 sin 120585119911 +119861 cos 120585119911 (39)
Substituting (39) into (38) and with 120593 = 0 one can have
sin 120585119897 = 0 (40)
120585 =
119899120587
119897
= radic120573 (41)
(1205723120573119872119909119910119888119868 + 1198631205724)
(11990320119872119909119910119888119860119891119868 minus 119866119869)
= 120573 (42)
Solving (42)
1198721198881199032 =
1198631205724120573 + 119866119869
(11990320119860119891 minus 1205723) 119910119888
119868 (43)
Because of the fact that 1198891198721198881199032119889120573 = 0 then
1205731198881199032
=
205
ℎ2
119908
1198991198881199032
=
119897radic1205731198881199032
120587
(44)
Table 1 Geometric dimension of examples
Number of example ℎ119908mm 119905
119908mm 119905
119891mm 119887
119891mm
1 1886 7 114 1002 1886 7 104 1003 1886 7 94 1004 1886 8 94 1005 1886 9 94 1006 150 8 94 1007 200 8 94 1008 200 8 94 809 200 8 94 120
When 119897radic1205731198881199032120587 is an integer the substitution of 120573
1198881199032into
(43) gets the lateral bending critical momentWhen 119897radic1205731198881199032120587
is not an integer a similar calculation mentioned before istaken to obtain the lateral bending critical moment
After getting the lateral bending buckling criticalmoment1198721198881199031
and the bending and torsional buckling critical moment1198721198881199032
the smaller one of these two is taken as the buckling loadof composite beam in negativemoment regionThe analyticalresults indicate that the composite beam in negative momentregion yields as a result of lateral bending and torsionalbuckling acting together Therefore in [11 12] the methodin which only one case is considered is questionable Thepresented work is an improvement to them
To sum up the calculation formula of buckling momentcan be expressed as
119872119888119903
= min1198641198681199101205731198881199031 + 11986312057221205731198881199031
(119860119891minus 1205721) 119910119888
119868
11986312057241205731198881199032 + 119866119869
(11990320119860119891 minus 1205723) 119910119888
119868
1205731198881199031
=
346
radic119864119868119910ℎ3
119908119863 + 037ℎ
4
119908
1205731198881199032
=
205
ℎ2
119908
1205721= 119905119908ℎ119908(
0086ℎ119908
119910119888
minus 037)
1205722=
12
ℎ3
119908
+
241205731198881199031
ℎ119908
+ 037ℎ1199081205732
1198881199031
1205723= 119905119908ℎ3
119908(
00036ℎ119908
119910119888
minus 00095)
1205724=
4
ℎ119908
+ 027ℎ1199081205731198881199032+ 00095ℎ
3
1199081205732
1198881199032
(45)
When 119897radic1205731198881199031120587 is not an integer substituting 120573
1198881199031that
corresponds to two integers of the left and right side of119897radic1205731198881199031120587 into (36) the smaller resulting value of 120573
1198881199031is
the desired value And when 119897radic1205731198881199032120587 is not an integer
substituting 1205731198881199032
that corresponds to two integers of the left
Advances in Materials Science and Engineering 7
Table 2 Calculation results
Number of example ANSYSMPa Literature method [4]MPa Literature method [15]MPa The proposed methodMPa1 581 times 102 221 times 102 838 times 102 583 times 102
2 546 times 102 205 times 102 832 times 102 551 times 102
3 513 times 102 187 times 102 827 times 102 518 times 102
4 613 times 102 213 times 102 122 times 102 628 times 102
5 703 times 102 246 times 102 173 times 102 769 times 102
6 632 times 102 256 times 102 192 times 102 629 times 102
7 608 times 102 206 times 102 109 times 102 609 times 102
8 480 times 102 138 times 102 108 times 102 489 times 102
9 789 times 102 292 times 102 104 times 102 824 times 102
and right side of 119897radic1205731198881199032120587 into (43) the smaller resulting
value of 1205731198881199032
is the desired value
42 Practical Example Analysis Nine cases of I-shape steel-concrete composite beam are shown in Table 1 The length ofspecimen in negative moment region is equal to 4000mmFinite element method is adopted to calculate these nineexamples by using ANSYS in which the concrete part ofcomposite beam is substituted by lateral restraint and the steelI-beam is simulated by SHELL 43 The method proposed byBritish Steel Construction Institution [4] and the bucklingmodel method [15] are also used to calculate the bucklingloads The obtained results are shown in Table 2
As shown in Table 2 the results obtained by bucklingmodel method [15] tend to be excessively unsafe The resultsby using the method proposed by British Steel ConstructionInstitution [4] are too conservative However the results ofthe proposed method are in good agreement with thoseof ANSYS which means that the proposed method is areasonable and effective method
5 Conclusions
Based on energy method a comprehensive and intensivestudy on rotation restraining rigidity 119896
120601and lateral restrain-
ing rigidity 119896119909which steel beam web to bottom plate of
steel-concrete composite beam in negative moment region isperformed in this paper The expressions of lateral bendingbuckling stress lateral bending and torsional buckling stressand buckling moments of steel-concrete composite beam innegative moment region are deduced Some conclusions aredrawn as follows
(1) Both the rotation constraint rigidity 119896120601and the lateral
constraint rigidity 119896119909show a linear relationship with
longitudinal compressive stress 1205901 at bottom flange
(2) The rotation constraint rigidity 119896120601and the lateral
constraint rigidity 119896119909could be negative When the
rotation constraint rigidity and the lateral constraintrigidity are negative that means the bottom flange ofsteel beam can be restrained by rotation or lateralconstraint when steel beam was in negative momentregion buckling
(3) Since ℎ2119908119896119909119896120593is not infinitesimal the lateral con-
straint rigidity of bottom flange to web cannot beneglected In other words in the calculation theequation 119896
119909= 0 cannot be used Therefore it is
proved theoretically that the lateral constraint rigidityof bottom flange to web cannot be approximated to bezero This point is different with the steel structure
(4) The results obtained by literatures [4 15] showsome deviations with the ANSYS results But theresults by the presented expressions agree well withthe ANSYS results The reason is that the proposedexpressions consider the lateral bending buckling andlateral bending-torsion buckling simultaneously Theproposed method is more reasonable and clearer inphysical concepts in comparison with those methodswhich consider only one buckling mode Besides theproposed expressions are more concise and suitablefor the engineering application
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge financial supportfrom Projects (51078355) supported by National NaturalScience Foundation of China Projects (IRT1296) supportedby the Program for Changjiang Scholars and InnovativeResearch Team in University (PCSIRT) and Project (12K104)supported by Scientific Research Fund of Hunan ProvincialEducation Department China Dr Jing-Jing Qi and DrWang-Bao Zhou have tried their best to help the authorsTheauthors express their sincere gratitude to Dr Jing-Jing Qi andDr Wang-Bao Zhou
References
[1] M Ma and O Hughes ldquoLateral distortional buckling ofmonosymmetric I-beams under distributed vertical loadrdquoThin-Walled Structures vol 26 no 2 pp 123ndash143 1996
[2] M A Bradford ldquoLateral-distortional buckling of continuouslyrestrained columnsrdquo Journal of Constructional Steel Researchvol 42 no 2 pp 121ndash139 1997
8 Advances in Materials Science and Engineering
[3] M L-H Ng and H R Ronagh ldquoAn analytical solution forthe elastic lateral-distortional buckling of I-section beamsrdquoAdvances in Structural Engineering vol 7 no 2 pp 189ndash2002004
[4] R M Lawson and J W RaekhamDesign of Haunched Compos-ite Beams in Buildings Steel Construction Institute 1989
[5] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987
[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985
[7] P Goltermann and S E Svensson ldquoLateral distortional buck-ling predicting elastic critical stressrdquo Journal of StructuralEngineering vol 114 no 7 pp 1606ndash1625 1988
[8] H R Ronagh ldquoProgress in the methods of analysis of restricteddistortional buckling of composite bridge girdersrdquo Progress inStructural Engineering and Materials vol 3 no 2 pp 141ndash1482001
[9] Z Diansheng and Y Xiaomin ldquoStudy on beam local buckling ofcold-formed thin-wall steel and concrete compositerdquo IndustrialArchitecture vol 38 supplement pp 539ndash542 2008
[10] L-Z Jiang L-J Zeng and L-L Sun ldquoSteel-concrete compositecontinuous beam web local buckling analysisrdquo Journal of CivilEngineering and Architecture vol 26 no 3 pp 27ndash31 2009
[11] L Jiang and L Sun ldquoThe lateral buckling of steel-concrete com-posite box-beamsrdquo Journal of Huazhong University of Scienceand Technology (Urban Science Edition) vol 25 no 3 pp 5ndash92008
[12] J Ye and W Chen ldquoBeam with elastic constraints of I-shapedsteelmdashconcrete composite distortional bucklingrdquo Journal ofBuilding Structures vol 32 no 6 pp 82ndash91 2011
[13] W-B Zhou L-Z Jiang and Z-W Yu ldquoAnalysis of beamnegative moment zone of the elastic distortional buckling ofsteel-concrete compositerdquo Journal of Central South University(Natural Science Edition) vol 43 no 6 pp 2316ndash2323 2012
[14] W-B Zhou L-Z Jiang and Z-W Yu ldquoThe calculation formulaof steel concrete composite beams under negative bendingzonedistortional buckling momentrdquo Chinese Journal of Compu-tational Mechanics vol 29 no 3 pp 446ndash451 2012
[15] P Zhu Steel-Concrete Composite Beam Design Theory ChinaArchitecture amp Building Press Beijing China 1989
[16] J Yao and J-G Teng ldquoWeb rotational restraint in elasticdistortional buckling of cold-formed lipped channel sectionsrdquoEngineering Mechanics vol 25 no 4 pp 65ndash69 2008
[17] W-B Zhou L-Z Jiang and Z-W Yu ldquoClosed-form solutionfor shear lag effects of steel-concrete composite box beamsconsidering shear deformation and sliprdquo Journal of CentralSouth University vol 19 no 10 pp 2976ndash2982 2012
[18] W-B Zhou L-Z Jiang and Z-W Yu ldquoClosed-form solutionto thin-walled box girders considering the effects of sheardeformation and shear lagrdquo Journal of Central South Universityvol 19 no 9 pp 2650ndash2655 2012
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Advances in Materials Science and Engineering 7
Table 2 Calculation results
Number of example ANSYSMPa Literature method [4]MPa Literature method [15]MPa The proposed methodMPa1 581 times 102 221 times 102 838 times 102 583 times 102
2 546 times 102 205 times 102 832 times 102 551 times 102
3 513 times 102 187 times 102 827 times 102 518 times 102
4 613 times 102 213 times 102 122 times 102 628 times 102
5 703 times 102 246 times 102 173 times 102 769 times 102
6 632 times 102 256 times 102 192 times 102 629 times 102
7 608 times 102 206 times 102 109 times 102 609 times 102
8 480 times 102 138 times 102 108 times 102 489 times 102
9 789 times 102 292 times 102 104 times 102 824 times 102
and right side of 119897radic1205731198881199032120587 into (43) the smaller resulting
value of 1205731198881199032
is the desired value
42 Practical Example Analysis Nine cases of I-shape steel-concrete composite beam are shown in Table 1 The length ofspecimen in negative moment region is equal to 4000mmFinite element method is adopted to calculate these nineexamples by using ANSYS in which the concrete part ofcomposite beam is substituted by lateral restraint and the steelI-beam is simulated by SHELL 43 The method proposed byBritish Steel Construction Institution [4] and the bucklingmodel method [15] are also used to calculate the bucklingloads The obtained results are shown in Table 2
As shown in Table 2 the results obtained by bucklingmodel method [15] tend to be excessively unsafe The resultsby using the method proposed by British Steel ConstructionInstitution [4] are too conservative However the results ofthe proposed method are in good agreement with thoseof ANSYS which means that the proposed method is areasonable and effective method
5 Conclusions
Based on energy method a comprehensive and intensivestudy on rotation restraining rigidity 119896
120601and lateral restrain-
ing rigidity 119896119909which steel beam web to bottom plate of
steel-concrete composite beam in negative moment region isperformed in this paper The expressions of lateral bendingbuckling stress lateral bending and torsional buckling stressand buckling moments of steel-concrete composite beam innegative moment region are deduced Some conclusions aredrawn as follows
(1) Both the rotation constraint rigidity 119896120601and the lateral
constraint rigidity 119896119909show a linear relationship with
longitudinal compressive stress 1205901 at bottom flange
(2) The rotation constraint rigidity 119896120601and the lateral
constraint rigidity 119896119909could be negative When the
rotation constraint rigidity and the lateral constraintrigidity are negative that means the bottom flange ofsteel beam can be restrained by rotation or lateralconstraint when steel beam was in negative momentregion buckling
(3) Since ℎ2119908119896119909119896120593is not infinitesimal the lateral con-
straint rigidity of bottom flange to web cannot beneglected In other words in the calculation theequation 119896
119909= 0 cannot be used Therefore it is
proved theoretically that the lateral constraint rigidityof bottom flange to web cannot be approximated to bezero This point is different with the steel structure
(4) The results obtained by literatures [4 15] showsome deviations with the ANSYS results But theresults by the presented expressions agree well withthe ANSYS results The reason is that the proposedexpressions consider the lateral bending buckling andlateral bending-torsion buckling simultaneously Theproposed method is more reasonable and clearer inphysical concepts in comparison with those methodswhich consider only one buckling mode Besides theproposed expressions are more concise and suitablefor the engineering application
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge financial supportfrom Projects (51078355) supported by National NaturalScience Foundation of China Projects (IRT1296) supportedby the Program for Changjiang Scholars and InnovativeResearch Team in University (PCSIRT) and Project (12K104)supported by Scientific Research Fund of Hunan ProvincialEducation Department China Dr Jing-Jing Qi and DrWang-Bao Zhou have tried their best to help the authorsTheauthors express their sincere gratitude to Dr Jing-Jing Qi andDr Wang-Bao Zhou
References
[1] M Ma and O Hughes ldquoLateral distortional buckling ofmonosymmetric I-beams under distributed vertical loadrdquoThin-Walled Structures vol 26 no 2 pp 123ndash143 1996
[2] M A Bradford ldquoLateral-distortional buckling of continuouslyrestrained columnsrdquo Journal of Constructional Steel Researchvol 42 no 2 pp 121ndash139 1997
8 Advances in Materials Science and Engineering
[3] M L-H Ng and H R Ronagh ldquoAn analytical solution forthe elastic lateral-distortional buckling of I-section beamsrdquoAdvances in Structural Engineering vol 7 no 2 pp 189ndash2002004
[4] R M Lawson and J W RaekhamDesign of Haunched Compos-ite Beams in Buildings Steel Construction Institute 1989
[5] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987
[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985
[7] P Goltermann and S E Svensson ldquoLateral distortional buck-ling predicting elastic critical stressrdquo Journal of StructuralEngineering vol 114 no 7 pp 1606ndash1625 1988
[8] H R Ronagh ldquoProgress in the methods of analysis of restricteddistortional buckling of composite bridge girdersrdquo Progress inStructural Engineering and Materials vol 3 no 2 pp 141ndash1482001
[9] Z Diansheng and Y Xiaomin ldquoStudy on beam local buckling ofcold-formed thin-wall steel and concrete compositerdquo IndustrialArchitecture vol 38 supplement pp 539ndash542 2008
[10] L-Z Jiang L-J Zeng and L-L Sun ldquoSteel-concrete compositecontinuous beam web local buckling analysisrdquo Journal of CivilEngineering and Architecture vol 26 no 3 pp 27ndash31 2009
[11] L Jiang and L Sun ldquoThe lateral buckling of steel-concrete com-posite box-beamsrdquo Journal of Huazhong University of Scienceand Technology (Urban Science Edition) vol 25 no 3 pp 5ndash92008
[12] J Ye and W Chen ldquoBeam with elastic constraints of I-shapedsteelmdashconcrete composite distortional bucklingrdquo Journal ofBuilding Structures vol 32 no 6 pp 82ndash91 2011
[13] W-B Zhou L-Z Jiang and Z-W Yu ldquoAnalysis of beamnegative moment zone of the elastic distortional buckling ofsteel-concrete compositerdquo Journal of Central South University(Natural Science Edition) vol 43 no 6 pp 2316ndash2323 2012
[14] W-B Zhou L-Z Jiang and Z-W Yu ldquoThe calculation formulaof steel concrete composite beams under negative bendingzonedistortional buckling momentrdquo Chinese Journal of Compu-tational Mechanics vol 29 no 3 pp 446ndash451 2012
[15] P Zhu Steel-Concrete Composite Beam Design Theory ChinaArchitecture amp Building Press Beijing China 1989
[16] J Yao and J-G Teng ldquoWeb rotational restraint in elasticdistortional buckling of cold-formed lipped channel sectionsrdquoEngineering Mechanics vol 25 no 4 pp 65ndash69 2008
[17] W-B Zhou L-Z Jiang and Z-W Yu ldquoClosed-form solutionfor shear lag effects of steel-concrete composite box beamsconsidering shear deformation and sliprdquo Journal of CentralSouth University vol 19 no 10 pp 2976ndash2982 2012
[18] W-B Zhou L-Z Jiang and Z-W Yu ldquoClosed-form solutionto thin-walled box girders considering the effects of sheardeformation and shear lagrdquo Journal of Central South Universityvol 19 no 9 pp 2650ndash2655 2012
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
8 Advances in Materials Science and Engineering
[3] M L-H Ng and H R Ronagh ldquoAn analytical solution forthe elastic lateral-distortional buckling of I-section beamsrdquoAdvances in Structural Engineering vol 7 no 2 pp 189ndash2002004
[4] R M Lawson and J W RaekhamDesign of Haunched Compos-ite Beams in Buildings Steel Construction Institute 1989
[5] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987
[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985
[7] P Goltermann and S E Svensson ldquoLateral distortional buck-ling predicting elastic critical stressrdquo Journal of StructuralEngineering vol 114 no 7 pp 1606ndash1625 1988
[8] H R Ronagh ldquoProgress in the methods of analysis of restricteddistortional buckling of composite bridge girdersrdquo Progress inStructural Engineering and Materials vol 3 no 2 pp 141ndash1482001
[9] Z Diansheng and Y Xiaomin ldquoStudy on beam local buckling ofcold-formed thin-wall steel and concrete compositerdquo IndustrialArchitecture vol 38 supplement pp 539ndash542 2008
[10] L-Z Jiang L-J Zeng and L-L Sun ldquoSteel-concrete compositecontinuous beam web local buckling analysisrdquo Journal of CivilEngineering and Architecture vol 26 no 3 pp 27ndash31 2009
[11] L Jiang and L Sun ldquoThe lateral buckling of steel-concrete com-posite box-beamsrdquo Journal of Huazhong University of Scienceand Technology (Urban Science Edition) vol 25 no 3 pp 5ndash92008
[12] J Ye and W Chen ldquoBeam with elastic constraints of I-shapedsteelmdashconcrete composite distortional bucklingrdquo Journal ofBuilding Structures vol 32 no 6 pp 82ndash91 2011
[13] W-B Zhou L-Z Jiang and Z-W Yu ldquoAnalysis of beamnegative moment zone of the elastic distortional buckling ofsteel-concrete compositerdquo Journal of Central South University(Natural Science Edition) vol 43 no 6 pp 2316ndash2323 2012
[14] W-B Zhou L-Z Jiang and Z-W Yu ldquoThe calculation formulaof steel concrete composite beams under negative bendingzonedistortional buckling momentrdquo Chinese Journal of Compu-tational Mechanics vol 29 no 3 pp 446ndash451 2012
[15] P Zhu Steel-Concrete Composite Beam Design Theory ChinaArchitecture amp Building Press Beijing China 1989
[16] J Yao and J-G Teng ldquoWeb rotational restraint in elasticdistortional buckling of cold-formed lipped channel sectionsrdquoEngineering Mechanics vol 25 no 4 pp 65ndash69 2008
[17] W-B Zhou L-Z Jiang and Z-W Yu ldquoClosed-form solutionfor shear lag effects of steel-concrete composite box beamsconsidering shear deformation and sliprdquo Journal of CentralSouth University vol 19 no 10 pp 2976ndash2982 2012
[18] W-B Zhou L-Z Jiang and Z-W Yu ldquoClosed-form solutionto thin-walled box girders considering the effects of sheardeformation and shear lagrdquo Journal of Central South Universityvol 19 no 9 pp 2650ndash2655 2012
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials