By Prof. Chen ShimingLecture Notes for Presentation

2015

Chapter 6:

Continuous composite Beams

连续组合梁

6.1 IntroductionAdvantages of continuous beams :

• i) greater load capacity due to redistribution ofmoments; ii) greater stiffness

• Disadvantages associated with continuous beams:i) increased complexity in designii) susceptibility to buckling in the negative moment region

• Design/ detailed calculations or ‘global analysis’ : to determine internal moments and forces in critical regions for the various loading cases and limit states

The concrete in the mid span region is in compression and the steel in tension.

Over the internal support this distribution reverses. The concrete cannot carry significant tensile strains and therefore cracks, the embedded reinforcement as effective in resisting moment.（混凝土开裂，钢梁易屈曲、失稳）

Distosional lateral buckling组合梁侧扭失稳

Local buckling 局部屈曲

• Internal moments in continuous composite beams can be determined by elastic analysis or, subject to certain conditions, by rigid-plastic analysis.

• Whether plastic analysis is appropriate depends on the susceptibility of the beam to local buckling：

internal moments in beam column connection

6.2 Cross Section Classification• In a continuous beam, the critical cross-sections

must be capable of developing/sustaining their plastic resistance moment, under increasing load, sufficient regions have fully-yield for a mechanism of plastic hinges to be present. (plastic design)

• The mechanism arises as a result of redistribution of moment. To ensure that the resulting strains can be accommodated without a reduction in resistance below the plastic moment, limitations must be placed on the slenderness of the elements of the cross-section which are in compression.

• Sections which can form a plastic hinge with the rotation capacity required for plastic global analysis are designated Class 1 or ‘plastic’ cross-sections.

• Eurocodes 3 and 4 introduce the concept of cross-section classification(截面分类） to determine whether or not local buckling limits the ability of the cross-section to develop its plastic moment resistance and the rotations necessary for the redistribution of internal moments.

• Class 1 cross-sections are those which may be considered capable of developing both the plastic moment resistance and the necessary rotation capacity before any local buckling.

• Class 2 cross-sections are able to develop their plastic moment resistance, but rotation capacity is limited by local buckling.

• The classification of a cross-section is determined by the classification of its constituent plate elements that are in compression, which in turn is determined by width-to-thickness ratios (or slenderness).

• By definition, Class 2 ‘compact 密实性’ cross-sectionscan develop the plastic moment capacity of the sectionalthough local buckling limits the rotation capacity andprevents full redistribution of moment at such sections.

• 根据钢梁腹板和受压翼缘板的宽厚比，来区分局部失稳类型。当钢梁腹板和受压翼缘板的宽厚比较小，组合梁局部失稳是在截面进入完全塑性状态后出现，当组合梁的腹板和受压翼缘板的宽厚比较大时，局部失稳会在截面尚处于弹性或刚进入屈服状态发生。

• 截面分类方法判定：Class 1, Class 2, Class 3 and Class 4。目前，评定组合梁局部失稳比较成熟的方法是欧洲组合结构规范（EC4）的截面分类方法。该方法根据组合梁试验的负弯矩截面弯矩-曲率特性曲线，将组合梁分为4类截面。

EC4根据截面转动能力，采用截面翼缘和腹板的宽厚比将组合梁截面分为4类：

• 第一类（Class 1)：可塑性截面，具有能使弯矩完全分配的具有足够塑性转动的截面，抗弯承载力达到塑性极限弯矩；

• 第二类（Class 2)：密实性截面，由钢梁的局部失稳所控制其塑性转动能力的截面，抗弯承载力达到塑性极限弯矩；

• 第三类（Class 3)：半密实性截面，钢梁的受压翼缘屈服，但局部失稳使之不能达到全塑性弯矩的截面；

• 第四类（Class 4)：柔细截面，钢梁的受压翼缘在达到屈服之前被局部失稳破坏的截面。

Classification for lower beam flanges in compression

Classification for web

6.3 Behaviour of Continuous Composite Beams

Moment-rotation curves at the internal support

• Initially, the behaviour is substantially linear, but as load increases reduction in flexural stiffness occurs.

• In mid-span regions, yielding occurs in the lower part of the steel section and crushing occurs in the top of the concrete slab, causing redistribution of moment to the supports.

• In hogging moment regions, fine cracks appear in the concrete at relative low levels of load. As the loading continues, cracking continues over an increasing length and yielding and later strain-hardening may occur in the lower part of the steel section.

• The support section may also develop flange buckle, eventually causing a loss of moment of resistance which initiates collapse .

6.4 Rigid-plastic design/analysis• What is “rigid-plastic global analysis? A well-

established method of analysis for determining internal moments and forces in continuous steel structures.

• The method is based on the assumption that the plastic regions are concentrated at discrete pointsand may be represented by a ‘plastic hinge’ at the ultimate state for the beam.

Plastic resistance moments• It is assumed that the effect of co-existent vertical

shear on the bending resistance can be neglected. However, when the shear force exceeds half the plastic shear resistance of the web of the steel section, allowance should be made for its effect on the resistance moment.

• 正弯矩塑性极限弯矩同简支组合梁截面。

• 负弯矩塑性极限弯矩：不计受拉区混凝土作用，All properly anchored reinforcing bars within the effective width are assumed to be stressed to their design yield strengths fr。

for a solid slab

for a slab formed with profiled steel sheeting

• At flexural failure, the whole of the concrete slab may be assumed to be cracked, whilst all the structural steel is at its design yield strength fr in tension or compression. The position of the neutral axis is determined by considering longitudinal equilibrium.

• The plastic neutral axis（P.N.A ） may be in the top flange or in the web. For each case an expression for the negative plastic resistance moment M’pl, can be determined by considering the moment of each rectangular stress block about the neutral axis.

Case 1 – P.N.A. in the flange of the steel section

Case 2 – P.N.A in the web of the steel section

Rotation capacity for plastic analysis• 连续梁的极限状态，假定塑性铰可充分自由转动，即塑性铰

具有充分弯矩分配所需的转动能力。

• The nature of composite beams implies that a large amount of redistribution may be required before the collapse mechanism is complete.

• In the early stages of loading the beam behaves substantially elastically, the bending moment at supports is up to twice as large as that at mid-span.

• However, the plastic resistance moment of mid-span regions ranges from being larger than at the support.

• A large amount of deformation isrequired with redistribution being either from or to the support.

• The rotation required on each side of the support to complete a plastic hinge mechanism increases as the resistance moment at the support decreases relative to that at mid-span.

Rotation capacity required

Rotation capacity required at a particular criticalcross-section will depend on:

• the relative length of each span;• the type and position of loading on each span• the patterns of load on the spans;• the relative magnitudes of the hogging and

sagging moments of resistance along the beam ;• the moment-rotation characteristics of cross-

sections along the beam.• Requirements for the satisfactory use of rigid-

plastic analysis are based on test results, supplemented by parametric studies undertaken by computer.

Requirements for rigid-plastic analysis• At each plastic hinge location, the cross-section

of the steel beam shall be symmetrical about theplane of its web;

• All cross-sections at plastic hinge locations are inclass 1; all other cross-sections are in class 1 or 2;

• Adjacent spans do not differ in length by morethan 50% of the shorter span; End spans do notexceed 115% of the length of the adjacent span;

• In any span in which more than half the totaldesign load is concentrated with a length of one-fifth of the span, then at any hinge locationwhere the concrete slab is in compression, notmore than 15% of the overall depth of themember should be in compression;

• The steel compression flange at a plastic hinge islaterally restrained.

Distribution of bending moments• Let the ratio of the negative to the positive moments

of resistance in a proposed section be y(y = M’pl / Mpl )• For the end span of a continuous composite beam,

under a uniformly distributed load of wf. the requiredvalue of Mpl is:

Mpl = wf β 2 L2 / 2

• For an internal span with equal support momentsMpl = wf L2 / 8( 1 + ψ )

[ ]1)1()1( 2/1 −ψ+ψ

=β

For the end span

For an internal span

Rigid-plastic analysis can be applied to continuouscomposite beams if the rotation capacity at each plastichinge location is sufficient to enable the required hingerotation to develop and no lateral-torsional bucklingoccurs.

For composite beams in buildings, the requirement concerning rotation capacity may be assumed to be satisfied when all cross-section at plastic hinge locations are in Class 1.

The plastic moment of resistance in hogging moment region can be determined by application of rectangular stress block theory to the structural steel section and ductile reinforcing steel within the effective cross-section.

The effective widths of the concrete flange can be determined from approximation to the sagging and hogging lengths of the beam.

Distribution of internal moments is dependent on the ratio of the negative (‘hogging’) moment of resistance to that in positive (‘sagging’)bending.

effective width of concrete flange 6.5 Elastic design

Elastic hogging bending resistance• Calculation of the elastic hogging bending resistance

of a composite cross-section is more simple than that of the sagging bending resistance. The concrete is considered cracked and only the steel section and the reinforcement will withstand the bending moment.

6.6 Elastic Analysis /ultimate state• Bending moment in continuous composite beams at

the ultimate limit state (ULS) may be determined byelastic analysis or, subject to certain conditions.Elastic analysis has the advantages of more generalapplication. It may also be more convenient to usethis approach.

• 弹性内力分析主要考虑梁的刚度分布和截面的强度特征。当

截面内力达到其弹性强度时，组合梁达到其弹性承载力极限

状态。

• 连续梁正弯矩区的刚度与相同截面的简支梁相同，负弯矩区

的刚度则取决于钢梁和钢筋所组成的组合截面。长期、短期

刚度等。

• Loss of stiffness due to cracking of concrete in negative moment regions /induce moment distribution

• The bending moment at an internal support at the serviceability limit state may be 15 to 30% lower than that given by an elastic analysis in which no account is taken of cracking.

• At the ultimate limit state yielding of steel will also influence the distribution of moments.

• A wide variation in flexural rigidity can occur along a composite beam of uniform cross-section, leading to uncertainty in the distribution of bending moments and hence the amount of cracking to be expected.

• EC 4 for the ultimate limit state permits two methods of elastic global analysis: the cracked section method and the uncracked section method.

• Both may be used in conjunction with redistribution of support moments, the degree of redistribution being dependent on the susceptibility of the steel section to local buckling.

• Design codes commonly permit negative moments at supports to be reduced, except at cantilevers, by redistribution to mid-span. The extent of the redistribution is dependent, in part, on the method of analysis.

• Two methods of elastic global analysis are given forULS:

cracked section method （开裂截面分析方法）

假定混凝土开裂区域采用负弯矩的截面惯性矩，正弯矩区采 用正弯矩区未开裂截面的换算截面惯性矩进行结构内力分析计算。

uncracked section method （未开裂截面分析法）

假定整个连续梁可以采用正弯矩区未开裂截面的换算截面惯性矩进行结构内力分析计算计算。

Cracked section analysis• a length of 15% of the span

on each side of internal supports, the section properties are those of the cracked section for negative moments. Outside the ‘15% length’, the section properties are those of the uncracked section.

Bending moments calculated assuming a cracked length of 15% would be correct to within 5% if any proportion of span between 8% ad 25% was in fact to be cracked. The simplifying assumption is therefore justified.

at the mid-span at the support

Redistribution due to cracking

Uncracked section method• The properties of the uncracked section are used

throughout.• For a continuous beam of uniform section the analysis

can be carried out without any prior calculation of the cross-section.

Redistribution of support moments for elastic analysisClass of cross section in hogging

moment region 1 2 3 4

For ‘uncracked’ elastic analysis 40 30 20 10

For ‘cracked’ elastic analysis 25 15 10 0

In order to determine the bending moment distribution, the following three procedures may be adopted, which are presented in order of decreasing difficulty:1. A non-linear analysis accounting for the tension

stiffening effect in the cracked zone, and consequent contribution to the section stiffness of the concrete between two adjacent cracks due to transferring of forces between reinforcement and concrete by means of bond. The effect of the slip between steel and concrete should also be taken into account in the case of partial shear connection.

2. An elastic analysis that assumes the beam flexural stiffness: in the negative moment zone of the beam, where the moment is higher than the cracking one, the “cracked” stiffness EIcr is used, while the “uncracked” stiffness EIun characterizes the remainder of the beam. In order to further simplify the procedure, the length of the cracked zone can be pre-defined as a percentage of the span l. Eurocode 4 recommends a cracking length equal to 0.15 l.

3. An elastic uncracked analysis, which considers for the whole beam the “uncracked” stiffness EIun and accounts for the effect of cracking by redistributing the internal forces between the negative and positive moment regions.

Negtive rebars

• To determine the section properties, allowance should be made shear lag by using the appropriate effective width of the concrete flange for sagging or hogging bending

Properties of cross-sections

)()( xdAdxA srr −=−

22 )()( rrss dxAxdAII −+−+=

• When concrete is cracked in negative bending and the slab reinforcement A r is included

ELASTIC RESISTANCE MOMENT• It is assumed that strain varies linearly over the full

depth of the composite cross-section ( no slip at the steel-concrete interface). Use is made of the theory of transformed sections, assuming that both concrete in compression and steel are linearly elastic materials.

•breadth of the equivalent steel slab depends upon the modular ratio η,

η = Ea/Ec

In hogging bending, the whole of the concrete slab is assumed to be cracked. The effective section therefore comprises the structural steel section and effectively anchored reinforcement within the effective width for hogging bending

6.7 Shear resistance in continuous beamsExperience shows that there is no significant reduction in the bending moment resistance - MRd due to shear as long as the design vertical shear force VSd does not exceed half of the shear resistance VRd.

6.8 Control of cracking in concrete• A cross-section of a negative-moment region of a

composite beam is shown in Figure. The maximumcrack width predicted as:

• crack-width control :

)08.09.1(7.2maxtes

sk dcE

wνρ

+σ

ψ=

skte

tkfσρ

−=ψ 65.01.1

ce

rte hb

A=ρ

limmax ww ≤

6.9 Design of shear connections• In hogging regions of a continuous composite beam

where concrete slab is in tension the connection is lessstiff, and the ultimate strength is slight reduced.

• To enable a full composite section, the numbers of shearconnectors required in a shear span of the hoggingmoment region in continuous composite beams are as thefollow:

d

l

PVN

β=

6.10 Lateral-torsional buckling

The length of the lower flange in compression can be considerable when only dead load acts on the span under consideration , which is prone to lateral buckling

• 在钢梁下翼缘设置横向支撑，可控制和减小钢梁下翼缘的侧向变形，提高组合梁的抗侧扭失稳承载力。

• Methods for the design of unrestrained steel beams against lateral-torsional buckling are not applicable to negative moment regions of continuous beams because in the former case it is assumed that each cross-section of the member rotates as a whole, without distortion.

• In the negative moment region of a composite member the restraint afforted to the upper flange results in distortion of the cross-section if the lower (compression) flange is to buckle laterally.

• The effect of the restraint to the compression flange resulting from the distortional stiffness of the cross-section can be accounted for by reducing the effective slenderness of the beam used in calculation of the buckling resistance moment.

• Approach : using beam on elastic restrained foundation theory

EIEI

受压翼缘侧向弹性约束连续分布压杆

压杆的临界荷载值Ncr可表示为：

式中EI为压杆的侧向抗弯刚度，k为侧向约束刚度。

kIEN 2cr =

23r

2

32

1

31

IEua

I3Eδ dfdd

++=δ1

=k

a

d d d

δ

u u 1 2 3

组合梁横向刚度模型

第一项为钢梁腹板产

生的横向水平位移；

第二项钢筋混凝土板

产生的水平位移；第

三项为钢筋混凝土板

与钢梁上翼缘连接区

产生的水平位移。

• 基于弹性侧向约束压杆理论的组合梁侧向失稳临界应力计算（BS5400:Part3）

• 临界荷载值转化为欧拉临界压力公式的形式，le为等效压杆的计算长度，BS5400:Part3给出le的表达式为：

• 其中k3为支承约束影响系数，当翼缘在支承处可自由水平转动时，取k3 = 1.0。

• 考虑构件的初始缺陷以及材料的非线性对临界压力的影响，BS5400引入了Perry-Robertson设计公式。根据等效压杆计算长度le，计算构件长细比l LT为：

25.0fs3 )(5.2 δIEkle =

2s

2cr / ef lIEN π=

ηνλ 4krl

y

eLT =

• 其中k4为截面影响系数，热轧I字和槽型截面取0.9，其余截面取1.0；η 为弯矩分布影响系数，纯弯曲取η= 1.0；ν 为截面形状对称性影响系数；ry =（If /bf tf）0.5 = （bf

2 /12）0.5，为受压翼缘截面惯性半径。

• 根据β = λ LT（fy为钢的屈服强度，N/mm2）可查临界应力曲线（BS5400:Part3 Figure 10），计算出失稳临界应力设计值σ li。

• 英国钢结构研究院(SCI）采用能量方程，求解组合梁临界失稳弯矩值Mcr。其推导计算方法特点是考虑了腹板的变形，截面的圣维南扭转(GJ)，侧向弯曲刚度(EIz)以及钢梁翼缘的受压变形，忽略混凝土板

的变形以及支座处的约束扭曲。并假设钢梁翼缘的压力沿梁长方向为均布。导得的临界失界弯矩Mcr为：

• 式中 L 为梁的跨长，D为钢梁截面的高， EIz为钢梁受压翼缘关于截面对称轴z的抗弯刚度。

22

23

2cr π42π

DLEt

DGJ

LDEIM wz ++=

Design methods for distortionallateral buckling(EC4)

C, is a property of the distribution of bending moment within length L

In the case of Class 3 or 4 cross-sections, the lateral-torsional buckling slenderness ratio is determined on the basis of the elastic moment resistance.

• 基于截面扭转特性的组合梁侧向失稳承载力计算方法（EC4）其特点是考虑了对组合梁失稳具有重要影响的三个主要因素：a〕腹板；b〕混凝土板以及c〕钢梁与混凝土板之间的连接变形。此外还考虑了弯矩分布对组合梁临界弯矩的影响。

• 根据截面扭转特性，EC4方法给出的失稳临界表达式为：

• 其中kc为截面对称性影响系数；C4为弯矩分布影响系数；L为梁无侧向约束自由长度；ks截面的抗侧扭刚度系数；G为材料剪切模量；Iat为截面St.Venant扭转常数；EsIf为钢梁受压翼缘关于截面对称轴z的抗弯刚度。

fs22

sta4

cr )/( IELkGILCkM c π+=

• Lateral restraintIt is necessary to ensure that such restraint issufficiently strong and stiff to be effective and thatthe pull-out strength of the shear connectors is notexceeded.It is usual to check the resistance of the restraintcomponents to a lateral force calculated as a smallpercentage of that in the compression flange.

6.11 Concluding summaryWhen elastic analysis is used to determine internal moments and forces, moment may be redistributed from internal supports allowing for concrete cracking and steel yielding.Extent of the redistribution is dependent on the classification of the steel section at internal supports and on the flexural rigidity in the negative moment regions.Uncracked section may be used to determine the flexural rigidity for every cross-section along the beam. Alternatively, it may be assumed that over a fixed length on each side of internal supports, the properties are those of the cracked section.

The concrete slab may be assumed to prevent the upperflange of the steel section from twisting or movinglaterally.

In negative moment regions the tendency of the lowerflange to buckle laterally is partially restrained by thedistortional stiffness of the cross-section. This results in areduction in the effective slenderness for lateral-torsionalbuckling.

Discrete lateral restraint may be provided to thecompression flange, for example by bracing or transverseweb stiffeners.