Category Archives: Composite Structures

This volume provides an introduction to the theory and design of composite structures of steel and concrete. Readers are assumed to be familiar with the elastic and plastic theories for bending and shear of cross-section of beams and columns of a single material, such as structural steel, and to have some knowledge of reinforced concrete. No previous knowledge is assumed of the concept of shear connection within a member composed of concrete and structural steel, nor of the use of profiled steel sheeting in composite slabs. Shear connection is covered in depth in Chapter 2 and Appendix A, and the principal types of composite member in Chapter 3, 4 and 5. All material of a fundamental nature that is applicable to both buildings and bridges is included, plus more detailed information and a worked example related to building. Subjects mainly relevant to bridges are covered in Volume 2. These include composite plate and box girders and design for repeated loading.

Introduction Continuous Beams And Slabs, And Beams In Frames

The definition of ‘continuous composite beam’ given in Eurocode 4: Part 1.1[12] is: A beam with three or more supports, in which the steel section is either continuous over internal supports or is jointed by full strength and rigid connections, with connections between the beam and each support such that it can be assumed that the support does not transfer significant bending moment to the beam. At the internal supports the beam may have either effective reinforcement or only nominal reinforcement. Beam-to-column connections in steelwork are classified in Eurocode 3: Part 1.1[11] both by stiffness, as:

nominally pinned,
rigid, or
semi-rigid
and by strength, as:
nominally pinned,
full-strength, or
partial-strengt

In Eurocode 4: Part 1.1, a ‘composite connection’ is defined as: A connection between a composite member and any other member in which reinforcement is intended to contribute to the resistance of the connection. The system of classification is as for steel connections, except that semi-rigid connections are omitted, because design methods for them are not ye sufficiently developed. A ‘full-strength and rigid’ connection has to be at least as stiff and strong as the beams connected, so a ‘continuous composite beam’ can be analyzed for bending moments as one long member without internal connections, by methods to be explained in Section 4.3. Bridge girders (Volume 2) are usually of this type. The example to be used here is a two-span beam continuous over a wall or supporting beam. In multi-bay plane frames, commonly used in structures for buildings, the beam-to-column connections are often ‘nominally pinned’. The beams are then designed as simply-supported, where full-strength connections are used, the frame should be analysed as a whole, and the beams are not ‘continuous’ as defined above. These beams are referred to here as ‘beams in frames’, as are those with partial-strength connections. In comparison with simple spans, beams in frames have the same advantages and disadvantages as continuous beams. The global analysis is more complex than for continuous beams, because the properties of columns and connections are involved, but the design of hogging moment regions of the beams is the same. In Section 4.3 on global analysis, only continuous beams are considered. For a given floor slab and design load per unit length of beam, the advantages of continuous beams over simple spans are:

higher span/depth ratios can be used, for given limits to deflections;
cracking of the top surface of a floor slab near internal columns can be controlled, so that the
use of brittle finishes (e.g. terrazzo) is feasible;
the floor structure has a higher fundamental frequency of vibration, and so is less susceptible
to vibration caused by movements of people;
the structure is more robust (e.g. in resisting the effects of or explosion).

The principal disadvantage is that design is more complex. Actions on one span cause action effects in adjacent spans, and the stiffness and bending resistance of a beam very along its length. It is not possible to predict accurately the stresses or deflections in a continuous beam for a given set of actions. Apart from the variation over time caused by the shrinkage and creep of concrete, there are the effects of cracking of concrete. In reinforced concrete beams, these occur at all cross-sections, and so have little influence on distributions of bending moment. In composite beams, significant tension in concrete occurs only in hogging regions. It is influenced by the sequence of construction of the slab, the method of propping used (if any), and by effects of temperature, shrinkage, and longitudinal slip.

The flexural rigidity of a fully cracked composite section can be as low as quarter of the ‘uncracked’ value, so a wide variation in flexural rigidity can occur along a continuous beam of uniform section. This leads to uncertainty in the distribution of longitudinal moments, and hence in the amount of cracking to be expected. The response to a particular set of actions also depends on whether it precedes or follows another set of actions that causes cracking in a different part of the beam.

For these reasons, and also for economy, design is based as far as possible on predictions of ultimate strength (which can be checked by testing) rather than on analyses based on elastic theory. Methods have to be developed from simplified models of behaviour. The limits set to be scope of some models seem arbitrary, as they correspond to the range of available research data, rather than to known limitations of the model.

Almost the whole of Chapter 3, on simply-supported beams and slabs, applies equally to the sagging moment regions of continuous members. The follows of hogging moment regions of beams are treated in Section 4.2, which applies also to cantilevers. Then follows the global analysis of continuous beams, and the calculations of stresses and deflections. Both rolled steel I- or H-sections and small plate or box girders are considered, with or without web encasement and composite slabs. It is always assumed that the concrete slab is above the steel member, because the use of slabs below steel beams with which they are composite is almost unknown in buildings, though it occurs in bridges.

Fire resistance of composite beam

Fire design, based on the 1993 draft Eurocode 4: Part 1.2, ‘Structural fire design’, is introduced in Section 3.3.7, the whole of which is applicable to composite beams, as well as to slabs, except Section 3.3.7.5.
Beams rarely have insulation or integrity functions, and have then to be designed only for the loadbearing function, R. The fire resistance class is normally the same as that of the slab that acts as the top flange of the beam, so only the structural steel section needs further protection. This may be provided by full encasement in concrete of a lightweight fire-resisting material. A more recent method is to encase only the web in concrete. This can be done before the beam is erected (except near end connections), and gives a cross-section of the type shown in Fig.3.31.
In a fire, the exposed bottom flange loses its strength, but the protected web and top flange do not.
For the higher load levels η *(defined in Section 3.3.7.2) and longer periods of fire resistance, minimum areas of longitudinal reinforcement within the encasement, ‘
sA , are specified, in terms
of the cross-sectional area f A of the steel bottom flange. The minimum depth a h and breadth f b of the steel I-section are also specified, for each standard fire resistance period.

The requirements of draft Eurocode 4; Part 1.2 for 60 minutes’ fire exposure (class R60) are shown in Fig.3.28. The minimum dimensions ha and bf increase with η *, as shown by the three lines in Fig.3.28(a). For other values of η *, interpolation may used.
The minimum ratios f A’ / A are zero for η* = 0.5 (ADE). For η* = 0.7 they are indicated  within the regions where they apply. To ensure that the additional reinforcement maintatins its strength for the period of fire exposure, minimum distances 1 a and 2 a are specified, in terms of Lmin b and the fire class. Those for class R60 are show in Fig.3.28(b).

The validity of tabulated data of this type is inevitably limited. The principal conditions for its use, given in the Eurocode, are as follows. The notation is as in Fig.3.15.
(a) The composite beam must be simply-supported with

(b) If the slab is composite, the voids formed above the steel beam by trapezoidal profiles must be filled with fire-resistant material.
(c) The web must be encased in normal-density concrete, held in place by stirrups, fabric, or stud connectors that pass through or are welded to the steel web.
The data given in Fig.3.28 are used for the design example in Section 3.11.4. The Eurocode also gives both simple and advanced calculation models, which are often less conservative than the trbulated data, and have wider applicability. These are outside the scope of this volume.

Prediction of fundamental natural frequency

In composite floors that need checking for vibration, damping is sufficiently low for its influence  on natural frequencies to be neglected. For free elastic vibration of a beam or one-way slab of uniform section, the fundamental natural frequency is

Values for other end conditions and multi-span members are given by Wyatt. The relevant flexural rigidity is El (pre unit width, for slabs), L is the span, and m the vibrating mass per unit length (beams) or unit area (slabs). Concrete in slabs should normally be assumed to be uncracked, and the dynamic modulus of elasticity should be used for concrete, in both beams and slabs. This modulus, Ecd, is typically about 8 kN/mm2 higher than the static modulus, for normal-density concrete, and 3 to 6 kN/mm2 higher, for lightweight-aggregate concretes of density not less than 1800kg/m3. For composite beams in sagging bending, approximate allowance for these effects can be made by increasing the value of I for variable loading by 10%.
Unless a more accurate estimate can be made, the mass m is usually taken as the mass of the characteristic permanent load plus 10% of the characteristic variable load. A convenient method of calculating f0 is to find first the midspan deflection, m δ say, caused by the weight of the mass m. For simply-supported members this is

For a single-span layout of the type shown in Fig.3.1, each beam vibrates as if simply-supported, so the length Leff of the vibrating area can be taken as the span, L. The width S of the vibrating area will be several times the beam spacing, s. A cross-section through this area is likely to be as shown in Fig. 3.27, with most spans of the composite slab vibrating as if fixed-ended. It follows from equation (3.95) that:
for the beam,

where m is the vibrating mass per unit area, and s is the spacing of the beams, and subscripts b and
s mean beam and slab, respectively.

Effects of shrinkage of concrete and of temperature

In the fairly dry environment of a building, an unrestrained concrete slab could be expected to shrink by 0.03% of its length (3 mm in 10 m) or more. In a composite beam, the slab is restrained by the steel member, which exerts a tensile force on it, through the shear connectors near the free ends of the beam, so it apparent shrinkage is less than the ‘free’ shrinkage. The forces on the shear connectors act in the opposite direction to those due to the loads, and so can be neglected in design.
The stresses due to shrinkage develop slowly, and to are reduced by creep of the concrete, but the increase they cause in the deflection of a composite beam may be significant. An approximate and usually conservative rule of thumb for estimating this deflection in a simply supported beam is to take it as equal to the long-term deflection due to the weight of the concrete slab acting on the composite member.
In the beam studied in Section 3.11, this rule gives an additional deflection of 9 mm, whereas the calculated long-term deflection due to a shrinkage of 0.03% (with a modular ration n=22) is 10mm.
In beams for buildings, it can usually be assumed that tabulated span/depth rations are sufficiently conservative to allow for shrinkage deflections; but the designer should be alert for situations where the problem may be unusually severe (e.g. thick slabs on small steel beams, electrically heated floors, and concrete mixes with high ‘free shrinkage’).
Composite beams also deflect when the slab is colder than the steel member. Such differential temperatures rarely occur in buildings, but are important in beams for bridges. Methods of calculation for shrinkage and temperature effects are given in Volume 2.

Vibration of composite floor structures

In British Standard 6472, ‘Evaluation of human exposure to vibration in buildings (1 Hz to 80 Hz)’, the performance of a floor structure is considered to be satisfactory when the probability of annoyance to users of the floor, or of complaints from then about interference with activities, is low. There can be no simple specification of the dynamic properties that would make a floor structure ‘serviceable’ in this space concerned, and the psychology of its users are all relevant.
An excellent guide to this complex subject is available. It and BS 6472 provided much of the basis for the following introduction to vibration design, which is limited to the situation in the design example a typical floor of an office building, shown if Fig.3.1.
Sources of vibration excitation
Vibration from external sources, such as highway or rail traffic, is rarely severe enough to influence design. If it is, the building should be isolated at foundation level.
Vibration from machinery in the building, such as lifts and traveling cranes, should be isolated at or near its source. In the design of a floor structure, it should be necessary to consider only sources of vibration on or near that floor. Near gymnasia or dance floors, the effects of rhythmic movement of groups of people can be troublesome; but in most buildings only two situations need be considered;

People waling across a floor with a pace frequency between 1.4 Hz and 2.5Hz; and an impulse, such as the effect of the fall of a heavy object.
Typical reactions on floors from people walking have been analysed by Fourier series. The basic fundamental component has an amplitude of about 240N. The second and third harmonics are smaller, but are relevant to design. Fundamental natural frequencies of floor structures (4.2Hz to 7.5 Hz). The number of cycles of this harmonic, as a person walks across the span of a floor, can be sufficient for the amplitude of forced vibration to approach is steady=state value. This situation will be considered in more detail later.
Pedestrian movement causes little vibration of floor structures with f0 exceeding about 7 Hz, but these should be checked for the effect of an impulsive load. The consequences that most influence human reactions are the peak vertical velocity of the floor, which is proportional to the impulse, and the time for the vibration to decay, which increases with reduction in the damping ratio of the floor structure. Design guidance is available for this situation, which is not further discussed here.
Human reaction to vibration
Models for human response to continuous vibration are given in BS 6472. For vibration of a floor that supports people who are standing or sitting, rather than lying down, the model consist of a base curve of root-mean-square (r.m.s) acceleration against fundamental natural frequency of the floor, and higher curves of similar shape. These are shown in the double logarithmic plot of Fig.3.26. Each curve represents an approximately uniform level of human response. The base curve denoted by R=1, where R is the response factor, corresponds to a ‘minimal level of adverse comment from occupants’ of sensitive locations such as hospital operating theatres and precision laboratories.

Curves for other values of R are obtained by multiplying the ordinates of the base curve of R. Those for R=4, 8 and 16 are shown. The appropriate value for R for use in design depends on the environment. The British Standard gives:
R=4 for offices
R=8 for workshops
with the comment that use of double those values ‘may result in adverse comment’, which ‘may increase significantly’ if the magnitudes of vibration are quadrupled.
Some relaxation is possible if the vibration is not continuous. Wyatt recommends that a floor subject to a person walking at resonant frequency once a minute could reasonably be permitted a response double the value acceptable for continuous oscillation.

Stresses and deflections in service

A composite beam is usually designed first for ultimate limit states. Its behaviour in service must then be checked. For a simply-supported beam, the most critical serviceability limit state is usually excessive deflection, which can govern the design where unpropped construction us used. Floor structures subjected to dynamic loading (e.g. as in a gymnasium) are also susceptible to excessive vibration (Section 3.11.3.2).
Cracking of concrete is a problem only in fully-encased beams. Which are rarely used, and in hogging regions of continuous beams (Section 4.2.5).
Some codes of practice limit stresses in service; but excessive stress is not itself a limit state. It may however invalidate a method of analysis (e.g.linear0elastic theory) that would otherwise be suitable for checking compliance with a serviceability criterion. No stress limits are specified in Eurocode 4; Part 1.1. The policy is to use elastic analysis, allowing for shear lag and creep; and to modify the results, where necessary, to allow for yielding of steel and, where partial shear connection is used, for excessive slip.
If yielding of structural steel occurs in service, in a typical composite beam for a building, it will be in the bottom flange, near midspan. The likelihood of this depends on the ratio between the characteristic variable and permanent loads, given by

where My is the bending moment at which yield of steel first occurs. For sagging bending, it is typically between 1.25 and 1.35 for propped construction, but can rise to 1.45 or above, for unpropped construction.
Deflections are usually checked for the rare combination of actions, given in equation (1.8). So for a beam designed for distributed loads gk and qk only, the ratio of design bending moments (ultimate/serviceability) is

The values given above show that this is unlikely for propped construction, but might occur for unpropped construction when a γ for structural steel is assumed to be 1.05 or less, rather than 1.10 as recommended in Eurocode 4.
Where the bending resistance of a composite section is governed by local buckling, as in c Class 3 section, elastic section analysis is used for ultimate limit states, and then stresses and/or deflections in service are less likely to influence design.
As shown below, elastic analysis of a composite section is more complex than plastic analysis, because account has to be taken of the method of construction and of the effects of creep. The following three types of loading then have to be considered separately:
load carried by the steel beam, short-term load carried by the composite beam long-term load carried by the composite beam

3.7.1 Elastic analysis of composite sections in sagging bending

It is assumed first that full shear connection is provided, so that the effect of slip can be neglected.
All other assumptions are as for the elastic analysis of reinforced concrete sections by the method of transformed sections. The algebra is different because the flexural rigidity of the steel section alone is so much greater than that of reinforcing bars.
For generality, the steel section is assumed to be asymmetrical (Fig.3.25) with cross-sectional area Aa, second moment of area Ia, and centre of area distance zs below the top surface of the concrete slab, which is of uniform overall thickness ht and effective width beff.
The modular ratio for short-term loading is

n= Ea / Ec

where the subscripts a and c refer to structural steel and concrete, respectively. For long-term loading, a value n/3 is a good approximation. For simplicity, a single value n/2 is sometimes used for both types of loading. From here onwards, the symbol n is used for whatever modular ration is appropriate, so it is defined by

n= Ea / E’c

where E’c
E is the relevant effective modulus for the concrete.
It is usual to neglect reinforcement in compression, concrete in tension, and also concrete between  the ribs of profiled sheeting, even when the sheeting spans longitudinally. The condition for the neutral-axis depth x to be less than he is

In global analyses, it is sometimes convenient to use values of I based on the uncracked composite section. The values of x and I are then given by equations (3.88) and (3.89) above, whether x exceeds hc or not. In sagging bending, the difference between the ‘cracked’ and ‘uncracked’ values of I is usually small.
Stresses due to a sagging bending moment M are normally calculated in concrete only at level I in Fig.3.25, and in steel at levels 3 and 4. these stresses are, with tensile stress positive:

Composite beams-longitudinal shear

Critical lengths and cross-sections

It will be show in Section 3.7 that the bending moment at which yielding of steel first occurs in a simply-supported composite beam can be below 70% of the ultimate moment. If the bending-moment diagram is parabolic, then at ultimate load partial yielding of the steel beam can extend over more than half of the span.
At the interface between steel and concrete, the distribution of longitudinal shear is influenced by yielding, and also by the spacing of the connectors, their load/slip properties, and shrinkage and creep of the concrete slab. For these reasons, no attempt is made in design to calculate this  distribution. Wherever possible, connectors are uniformly spaced along the span.

It was shown is Section 3.5.3 that this cannot always be done. For beams with all critical sections in Class 1 or 2, uniform spacing is allowed by Eurocode 4 along each critical length, which is a length of the interface between two adjacent critical cross-sections. These are defined as sections of maximum bending moment. supports.
sections subjected to heavy concentrated loads.
Places where there is a sudden change of cross-section of the beam, and free ends of cantilevers.
There is also a definition for tapering members.
Where the loading is uniformly-distributed, a typical design procedure, for half the span of a beam, whether simply-supported or continuous, would be as follows.
(1) Determine the compressive force required in the concrete slab at the section of maximum sagging moment, as explained in Section 3.5.3. Let this be Fc.
(2) Determine the tensile force in the concrete slab at the support that is assumed to contribute to the bending resistance at that section (i.e. zero for a simple support, even if crack-control reinforcement is present; and the yield force in the longitudinal reinforcement, if the span is designed as continuous). Let this force be F1.
(3) If there is a critical cross-section between these two sections, determine the force in the slab at that section. The bending moment will usually be below the yield moment, so elastic analysis of the section can be used.
(4) Choose the type of connector to be used, and determine its design resistance to shear, PRd, as explained in Section 2.5.
(5)The number of connectors required for the half span is

The number required within a critical length where the change in longitudinal force is Rd ΔF / P .
An alternative to the method of step (3) would be to use the shear force diagram for the half span considered. Such a diagram is shown if Fig.3.18 for the length ABC of a span AD, which is continuous at A and has a heavy point load at B. The critical sections are A, B, and C. The total number of connectors is shared between lengths AB and BC in proportion to the area of the shear force diagram, OEFH and GJH.
In practice it might be necessary to provide a few extra connectors along BC, because codes limit the maximum spacing of connectors, to prevent uplift of the slab relative to the steel beam, and to ensure that the steel top flange is sufficiently restrained from local and lateral bucking.

Dactile and non-ductile connectors

The use of uniform spacing is possible because all connectors have some ductility, of slip capacity.

This term has no standard definition, but is typically assumed to be the maximum slip at which a connector can resist 90% of its characteristic shear resistance, as defined by the falling branch of a load slip curve obtained in a standard push test.
Slip enables longitudinal shear to be redistributed between the connectors in a critical length, before any of them fail. The slip required for this purpose increases at low degrees of shear connection, and as the critical length increases (a scale effect). A connector that is ‘ductile’ (has sufficient slip capacity) for a short span becomes ‘non-ductile’ in a long span, for which a more conservative design method must be used.
The definitions of ‘ductile’ connectors given in Eurocode 4 for headed studs welded to a steel beam with equal flanges are shown in Fig.3.19. The more liberal definition. The push test specified in British codes since 1965 is, in any case, unsuitable for this purpose, because the reinforcement in the slabs is insufficient to prevent longitudinal splitting (Section 2.5).

Where non-ductile connectors are used, design for full shear connection is the same (in Eurocode 4) as for ductile connectors, but for N/Nf<1, the two methods shown in Fig.3.16 are replaced by methods that require higher ratios N/Nf for a given value of  Mpl Rd . . These are approximations to elastic behaviour, and so rely less on slip capacity. They are explained in Reforence 15. There is also a rule in Eurocode 4 that limits the use of uniform spacing of non-ductile connectors.

Transverse reinforcement

The reinforcing bars shown in Fig.3.20 are longitudinal reinforcement for the concrete slab, to enable it to span between the beam shown and those either side of it. They also enhance the resistance to longitudinal shear of vertical cross-sections such as B-B. Bars provided for that purpose are known as ‘transverse reinforcement’, as their direction is transverse to the axis of the composite beam. Like stirrups in the web of a reinforced concrete T-beam, they supplement the shear strength of the concrete, and their behaviour can be represented by a truss analogy.

The design rules for these bars are extensive; as account has to be taken of many types and arrangements of shear connectors, of haunches, of the use of precast or composite slabs, and of interaction between the longitudinal shear per unit length on the section considered, vSd, and the transverse bending moment, show as Ms in Fig.3.20. The loading on the slab also causes vertical shear stress on planes such as B-B; but this is usually so much less than the longitudinal shear stress vsd/Acv on the plane, that it can be neglected.

The notation here is that of Eurocode 4; Part 1.1; Acv is the cross-sectional area per unit length of beam of the concrete shear surface being considered. The word ‘surface’ us used here because EFGH in Fig.3.20, although not a plane, is another potential surface of shear failure. In practice, the rules for minimum height of shear connectors ensure that in slabs of uniform thickness, planes such as B-B are more critical; but this may not be so for haunched slabs, considered later.
The design longitudinal shear per unit length for surface EFGH is the same as that for the shear connection, and in a symmetrical T-beam half of that value is assumed to be transferred through each of planes B-B and D-D. For an L-beam or where the flange of the steel beam is wide (Fig.3.21), the more accurate expressions should be used:

For planes such as B-B in Fig.3.20, the effective area of transverse reinforcement per unit length of beam, Ac, is the whole of the reinforcement that is fully anchored on both sides of the plane (i.e. able to develop its yield strength in tension). This is so even where the top bars are fully stressed by the bending moment Ms, because this tension is balanced by transverse compression, which  transverse compression, which enhances the shear resistance in the region CJ by an amount at least equivalent to the contribution the reinforcement would make, in absence of transverse bending.
Effective areas are treated in more detail in Volume 2.

3.6.3.1 Design rules for transverse reinforcement in solid slabs

Part of a composite beam is shown is plan in Fig.3.22. The truss model for transverse reinforcement is illustrated by triangle ACE, in which CE represents the reinforcement for a unit length of the beam, Ac, and v is the design shear force per unit length. The force v, applied at some point A, is transferred by concrete struts AC and AE, at 45° to the axis of the beam. The strut force at C is balanced by compression in the slab and tension in the reinforcement. The model fails when the reinforcement yields. The tensile force in it is equal to the shear on a plane such as B-B caused by the force v, so the model gives a design equation of the from

 

 

Resistance to sagging bending

 Cross-sections in Class 1 or 2

The methods of calculation for sections in Class 1 or 2 are in principle the same as those for composite slabs, explained in Section 3.3.1, to which reference should be made. The main assumptions are as follows:
the tensile strength of concrete is neglected;
plane cross-sections of the structural steel and reinforced concrete parts of a composite section each remain plane;
and, for plastic analysis of sections only;
the effective are of the structural steel member is stressed to its design yield strength f y/γa in tension or compression;
the effective area of concrete in compression resists a stress of 0.85  fy /γa constant over the whole depth between the plastic neutral axis and the most compressed fibre of the concrete. In deriving the formulae below, it is assumed that the steel member is a rolled 1-section, of cross-sectional area Aa, and the slab is composite, with profiled sheeting that spans between adjacent steel members. The composite section is in Class 1 or 2, so that the whole of the design load can be assumed to be resisted by the composite member, whether the construction was propped or unpropped. This is because the inelastic behaviour that precedes flexural failure allows internal redistribution of stresses to occur.
The effective section is shown in Fig.3.15(a). As for composite slabs, there are three common situations, as follows. The first two occur only where full shear connection is provided.
(1) Neutral axis within the concrete slab The stress blocks are shown in Fig.3.15(b). the depth x, assumed to be the position of the plastic neutral axis, is found by resolving longitudinally:

Taking moments about the line of action of the force in the slab.

where g h defines the position of the centre of area of the steel section, which need not be symmetrical about its major (y-y) axis.
(2) Neutral axis within the steel top flange
The force Ncf, given by

(3) Partial shear connection
The symbol Nct was used in paragraphs (1) and (2) above for consistency with the treatment of composite slabs in Eurocode 4 and in Section 3.3.1. In design, its value us always the lesser o the two values given by equations (3.56) and (3.58). It is the force which the shear connectors between the section of maximum sagging moment and each free end of the beam (a ‘shear span’) must be designed to resist, if full shear connection is to be provided. In draft Eurocode 4; Part 1.1 the symbol used in the clause on partial shear connection in beams is Fcf, so in this explanation it is used in place of Ncf.
Let us suppose that the shear connection is designed to resist a force Fc, smaller than Fcf. If each  connector has the same resistance to shear, and the number in each shear span is N, then the  degree of shear connection is defined by:

Where Nf is the number of connectors required for full shear connection. The plastic moment of resistance of a composite slab with partial shear connection had to be derived in Section 3.3.1(3) by an empirical method, because the flexural properties of profiled sheeting are to complex. For composite beams, simple plastic theory can be used.

The depth of the compressive stress block in the slab, xc, is given by

and is always less than hc. The distribution of longitudinal strain in the cross-section is intermediate between the two distributions shown (for stress) in Fig.2.2(c), and is shown in Fig.3.15(d), in which C means compressive strain. The neutral axis in the slab is at a depth xn greater than xc, as shown.
In design of reinforced concrete beams and slabs it is generally assumed that xa / xc is between 0.8 and 0.9. The less accurate assumption  xa = xc is made for composite beams and slabs to  avoid the complexity that otherwise occurs in design when xa = hc or, for beams with non-composite slabs, a c x ≈ h . This introduces as error in pl M that is on the unsafe side, but is negligible for composite beams. It is not negligible for composite columns, where it I allowed for (Section 5.6.5.1).
There is a second neutral axis within the steel 1-section. If it lies within the steel top flange, the stress block are as shown in Fig.3.15(c), except that the block for the force Nct is replaced by a shallower one, for force Fe. By analogy with equation (3.62) the resistance moment is

The curve is not valid for very low degrees of shear connection, for reasons explained in Section  3.6.2. Where it is valid, it is evident that a substantial saving in the cost of shear connectors can be slightly below pl Rd M . .
Where profiled sheeting is used, there is sometimes too little space in the troughs for Nt connectors to be provided within a shear span, and then partial-connection design becomes essential.
Unfortunately, curve ABC in Fig.3.16 cannot be represented by a simple algebraic expression. In practice, it is therefore sometimes replaced (conservatively) by the line AC, given by

where PRd is the design resistance of one connector.
The design of shear connection I considered in greater depth in Section 3.6.
Variation in bending resistance along a span In design, the bending resistance of a simply-supported beam is checked first at the section of maximum sagging moment, which is usually at midspan. For a steel beam of uniform section, the
bending resistance is then obviously sufficient, elsewhere within the span; but this may not be so for a composite beam. Its bending resistance depends on the number of shear connectors between the nearer end support and the cross-section considered. This is shown by curve ABC IN Fig.3.16, because the x coordinate is proportional to the number of connectors.
Suppose, for example, that a beam of span L is deigned with partial shear connection and N / Nt== 0,5 idspan. Curve ABC is redrawn in Fig.3.17(a), with the bending resistance at  midspan, MPl Rd . , denote by B. If the connectors are uniformly spaced along the span, as is usual

in buildings, then the axis N/Nf is also an axis x/L, where x is the distance from the nearer support,
and N is the number of connectors effective in transferring the compression to the concrete slab
over a length x from a free end. Only these connectors can contribute to the bending resistance
pl Rd M . , at that section, denoted E in Fig.3.17(b). In other words, bending failure at section E
would be caused (in the design mode) by longitudinal shear failure along length DE of the
interface between the steel flange and the concrete slab…

 

 

Classification of steel elements in compression

Because of local buckling, the ability of a steel flange or web to resist compression depends on its slenderness, represent by its breadth/thickness ration. In design to Eurocode 4, as in Eurocode 3, each flange or web in compression is placed in one of four classes. The highest (least slender) class is Class 1(plastic). The class of a cross-section of a composite beam is the lower of the classes of its web and compression flange, and this class determines the design procedures that are available.
This well-established system is summarized in Table 3.1. The Eurocodes allow several methods of plastic global analysis, of which rigid-plastic analysis (plastic hinge analysis) is the simplest. This is considered further in Section 4.3.3.
The Eurocodes give several idealized stress-strain curves for use in plastic section analysis, of which only the simplest (rectangular stress blocks) are use in this volume.
Table 3.1 Classification of sections, and methods of analysis.

Notes: (1) bole in the web method enables plastic analysis to be used;
(2) with reduced effective width of yield strength;
(3) for Grsde 50 steel (fy-355N/mm2), c is half the width of a flange of thickness
(4) elastic analysis may be used, but is more conservative.
The class boundaries are defined by limiting slenderness ration that are proportional to (fy)-0.5, where fy is the nominal yield strength of the steel. This allows for the influence of yielding on loss of resistance during buckling. The ratios is Eurocode 4; Part 1.1 for steel with fy=355N/mm2 are given in Table 3.1 for uniformly compressed flanges of rolled 1-sections, of overall width 2c.
Encasement of webs in concrete, illustrated in Fig.3.31, is done primarily to improve resistance to fire (Section 3.10). It also prevents rotation of a flange towards the web, which occurs during local buckling, and so enables higher c/t rations to be used at the class 2/3 and 3/4 boundaries, as shown.
At the higher compressive strains that are relied on in plastic hinge analysis, the encasement is weakened by crushing of concrete, so the c/t ration at the class 1/2 boundary is unchanged.
The class of a steel web is strongly influenced by the proportion of its clear depth, d, that is in  compression, as shown in Fig.3.14. For the class 1/2 and 2/3 boundaries, plastic stress blocks are used, and the limiting d/t ratios are given in Eurocodes 3 and 4 as functions of α , defined in Fig.3.14. the curves show, for example, that a web with d/t=40 moves from Class 1 to Class 3 when α increases from 0.7 to 0.8. This high rate of change is significant in the design of continuous beams (Section 4.2.1).
For the class 3/4 boundary, elastic stress distributions are used, defined by the ration ψ = −1.
The elastic neutral axis is normally higher, in a composite T-beam, than the plastic neutral axis, and its position for propped and unpropped construction are different, so the curve for the class 3/4 boundary is not comparable with the others in Fig.3.41.

For simply-supported composite beams, the steel compression flange is restrained from local bucking (and also from lateral buckling) by its connection to the concrete slab, and so is in Class 1. The plastic neutral axis for full interaction is usually within the slab or steel top flange, so the web is not in compression, when flexural failure occurs, unless partial shear connection (Section 3.5.3.1) is used. Even then, α us sufficiently small for the web to be in Class 1 or 2. (This may not be so for the much deeper plate or box girders used in bridges.)
During construction of a composite beam, the steel beam alone may be in a lower slenderness class than the completed composite beam, and may be susceptible to lateral buckling. Design for this situation is governed by a code for steel structures (e.g. Eurocode 3).

Effective cross-section

The presence of profiled steel sheeting in a slab is normally ignored when the slab is considered as
part of the top flange of a composite beam. Longitudinal shear in the slab (explained in Section 1.6)
causes shear strain in its plane, with the result that vertical cross sections through the compo-sited
T-beam do not remain plane, when it is loaded. At a cross-section, the mean longitudinal bending
stress through the thickness of the slab varies as sketched in Fig.3.13.

For beams in buildings, it is usually accurate enough to assume that the effective width is lo/8 on each side of the steel web, where lo is the distance between points of zero bending moment. For a simply-supported beam, this is equal to the span, L, so that

provided that a width of slab L/8 is present on each side of the web.
Where profiled sheeting spans at right angles to the span of the beam (as in the worked example here), only the concrete above its ribs can resist longitudinal compression (e.g. its effective thickness in Fig.3.9 is80 mm). Where ribs run parallel to the span of the beam, the concrete within ribs can be included, though it is rarely necessary to do so.
Longitudinal reinforcement within the slab is usually neglected in regions of sagging bending.