For instance, in an overhanging cantilever beam, the multiplier for self-weight in the cantilever section will be 1. When a design involves geotechnical action, a number of approaches are given in EN , and the choice of the method is a Nationally Determined Parameter.
For details, reference should be made to EN and EN Accidents are unintended events such as explosions, fire or vehicular impact, which are of short duration and which have a low probability of occurrence.
Also, a degree of damage is generally acceptable in the event of an accident. The loading model should attempt to describe the magnitude of other variable loads which are likely to occur in conjunction with the accidental load. Accidents generally occur in structures in use. Therefore, the values of variable actions will be less than those used for the fundamental combination of loads in 1 above. Where the dominant action is not obvious, each variable action present is in turn treated as dominant.
Serviceability limit state 3 Characteristic combination. It might be appropriate for checking states such as micro cracking or possible local non-catastrophic failure of reinforcement leading to large cracks in sections. Partial safety factors for actions in building structures — ultimate limit state in accordance with the UK National Annex Action Combination Fundamental Accidental Permanent actions caused by structural and non-structural components Stability check Unfavourable 1.
It should be realized that the above combinations describe the magnitude of loads which are likely to be present simultaneously. The actual arrangement of loads in position and direction within the structure to create the most critical effect is a matter of structural analysis e.
However, in practice simplified methods given in Section 2. Also, in practical examples the dominant loads are likely to be fairly obvious, and therefore the designer will generally not be required to go through all the combinations.
For normal building structures, expressions 6. Material properties 2. Generally, in design, only one lower characteristic value will be of interest. However, in some problems such as cracking in concrete, an upper characteristic value may be required, i. Design values In order to account for the differences between the strength of test specimens of the structural materials and their strength in situ, the strength properties will need to be reduced.
Uncertainties in the resistance models are also Table 2. The values of partial safety factors for material properties are shown in Table 2. Geometric data The structure is normally described using the nominal values for the geometrical parameters.
Variability of these is generally negligible compared with the variability associated with the values of actions and material properties. In special problems such as buckling and global analyses, geometrical imperfections should be taken into account.
The code specifies values for these in the relevant sections. Traditionally, geometrical parameters are modified by factors which are additive. Verification Ultimate limit state 1 When considering overall stability, it should be verified that the design effects of destabilizing actions are less than the design effects of stabilizing actions. Durability As one of the fundamental aims of design is to produce a durable structure, a number of matters will need to be considered early in the design process.
The environmental conditions should be considered at the design stage to assess their significance in relation to durability and to enable adequate provisions to be made for protection of the materials. Example 2. Assume office use for this building. Note that the load combinations for the design of elements could be different. Case 2 Treat the imposed load on roof as dominant load Fig. Case 3 Treat the imposed load on floors as the dominant load Fig. Note When the wind loading is reversed, another set of combinations will need to be considered.
However, in problems of this type, the designer is likely to arrive at the critical combinations intuitively rather than searching through all the theoretical possibilities. Frame Example 2. Four-span continuous beam Example 2. Assume that spans and are subject to domestic use, and spans and are subject to parking use. The load cases to be considered Fig. Assume that the beam is subject to dead and imposed loads and a point load at the end of the cantilever arising from dead loads of the external wall.
Calculate the design lateral loads for the ultimate limit state. In this example, it is assumed that the operational depth h of water has been determined on this basis. Load cases Example 2. In this example, this condition is treated as an accidental design situation. These two cases are shown in Figs 2. P Fig. Continuous beam Example 2. Water tank Example 2. Pressure diagram Example 2. Alternative design loading Example 2.
Introduction The purpose of analysis is the verification of overall stability and establishment of action effects, i. In turn, this will enable the calculation of stresses, strains, curvature, rotation and displacements.
In certain complex structures the type of analysis used e. To carry out the analysis, both the geometry and the behaviour of the structure will need to be idealized. Commonly, the structure is idealized by considering it as made up of elements depicted in Fig. The first two models are common for slabs and frames, and plastic analysis is popular in the design of slabs; non-linear analysis is very rarely used in day-to-day design. The above methods, with the exception of plastic analysis, are suitable for both serviceability and ultimate limit states.
Plastic methods can be used generally only for ultimate limit state i. In addition to global analysis, local analyses may also be necessary, particularly when the assumption of linear strain distribution does not apply.
In these cases strut and tie models a plastic method are commonly employed to analyse the structures. Load cases and combination In the analysis of the structure, the designer should consider the effects of the realistic combinations of permanent and variable actions.
Within each set of combinations e. Definition of structural elements for analysis. Account is taken of the probability of loads acting together, and values are specifed accordingly. The EN code for actions Eurocode 1 specifies the densities of materials to enable the calculation of permanent actions and surcharges , and values of variable action such as imposed gravity, wind and snow loads.
It also provides information for estimating fire loads in buildings, to enable fire engineering calculations to be carried out. Although EN forms part of a suite of codes including those which specify loads, there is no reason why the Eurocode cannot be used in conjunction with other loading codes.
The definition of the characteristic value will affect the overall reliability. While the general requirement is that all relevant load cases should be investigated to arrive at the critical conditions for the design of all sections, EN permits simplified load arrangements for the design of continuous beams and slabs. The arrangements to be considered are: Clause 5. Although not stated, the above arrangements are intended for braced non-sway structures. They may also be used in the case of sway structures, but the following additional load cases involving the total frame will also need to be considered: 1 all spans loaded with the design permanent loads 1.
Clause 5. No further redistribution should be carried out. In practice, the designer need not actually calculate these additional deformations to carry out this check.
When the spans are short, EN provides alternative design models e. The contribution of axial loads to deflections may be neglected if the axial stresses do not exceed 0. Imperfections 3. General Perfection in buildings exists only in theory; in practice, some degree of imperfection is unavoidable, and designs should recognize this, and ensure that buildings are sufficiently robust to withstand the consequences of such inaccuracies.
For example, load-bearing elements may be out of plumb or the dimensional inaccuracies may cause eccentric application of loads. Most codes allow for these by prescribing a notional check for lateral stability.
The exact approach adopted to achieve this differs between codes. EN has a number of provisions in this regard, affecting the design of 1 the structure as a whole, 2 some slender elements and 3 elements which transfer forces to bracing members. This is then modified for height and for the number of members involved. This factor recognizes that the degree of imperfection is statistically unlikely to be the same in all the members.
As a result of the inclination, a horizontal component of the vertical loads could be thought of being applied at each floor level, as shown in Figs 3.
Minimum tie force for perimeter columns forces should be taken into account in the stability calculation. This is in addition to other design horizontal actions, such as wind. Design of slender elements In the design of slender elements, which are prone to fail by buckling e.
Members transferring forces to bracing elements In the design of these elements such as a floor diagram , a force to account for the possible imperfection should be taken into account in addition to other design actions. This additional force is illustrated in Fig. This force need not be taken into account in the design of the bracing element itself.
Second-order effects As structures subject to lateral loads deflect, the vertical loads acting on the structure produce additional forces and moments.
These are normally referred to as second-order effects. Consider a cantilever column shown in Fig. EN requires second-order effects to be considered where they may significantly affect the stability of the structure as a whole or the attainment of the ultimate limit state at critical sections. Second-order effect order bending moments i. Although this would suggest that the designer would first have to check the second-order bending moment before ignoring it, the code provides some simplified criteria, to verify if second-order analysis is required.
These tests are summarized in clause 5. These checks essentially ensure that adequate Clause 5. Time-dependent effects The main effects to be considered are creep and shrinkage of concrete and relaxation of prestressing steel. EN provides information in clause 7. Design by testing EN itself does not provide any useable guidance on this subject. However, there is some guidance in EN Annex D of EN provides further information, particularly regarding the statistics.
Design entirely based on testing is not common in building structures. However, this is an accepted method in some other fields e. In such work, test programmes should be designed in such a way that an appropriate design strength can be established, which includes proper allowance for the uncertainties covered by partial safety factors in conventional design.
When testing is carried out on elements which are smaller than the prototype, size effects should be considered in the interpretation of results e. Test methods and procedures will obviously be different in each case. Before planning a test, the precise nature of the information required from the test together with criteria for judging the test should be specified.
Structural analysis 3. Elastic analysis with or without redistribution General EN provides limited guidance on analysis. Elastic analysis remains the most popular method for frame e. Braced frames may be analysed as a whole frame or may be partitioned into subframes Fig. The subframes may consist of beams at one level with monolithic attachment to the columns.
As a further simplification, beams alone can be considered to be continuous over supports providing no restraint to rotation. Clearly this is more conservative. In unbraced structures, it is generally necessary to consider the whole structure, particularly when lateral loads are involved.
A simplified analysis may be carried out, assuming points of contraflexure at the mid-lengths of beams and columns Fig. Stiffness parameters Clause 5. However, when computing the effects of deformation, shrinkage and settlement reduced stiffness corresponding to cracked cross section should be used. Effective spans Calculations are performed using effective spans, which are defined below. The principle is to identify approximately the location of the line of reaction of the support. Typical conditions are considered below.
Discontinuous end support on bearings Fig. Effective flange width: a T beam; b L beam. The calculation of the variation is a complex mathematical problem. A uniform distribution of stress is assumed over the effective width. For building structures the effective widths shown in Fig. In analysis, EN permits the use of a constant flange width throughout the span.
This should take the value applicable to span sections. This applies only to elastic design with or without redistribution and not when more rigorous non-linear methods are adopted.
Redistribution of moments The moment-curvature response of a true elasto-plastic material will be typically as shown at Fig. The long plateau after Mp is reached implies a large rotation capacity. When a bending moment in a critical section usually at a support reaches Mp, a plastic hinge is said to be formed. The structure will be able to withstand further increases in loading until sufficient plastic hinges form to turn the structure into a mechanism.
Had the beam been elastic, then the bending moment will be as shown in Fig. The process illustrated is plastic analysis.
Clearly, to exploit plasticity fully, the material must possess adequate ductility rotation capacity. Concrete has only limited capacity in this regard. The moment redistribution procedure is an allowance for the plastic behaviour without carrying out plastic hinge analysis.
Indirectly, it also ensures that the yield of sections under service loads and large uncontrolled deflections are avoided. No redistribution is permitted in sway frames or in situations where rotation capacity cannot be defined with confidence. Moment-curvature: idealization Fig. Thus, where support moments are reduced, the moments in the adjacent spans will need to be increased to maintain equilibrium for that particular arrangement of loads.
In the case of continuous beams or slabs, subject predominantly to flexure and in which the ratio of the lengths of the adjacent spans lie in the range 0. These requirements are presented graphically in Fig. Table 3. It should be noted that the above limitations try to ensure sufficiently ductile behaviour. Plastic analysis Apart from moment redistribution, EN allows the use of plastic analysis without Clause 5.
The first of the conditions can be expressed as a reinforcement percentage for a balanced section, i. Note that the stress block is a function of fck for concrete strength higher than 50 MPa. For details of the method, standard textbooks should be consulted. In conjunction with plastic analysis, an appraisal of the behaviour of the structure at serviceability should be undertaken. Non-linear analysis This takes into account the non-linear deformation properties of reinforced-concrete sections.
This method is seldom used in practical design as it is complex, requiring a computer and prior knowledge of reinforcement details throughout the structures. It could be useful in appraising the capacity of existing structures or when a large repetition of a particular structure is considered e.
The method involves plastic hinge analysis see redistribution of moments, p. A hinge is said to have formed when the steel starts to yield. Consider an encastre beam, as shown in Fig. The procedure to predict the ultimate load capacity may be as follows: 1 Knowing the reinforcement, section properties and concrete grade, calculate the Myk at A and C.
Myk is the bending moment which produces a stress fyk in the tension reinforcement. At each stage calculate a the rotation of the hinges at A and B by integrating the curvatures of the beam between hinges this will require the beam to be divided into a number of sections.
At each section the curvature should be calculated using equation 7. Note that the rotations are calculated using mean values of mechanical properties whereas the strengths are calculated using design properties, i.
It must be clear that the analysis of a continuous beam will be fairly complex even with a computer. Also, a number of boundary conditions will need to be imposed in the analysis. In practice, therefore, this procedure is rarely used, and elastic analysis with moment redistribution is preferred. Strut-and-tie models Strut-and-tie models utilize the lower-bound theorem of plasticity, which can be summarized as follows: for a structure under a given system of external loads, if a stress distribution throughout the structure can be found such that 1 all conditions of equilibrium are satisfied and 2 the yield condition is not violated anywhere, then the structure is safe under the given system of external loads.
This approach particularly simplifies the analysis of the parts of the structure where linear distribution of strain is not valid. Typical areas of application are noted in Section 3. Typical models are shown in Fig. As can be seen, the structure is thought of as comprising notional concrete struts and reinforcement ties. Occasionally, concrete ties may also be considered e. While there might appear to be infinite freedom to choose the orientation of struts and ties, this is not so. Concrete has only a limited plastic deformation capacity; therefore, the model has to be chosen with care to ensure that the deformation capacity is not exceeded at any point before the assumed state of stress is reached in the structure.
In this context, the deformation of the struts may be neglected, and the model optimized by minimizing the expression Fig. Having idealized the structure as struts and ties, it is then a simple matter to arrive at the forces in them based on equilibrium with external loads.
Stresses in the struts and ties should be verified including those at nodes, where a number of members meet. Schlaich and Schafer1 recommend a procedure for checking the node regions. Bearing in mind that strut and tie models fall under plastic analysis, it is interesting to note that the code does not impose any conditions similar to those noted in Section 3. This is an inconsistency. It is therefore important to follow the theory of elasticity fairly closely in choosing the model as already discussed above.
Design aids and simplifications 3. One-way spanning slabs and continuous beams The bending moment coefficients shown in Fig. This assumes supports which do not offer rotational restraint.
The coefficients for locations may be used without sacrificing too much economy even when continuity with internal columns is to be taken into account. However, continuity with external columns has a significant effect on the bending moments at locations 1 and 2. The following modifications may be made, depending on the ratio of the total column wall stiffness to the slab stiffness.
It should be noted that the values of bending moments in Fig. It will not, therefore, be possible to redistribute the bending moments, as the loading case applicable will vary with each span.
Under these conditions the bending moments and shear forces may be obtained using Table 3. No redistribution of moments is permitted to the bending moments obtained using the above table.
Two-way spanning slabs General For rectangular slabs with standard edge conditions and subject to uniformly distributed loads, normally the bending moments are obtained using tabulated coefficients.
Such coefficients are provided later in this section. Bending moment coefficients: a continuous slab on point supports; b continuous slabs with continuity at external columns Table 3. Ultimate bending moment and shear forces in one-way spanning slabs continuous beams and flat slabs End support and end span Simple Monolithic At first Middle At outer Near middle At outer Near middle interior of interior Interior support of end span support of end span support spans supports Moment 0 0.
Conditions for the use of the tabular value are: 1 The table is based on the yield line i. Therefore, conditions 1 - 3 in Section 3. In the corner area shown: 1 provide top and bottom reinforcement 2 in each layer provide bars parallel to the slab edges Table 3.
Consider panels 1 and 2 in Fig. As the support on grid A for panel 1 is discontinuous and support on grid C for panel 2 is continuous, the moments for panels 1 and 2 for the support on grid B could be significantly different. Corner reinforcement: two-way spanning slabs Fig. Loads on beams supporting two-way spanning slabs slab may be reinforced throughout for the worst case span and support moments.
However, this might be uneconomic in some cases. In such cases, the following distribution procedure may be used: 1 Obtain the support moments for panels 1 and 2 from Table 3.
Treating M-1 and M-2 as fixed end moments, the moments may be distributed in proportion to the stiffnesses of span lx in panels 1 and 2. Loads on supporting beams Loads on supporting beams may be obtained using either Fig. Flat slabs Flat slab structures are defined as slabs solid or coffered supported on point supports. Unlike two-way spanning slabs on line supports, flat slabs can fail by yield lines in either of the two orthogonal directions Fig.
For this reason, flat slabs must be capable of carrying the total load on the panel in each direction. Methods of analyses Many recognized methods are available. These include: 1 the equivalent frame method 2 simplified coefficients 3 yield line analysis 4 grillage analysis. For other approaches specialist literature should be consulted e.
The width of the slab to be used for assessing the stiffness depends on the aspect ratio of the panels and whether the loading is vertical or horizontal. Vertical loading. When the aspect ratio is less than 2 the width may be taken as the distance between the centre lines of the adjacent panels. For aspect ratios greater than 2, the width may be taken as the distance between the centre lines of the adjacent panels when considering bending in the direction of longer length spans of the panel and twice this value for bending in the perpendicular direction.
See Fig. The width of beams for frame analysis is as follows: Table 3. Possible failure modes of flat slabs Fig. Horizontal loading in the frame will be considered only in unbraced structures. In these cases, the question of restraint to the columns, and hence the effective length of columns, is a matter of judgement. If the stiffness of the slab framing into the column is overestimated, the effective length of the column will reduce correspondingly.
As the stiffness at slab-column junctions is a grey area, codes of practice adopt a cautious approach. For the slab, half the stiffness applicable to vertical loading is used. Additional stiffening effects of drops or solid concrete around columns in coffered slabs may be included, but this will complicate hand calculations. Analysis The equivalent frames may be analysed using any of the standard linear elastic methods such as moment distribution see Section 3.
Braced structures may be partitioned into subframes consisting of the slab at one level continuous with columns above and below. The far ends of the columns are normally taken as fixed unless this assumption is obviously wrong e. The load combinations given in Section 3. Simplified coefficients In braced buildings with at least three approximately equal bays and both slabs subject predominantly to uniformly distributed loads, the bending moments and shear forces may be obtained using the coefficients given in Section 3.
Lateral distribution of moments in the width of the slab In order to control the cracking of the slabs under service conditions, the bending moments obtained from the analysis should be distributed taking into account the elastic behaviours of the slab.
As can be imagined, the strips of the slab on the lines of the columns will be stiffer than those away from the columns. Thus, the strips closer to the column lines will attract higher bending moments. Division of panels Flat slab panels should be divided into column and middle strips, as shown in Fig. The design moments to be resisted by the column strip may be decreased by an amount such that the total positive and the total negative design moments resisted by the column strip and middle strip together are unchanged.
As drawn, Fig. The width of the middle strip should be adjusted accordingly. Allocation of moments between strips The bending moments obtained from the analysis should be distributed between the column and middle strips in the proportions shown in Table 3.
In some instances the analysis may show that hogging moments occur in the centre of a span e. When the above condition is not met, the moment is concentrated more in the middle strip. Moment transfer at edge columns As a result of flexural and torsional cracking of the edge and corner columns, the effective width through which moments can be transferred between the slabs and the columns will be much narrower than in the case for internal columns.
If Mt max is less than these limits, the structure should be redesigned. When the bending moment at the outer support obtained from the analysis exceeds Mt max, then the moment at the outer support should be reduced to Mt max, and the span moment should be increased accordingly.
In the latter case, it is essential to provide torsional links along the edge of the slab. However, U bars as distinct from L bars with longitudinal anchor bars in the top and bottom may be assumed to provide the necessary torsional reinforcement Fig. Effective slab widths for moment transfer Fig. Flat slabs: detailing at outer supports Fig. ANALYSIS Bending moments in excess of Mt max may be transferred to the column only if an edge beam which may be a strip of slab is suitably designed to resist the tension.
Beams For the design of beams, the bending moment coefficients given in Fig. Simplifications EN permits the following simplifications regardless of the method of analysis used: 1 At a support assumed to offer no restraint to rotation e.
This recognizes the effect of the width of support and arbitrarily rounds off the peak in the bending moment diagram. This provision is quite reasonable, as failure cannot occur within the support. This ensures a minimum design value for the support moment, particularly in the case of wide supports. This is reasonable; but the effect of continuity should be considered when designing the support such as columns or walls. Concrete 4. General EN covers concrete of strength up to 90 MPa in normal-weight concrete and 80 MPa in lightweight concrete.
It relies on EN for the specification and production of concrete. Strength In EN the compressive strength of concrete is denoted by concrete strength classes, which relate to the characteristic cylinder strength fck or the cube strength fck, cube.
The relationship between the cylinder and cube strengths is given in the code, and is reproduced in Table 4. The code provides models for strength development with time, and these should be used when concrete strength need to be calculated. Table 4. Elastic deformation EN recognizes that deformation properties are crucially dependent on the composition of concrete, and in particular on the aggregates.
Therefore, it provides only indicative information, which should be sufficient for most normal structures. However, in structures that are likely to be sensitive to deformation, it is advisable to determine the properties by controlled testing and appropriate specification.
The values given apply to concretes with quartzite aggregates. The reduction factors for other Table 4. Strength classes and associated properties fck, cy MPa 15 20 25 30 35 40 45 50 55 60 70 80 90 fck, cube MPa 20 25 30 37 45 50 55 60 67 75 85 95 fcm MPa 24 28 33 38 43 48 53 58 63 68 78 88 98 fctm MPa 1.
Creep and shrinkage The creep and shrinkage of concrete depend on a number of factors, including the ambient humidity, the dimensions of the element, the composition of the concrete and the age of concrete at the time of loading. When the compressive stress at time t0 exceeds 0.
Drying shrinkage is the result of expiration of moisture from the concrete to the surrounding air. Tables 4. Values at 30, and days are given in Table 4. Stress-strain relationships A distinction should be made between the stress-strain curve used for analysis and that used for the design of cross-sections. Figure 4. For numerical values, see EN EN offers two alternatives for the design of cross-sections. These are parabola- rectangle Fig. For numerical values of the parameters, see EN Creep coefficients for concrete in outdoor conditions Table 4.
This is shown in Fig. EN recommends a value of 1. Multipliers kh to shrinkage coefficient for size of members h0 mm kh 1. Parabola-rectangle diagram for concrete under compression sc fck fcd 0 ec3 ecu3 ec Fig.
Rectangular stress distribution for concrete between the predicted strength and that obtained in experiments. See the background paper to the UK National Annex. These are functions of concrete strength, but have constant values of 0. Lightweight concrete 4. General EN deals with additional requirements for lightweight concrete in Chapter The requirements for normal-weight concrete are generally applicable to lightweight concrete unless specifically varied.
Density classes Six density classes are defined in EN In each class, a range is given for the density and the nominal density to be used in design calculations.
Density classes for lightweight concrete Density class 1. Stress-strain diagrams of typical reinforcing steel. The creep strains so derived should be further multiplied by 1. The UK National Annex proposes to adopt the same value.
Reinforcing steel EN rules are valid for ribbed bars, de-coiled rods, welded fabrics and lattice girders. The range of essential properties is listed in the normative Annex C of the code. Although the reinforcement standard EN will be published soon, it will confine itself to the testing requirements of properties and not define the properties themselves. Strength fyk The grade of reinforcement steel denotes the specified characteristic yield stress fyk.
It is obtained by dividing the characteristic yield load by the nominal cross-section area of the bar. For products without a pronounced yield stress, the 0.
Typical and idealized stress—strain diagrams are given in Fig. EN is valid for yield stress of reinforcement in the range MPa. In the UK, the characteristic strength fyk of the commonly used grade of reinforcement is likely to be MPa. In general, ductility is inversely related to yield stress. Therefore, in applications where ductility is critical e. Stress-strain diagram for typical prestressing steel Table 4. Limits for the ratio of the actual yield stress to the specified strength are specified for such applications.
Currently, no UK standard gives any guidance. There is no technical reason why other types of reinforcement should not be used in conjunction with EN , provided suitable allowance is made for the behaviour. Relevant authoritative publications should be consulted when other types of reinforcement are used.
Ductility is an essential property if advantage is to be taken of the plastic behaviour of structures. The greater the ductility, the greater the elongation in axially loaded members, and the greater the rotation capacity in members subjected to flexure.
When the ultimate strength is controlled by the strain in concrete reaching the limiting value, the length of the plastic zone influences the rotation. The longer this length, the greater is the rotation. These are shown in Table 4. Prestressing steel EN states that its requirements will be satisfied if the prestressing steel complies with EN The definition of the relevant parameters is shown in Fig. Basic assumptions The basic assumptions about section behaviour are very similar to those adopted by many, if not most, modern codes of practice.
The assumptions define the stress-strain responses to be assumed for steel and concrete, and the assumptions to be made about the strains at the ultimate limit state. It is these assumptions about the strain that define failure. Stress-strain curves The information required to obtain the design stress-strain curves for concrete, ordinary Clause 3.
Clause 3. These are given in clauses 3. In each case, it is possible to choose between two possible bi-linear Clause 3. These curves are shown in Fig. Where the horizontal top branch is used, no limit on the tensile strain is imposed; however, the characteristics of the inclined top branch depend on the ductility class of the reinforcement. For LWAC bundles of bars should not consist of more than two bars and the equivalent diameter should not exceed 45 mm. Step by step or approximate methods may be used in these calculations.
The action should include for dynamic effects. The fall may occur in any construction stage. The recommended value of k is 1,0.
In general, the methods given in Section B. It should be noted that the guidance in this Annex has been verified by site trials and measurements. For background information reference can be made to the following: Le Roy, R. Toutlemonde, F. Le Roy, R. Section B. The alternative approach takes account of the effect of adding silica fume and significantly improves the precision of the prediction. When concrete is to be loaded at earlier ages, with significant strength development at the beginning of the loading period, specific determination of the creep coefficient should be undertaken.
This should be based on an experimental approach and the determination of a mathematical expression for creep should be based on the guidelines included in Section B. Extrapolating such results for very long-term evaluations e. When safety would be increased by overestimation of delayed strains, and when it is relevant in the project, the creep and shrinkage predicted on the basis of the formulae or experimental determinations should be multiplied by a safety factor, as indicated in Section B.
For HSC without silica fume, creep is generally greater than predicted in the average expressions of Section B. Two expressions for shrinkage and two for creep, are given in this clause. This distinguishes phenomena which are governed by different physical mechanisms.
Therefore the hardening rate controls the progress of the phenomenon. Shrinkage appears to be negligible for maturity less than 0,1. For ages beyond 28 days, the variable governing the evolution of autogeneous shrinkage is time.
In cases where this strength fcm t is not known, it can be evaluated in accordance with 3. Furthermore, the younger the concrete at loading, the faster the deformation. However this tendency has not been observed for non silica-fume concrete.
For this material, the creep coefficient is assumed to remain constant at a mean value of 1,4. The following procedure, based on the experimental determination of coefficients altering the formulae of Section B. The measurements should be obtained under controlled conditions and recorded for at least 6 months.
However, when safety would be increased by overestimation of delayed strains, and when it is relevant in the project, the creep and shrinkage predicted on the basis of the formulae or experimental determinations should be multiplied by a safety factor. The following clauses of EN apply. NOTE The minimum reinforcement is obtained if the directions of reinforcement are identical to the directions of the principal stresses. In order to avoid unacceptable cracks for the serviceability limit state, and to ensure the required deformation capacity for the ultimate limit state, the reinforcement derived from Expressions F.
In no case should the distance to the free edge be taken as less than 50 mm. This minimum value depends on the strength of the concrete at the time of tensioning. The cross section of the prism associated with each anchorage is known as the associate rectangle. The axis of the prism is taken as the axis of the tendon, its base is the associate rectangle and its depth behind the anchorage is taken as 1,2. The arrangement of the reinforcement should be modified if it is utilised to withstand the tensile forces calculated according to 8.
For higher compressive stresses, non-linear creep effects should be considered. Any variation in restraint conditions during the construction stages or the lifetime of the structure should be taken into account in the evaluation. Particularly useful for verification at intermediate stages of construction in structures where properties vary along the length e. Methods based on the theorems of linear Applicable to homogeneous structures with rigid viscoelasticity restraints.
The ageing coefficient method This method will be useful when only the long -term distribution of forces and stresses are required. Applicable to bridges with composite sections precast beams and in-situ concrete slabs. Simplified ageing coefficient method Applicable to structures that undergo changes in support conditions e.
Brief outline details of some of the methods are given in the following sections. The second term represents the creep due to this stress. The third term represents the sum of the instantaneous and creep deformations due to the variation in stresses occurring at instant ti. The fourth term represents the shrinkage deformation. When the stress in pre- stressing steel is greater than 0,5fpmax relaxation and a variable state of deformation should be taken into account.
This is accounted for by a step-by-step process. Structural analysis is carried out at successive time intervals maintaining conditions of equilibrium and compatibility and using the basic properties of materials relevant at the time under consideration.
The deformation is computed at successive time intervals using the variation of concrete stress in the previous time interval. In particular, on a section level, the changes in axial deformation and curvature due to creep, shrinkage and relaxation may be determined using a relatively simple procedure.
Relaxation at variable deformation may be evaluated in a simplified manner at infinite time as being the relaxation at constant length, multiplied by a reduction factor of 0, The eight components of internal forces are listed below and shown in Figure LL. It may be appropriate to take into account the multiaxial compression state in the definition of fcd. The thickness of the different layers should be established by means of an iterative procedure see rules to In particular clause 6. In expression 6.
This will cause the reinforcement to become eccentric in the layer; as a consequence two internal bending moments arise, and these should be in equilibrium within the shell element. The internal layer should be checked for an additional out of plane shear corresponding to the force transferred between the layers of reinforcement. The following simplifications to the general model may be introduced for the purpose of this application Figure MM.
If the resulting reinforcement is eccentric within the two plates, the Expressions LL. The procedure is based on the fatigue load models given in EN For the calculation of damage equivalent stress ranges for steel verification, the axle loads of fatigue load model 3 shall be multiplied by the following factors: 1,75 for verification at intermediate supports in continuous bridges 1,40 for verification in other areas. It can be calculated by Equation NN. The values have been calculated on the basis of a constant ratio of bending moments to stress ranges.
The values given for mixed traffic correspond to the combination of train types given in Annex F of EN For structures carrying multiple tracks, the fatigue loading shall be applied to a maximum of two tracks in the most unfavourable positions see EN The effect of loading from two tracks can be calculated from Equation NN. The effect of loading from two tracks may be calculated from Equation NN. The forces from the upper flange result in forces being applied to the diaphragm and these determine the design of the element.
Figures OO. In these circumstances, it is then only necessary to check the support nodes. The nodes at the bearings must be checked using the criteria given in 6. Figure OO. Strut and tie model Reinforcement should be designed for the tie forces obtained from the resistance mechanisms adopted, taking account of limitations on tension in the reinforcement indicated in 6. In general, due to the way in which vertical shear is transmitted, it will be necessary to provide suspension reinforcement.
If inclined bars are used for this, special attention should be paid to the anchorage conditions Figure OO. A Reinforcement Figure OO. Anchorage of the suspension reinforcement If the suspension reinforcement is provided in the form of closed stirrups, these must enclose the reinforcement in the upper face of the box girder Figure OO. Links as suspension reinforcement In cases where prestressing is used, such as post-tensioned tendons, the design will clearly define the order in which these have to be tensioned diaphragm prestressing should generally be carried out before longitudinal prestressing.
P Bhatt, T. MacGinley, B. Although the detailed design methods are generally according to European Standards EuroCodes , Codes of practice, including Eurocode 2 [5], provide guidance for design using linear-elastic analysis, non-linear analysis, plastic methods, strut-and-tie modelling, stress analysis and more approximate methods based on moment It is implied but not clearly stated in Eurocode 2 : Part 3 that the cracking check may be carried out under quasipermanent loading.
Eurocode : Basis of Structural Design. Forthcoming : provisional. Designers ' Guide to EN Eurocode 2 : Design of Concrete The book contains many worked examples to illustrate the various aspects of design that are presented in the text.
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