## Strut And Tie Method Example

Figure 2.2 — Definition of dimensions

Figure 2.2 — Definition of dimensions

(4) The distance lo between points of zero moment may be obtained from Figure 2.3 for typical cases. Figure 2.3 — Approximate effective spans for calculation of effective breadth ratios

The following conditions should be satisfied:

i) The length of the cantilever should be less than half the adjacent span.

ii) The ratio of adjacent spans should lie between 1 and 1.5.

(5) For the dispersion of prestressing forces in T beams see 4.2.3.5.3. 2.5.2.2.2 Effective span of beams and slabs

(1) The effective span (leff) of a member may be calculated as follows:

where:

ln is the clear distance between the faces of the supports values for a1 and a2, at each end of the span, may be determined from the appropriate ai values in Figure 2.4. 2.5.3 Calculation methods

Fv Vertical force acting on a corbel

Fsd,sup Design support reaction

Hc Horizontal force acting at the bearing on a corbel

AMsd Reduction in the design support moment for continuous beams or slabs, due to the support reaction Fsd.sup, when the support provides no restraint to rotation ac Distance between the point of application of the applied vertical load and the face of the supporting member (corbel design)

bsup Breadth of a support hc Overall depth of a corbel at the face of the supporting member

5 Ratio of redistributed moment to the moment before redistribution v Coefficient relating the average design compressive stress in struts to the design value of concrete compressive strength (fcd)

2.5.3.1 Basic considerations

P(1) All methods of analysis shall satisfy equilibrium.

P(2) If compatibility conditions are not checked directly for the limit states considered, then measures shall be taken to ensure that, at the ultimate limit states, the structure has sufficient deformation capacity and that unsatisfactory performance under service conditions is avoided.

P(3) Normally, equilibrium will be checked on the basis of the undeformed structure (first order theory). However, in cases where deformations lead to a significant increase in the internal forces and moments, equilibrium shall be checked considering the deformed structure (second order theory) (see 2.5.1, 4.3.5). P(4) Global analysis for imposed deformations, such as temperature and shrinkage effects, may be omitted where structures are divided, by joints, into sections chosen to accommodate the deformations. (5) In normal cases, such sections should not exceed | 30 m |

2.5.3.2 Types of structural analysis

2.5.3.2.1 Serviceability analysis

P(1) Analyses carried out in connection with serviceability limit states will normally be based on linear elastic theory.

(2) In this case it will normally be satisfactory to assume a stiffness for members based on the stiffness of the uncracked cross-section and an elastic modulus as defined in 3.1.2.5.2. Allowance for time-dependent effects should be made if these are likely to be significant (see 3.1 and 3.3).

P(3) Where cracking of the concrete has a significant unfavourable effect on the performance of the structure or member considered, it shall be taken into account in the analysis. Where the effect is favourable, it may be taken into account provided compatibility conditions are satisfied.

### 2.5.3.2.2 Ultimate limit states

P(1) Depending on the specific nature of the structure, the limit state being considered and on the specific conditions of design or execution, analysis for the ultimate limit states may be linear elastic with or without redistribution, non-linear or plastic.

P(2) The method used should be formulated so that, within its defined field of validity, the level of reliability generally required by this code is achieved, taking account of the particular uncertainties associated with the method. See, for example, 2.5.3.4.2.

P(3) In this section, the term "non-linear analysis" relates to analyses which take account of the non-linear deformation properties of reinforced or prestressed concrete sections. Analyses which take account of non-linear behaviour resulting from the deformation of elements are termed "second order analyses" (thus a "non linear second order analysis" takes account of both effects).

(4) The application of linear elastic theory will normally require no specific measures to ensure adequate ductility, provided that very high percentages of reinforcement are avoided at critical sections. However, where the moments obtained from linear elastic analysis are redistributed, it is necessary to ensure that critical sections have a rotation capacity sufficient to accommodate the amount of redistribution carried out.

(5) The plastic approach to analysis may be used only for very ductile structural elements where high ductility steel is used (see 3.2.4.2).

(6) Wherever possible, reinforcement splices should be located away from critical sections. If this is not possible, the deformation or rotation capacity of the splice region should be assessed on the basis of the total amount of reinforcement present.

### 2.5.3.3 Simplifications

P(1) Simplified methods or design aids based on appropriate simplifications may be used for analysis provided they have been formulated to give the level of reliability implicit in the methods given in this code over their stated field of validity. Redistribution is limited to that permitted by the assumptions implicit in the chosen simplified method.

(2) A value of zero may be taken for Poisson's ratio in place of the value given in 3.1.2.5.3.

(3) Continuous slabs and beams may generally be analysed on the assumption that the supports provide no rotational restraint.

(4) Regardless of the method of analysis used, where a beam or slab is continuous over a support which may be considered to provide no restraint to rotation, the design support moment, calculated on the basis of a span equal to the centre-to-centre distance between supports, may be reduced by an amount AMsd where:

Fsd.sup is the design support reaction bsup is the breadth of the support

(5) Where a beam or slab is cast monolithically into its supports, the critical design moment at the support may be taken as that at the face of the support but not less than the value given by 2.5.3.4.2(7).

(6) The loads applied to supporting members by the reactions from one-way spanning slabs, ribbed slabs and beams (including T beams) may be calculated on the assumption that the members supported are simply supported. Continuity should, however, be taken into account at the first internal support and at other internal supports if the spans on either side of the support differ by more than 30 %.

2.5.3.4 Structural analysis of beams and frames

2.5.3.4.1 Acceptable methods of analysis

P(1) Any of the methods given in 2.5.3.2.2(1) may be used.

2.5.3.4.2 Linear analysis with or without redistribution

P(1) The possible influence on all aspects of the design of any redistribution of the moments shall be taken into account. These aspects include bending, shear, anchorage and curtailment of the reinforcement and cracking.

P(2) The moments calculated using a linear elastic analysis may be redistributed provided that the resulting distribution of moments remains in equilibrium with the applied loads.

(3) In continuous beams where the ratio of adjacent spans is less than 2, in beams in non-sway frames and in elements subject predominantly to flexure, an explicit check on the rotation capacity of critical zones may be omitted provided that the conditions a) and b) given below are satisfied.

a) for concrete grades not greater than C 35/45

for concrete grades greater than C 35/45 5 T 0.56 + 1.25 x/d b) For high ductility steel, 5 T 0.7 For normal ductility steel, 5 T 0.85 where:

5 is the ratio of the redistributed moment to the moment before redistribution x is the neutral axis depth at the ultimate limit state after redistribution d is the effective depth.

For definitions of steel classes, see 3.2.4.2.

(4) In general, no redistribution is permitted in sway frames.

(5) In elements as defined in (3), where no redistribution has been carried out, the ratio of x/d should not exceed x/d = 0.45 for concrete Grades C12/15 to C35/45 x/d = 0.35 for concrete Grade C40/50 and greater at the critical section unless special detailing provisions (e.g. confinement) are made.

(6) Redistribution should not be carried out in circumstances where the rotation capacity cannot be defined with confidence (e.g. in the corners of prestressed frames).

(7) To allow for approximations in the idealisation of the structure and for possible unconsidered differences in the structural form during construction, the design moment at the faces of rigid supports in continuous spans should not be less than 65 % of the support moment calculated assuming full fixity at the faces of the rigid supports.

2.5.3.4.3 Non-linear analysis See Appendix 2.

2.5.3.4.4 Plastic Analysis See Appendix 2

2.5.3.5 Analysis of slabs

P(1) This section applies to slabs as defined in Section 2.5.2.1 subjected to bi-axial internal moments and forces. It may be extended to non solid slabs (ribbed, hollow, waffle slabs) if their response is similar to that of a solid slab, particularly with regard to the torsional stiffness.

(2) One-way spanning slabs subjected mainly to uniformly distributed loading may be considered as beams and be analysed according to 2.5.3.4.

(3) For flat slabs, a minimum design moment should be provided over supports to ensure the validity of the design for punching shear (see 4.3.4.5.3).

2.5.3.5.2 Determination of action effects P(1) 2.5.3.1, paragraphs (1) and (2) apply.

2.5.3.5.3 Acceptable methods of analysis

P(1) The following methods of analysis may be used:

a) linear analysis with or without redistribution;

b) plastic analysis based either on the kinematic method (upper bound) or on the static method (lower bound);

c) numerical methods taking account of the non-linear material properties;

(2) The application of linear methods of analysis is suitable for the serviceability limit states as well as for the ultimate limit states. Plastic methods, with their high degree of simplification, should only be used in the ultimate limit states.

(3) Current methods of plastic analysis are: the yield line theory (kinematic method) and the strip method (lower bound or static method).

2.5.3.5.4 Linear analysis with or without redistribution

P(1) For linear analysis with or without redistribution the same conditions as given for beams and frames in 2.5.3.4.2(2) apply.

(2) The bending moment at continuous edges may be reduced as defined in 2.5.3.4.2(2) and (3).

(3) For checking shear, torsion and bearing reactions, a linear interpolation between the action effects of a fully restrained edge and those of a simply supported edge may be used.

(4) See Appendix 2 for reinforcement proportioning in situations where the directions of the principal moments do not coincide with those of the reinforcement.

### 2.5.3.5.5 Plastic methods of analysis

P(1) Plastic analysis without any direct check on rotation capacity may be used for the ultimate limit state if appropriate ductility conditions are met.

(2) When using plastic analysis, the area of tensile reinforcement should not exceed, at any point or in any direction a value corresponding to x/d = 0.25

(3) A check on the rotational capacity is not necessary for high ductility reinforcing steel (see 3.2.4.2). Normal ductility steel should not be used unless its application can be justified.

(4) For the kinematic method, a variety of possible mechanisms should be examined assuming the design values of the material properties appropriate for the ultimate limit state.

(5) The ratio of the moments over continuous edges to the moments in the span should be between: I 0.5 and 2.0 I

(6) When static methods of plastic analysis are used, it may be convenient to determine the distribution of moments on the basis of a linear analysis and to design the necessary reinforcement on a plastic interpretation of this distribution by satisfying the equilibrium conditions. (See Appendix 2 for reinforcement proportioning).

2.5.3.5.6 Numerical methods of non-linear analysis See Appendix 2.

2.5.3.5.7 Analysis of prestressed slabs

(1) The rules given in (2)—(4) below complement those given in Section 2.5.4.

(2) Regardless of the type of tendons used (e.g. bonded or unbonded), the contact forces due to the curvature and friction of the tendons and the forces acting on the anchorage devices may be treated as external loads in the serviceability limit states.

(3) For the ductility classification of prestressed tendons see 3.3.4.3(3)

(4) Plastic analysis should not be applied to members where pretensioned tendons are used, unless justified.

2.5.3.6 Structural analysis of walls and plates loaded in their own plane

### 2.5.3.6.1 Permitted methods of analysis

P(1) This section applies to elements for which the assumption of a linear strain distribution is not valid. P(2) The following methods may be used for the determination of the internal forces and moments:

a) Methods based on linear analysis (see 2.5.3.6.2)

b) Methods based on plastic analysis (see 2.5.3.6.3)

c) Methods based on non-linear material behaviour (see Appendix 2).

P(3) Independently from the procedure adopted in the ultimate limit state, possible model uncertainty related to the overall response of the structure shall be taken into account.

P(4) Corbels and deep beams are special cases and are treated in 2.5.3.7.2 and 2.5.3.7.3 respectively.

### 2.5.3.6.2 Linear analysis

P(1) Linear analysis may be used for both the serviceability limit states and the ultimate limit states. However analysis for the ultimate limit states requires detailing of the reinforcement which is able to withstand the totality of the design tensile stresses in the concrete and which satisfies the equilibrium conditions in the ultimate limit states.

P(2) Imposed deformation (e.g. thermal effects, settlements of the supports) and second order effects shall be considered when significant.

P(3) When using numerical methods based on the theory of elasticity, cracking effects in regions of high stress concentration shall be taken into account.

(4) The effects of high stress concentrations may be allowed for by a reduction in the rigidity of the relevant zones.

2.5.3.6.3 Plastic analysis

P(1) Methods based on lower bound plastic solutions may be used provided appropriate measures are taken to ensure that ductility conditions are satisfied.

(2) Elements may be idealised as statically determinate trusses consisting of straight notional struts (carrying the compressive forces in the concrete) and ties (the reinforcement). The forces in the members of the truss are established from considerations of equilibrium. Sufficient reinforcement is then provided to carry the tension in the ties and a check is performed to ensure that the compressive stresses in the struts are not excessive. Detailing requirements should then be checked, with particular regard to anchorage of all reinforcement, and to local bearing stresses due to concentrated forces.

(3) In order to ensure approximate compatibility, the location and orientation of the struts and ties should reflect the distribution of internal forces resulting from an elastic analysis of the member.

(4) In checking concrete stresses in the struts, consideration should be given to a possible reduction in strength due to transverse tensile stresses, or cracking or the influence of shear. The average design compressive stress in the struts may be taken as v.fcd. In the absence of other data, v may be taken as | 0.61, including an allowance for sustained loading. Higher values for v (even v > 1) may be justified based on a triaxial state of compressive stress, provided it can be shown that the complementary transverse compression can be realised in practice (See 5.4.8.1).

(5) The design stress in the ties is limited to fyd.

(6) Detailing should comply with 5.4.

2.5.3.6.4 Non-linear analysis (1) See Appendix 2.

2.5.3.7 Corbels, deep beams, and anchorage zones for post-tensioning forces

2.5.3.7.1 General

(1) These types of elements may be analysed, designed and detailed in accordance with 2.5.3.6.3.

2.5.3.7.2 Corbels

(1) Corbels with 0.4 hc r ac r hc (See Figure 2.5) may be designed using a simple strut and tie model.

(2) For deeper corbels (ac < 0.4 hc), other adequate strut and tie models may be considered.

(3) Corbels for which ac > hc may be designed as cantilever beams

(4) Unless special provision is made to limit horizontal forces on the support, or other justification is given, the corbel should be designed for the vertical force Fv, and a horizontal force Hc T I 0.2 Fv | acting at the bearing area.

(5) The overall depth (hc) of the corbel should be determined from considerations of shear (see 4.3.2).

(6) The local effects due to the assumed strut and tie system should be considered in the overall design of the supporting member.

P(7) The detailing requirements of Chapter 5 generally, and 5.4.4 in particular, shall be met. Figure 2.5 — Example of a corbel, with a strut and tie model

2.5.3.7.3 Deep beams

(1) Deep beams under a concentrated load may be designed using a simple strut and tie model.

(2) In some cases, e.g. lower depth/span ratios, distributed loads, more than one concentrated load, etc., models combining strut and tie action with truss action may be used.

(3) Continuous deep beams are sensitive to differential settlement. A range of support reactions, corresponding to possible settlements, should therefore be considered.

P(4) The detailing requirements of Chapter 5 generally, and 5.4.5 in particular, shall be met.

2.5.3.7.4 Zones subjected to concentrated forces

P(1) Such zones shall be analysed and designed to take account of:

— the overall equilibrium of the zone;

— the transverse tensile effects due to the anchorages, individually and as a whole;

— compression struts, which develop in the anchorage zone of post-tensioned members, and local bearing stresses under the anchorages.

(2) Such zones in post-tensioned members may be designed by using adequate strut and tie models based on 2.5.3.6.3.

(3) Three-dimensional models should be considered, where the dimensions of the bearing area are small compared with the cross-section of the anchorage zone.

P(4) The detailing requirements of Chapter 5 generally, and 5.4.6 in particular, shall be met. 2.5.4 Determination of the effects of prestressing 2.5.4.0 Notation (See also 1.6 and 1.7)

Pd Design value of the prestressing force at the ultimate limit state (Assuming that Pmt is equal to

Pk inf Lower characteristic value of the prestressing force for serviceability calculations Pk,sup Upper characteristic value of the prestressing force for serviceability calculations

Po Initial force at the active end of the tendon immediately after stressing o Mean value of the prestressing force immediately after stressing (post-tensioning) or transfer (pre-tensioning) at any point distance x along the member (i.e. the force after immediate losses have occurred)

Pm,t Mean value of the prestressing force at time t, at any point distance x along the member

Pm, z Mean value of the prestressing force, after all losses have occurred, at any point distance x along the member

Po,max Maximum permissible value of Po

APc Loss of prestress due to elastic deformation of the member at transfer

### APsl Loss of prestress due to anchorage slip

APt(t) Loss of prestress due to creep, shrinkage and relaxation at time t (Note: in 4.2.3.5.5, the loss of stress from which APt(t) is calculated is represented by %ap, c + s + r)

APu(X) Loss of prestress due to friction rinf } Respectively, coefficients used to determine lower and upper characteristic values of the rsup } prestressing force at the serviceability limit state

2.5.4.1 General

P(1) This section relates to structures where prestress is provided by fully bonded internal tendons. P(2) The effects to be considered are:

— local effects around anchorages and where tendons change direction

— direct effects in determinate structures

— direct and secondary indirect effects due to redundant restraints in indeterminate structures.

(3) Members containing permanently unbonded tendons are covered in Part 1D.

(4) Members containing tendons which are temporarily unbonded during construction may be treated using simplified assumptions. In general, they may be treated as members with bonded tendons, except that at the ultimate limit state, the stress in tendons is assumed not to have increased due to loading.

2.5.4.2 Determination of prestressing force

P(1) The mean value of the prestressing force is given by a) or b) below, whichever is appropriate:

a) for pre-tensioned members:

APu(x) may require consideration where deflected tendons are used.

b) for post-tensioned members:

where:

Pm,t is the mean value of the prestressing force at time t and at a particular point along the member

Po is the initial force at the active end of the tendon immediately after stressing

APu(x) is the loss due to friction

APsl is the loss due to anchorage slip

APc is the loss due to elastic deformation of the member at transfer

APt(t) is the loss due to creep, shrinkage and relaxation at time t

(2) For limits on the initial prestress and methods of calculating losses, see 4.2.3. For transmission lengths and the dispersion of prestress, see 4.2.3.5.

P(3) For serviceability calculations, allowance shall be made for possible variations in prestress. Two characteristic values of the prestressing force at the serviceability limit state are estimated from: Pi = r p x k.sup xsup x m,t

= (2.20) Pk.inf = rinf Pm,t where Pk.sup and Pk.inf are respectively the upper and lower characteristic values. Pmt is the mean prestressing force estimated using the mean values for the deformation properties and the losses calculated in accordance with 4.2.3.

(4) The coefficients rsup and rinf may be taken as 11.11 and | 0.9 | respectively in absence of a more rigorous determination and provided that the sum of the losses due to friction and time dependent effects is r 30 % of the initial prestress.

(5) The values of Pmt which will generally be used in design are: Pm,o - the initial prestress at time t = 0

Pm, Z - the prestress after occurrence of all losses. P(6) At the ultimate limit state, the design value of prestress is given by:

(7) Values for Yp are given in Table 2.2.

P(8) For considering local effects at the ultimate limit state, the prestressing force shall be taken as equal to the characteristic strength of the tendons.

(9) This applies when checking the influence of concentrated forces or bursting effects at anchorages or where tendons change direction. (See 4.2.3).

2.5.4.3 Effects of prestressing under service conditions

P(1) The statically determinate and indeterminate internal forces and moments caused by prestressing shall be calculated by elastic theory.

(2) For normal buildings where the calculation of crack width is not considered necessary, the mean values of prestress may be used.

(3) In other cases, where the structural response is highly sensitive to the influence of prestress, the effects of prestress may be determined according to a) or b) below, as appropriate.

a) for checking cracking or decompression (see 4.4.2), the opening ofjoints between precast elements and fatigue effects, the relevant estimated characteristic values of the prestress are used.

b) for checking compressive stresses (see 4.4.1), the mean values of prestress are used.

2.5.4.4 Effects of prestressing at the ultimate limit states

2.5.4.4.1 Structural analysis — linear methods

P(1) Statically determinate and indeterminate effects of prestress shall be calculated using the appropriate ultimate design value of the prestressing force.

(2) In linear structural analysis, Yp may be taken as 1.0.

P(3) Where linear analysis with redistribution is used, the moments to which the redistribution is applied shall be calculated including any statically indeterminate effects of prestress.

2.5.4.4.2 Structural analysis — non-linear or plastic methods

2.5.4.4.3 Design of sections

P(1) When assessing the behaviour of a section at the ultimate limit state, the prestressing force acting on the section is taken as the design value, Pd. The prestrain corresponding to this force shall be taken into account in the assessment of section strength.

(2) The prestrain may be taken into account by shifting the origin of the design stress-strain diagram for the prestressing tendons by an amount corresponding to the design prestress.

(3) Yp may be taken as 1.0 provided the following conditions are both met:

a) not more than 25 % of the total area of prestressed steel is located within the compression zone at the ultimate limit state.

and b) the stress at ultimate in the prestressing steel closest to the tension face exceeds fp0.1k/Ym.

If the conditions are not met, the lower value of Yp given in Table 2.2 should be applied to all tendons.

(4) For the effects of inclined tendons, see 4.3.2.4.6(2).

(5) Any indirect prestressing moments due to redundant restraints should be taken at their characteristic values.

2.5.5 Determination of the effects of time dependent deformation of concrete

Ec,eff Effective tangent modulus of elasticity of the concrete at a stress Bc = 0

£n(t) An imposed deformation, independent of stress (e.g. due to shrinkage or temperature effects)

etot(t,to) Total strain in the concrete subject to initial loading at time to with a stress a(to) and subject to subsequent stress variations Aa(ti)

a(t) ) Compressive stress in the concrete at time t and to respectively B(to) )

x Ageing coefficient, dependent on strain development with time

### 2.5.5.1 General

P(1) The accuracy of the procedures for the calculation of the effects of creep and shrinkage of concrete shall be consistent with the reliability of the data available for the description of these phenomena and the importance of their effects on the limit state considered.

P(2) In general, the effects of creep and shrinkage shall be taken into account only for the serviceability limit states. An important exception concerns second order effects (see Appendix 3).

P(3) Special investigations shall be considered when the concrete is subjected to extremes of temperature.

(4) The effects of steam curing may be taken into account by means of simplified assumptions.

(5) The following assumptions may be adopted to give an acceptable estimate of the behaviour of a concrete section if the stresses are kept within the limits corresponding to the normal service conditions:

— creep and shrinkage are independent

— a linear relationship is assumed between creep and the stress causing the creep.

— non-uniform temperature or moisture effects are neglected.

— the principle of superposition is assumed to apply for actions occurring at different ages.

— the above assumptions also apply to concrete in tension.

P(6) For the evaluation of the time dependent losses of prestress, the effects of creep, shrinkage and relaxation of the tendons shall be taken into account. (See 4.2.3.5). (7) The creep function is given by the relationship:

where:

to is the time at initial loading of the concrete t is the time considered

J(t,to) is the creep function at time t

Ec(to) is the tangent modulus of elasticity at time to

Ec28 is the tangent modulus of elasticity at 28 days

0(t,to) is the creep coefficient related to the elastic deformation at 28 days related to Ec2g-

Values are given in section 3.1 for final creep coefficients 0(z,to) for typical situations. It should be noted, however, that the definitions of Ec(to) and Ec28 above, as well as in Appendix 1, differ from that in 3.1.2.5.2 where the secant modulus Ecm is defined. Hence, where the creep coefficients 0( ,to) of Table 3.3 are used in connection with Equations 2.21-2.24, and where creep deformations are significant, the values of Table 3.3 should be multiplied by 1.05.

(8) Values for the final shrinkage strains are given for typical conditions in 3.1.

(9) On the basis of the assumptions listed in (5) above the total strain for concrete subjected to initial loading at time to with a stress B(to) and subjected to subsequent stress variations Aa(ti) at time ti may be expressed as follows:

(tot (t,to) = (n(t) + B(to) J(t,to) + C J(t,ti) AB(ti) (2.22)

In this expression (n(t) denotes an imposed deformation independent of the stresses (e.g. shrinkage, temperature effects).

(10) For the purpose of structural analysis, equation (2.22) may be written as follows:

where the ageing coefficient x depends on the development of strain with time.

(11) In normal cases, x may be taken as 0.8. This simplification is good in the case of pure relaxation of the effects of a constant imposed deformation but is also adequate in cases where only long term effects are considered.

(12) If the stresses in the concrete only vary slightly, the deformations may be calculated using an effective modulus of elasticity:

For the notation see (7) above.

(13) For a more accurate analysis of the effects of time dependent deformation of concrete, see Appendix 1.

3 Material properties 3.1 Concrete

fc strength of concrete (Figure 3.1)

fck cube Characteristic compressive cube strength of concrete at 28 days fctk 0.05 Lower characteristic tensile strength (5 % fractile)

fctk 0.95 Upper characteristic tensile strength (95 % fractile)

fctax Axial tensile strength of concrete fct fl Flexural tensile strength of concrete fct,sp Splitting tensile strength of concrete

(c1 Compressive strain in the concrete at the peak stress fc

(cs Z Final shrinkage strain for normal weight concrete

£cs Basic shrinkage strain for normal weight concrete ecu Ultimate compressive strain in the concrete

0(z, to) Final creep coefficient of concrete

### 3.1.1 General

P(1) This section applies to concrete as defined in ENV 206, (Section 3, definitions 3.6 to 3.8)9) i.e. to concrete having a closed structure made with specified aggregates, so composed and compacted as to retain no appreciable amount of entrapped air other than entrained air. (See ENV 206-5.2)

P(2) For the production of plain, reinforced or prestressed structures, concrete as defined in P(1) above shall be used.

P(3) Concrete technology specifications shall satisfy the corresponding Clauses of ENV 206, as relevant to this Code.

(4) The concrete may be considered to have a closed structure, if the amount of entrapped air, after compaction, is not more than the limits given in paragraph 5.2 of ENV 206, entrained air and pores of the aggregate excepted.

(5) This section applies also to concrete subjected to heat treatment during the hardening process as defined in Clause 10.7 of ENV 206.

3.1.2 Normal weight concrete

3.1.2.1 Definitions

P(1) Normal weight concrete is a concrete having an oven dry (105 °C) density greater than 2 000 kg/m3, but not exceeding 2 800 kg/m3.

P(2) The density of hardened concrete shall be determined in accordance with ENV 206-7.3.2.

### 3.1.2.2 Compressive strength of concrete

P(1) This Eurocode is based on the characteristic compressive cylinder strength, fck, defined as that value of strength below which 5 % of all possible strength test results for the specified concrete may be expected to fall.

(2) The compressive strength of concrete should be determined by means of standard tests in accordance with Clause 7.3.1.1 of ENV 206, either on concrete cylinders or concrete cubes.

(3) The design rules of this Code are based solely on the characteristic 28 day strength, fck, of cylinders; cube strength, fck, cube, is mentioned but only as an alternative method to prove compliance.

(4) For some applications, it may be necessary to establish a minimum compressive strength at ages earlier or later ages than 28 days, or from specimens stored under different conditions to those defined in ISO 2736.

(5) Where necessary, direct tests should be carried out to determine conversion factors for strength, under any of the following circumstances:

— test specimens whose size or shape is different from those given in ENV 206.

— specimens are stored under non-standard conditions

— a measure of strength at different ages is required.

3.1.2.3 Tensile strength

P(1) The term tensile strength relates to the maximum stress which concrete can withstand when subjected to uniaxial tension.

P(2) The actual value of the tensile strength should be determined in accordance with Clause 7.3.1.2 of ENV 206.

9) ENV 206, "Concrete — Performance, Production, Placing and Compliance Criteria". Final draft, February 1989, (BSI Document 89/11639)

(3) If the tensile strength is measured as splitting tensile strength, fctsp, or as flexural tensile strength, fct,fl, the axial tensile strength fctax may be derived approximately from those values by using the following conversion factors:

fct,ax = I 09 I fct,sp or fct,ax = 105.1 fct,fl (3.1)

(4) In the absence of more accurate data, the mean and characteristic tensile strength of the concrete may be derived from the following equations.

fc ctm

fctm mean value of the tensile strength fck characteristic cylinder compressive strength of the concrete fctk 0.05 lower characteristic tensile strength (5 %-fractile). fctk 0.95 upper characteristic tensile strength (95 %-fractile).

The corresponding mean and characteristic values for the different concrete strength classes are given in Table 3.1.

3.1.2.4 strength classes of concrete

P(1) Design shall be based on a strength class of concrete which corresponds to a specified value of the characteristic compressive strength.

(2) The compressive strength of concrete is classified by concrete strength classes which relate to the cylinder strength, fck, or the cube strength fck cube, in accordance with ENV 206, Clauses 7.3.1.1 and 11.3.5.

(3) In Table 3.1, the characteristic strength fck and the corresponding tensile strength are given for the different strength classes of concrete.

Concrete strength classes less than C12/15, or higher than C50/60, should not be used for reinforced and prestressed concrete work unless their use is appropriately justified. For plain unreinforced concrete, see also Part 1A to this code.

Table 3.1 — Concrete strength classes, characteristic compressive strengths fck (cylinders) mean tensile strength fctm, and characteristic tensile strengths fctk of the concrete (in N/mm2). (The classification of concrete eg, C20/25 refers to cylinder/cube strength as defined in Section 7.3.1.1 of ENV 206)

Strength Class of Concrete

C12/15

C16/20

C20/25

C25/30

C30/37

C35/45

C40/50

C45/55

C50/60

fck

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