The following loading conditions should usually be considered in the design of continuous beams and slabs: (1) the design ultimate load of 1.4Gk + 1.6£k on all spans; and (2) the design ultimate load as (1) on alternate spans with I(/ k on intermediate spans. When moments at sections are determined by elastic analysis, the maximum moment may be reduced by redistribution provided that the calculated depth of the neutral axis is not greater than (/?b - 0.4)d where d is the effective depth and /?b is: moment at the section after redistribution moment at the section before redistribution and that the resistance moment at any section is not less than 70% of the moment at that section from elastic analysis. 12.4.5 Continuous and two-way solid slabs Slabs which are continuous in extent in one or two directions may be designed as simply supported, provided that continuous ties that may be required for overall stability of the structure are incorporated in the construction. In such cases, cracking will develop in the top surface of the floors at their supports and some provision will be needed for dealing with this in applying floor finishes. Where slabs are required to span in one direction over a number of supports, they should be designed for moments and shears, calculated in similar manner to those for continuous beams. If solid slabs are required to span in two directions, yield line analysis or the strip method of design may be used. British Standard 8110, however, gives simple methods for the design of rectangular slabs for simply supported two-way panels and twoway continuous or restrained slabs. 12.4.6 Flat slab construction Flat slab construction usually consists of a slab which spans between columns in two directions without supporting beams. Drops may be provided over the columns by increasing the depths of the slab and sometimes the column heads may be flared to reduce shear stresses. The slabs may be solid or ribbed in two directions. British Standard 8110 offers a method of design but does not exclude the use of other methods such as finite element analysis or other procedures. In the BS method, it is assumed that the slab is supported by a rectangular grid of columns in which the ratio of the longer spans to the shorter spans is not greater than 2. The slabs are divided longitudinally and transversely into column strips and middle strips; the columns and column trips are designed as frames spanning in each direction. Each frame is then analysed elastically; a simplified method is given for the situation where the structure is braced against lateral loading and the column grid has a regular layout. Procedures are given for determining the widths of column strips and for the treatment of drops. 12.4.7 Frames The loads to be adopted in the design of frames with their factors have already been given in Table 12.3. When considering the ultimate limit state, the forces, shears and moments calculated for design should be the worst combinations of loading regarded as feasible. British Standard 8110 gives some simplified procedures, which may be used for a number of common forms of construction. These analyse frameworks by breaking them down into subframes and make some provision for

redistribution of moments. Two types of frame are dealt with the no-sway frame, in which bracing, such as shear walls and lift or stair wells, are used to restrain sidesway, and sway frames, in which the frame itself provides the lateral restraint. For the latter, the amount of moment redistribution allowed is restricted with further restrictions on frames of four or more storeys in height to avoid excessive deflection and the possibility of frame instability. 12.4.8 Columns and walls The determination of the loads and moments on columns is given in BS 8110 to which reference should be made for details. A column is described as slender when the ratio of the effective length to the corresponding breadth with respect to either axis is greater than 12 (10 for lightweight aggregate concrete); if the ratio is less than 12, the column is said to be short. The effective length is dependent on the length of the column and on the degree of restraint at the top and bottom connections with the structure. Generally, the slenderness ratio for a column should not be greater than 60. A distinction is made between braced and unbraced columns, a column being described as braced when the lateral stability of the whole structure is ensured by providing walls or bracing to resist all horizontal forces. The procedures for dealing with walls in BS 8110 have much in common with those for columns. A concrete component is defined as a wall when the greater of the lateral dimensions is at least 4 times the smaller. For plain walls, however, the ratio may be less (since columns without reinforcement are not recognized) and reduction factors are then applied. To be described as a reinforced wall, the area of vertical reinforcement should not be less than 0.4% of the cross-sectional area of concrete; if the amount of reinforcement is less, the wall should be designed as a plain wall. Some reinforcement may be required in plain walls to control cracking. A stocky wall is one in which the ratio of effective length to thickness does not exceed 12 (10 for lightweight concrete), otherwise the wall should be treated as being slender. As for columns, the effective length is dependent on the height and conditions of end-restraint. Methods for calculating the loads and moments on walls (as for columns) are also given in some detail in BS 8110 to which reference should be made. Provided the recommendations in the British Standards are followed, the deflections of columns and walls should not be excessive.

12.5 Reinforced concrete 12.5.1 General In the design of reinforced concrete to meet the requirements of the Code, BS 8110, it will usually be most appropriate to consider the ultimate limit state first and then check the design against the requirements for cracking and deflection. This might be inappropriate in exceptional circumstances, e.g. where steels of characteristic strengths in excess of 500 N/mm 2 are being used or where spans were exceptionally long: in these cases cracking or deflection might govern design. In the sections that follow, design will be treated on the assumptions that normal conditions obtain. For these the Code gives simplified treatments for dealing with both cracking and deflection. It also gives methods more suited to the exceptional cases for which reference to the Code should be made. 12.5.2 Beams 72.5.2.7 Bending Ultimate resistance in bending is calculated by assuming that:

(1) Sections which are plane before bending remain plane after bending. (2) Stresses in the concrete may be determined using the stressstrain curve in Figure 12.4 (as assessed in the preparation of the design charts in Part 3 of the BS 8110), or may be taken as uniformly distributed across the most stressed 90% of the compression zone as indicated in Figure 12.7(a) with a value of 0.67/cu/ym, i.e. 0.45/cu for deriving simplified formulae. Ultimate compressive strain in the concrete for analysis of sections is 0.0035. (3) The strength of the concrete in tension is ignored. (4) The stress in the steel is derived from the stress-strain relationships in Figure 12.5 with a value not greater than fy/ym, i.e. 0.87/y in tension and not greater than 0.83/y/ym in compression, i.e. 0.72/y. The simplified assumptions may be used to derive design formulae, which are shown in Figure 12.7(a-d). For beams reinforced in tension only:

C=QAfcubdc but not greater than 0.2fcubd T not greater than 0.87/y^s

(12.5)

If dc is not to exceed Q.5d as a practical limit, then: Mu = 0.87/^d(I -0.97/y4//cuZ>d) and not greater than 0.156/c>/2

(12.6)

For beams reinforced in tension and compression: Q = 0-4/cuK but not greater than 0.2/cufo/

(12.7)

Cs = 0.0035[(4-4)MK£s but not greater than 0.12A'Jy (12.8) r=0.0035[(rf- dJ/d]AtE% but not greater than 0.87/y,4s (12.9)

0.67£,

Compression = 0.72 fy

-Tf * a45/Concrete

(12.10)

then:

Steel Strain

If dc is not greater than 0,5d and d' is not greater than 0.5JC where dc = [(T-Cs)/Cc]d

Stress

Stress

fy Tension = — = 0.87 f..y 'm

Strain 0.0035

M11 =Cc(d- 0.454) + C,(2-32)

where A/p is the total relaxation loss in the steel, S is the shrinkage strain and is the creep strain for unit stress. Using these formulae, the value of P is obtained from:

P-f^

(12.33)

where/p is the stress in the tendon at the stage considered. P is then substituted in the appropriate expressions in Figure 12.13 to obtain the required stress conditions immediately after transfer and subsequently under the loads for serviceability limit states. In members subjected to bending in one direction only it will be normal to locate the centre of the prestressing tendons at an eccentricity which will provide maximum compression at what will become the tension face with a small amount of tension or compression at what will become the compression face under load. For members subjected to loads from any transverse direction, the tendons will be placed concentrically to give a uniform prestress. The formulae given apply to either case. 12.6.3 Losses of prestress In making these calculations for the stress conditions during manufacture and in service, quantitative allowances must be made for the elastic contraction, shrinkage and creep of concrete, and relaxation of the steel. These characteristics are variable and are much influenced by the nature of the materials used, methods of production and the service conditions. Part 1 of BS 8110 gives values for the calculation of these losses for general use, while Part 2 amplifies this information for special circumstances; it is recommended that specialist literature should be consulted for very unusual conditions of temperature or exposure. The shrinkage of concrete, so far as it affects the loss of prestress, depends on the quality of the concrete, the size of the component, the nature of the aggregate, age at transfer and the conditions of exposure. It is usually reasonable to assume that the shrinkage of concrete may be taken as 100 x 10 6 for external exposure in the UK and 300 x 10~ 6 for indoor conditions. Experimental evidence shows that the creep of concrete is proportional to the applied stress for the stresses generally applied during transfer and, as for shrinkage, it is considerably affected by circumstances. For the calculation of the loss of prestress, it is convenient to define the amount of creep as a multiple of the elastic contraction at transfer, and to adopt a factor of 1.8 for transfer within 3 days, reducing to 1.4 for transfer at 28 days for outdoor exposure in the UK. These values may also be used for class 1 and 2 structures for internal conditions. Further advice for other conditions and for class 3 structures is given in Part 2 of BS 8110; for other problems specialist publications should be consulted. If it is necessary to estimate the amount of creep at some intermediate stage in the life of a structure, it is often sufficient to assume that about half occurs during the first month after transfer and that about three-quarters of the total occurs during the first 6 months following transfer. The relaxation of prestressing tendons due to creep of steel is dependent on the type of steel and method of manufacture. The relevant BS Standards (Table 12.7) require a 1000-h test for relaxation of tendons at different levels of initial stress as part of the acceptance test procedure. The relaxation loss used in design is obtained from the value in the manufacturers' test certificate corresponding to the initial prestress multiplied by a relaxation factor. The values in BS 8110 are quoted in Table 12.10. If, at the design stage, the steel supplier is not known, it will usually be sufficient to base calculations on specified values. For many forms of repetitive construction, Once the losses of prestress have been established, it may be possible to express the total loss as a percentage of the initial stress in the steel at transfer. Values for total loss, due to elastic contraction, shrinkage and creep of concrete, and relaxation of steel, of 20% for

Table 12.10 Relaxation factors for different types of tendons Type of tendon Wire and strand relaxation class 1 relaxation class 2 Bar

Relaxation factor Pretensioning Post- tensioning 1.5 1.2 -

2.0 1.5 2.0

pretensioning and 15% for post-tensioning for an initial stress in tendons of 70% of their characteristic strength have been found to be appropriate. If such an assumption is made, however, detailed refinement in design should not be attempted. Other sources of loss also need to be taken into account with post-tensioning. These arise through the movement of the tendons in the anchorage during the process of transfer, which needs to be determined by measurement and should be given by the manufacturer of the system, and through the development of friction between the tendon and its surroundings. The profiles of the cables or bars may be curved to provide for counteracting variation of moment due to dead and imposed loads or due to continuity. As a result, friction develops during stressing between the cables or bars and the inner surfaces of ducts or tendon deflectors. The amount of friction depends on the construction of the cable, the materials in sliding contact and the angular displacement. For long cables, the actual profiles are likely to deviate from their correct position to such an extent that they have an effective additional curvature, which causes considerable frictional effects. Then the force in a tendon, Px, at a distance ;c from the jacking point is given by: Px = P0 exp - [fax/rj + Kx]

(12.36)

where P0 is the force in the tendon at the jacking end, // is the coefficient of friction from Table 12.11, rps is the radius of curvature, x/rps is the angle of deviation over length x and K is the constant and the form of tendon and duct which has a usual value of 33 x 10~4/m but may be reduced to 17 x 10~4/m for rigid sheaths or rigidly fixed duct formers or to 25 x 10~4/m for greased strands in plastic sheaths.

Table 12.11 Values for coefficient of friction JL/ Condition

/j,

Lightly rusted strand on unlined concrete duct Lightly rusted strand on lightly rusted steel duct Lightly rusted strand on galvanized duct Bright strand on galvanized duct Greased strand on plastic sleeve

0.55 0.30 0.25 0.20 0.12

12.6.4 Stress limitations at transfer and for serviceability conditions Limits need to be set on the stresses in the steel and concrete at transfer to ensure that the deformation of the materials is not excessive since this would lead to high losses of prestress, severe cracking and undue distortion of components. Limits also need to be imposed on the stresses likely to occur in service to keep deflections within acceptable bounds and to control cracking to required limits. All calculations of stresses for these two sets of conditions are based on the assumptions that the section is uncracked and that the strains due to applied stresses are proportional to those stresses. The stress in tendons during the initial stressing operations should not normally be more than 75% of the characteristic strength but may be as much as 80% provided special care is taken. The stress at transfer should not normally be more than 70% and never more than 75% of the characteristic strength. The allowable maximum limit for compressive stress in concrete at transfer is 0.5/ci at the extreme compression face or 0.4/ci for a nearly uniform prestress, where/ cj is the strength at the time of transfer. For the serviceability limit state, the compressive stress in the concrete should not be more than 0.33/cu at the extreme compression face but for continuous construction this limit may be raised to 0.4/cu within the negative moment zone. The stress in direct compression should not be greater than 0.25/cu. Flexural tensile stresses in concrete are defined according to the class of structure decided at the outset of design. For class 1 structures, the maximum tensile stress at transfer is limited to 1 N/mm 2 and no tensile stresses are allowed for serviceability limit states. For class 2 structures, in which some flexural tensile stresses are allowed up to the tensile strength of the concrete for pretensioning and up to 0.8 times the tensile strength of the concrete for post-tensioning, the tensile strength is assumed to be 0.45V/d for transfer and QA5Jfm for tne serviceability limit states. Where a design service load is only likely to be rarely imposed and the concrete is normally stressed in compression so that any cracks that might occur are closed, the allowable tensile stress may be increased by 1.7 N/mm2 provided that pretensioned tendons are well distributed throughout the concrete stressed in tension and post-tensioned tendons are supplemented by secondary reinforcement. Although cracking is permitted in class 3 structures, the section is assumed to be uncracked and limits are set for notional tensile stresses for use in calculations for service loading to impose some restriction on the widths of cracks. At transfer, the limits set for tensile stresses are, however, the same as those for class 2 structures. The values for allowable notional tensile stresses are obtained from Table 12.12, which are multiplied by the factors in Table 12.13 to allow for the effect of the depth of section on cracking.

Table 12.12 Class 3 members - limits for notional tensile stresses Group

Pretensioned tendons Grouted post-tensioned tendons Pretensioned tendons distributed in tensile zone and close to the concrete tension face

Limiting crack width (mm)

Design stress (N/mm2) for concrete of grade 30

40

0.1 0.2 0.1 0.2 0.1 0.2

3.2 3.8 -

4.1 5.0 4.1 5.0 5.3 6.3

50 and over

4.8 5.8 4.8 5.8 6.3 7.3

Table 12.13 Class 3 members - depth factors Depth of member including composite members (mm)

Factor

200 and under 400 600 800 1000 and over

1.1 1.0 0.9 0.8 0.7

These stresses may be exceeded in certain circumstances for class 3 structures as indicated in BS 8110.

12.6.5 Beams 12.6.5.1 Flexural strength The methods of calculation of the ultimate flexural strength of prestressed concrete beams are similar to those for reinforced concrete beams with the additional need that allowance must be made for the effect of the condition of prestress. The assumptions made are: (1) Sections which are plane before remain plane after bending. (2) The stresses in the concrete may be determined from the stress-strain curve in Figure 12.4 or, more normally, may be taken as uniformly distributed across the compression zone to a depth of 0.9 times that of the neutral axis with a value of 0.45y^u for deriving simple formulae as in Figure 12.14(a). As for reinforced concrete, the ultimate compressive strain for the concrete is taken as 0.0035. (3) The tensile strength of the concrete is ignored. (4) The strains at ultimate in pretensioned tendons and in posttensioned and bonded tendons are assumed to conform generally with the strains in the concrete so that the stresses may be determined from the stress-strain relationships for steel given in Figure 12.6 making allowance for the initial stress condition in the steel after all losses. As a simple alternative, the stresses at ultimate may be obtained from the curves in Figure 12714(b) where allowance is made for the strength of the steel and the concrete, their respective proportions and the initial stress in the steel. (5) The strains at ultimate in unbonded post-tensioned tendons do not conform directly with the compressive strains in the adjacent concrete but are directly influenced by the increase in separation between the end anchorages. A method of calculation of the stress in the steel is given in BS 8110 or the stress may be determined by test or analysis. (6) Any additional steel reinforcement close to the tension face should be assumed to be stressed to its characteristic yield stress at ultimate. Such reinforcement in the compression zone should normally be ignored. If the compression zone at failure is not rectangular in shape, the ultimate strength should be calculated from first principles using the stress-strain relationships shown in Figures 12.4 and 12.6. 12.6.5.2 Deflection Control of deflection in the design of reinforced concrete beams is governed in BS 8110 by limitations on span:depth ratio, but the method is not appropriate for prestressed concrete beams. For normal construction no specific requirement is given, the limitations on stresses for service conditions usually being sufficient to avoid deflections becoming excessive. For special construction, where checks are required however,

Figure 12.14 Flexural strength of beams - approximate method (pretensioning and post-tensioning with bond) some guidance is provided in Part 2 of BS 8110. It should then be assumed in the calculation of the short- and long-term deflections of class 1 and class 2 prestressed beams that behaviour is elastic and that the properties of the section are those for the concrete with the deformation characteristics appropriate to the nature of the loading. For long-term loading, an effective modulus should be used in the calculations, which may be derived from the data given on creep. The same approach to the calculation of deflection may be adopted for class 3 prestressed beams provided that the section is not cracked. If, however, it is cracked under the load being considered, deflection is more likely to require limitation and it should then be calculated from the moment-curvature relationship determined from the properties of the materials and the characteristics of the section. It should be noted that for members, such as precast massproduced units with pretensioned steel with a uniform eccentric prestress along their length, the upward deflection at transfer and later in service, if the permanent loading is light, may need to be checked by calculation to ensure that it is not excessive. If such members are heavily loaded, it should be remembered that the regions near their ends are subjected to the effects of reversed bending due to the prestress which will tend to reduce the central deflection. 12.6.5.3 Shear Shear in prestressed concrete beams needs only to be considered

for ultimate conditions. Then, sections subjected to shear remote from regions of maximum bending are likely to be uncracked but those subjected to both bending and shear will usually be cracked in flexure. These two situations give rise to substantially different distributions of stress and therefore require different methods of analysis; each is dealt with in BS 8110. In each case, the contribution of the concrete to shear strength is calculated and may be taken into account when provision is made for shear reinforcement. Firstly, irrespective of the situation and amount of shear reinforcement, a limitation is set on the maximum shear that a section may sustain. This maximum shear strength is defined by limiting the maximum shear stress (V/bv-d) for cracked or uncracked sections to the lesser of 0.8V/CU or 5 N/mm2. For uncracked sections, the ultimate resistance of the concrete (Kco) is given by: PcO=V 6 0 -VA

(12.39)

Ultimate shear stress in concrete for uncracked beams