6. Reinforced Concrete Beams in Flexure

At relatively low values of applied bending moment, beams are designed with tensile reinforcement only, and are said to be singly reinforced, and those requiring tensile and compression reinforcement to resist much higher applied loading are denoted as doubly reinforced. 

In addition, the shear induced by the applied loading also needs to be resisted. 

This resistence is commonly provided by the use of shear links, although other approaches such as bent-up bars or a combination of bent-up bars and shear links may be used.

6.1 Behaviour of Beams in Flexure

Moment-curvature diagrams showing effect of reinforcement percentage

There appears two distinct pattern of behavior.

The first pattern is where there is a considerable increase in curvature after the attainment of the maximum moment and where the behaviour exhibits a characteristic close to elastic-perfectly plastic behaviour.

the second type of behaviour is where failure occurs as the maximum moment is achieved with no subsequent ductility. 

The first type of behaviour clearly 

4.1.1 Ultimate Limit State

Whatever the method used to determine the internal forces, the actions applied to the structure must be factored using those values pertaining to ULS, whether for single or multiple variable loads. Full plastic analysis may not be used for frames unless the plastic hinges, other than the last one to form, occur in the beams. 

This is because columns with high axial loads will not have any ductile capacity (and will not therefore satisfy the maximum limits for neutral axis depth to effective depth ratios (x/d).

This therefore means that frames are normally analysed elastically and the moment field then adjusted using redistribution to simulate plastic behaviour.

For slabs, plastic analysis using the Johansen Yield Line methods may be used, but there are severe restrictions on the x/d ratio to ensure ductility.

In practice, however, due to the relatively low levels of reinforcement, the x/d ratio, is rarely critical. 

It must be noted the maximum amount of redistribution is reduced and plastic analysis prohibited if Class A reinforcement is used.

In elastic analysis, the section properties are generally calculated on the gross concrete section (with no allowance for the reinforcement), and Young's modulus taken as the secant modulus, Ecm (in GPa), 

where fck (MPa) is the concrete grade strength.

Serviceability Analysis

This must be performed using an elastic analysis and contain all actions applied to the structure  including any deformations or settlement effects. 

It is not generally necessary to consider time-dependant effects such as creep, except where they will cause substantial changes in internal forces.

This last will occur in hyperstatic prestressed concrete structures such as continuous bridge decks. 

Although normal reinforced concrete undergoes creep, the effect is less as the stress levels are  potentially lower and thus is effectively ignored.

Generally a full serviceability analysis (and design) will be unnecessary. For most structures, it is sufficient to use "deemed to satisfy" approaches for, say, deflection or cracking.
Exceptions to this approach are where there is an explicit requirement to determine crack widths in liquid-retaining structures or bridgeworks, or a design which fails the "deemed to satisfy" clauses, but needs to be shown to be adequate.
It is accepted that "deemed to satisfy" clauses are conservative.

2. Mechanical Properties of Reinforced Concrete

Characteristic Strength

The properties of manufactured materials vary because the particles of the material are not uniform and because of inconsistencies in the manufacturing process which are dependent on the degree of control. These variations must be recognized and incorporated into the design process. 

For reinforced concrete, the material property that is of most importance is strength.

A strength to be used as a basis for design must be selected from the variation in values.

This strength, when defined, is called the characteristic strength. If the characteristic strength is defined as the mean strength, then 50% of the material is below this value and this is unsafe. Ideally the characteristic strength should include 100% of the samples, but this is impractical because it is a low value and results in heavy and costly structures.

The characteristic strength is calculated from the equation 

Where for n samples the standard deviation 

Characteristic Strength of Concrete

Concrete is a composite material which consists of coarse aggregate, fine aggregate and a binding paste mixture of cement and water. 

The mean strength which must be achieved in the mixing process can be determined if the standard deviation is know from previous experience. 

The strength of concrete increases with age and it is necessary to adopt a time after casting as a standard. The characteristic compressive strength for concrete is the 28-day cylinder or cube strength, i.e. the crushing strength of a standard cylinder, or cube, cured under standard conditions for 28 days as described in BS EN12390-1 (2000). The 28-day strength is approximately 80% of the strength at one year, after which there is very little increase in strength. Strengths higher than the 28-day strength are not used for design unless there is evidence to justify the higher strength for a particular concrete.

The characteristic tensile strength of concrete is the uniaxial tensile strength (fct,ax) but in practice this value is difficult to obtain and the split cylinder strength (f ct,sp) is more often used. 

The relation between the two values is (fct,ax)=0,9 (fct,sp). 
In the ebsence of test values of the tensile strength it may be assumed that the mean value of the tensile strength (fctm)=0,3 (fck)^2/3, where fck is the characteristic cylinder compressive strength. In situations where the tensile strength of concrete is critical, e.g. shear resistence, the mean value may be unsafe and a 5% fractile (f ctk0,05=0,7 fctm) is used. In other situations where the tensile strength is not critical, a 95% fractile (f ctk 0,95= 1,3 fctm) is used. 

BS EN 206-1 (2000) specifies the mix, transportation, sampling and testing of concrete. 
Concrete mixes are either prescribed, i.e. specified by mix proportions, or designed, i.e. specified by characteristic strength.
For example, Grade C30P denotes a prescribed mix which would normally give a strength of 30 MPa; Grade C25/30 denotes a designed mix for which a cylinder strength of 25 MPa or cube strength of 30 MPa is guaranteed. 
The grades recommended by EN are from 12/15 to 50/60 in steps approximately 5 MPa for normal weight aggregates.
The lowest Grades recommended for prestressed concrete are C30 for post-tensioning and C40 for pretensioning.

Characteristic Strength of Steel

The characteristic axial tensile yield strength of steel reinforcement (fyk) is the yield stress for hot rolled steel and 0,2% proof steel for steel with no pronounced yield stress. The value recommended in the European Code is :
High yield steel (hot rolled or cold worked) 500MPa. 
For prestressing tendons, the characteristic strength (fpk) at 0,1% proof stress, varies from 1000 to 2000 MPa depending on the size of tendon and type of steel.

DESIGN STRENGTH

The design strength allow for reduction in strength between the laboratory and site. 
Laboratory samples of the material are processed under strictly controlled standard conditions. Conditions on site vary and in the case of concrete: segregation can occur while it is being transported; conditions for casting and compaction differ; there may be contamination by rain; and curing conditions vary especially especially in hot or cold weather.

The design strength of a material is therefore lower than the characteristic strength, and is obtained by dividing the characteristic strength by a partial safety factor (ϒm),
The value chosen for a partial safety factor depends on the susceptibility of the material to variation in strength, e.g. steel reinforcement is less effected by site conditions than concrete.

Design Strength of Steel

The fundamental partial safety factor for steel ϒs = 1,15. For accidental loading, e.g. exceptional loads, fire or local damage, the value reduces to ϒs = 1,0. 

For earthquake and fatigue conditions values are increased.

Design Strength of Concrete

The fundamental partial safety factor for concrete ϒc = 1,5. For accidental loading, the value reduces to ϒc = 1,3. For earthquakes and fatigue conditions, values are increased.

Stress-Strain Relationship for Steel

A shape of the stress-strain curve for steel depends upon the type of steel and the treatment given to it during manufacture. For reinforcement steel, actual curves for short term tensile loading have the typical shapes, Curves for compression are similar.

Typical stress-strain relationships for steel reinforcement

It can be seen that hot rolled reinforcing steel yields, or becomes significantly plastic, at stresses well below the failure stress and at strains well below the limiting strain for concrete (0,0035). In a reinforced concrete member, therefore, the steel reinforcement may undergo considerable plastic deformation before the Ultimate Limit State (ULS) is reached, but will not fracture. However large steel strains are accompanied by the formation of cracks in the concrete in the tensile zone, and these may become excessive and result in serviceability failure at loads below the (ULS). 

For normal design purposes, the European Code recommends a single, idealized stress-strain curve for both hot rolled and cold worked reinforcement.

Design stress-strain relationship for steel reinforcement

Modulus of Elasticity for Steel

The modulus of elasticity for steel (Es) is obtained from the linear part of the relationship between stress and strain. This is a material property and values from a set of samples vary between 195 and 205 GPa. For design purposes, this variation is small and the European Code adopts a mean value of Es =200 GPa. The modulus of elasticity for prestressing steel varies with the type of steel from 175 to 195 GPa. Values of the moduli of elasticity are required in calculations involving deflections, loss of prestress and for the analysis of statically indeterminate structures.

Stress-Strain Relationship for Concrete

A curve representing the stress-strain relationship for concrete.

Stress-strain relationship for uniaxial compression of concrete

The important values of peak stress, peak strain and ultimate strain vary with the strength of concrete, rate of loading, age of the concrete, temperature and shrinkage. It is not easy to deide on design values that are safe and realistic.
The maximum stress is reached at a strain of approximately 0,002, after which the stress starts to fall. Disintegration of the concrete does not commence, however, until the strain reaches 0,0035, which is therefore taken as the limiting strain fro concrete at the ultimate limit state.
The maximum stress is the characteristic cylinder strength (fck).

The preferred idealization of the cross section design. 

Parabolic-rectangular stress-strain relationship for concrete in compression

Modulus of Elasticity for Concrete

The value of the modulus of elasticity is related to the type of aggregate and the strength of the concrete. Since the stress-strain curve for concrete is non-linear, a secant or static modulus is used. For normal weight concrete the modulus of elasticity

where the mean cylinder compressive strength fcm = fck + 8 MPa.

1.4.4. Limit State Design

It is self evident that a structure should be "safe" during its lifetimes, i.e. free from the risk of collapse. There are, however, other risks associated with a structure and the term safe is now replaced by the term "serviceable". 

A structure should not during its lifetime become "unserviceable", i.e. it should be free from risk of collapse, rapid deterioration, fire, cracking, excessive deflection etc.

A structure should be designed to be safe under all conditions of its useful life and to ensure that this is accomplished certain distinct performance requirements, called "limit states", have been identified. The method of limit state design recognizes the variability of loads, materials construction methods and approximations in the theory and calculations.

Limit state may be at any stage of the life of a structure, or at any stage of loading. The limit states which are important for the design of reinforced concrete are at ultimate and serviceability. Calculations for limit state involve loads and loads factors, and material factors and strengths.

Stability, an ultimate limit state, is the ability of a structure or part of a structure, to resist overturning, overall failure and sway. Calculations should consider the worst realistic combination of loads at all stages of construction.

Deflection is a serviceability limit state. Deflections should not impair the efficiency of a structure or its components, nor cause damage to the finishes. Generally the worst realistic combination of unfactored imposed loads is used to calculate elastic deflections. These values are compared with limit states of deformation.

Dynamic affects to be considered at the serviceability limit state are vibrations caused by machines, and oscillations caused by harmonic resonance, e.g. wind gusts on buildings. The natural frequency of the building should be different from the exciting source to avoid resonance.

1.4.3 Statistical Basis for Design

When a material such as concrete is manufactured to a specified mean strength, tests on samples show that the actual strength deviates from the mean strenght to a varying degree depending on how closely the process is controlled and on the variation in strength of the component materials. It is found that the spread of results approximates to a normal distribution curve. 

Normal distribution curve

The probability of a test result falling between two values of strength such as x1 and x2 is given by the area under the curve between the two values, i.e. the shaded area in the figure. The area under the whole curve is thus equal to unity.

The effect of quality control is as shown, poor control producing a flatter curve. The probability of a test falling below a specified value is clearly greater when the quality of control is reduced.

The equation of the normal distribution curve is:

y= 1/[σ(2π)˄0,5] exp [-(x-x)˄2/(2σ˄2)]

which shows that the curve is fully defined by the mean x and the standard deviation σ of the variable x.

For a set of n values of x, the mean is given by:

X= Σx/n

The standard deviation which is a measure of the dispersion, is the root mean square of the deviations of x from the mean, given by:

σ= [Σ(x-x)˄2/n]˄0,5

In practice, it is not usually possible to obtain all of the values of x that would theoretically be available. For example, it would not be possible to test all the concrete in a structure and it is therefore necessary to obtain estimates of x and σ by sampling. In this case, the best estimate of the mean is still that given by X= Σx/n, but the best estimation for the standard deviation is given by:

σ= [Σ(x-x)˄2/(n-1)]˄0,5

where n is the number of the test results in the sample.

For hand calculations, a more convenient form is 

σ=[(Σx˄2 - nx˄2) / (n - 1)]˄0,5

Statistical distributions can also be obtained to show the variation in strength of other structural materials such as steel reinforcement and prestressing tendons. It is also reasonable to presume that if sufficient statistical data were available, distributions could be defined for the loads carried by a structure. It follows that it is impossible to predict with certainty that the strength of a structural member will always be greater than the load applied to it or that failure will not occur in some other way during the life of the structure. The philosophy of limit state design is to establish limits, based on statistical data, experimental results and engineering experience and judgement, that will ensure an acceptably low probability of failure. At present there is insufficient information to enable distributions of all the structural variables to be defined and it is unlikely that is will ever be possible to formulate general rules for the construction of a statistical model of anything so complicated as an actual structure.

1.4 Structural Design

The Object of Structural Design

The requirements of the client and user may include the following:

• The structure should not collapse locally or overall
• It should not be so flexible that deformations under load are unsightly or alarming, or cause damage to the internal partitions and fixtures; neither should any movement due to live loads, such as wind, cause discomfort or alarm to the occupants/users.
• It should not require excessive repair or maintenance due to accidental overload or because of the action of weather.
• In the case of a building, the structure should be sufficiently fire resistant to, give the occupants time to escape, enable the fire brigade to fight the fire in safety, and to restrict the spread of fire to adjacent structures.
• The working life of the structure should be acceptable (generally varies between 10 and 120 years).

The designer should be conscious of the costs involved which include:

• Initial cost which includes fees, site preparation, cost of materials and construction.
• Maintenance costs, e.g. decoration and structural repair.
• Insurance chiefly against fire damage.
• Eventual demolition.