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.