"Gheorghe Asachi" Technical University of Iai Faculty of Civil Engineering and Building Services

REINFORCED CONCRETE

Course

Chapter 4 Deformations of concrete Reinforced Concrete Students notes

Capitolul 4. Deformation of concrete

During its service life, concrete is subjected to many types of deformation. Some deformations are induced by the modifications which take place into the structure during hardening other by the external loads.

4.1 Deformation under short term loading 4.1.1 Mechanical behaviour

The concrete deformations to compression are developing on four stages as follows:

Fig. 41 Characteristic stress-strain diagram (curve) of concrete to compression and tension

1. First stage - up to a stress level of (0,30,5)fc the behavior of concrete is considered elastic, therefore the stress- strain relation is linear. There are small plastic deformations which have low values compared to the elastic ones. 2. Second stage - between (0,30,75)fc, the micro-cracks are developing and form a network that affects the compactness; the effects are:

- plastic deformations increase but are stable

- Poissons ratio increases - concrete volume increases.

Chapter 4 Deformations of concrete Reinforced Concrete Students notes

3. Third stage - the micro-cracks strogly develop into the matrix and get united with that from the separation surface. In this stage the deformation modulus decreases, Poissons ratio exceeds 0,5; the volume of specimen increases, deformations are instable. d) last stage corresponds to the descending part of stress-strain curve; the structure of concrete is destroyed by a macro-cracking mesh.

Fig. 42 Compression stress-strain diagrams for concrete with different strengths

The limit strains, i.e. the strain at reaching the maximum strength c,lim and the ultimate strain cu are affected by the strength of concrete as follows:

- the greater the compressive strength, the greater the strain at reaching the maximum strength c,lim.

- the greater the compressive strength, the smaller the ultimate strain cu.

The behavior of the concrete under compression loading is determined by: - its character of composite, micro-porous and micro-cracked material - the separation surface between matrix and aggregate (interface transition zone, ITZ) - the bond, prevalent physical between both component, as well as by their properties

Chapter 4 Deformations of concrete Reinforced Concrete Students notes

4.1.2 Moduli of deformation

To calculate the stiffness or expected deflection of structural members, it is necessary that a modulus of elasticity to be established. For concrete three deformation modulus can be calculated. Conventionally one of them is considered in calculation as modulus of elasticity. The deformation modulus can be calculated using the stress strain curve of concrete in compression.

Fig. 43 Moduli of deformation of concrete in compression

i. Elastic tangent modulus It is given by a tangent line to the

stress-strain diagram into the zero point. = =

[N/mm2]

ii. Secant modulus; it is given by a secant that cuts the curve into the zero point and a point given by a level of stress equal to 0,4 , see Fig. 43. According to Eurocode 2 (EC2) the secant modulus Ecm is considered elasticity modulus of concrete. =

= .

. [N/mm2] iii. Tangent modulus; it is given buy a tangent to curve, see Fig. 43

= =

Chapter 4 Deformations of concrete Reinforced Concrete Students notes

4.1.3 Strength and modulus of elasticity with time

The rate at which concrete strength increases with time depends on a variety of parameters as: - type and strength class of the cement; - type and amount of admixtures and additions; - the ratio W/C; - environmental conditions.

According to Eurocode 2 (EC2), the development of compressive strength with time may be estimated as () = () () = {[1 (28 )

.]} where

() mean compressive strength of concrete at an age t; - value at 28 days

() - function to describe the development of fcm with time concrete age coefficient which depends on the strength class of cement Strength class of concrete 32.5 32.5R, 42.5 42.5R, 52.5

s 0.38 0.25 0.20

The modulus of concrete develops more rapidly than does the compressive strength. According to Eurocode 2 (EC2), variation of the modulus of elasticity with time, can be estimated by: () = ( () )

. , - values at 28 days

Chapter 4 Deformations of concrete Reinforced Concrete Students notes

4.1 Deformation due to thermal expansion of concrete The concrete undergoes deformations of volume due to the action of variations in external temperature

and the hydration heat of the cement. The volume of a concrete member increases as its temperature increases. The corresponding length change l depends on the initial length l, the change in temperature T and the

coefficient of thermal expansion acc. to the equation =

and the thermal strain is =

Note. The linearity between thermal strain and temperature above mentioned holds true only for temperatures in the range of about 0oC to 60oC.

The coefficient of thermal expansion depends on the coefficients of thermal expansion of the aggregate and of the hydrated cement paste and as well as the moisture state of concrete.

= + Tabel 4-1: Typical values of the coefficient of thermal expansion of concrete [Deitling 1962]

Type of aggregate 105 [1]

Quartzitic rock, sand, gravel 1.2 -1.4 Granite, gneiss 0.9 1.2

Basalt, gabro, diorite 0.85 1.1 Dense limestone 0.65 0.9

According to EC2 for general calculation, unless more accurate information is available, the linear coefficient of thermal expansion may be taken equal to 1 10[].

Chapter 4 Deformations of concrete Reinforced Concrete Students notes

4.2 Design compressive and tensile strengths 4.2.1 Design compressive and tensile strengths

Design strengths are obtained by combining partial safety factors for materials with their characteristics values. The design compressive strength for concrete fcd is defined as follows

=

- is the partial safety factor for concrete; =1,5 - is the coefficient taking account of long term effects on the compressive strength and of

unfavourable effects resulting from the way the load is applied. = 1,0 according to National Annex Note: There is a lack of information concerning coefficient. In some situations 0,85 (bridge

design) is the value adopted in order to prevent an overestimation of the flexural resistance The design tensile strength for concrete fctd is defined as follows

= ,,

- is the partial safety factor for concrete; - coefficient taking account of long term effects on the tensile strength, and of unfavorable

effects resulting from the way the load is applied. = 1,0 according to National Annex

Chapter 4 Deformations of concrete Reinforced Concrete Students notes

Tabel 4-2: Strength and deformation characteristics for concrete according to Eurocode 2, Table 3.1

Poissons ratio =

,,

According to EC2 Poissons ratio may be taken as: 0.2 for uncraked concrete 0 for craked concrete

Chapter 4 Deformations of concrete Reinforced Concrete Students notes

4.2.2 Stress-strain relations for the design of sections

EC2 makes a distinction between the requirements for stress-strain relationships for use in the verification of cross-sections and for use non-linear analysis. The former is discussed here and the last at the subsequent heading.

The cross-section design provide three alternative stress-strain diagrams: i. parabolic-rectangular ii. bilinear iii. simplified rectangular.

They are for ultimate limit state (ULS) design only, not for serviceability limit state (SLS). i. Parabolic-rectangular diagram

Fig. 44 a Parabolic rectangular distribution

Stress-strain relationships

= 1 1 for 0

= for Where n is the exponent regarding the strength of concrete n=2 for normal strength class, i.e. C12/15..C50/60 - is the strain at reaching the maximum strength, see Fig. 44a - is the ultimate strain, see Fig. 44a.

Note. Designing process based on the parabolic-rectangular diagram is accurate but is not suitable for hand calculation.

Chapter 4 Deformations of concrete Reinforced Concrete Students notes

ii. Bilinear diagram

Fig. 44 b Bilinear diagram

Stress-strain relationships =

for 0 (linear)

= for (constant) - is the strain at reaching the maximum strength fcd, see Fig. 44b - is the ultimate strain, see Fig. 44b.

iii. Equivalent rectangular diagram

Fig. 44 c Equivalent rectangular diagram

= 0 for 0 (1 ) = for (1 ) (constant) Where could be shape coefficients = 0,8 and = 1,0 for 50 = 0,8 ( 50) 400 and = 1,0 ( 50) 200 for 50 < 90 - is the strain at reaching the maximum strength fcd, see Tabel 4-1. For design process, the equivalance is considered that:

Chapter 4 Deformations of concrete Reinforced Concrete Students notes

4.2.3 Stress-strain relation for non-linear structural analysis

Fig. 45 Schematic representation of the stress-strain relation for structural analysis (the use 0,4fcm for the definition of Ecm is approximate)

Stress-strain relationships

=

() for 0 || ||

where = is the strain at peak stress, see Tabel 4-1 or Table 3.1 of Eurocode 2 (SR EN 1992-1-1) = 1,05 || - compressive strength mean value - elasticity modulus of concrete (secant modulus), see Tabel 4-1 or Table 3.1 of Eurocode 2 (SR EN 1992-1-1)

Note. The non-linear analysis are not involved in current situations as concrete structures with ordinary spans and bays. Instead, for bridges of great span is advisable to use such type of analysis.

Chapter 4 Deformations of concrete Reinforced Concrete Students notes

4.2.4 Confined concrete

In situations where a concrete member is under tri-axial stress state, the EC2 allows enhancement of the characteristic compressive strength fck and ultimate limits, i.e. higher strength and higher critical strains are achieved.

Such confinement may be provided by stirrups or helix reinforcement, but no guidance on detailing is given in the code (EC2). It was not intended that this rule be invoked for general calculations on bending and axial force.

Fig. 46 Stress-strain relationship for confined concrete

In the absence of more precise data, the stress-strain relation shown in Fig. 46 may be used. The increase in strength and strain is as follows

, = 1,0 + 5

for 0,05

, = 1,125 + 2,5

for > 0,05

, = ,

, = + 0,2

= - effective lateral compressive stress at the ULS due to confinement

Chapter 4 Deformations of concrete Reinforced Concrete Students notes

4.3 Shrinkage and swelling of the concrete 4.3.1 Basic considerations

Because of the humidity of the environment during hardening, concrete shows permanent volume modifications. If the hardening takes place in air, then its volume decreases (loosing water), this beeing called shrinkage. If the concrete is kept in water or in high humidity, then an increase of its volume occurs, this being called swelling.

There are several types of shrinkage deformations as: Plastic shrinkage Autogenous shrinkage Drying shrinkage

These phenomena increase in time (time-dependent phenomenon), first quickly and then slowly, and after 35 years these deformations are consumed, particularly the drying shrinkage, see Fig. 47.

Fig. 47 Time development of total shrinkage and swelling in normal strength concrete These deformations are partially reversible to changing of the environment conditions (relative humidity). When water moves out of a porous body which is not fully rigid, contraction takes place. In concrete, from its fresh state to later in life, such movement of water generally occurs.

Chapter 4 Deformations of concrete Reinforced Concrete Students notes

Plastic shrinkage Plastic shrinkage occurs when water is rapidly lost from concrete by evaporation while it still is in its plastic stage. The greater the loss of water per hour at the placing of concrete, the greater the plastic shrinkage.

The greater the temperature and the speed of wind, the greater the plastic shrinkage. If the water loss is very high, then the cracking occurs

For hardening concrete the standard EC2 splits the shrinkage into two components: Autogenous shrinkage (short term shrinkage) occurs during hydration and hardening of concrete

without loss of moisture and the strain depends only on concrete strength. It is also called self-desiccation shrinkage. The majority of this component therefore occurs relatively quickly and is substantially complete in few months.

Autogenous shrinkage results from the volume reduction during the hydration of cement, i.e. the volume of the hardened cement paste is less than the sum of the volume of water and the volume of cement prior to the chemical reaction.

Drying shrinkage (long term shrinkage) it is considered the most important parameter influencing the magnitude of total shrinkage.

Drying shrinkage is associated with movement of water through and out of the concrete section and therefore depends on relative humidity and effective section thickness as well as concrete composition.

Therefore drying shrinkage increases with: - increasing water content of the concrete; the smaller W/C ratio, the smaller the shrinkage. - decreasing relative humidity RH of the surrounding environment; Total shrinkage is proportional with: cement paste content, fineness of cement and content ao alkalis. Shrinkage of concrete decreases with increasing modulus of aggregate and section size of the member

Chapter 4 Deformations of concrete Reinforced Concrete Students notes

4.3.2 Prediction of shrinkage

Shrinkage is a characteristic of concrete that was strongly studied in the last century, particularly by experiments. Relationships were developed and according to EC2 the value of the total shrinkage is

= + where is the drying shrinkage strain is the autogenous shrinkage strain Drying shrinkage () = (, ) , where

, - nominal unrestrained drying shrinkage; depends on RH and strength class; Table 3.2 EC2 - is a coefficient depending on the estimated (notional) size ho (size of the section of concrete member),

which is taken from Table 3.3 EC2; the greater ho the smaller kh coefficient. (, ) =

().

where

(, ) factor to calculate rate of shrinkage with time is the age of the concrete at the moment considered, in days is the age of the concrete (days at the beginning of drying shrinkage (or swelling). Normally this is at the

end of curing duration of drying (days Whenever the size ho cannot be estimated, use the relation = 2 where is the cross-section area of the member is the perimeter of that part of the cross section which is exposed to drying.

Chapter 4 Deformations of concrete Reinforced Concrete Students notes

Autogenous shrinkage The autogenous shrinkage is independent of the ambient humidity RH and of member of size and

that it develops more rapidly than drying shrinkage. () = () () where

() = 2.5( 10)10

() = 1 ..

- time in days

Chapter 4 Deformations of concrete Reinforced Concrete Students notes

Capitolul 5.

Capitolul 4. Deformation of concrete4.1 Deformation under short term loading4.1.1 Mechanical behaviour4.1.2 Moduli of deformation4.1.3 Strength and modulus of elasticity with time

4.1 Deformation due to thermal expansion of concrete4.2 Design compressive and tensile strengths4.2.1 Design compressive and tensile strengths4.2.2 Stress-strain relations for the design of sections4.2.3 Stress-strain relation for non-linear structural analysis4.2.4 Confined concrete

4.3 Shrinkage and swelling of the concrete4.3.1 Basic considerations4.3.2 Prediction of shrinkage