Properties of hardened concrete

Shrinkage

Shrinkage is the time-dependent reduction in volume of concrete, without any applied load, caused by loss of moisture and by chemical reaction (hydration and carbonation). Typical long-term values are 0.0003 to 0.0008 (300 to 800 microstrain).

Types of shrinkage

Effects: surface and through cracks, warping of slabs, loss of prestress, and extra deflection. Shrinkage is reduced by low W/C, low paste content, large well-graded aggregate, proper curing, and provision of joints (IS 456, Cl. 6.2.4).

Creep

Creep is the gradual increase in strain of concrete with time under a constant sustained stress, over and above the instantaneous elastic strain. It is caused mainly by seepage of gel water out of the C-S-H layers and by slip of gel particles. Creep continues for years, with about 50% occurring in the first 3 months and 75 to 80% in the first year.

Creep coefficient and factors. Creep coefficient $\theta = \varepsilon_{creep}/\varepsilon_{elastic}$. IS 456 (Cl. 6.2.5.1) gives $\theta$ = 2.2 (loaded at 7 days), 1.6 (28 days) and 1.1 (1 year). It increases with higher W/C and paste content, lower strength at loading, higher stress (up to about 40 to 50% of $f_c$), lower humidity, smaller members and lower-modulus aggregates. It decreases with later age of loading, good curing and stiff aggregates.

Effects: increases deflection of beams and slabs, loss of prestress, redistribution of stress from concrete to steel in columns, but also relieves stress concentrations and reduces cracking due to shrinkage.

 strain
   |                          creep recovery
   |             ______________  (slow, partial)
   |          .-'              \
   |       .-'   creep          \
   |     .'                      \___________
   |   .'                  instantaneous  ___ permanent
   |  |  elastic strain     recovery        deformation
   |  |
   |__|______________________|__________________ time
   0  load applied           load removed

• On loading, an instantaneous elastic strain appears.
• Under sustained load, strain keeps rising with time (creep), fast at first, then slowly.
• On unloading, an instantaneous (elastic) recovery occurs, nearly equal to the elastic strain on loading (slightly less because $E$ rises with age).
• Then a slow creep recovery follows, which is only a part of the creep strain.
• The remaining strain is the permanent (irrecoverable) deformation.

Modulus of elasticity of concrete. Concrete is not perfectly elastic: the stress-strain curve is curved from the start, so several moduli are defined.

 stress
   |            .--'-.
   |          .'      \
   |        .'  secant
   |      .'  _.-'
   |    .' .-'  tangent at point
   |  .'.-'
   | /-' initial tangent
   |/___________________ strain

IS 456 (Cl. 6.2.3.1): $E_c = 5000\sqrt{f_{ck}}$ N/mm$^2$ (short-term static modulus, secant). For M25: $E_c = 5000\times 5 = 25000$ N/mm$^2$. Dynamic modulus is found from UPV: $E_d = \rho v^2 \frac{(1+\mu)(1-2\mu)}{1-\mu}$.

Hardened concrete properties decide the performance of a structure in service. The main ones:

1. Strength

• Compressive strength: the basic property, tested on 150 mm cubes at 28 days (IS 516); characteristic strength $f_{ck}$ is the value below which not more than 5% of results fall (IS 456, Cl. 2.2). Grades M20 to M40 are common.
• Tensile strength: direct tension is difficult to find, so splitting test is used; $f_{ct} \approx 0.7\sqrt{f_{ck}}$ is the flexural strength (IS 456, Cl. 6.2.2), while split tensile strength is about $0.5\sqrt{f_{ck}}$ to $0.6\sqrt{f_{ck}}$.
• Shear and bond strength (design shear stress $\tau_c$ from IS 456 Table 19; bond stress $\tau_{bd}$ from Cl. 26.2.1.1).

2. Elasticity

Stress-strain curve is non-linear; short-term modulus $E_c = 5000\sqrt{f_{ck}}$ N/mm$^2$ (IS 456, Cl. 6.2.3.1). Poisson's ratio is 0.15 to 0.20.

3. Volume change and time-dependent behaviour

• Shrinkage: strain about 0.0003 (Cl. 6.2.4).
• Creep: creep coefficient 1.1 to 2.2 (Cl. 6.2.5.1).
• Thermal: coefficient of expansion about $10\times10^{-6}$ per °C.

4. Durability and permeability

Resistance to water penetration, chemicals, freeze-thaw, abrasion and carbonation; it depends on W/C, cement content and compaction (IS 456 Tables 4 and 5).

5. Other properties

• Density: 2400 kg/m$^3$ plain, 2500 kg/m$^3$ reinforced (IS 456 Cl. 19.2.1).
• Fatigue and impact resistance, fire resistance, and unit weight.

Strength increases with age, lower W/C, good compaction and curing.

Elastic properties

Concrete is a composite of paste, aggregate and interface, and so it is not truly elastic. On loading, the stress-strain curve is nearly straight up to about 30 to 40% of the ultimate stress, then curves as microcracks grow. For design, concrete is treated as elastic at working stress. The elastic properties are the modulus of elasticity $E$, Poisson's ratio $\mu$ (0.15 to 0.20 for concrete) and modulus of rigidity $G = E/[2(1+\mu)]$.

Modulus of elasticity (Young's modulus)

It is the ratio of stress to strain within the elastic range:

$$E = \frac{\sigma}{\varepsilon}$$

• Initial tangent, tangent, secant and chord moduli are used (secant at about 33% of $f_c$ is the static modulus).
• Dynamic modulus from UPV or resonant frequency is higher than static modulus.
• IS 456 (Cl. 6.2.3.1): $E_c = 5000\sqrt{f_{ck}}$ N/mm$^2$. For M20: $E_c = 5000\times4.472 = 22360$ N/mm$^2$.

Factors affecting $E$

• Higher strength (lower W/C) gives higher $E$.
• Stiffer aggregates (granite) and higher aggregate content give higher $E$; light-weight aggregate gives lower $E$.
• Moist condition gives higher $E$ than dry; porosity reduces $E$.
• It increases with age and with slower rate of loading giving a lower apparent value (creep effect).

$E$ is used to calculate deflection, modular ratio $m = 280/(3\sigma_{cbc})$ (IS 456, Annex B) and to find stress distribution in composite sections.

Powers' model gives the strength of concrete in terms of the gel/space ratio $x$:

$$x = \frac{\text{volume of gel}}{\text{volume of gel} + \text{capillary pores}} = \frac{0.657\,\alpha}{0.319\,\alpha + w/c}$$

where $\alpha$ is the degree of hydration and $w/c$ the water-cement ratio by mass. The numbers come from cement of specific gravity 3.15: 1 g of cement has 0.319 cm$^3$ volume and produces gel of $2.06\times0.319 = 0.657$ cm$^3$ (including gel pores). Compressive strength:

$$f_c = 240\,x^3\ \text{MPa}$$

(240 MPa is the intrinsic strength of the gel; it varies from about 234 to 255 MPa for different cements.) The ratio $x$ does not depend on the mass of cement; the mass is only needed to give volumes.

Given

Cement = 550 g; $w/c$ = 0.45, so water = $0.45\times550 = 247.5$ g.

Full hydration ($\alpha = 1$)

Volume of cement $= 550\times0.319 = 175.5$ cm$^3$ (about 550/3.15). Gel volume $= 550\times0.657 = 361.4$ cm$^3$. Space available $= 175.5 + 247.5 = 423.0$ cm$^3$.

$$\begin{aligned} x &= \frac{0.657\times1}{0.319\times1+0.45} = \frac{0.657}{0.769} = 0.854 \\ f_c &= 240\times(0.854)^3 = 149.7\ \text{MPa} \end{aligned}$$

60% hydration ($\alpha = 0.6$)

Gel $= 361.4\times0.6 = 216.8$ cm$^3$; space $= 175.5\times0.6 + 247.5 = 352.8$ cm$^3$.

$$\begin{aligned} x &= \frac{0.657\times0.6}{0.319\times0.6+0.45} = \frac{0.3942}{0.6414} = 0.615 \\ f_c &= 240\times(0.615)^3 = 55.7\ \text{MPa} \end{aligned}$$

Answer: gel/space ratio 0.854 and strength 149.7 MPa at full hydration; gel/space ratio 0.615 and strength 55.7 MPa at 60% hydration. (Powers' theoretical strengths are for the paste; actual concrete strength is lower.)

Powers' model gives the strength of concrete in terms of the gel/space ratio $x$:

$$x = \frac{\text{volume of gel}}{\text{volume of gel} + \text{capillary pores}} = \frac{0.657\,\alpha}{0.319\,\alpha + w/c}$$

where $\alpha$ is the degree of hydration and $w/c$ the water-cement ratio by mass. The numbers come from cement of specific gravity 3.15: 1 g of cement has 0.319 cm$^3$ volume and produces gel of $2.06\times0.319 = 0.657$ cm$^3$ (including gel pores). Compressive strength:

$$f_c = 240\,x^3\ \text{MPa}$$

(240 MPa is the intrinsic strength of the gel; it varies from about 234 to 255 MPa for different cements.) The ratio $x$ does not depend on the mass of cement; the mass is only needed to give volumes.

Given

Cement = 600 g, $w/c = 0.45$, so water = $0.45\times600 = 270$ g. Volume of cement $= 600\times0.319 = 191.4$ cm$^3$; gel at full hydration $= 600\times0.657 = 394.2$ cm$^3$.

Full hydration ($\alpha = 1$)

$$\begin{aligned} x &= \frac{394.2}{191.4+270} = \frac{394.2}{461.4} = 0.854 \\ f_c &= 240\times0.854^3 = 149.7\ \text{MPa} \end{aligned}$$

60% hydration ($\alpha = 0.6$)

$$\begin{aligned} x &= \frac{0.6\times394.2}{0.6\times191.4+270} = \frac{236.5}{384.8} = 0.615 \\ f_c &= 240\times0.615^3 = 55.7\ \text{MPa} \end{aligned}$$

Answer: gel/space ratio 0.854 and strength 149.7 MPa at full hydration; 0.615 and 55.7 MPa at 60% hydration. The result is independent of the 600 g mass because only the ratio $w/c$ matters.

Creep is the gradual increase in strain of concrete with time under a constant sustained stress, over and above the instantaneous elastic strain. It is caused mainly by seepage of gel water out of the C-S-H layers and by slip of gel particles. Creep continues for years, with about 50% occurring in the first 3 months and 75 to 80% in the first year.

 strain
   |                          creep recovery
   |             ______________  (slow, partial)
   |          .-'              \
   |       .-'   creep          \
   |     .'                      \___________
   |   .'                  instantaneous  ___ permanent
   |  |  elastic strain     recovery        deformation
   |  |
   |__|______________________|__________________ time
   0  load applied           load removed

• On loading, an instantaneous elastic strain appears.
• Under sustained load, strain keeps rising with time (creep), fast at first, then slowly.
• On unloading, an instantaneous (elastic) recovery occurs, nearly equal to the elastic strain on loading (slightly less because $E$ rises with age).
• Then a slow creep recovery follows, which is only a part of the creep strain.
• The remaining strain is the permanent (irrecoverable) deformation.

Mechanism: (i) seepage of adsorbed and gel water from the C-S-H under sustained stress, (ii) viscous slip of gel particles, (iii) microcracking at high stress and (iv) delayed elastic effect of aggregates. Specific creep is the creep strain per unit stress.

Creep coefficient and factors. Creep coefficient $\theta = \varepsilon_{creep}/\varepsilon_{elastic}$. IS 456 (Cl. 6.2.5.1) gives $\theta$ = 2.2 (loaded at 7 days), 1.6 (28 days) and 1.1 (1 year). It increases with higher W/C and paste content, lower strength at loading, higher stress (up to about 40 to 50% of $f_c$), lower humidity, smaller members and lower-modulus aggregates. It decreases with later age of loading, good curing and stiff aggregates.

Effects: increases deflection of beams and slabs, loss of prestress, redistribution of stress from concrete to steel in columns, but also relieves stress concentrations and reduces cracking due to shrinkage.

Shrinkage is the time-dependent reduction in volume of concrete, without any applied load, caused by loss of moisture and by chemical reaction (hydration and carbonation). Typical long-term values are 0.0003 to 0.0008 (300 to 800 microstrain).

Types of shrinkage

Factors: higher water and cement content, higher W/C, small aggregate size, low-modulus aggregate, low humidity, thin members and poor curing all increase shrinkage. Control: low W/C, large well-graded aggregate, proper curing, contraction joints and sufficient reinforcement. IS 456 (Cl. 6.2.4) gives an approximate shrinkage strain of $0.0003$ for design when no test data are available.

Elastic deformation

Immediate strain that appears on loading and disappears on unloading. For concrete it follows a curved stress-strain path, but within working stress it is treated as linear with $E_c = 5000\sqrt{f_{ck}}$ N/mm$^2$ (IS 456, Cl. 6.2.3.1). Elastic strain $\varepsilon_e = \sigma/E_c$.

Shrinkage

Shrinkage is the time-dependent reduction in volume of concrete, without any applied load, caused by loss of moisture and by chemical reaction (hydration and carbonation). Typical long-term values are 0.0003 to 0.0008 (300 to 800 microstrain). Main types are plastic, drying, autogenous and carbonation shrinkage. It is reduced by low W/C, large aggregate, good curing and joints. IS 456 (Cl. 6.2.4.1) gives design strain of 0.0003.

Creep

Creep is the gradual increase in strain of concrete with time under a constant sustained stress, over and above the instantaneous elastic strain. It is caused mainly by seepage of gel water out of the C-S-H layers and by slip of gel particles. Creep continues for years, with about 50% occurring in the first 3 months and 75 to 80% in the first year. Creep coefficient values are 2.2, 1.6 and 1.1 for loading at 7 days, 28 days and 1 year (IS 456, Cl. 6.2.5.1). Creep raises deflections and causes loss of prestress; part of it is recoverable on unloading.

Concrete has three phases (aggregate, paste and the weak interfacial transition zone). Even before loading, bond microcracks exist at the interface because of bleeding, shrinkage and thermal strain. Their growth with load controls the shape of the curve.

 stress
 f_c |             _.-'''-._
 0.9 |        _.-''         '-.
 0.75|     .-'  III            \
 0.5 |   .'    II               \
 0.3 | .'  I                     \
     |/                           \__
     |_______________________________ strain

Stage I: up to about 30% of $f_c$

Stress-strain is nearly linear. Pre-existing interface cracks stay stable; no new cracks form. The energy is stored elastically.

Stage II: 30% to about 50%

Curve starts to bend. Bond cracks at the aggregate-paste interface begin to grow in number and length. Mortar cracks are still absent; the system remains stable.

Stage III: 50% to about 75%

Cracks start to appear in the mortar (matrix) and link with the bond cracks. Crack growth is stable only while the stress is not raised; this causes the curve to flatten. About 75% of $f_c$ is called the critical stress or onset of unstable crack propagation, also the long-term strength.

Stage IV: 75% to 100%

Crack propagation becomes unstable; the matrix cracks join the bond cracks into continuous crack paths. Volumetric strain reverses (the concrete starts to dilate), strain rises quickly and the peak stress is reached.

Post-peak

Strain softening: the descending branch; failure by shear or splitting along a continuous crack surface, with the strain at failure about 0.003 to 0.004.

Result: the curve is non-linear even in the early part because progressive microcracking and gel creep increase the strain faster than the stress.

Assumptions: specific gravity of cement 3.15; 1 cm$^3$ of cement gives 2 cm$^3$ of hydrated products (gel, with gel pores); capillary pores are the space not occupied by hydrated products or unhydrated cement. Take 1 cm$^3$ of cement as the basis. Degree of hydration at 28 days under standard curing is taken as complete ($\alpha = 1$), since no value is given.

$$\begin{aligned} \text{Volume of water} &= 0.5\times3.15 = 1.575\ \text{cm}^3 \\ \text{Total paste volume} &= 1 + 1.575 = 2.575\ \text{cm}^3 \\ \text{Hydrated products} &= 2\times1.0 = 2.00\ \text{cm}^3 \\ \text{Unhydrated cement} &= 1 - 1.0 = 0.00\ \text{cm}^3 \\ \text{Capillary pores} &= 2.575 - 2.00 - 0.00 = 0.575\ \text{cm}^3 \\ \text{Capillary porosity} &= \frac{0.575}{2.575}\times100 = 22.3\% \end{aligned}$$

Answer: capillary porosity = 22.3% of the paste volume (if the hydration at 28 days is less than complete, the porosity is higher; e.g. 32.0% at 75% hydration).

Assumptions: specific gravity of cement 3.15; 1 cm$^3$ of cement gives 2 cm$^3$ of hydrated products (gel, with gel pores); capillary pores are the space not occupied by hydrated products or unhydrated cement. Take 1 cm$^3$ of cement as the basis. Degree of hydration $\alpha = 0.75$.

$$\begin{aligned} \text{Volume of water} &= 0.5\times3.15 = 1.575\ \text{cm}^3 \\ \text{Total paste volume} &= 1 + 1.575 = 2.575\ \text{cm}^3 \\ \text{Hydrated products} &= 2\times0.75 = 1.50\ \text{cm}^3 \\ \text{Unhydrated cement} &= 1 - 0.75 = 0.25\ \text{cm}^3 \\ \text{Capillary pores} &= 2.575 - 1.50 - 0.25 = 0.825\ \text{cm}^3 \\ \text{Capillary porosity} &= \frac{0.825}{2.575}\times100 = 32.0\% \end{aligned}$$

Answer: capillary porosity = 32.0% of the paste volume.

Properties of hardened concrete

1. Strength: compressive, tensile (direct, splitting, flexural), shear and bond strength.
2. Elastic properties: modulus of elasticity and Poisson's ratio.
3. Dimensional changes: shrinkage, creep and thermal movement.
4. Durability: permeability, resistance to chemicals, freeze-thaw, abrasion and carbonation.
5. Density and unit weight.
6. Fatigue, impact and fire resistance.
7. Water tightness and porosity.

Elastic behaviour

Concrete is a heterogeneous material with a curved stress-strain relation, so it is only approximately elastic.

• Linear up to about 30% of $f_c$, then curved because of microcrack growth; peak strain about 0.002, ultimate strain 0.0035 (IS 456, Cl. 38.1).
• On unloading, part of the strain is recovered (elastic) and the rest is permanent.
• Different moduli are defined: initial tangent, tangent, secant, chord and dynamic.
• IS 456 (Cl. 6.2.3.1): $E_c = 5000\sqrt{f_{ck}}$ N/mm$^2$ (secant, short-term). Poisson's ratio is 0.15 to 0.20.
• Stiffer aggregate, higher strength, lower W/C and older age give a higher modulus; creep reduces the effective value in long-term loading ($E_{ce} = E_c/(1+\theta)$, Annex C).

Shrinkage is the time-dependent reduction in volume of concrete, without any applied load, caused by loss of moisture and by chemical reaction (hydration and carbonation). Typical long-term values are 0.0003 to 0.0008 (300 to 800 microstrain).

Types of shrinkage

Creep, instantaneous recovery and creep recovery

Creep is the gradual increase in strain of concrete with time under a constant sustained stress, over and above the instantaneous elastic strain. It is caused mainly by seepage of gel water out of the C-S-H layers and by slip of gel particles. Creep continues for years, with about 50% occurring in the first 3 months and 75 to 80% in the first year.

 strain
   |                          creep recovery
   |             ______________  (slow, partial)
   |          .-'              \
   |       .-'   creep          \
   |     .'                      \___________
   |   .'                  instantaneous  ___ permanent
   |  |  elastic strain     recovery        deformation
   |  |
   |__|______________________|__________________ time
   0  load applied           load removed

• On loading, an instantaneous elastic strain appears.
• Under sustained load, strain keeps rising with time (creep), fast at first, then slowly.
• On unloading, an instantaneous (elastic) recovery occurs, nearly equal to the elastic strain on loading (slightly less because $E$ rises with age).
• Then a slow creep recovery follows, which is only a part of the creep strain.
• The remaining strain is the permanent (irrecoverable) deformation.

Maturity of concrete is the product of curing time and temperature above a datum temperature. It lets the strength of concrete cured at different temperatures be compared, since strength is a function of maturity (the Saul's rule).

$$M = \sum (T - T_0)\,\Delta t$$

where $T$ = mean temperature of concrete during the interval $\Delta t$ (°C), $T_0$ = datum temperature (about $-10.5$ °C as per Saul; $-11$ °C is often used), and the unit of $M$ is °C-hours (or °C-days).

Concrete of the same mix having the same maturity has about the same strength, whatever the temperature history.

Example. Concrete cured at 20 °C for 28 days has maturity $= (20+10)\times 28\times 24 = 20160$ °C-h. Concrete cured at 10 °C reaches the same maturity in $20160/((10+10)\times 24)= 42$ days. Concrete at 30 °C needs only $20160/((30+10)\times 24) = 21$ days.

Use: estimating in-situ strength to decide stripping time, hot/cold weather planning and steam-cured precast products. The rule works well only for the early age and when the concrete is not dried out.

Maturity of concrete is the product of curing time and temperature above a datum temperature. It lets the strength of concrete cured at different temperatures be compared, since strength is a function of maturity (the Saul's rule).

$$M = \sum (T - T_0)\,\Delta t$$

where $T$ = mean temperature of concrete during the interval $\Delta t$ (°C), $T_0$ = datum temperature (about $-10.5$ °C as per Saul; $-11$ °C is often used), and the unit of $M$ is °C-hours (or °C-days).

Concrete of the same mix having the same maturity has about the same strength, whatever the temperature history.

Example. Concrete cured at 20 °C for 28 days has maturity $= (20+10)\times 28\times 24 = 20160$ °C-h. Concrete cured at 10 °C reaches the same maturity in $20160/((10+10)\times 24)= 42$ days. Concrete at 30 °C needs only $20160/((30+10)\times 24) = 21$ days.

Shrinkage

Shrinkage is the time-dependent reduction in volume of concrete, without any applied load, caused by loss of moisture and by chemical reaction (hydration and carbonation). Typical long-term values are 0.0003 to 0.0008 (300 to 800 microstrain). Types: plastic, drying, autogenous, carbonation.

Creep

Creep is the gradual increase in strain of concrete with time under a constant sustained stress, over and above the instantaneous elastic strain. It is caused mainly by seepage of gel water out of the C-S-H layers and by slip of gel particles. Creep continues for years, with about 50% occurring in the first 3 months and 75 to 80% in the first year.

Fatigue strength

Fatigue is the progressive internal damage and eventual failure of concrete under repeated (cyclic) loading, at a stress level well below the static strength.

• Fatigue strength is the maximum stress range that concrete can carry for a given number of cycles (usually 10 million). For plain concrete it is about 55% of the static compressive strength at 10 million cycles (for flexure about 55% of modulus of rupture) for 0 to maximum stress.
• Results are plotted as an S-N curve (stress ratio against log of number of cycles). Concrete shows no clear endurance limit, but the curve flattens near $10^7$ cycles.
• Failure occurs by growth of microcracks at the aggregate-paste interface, with increased strain and lower stiffness before final rupture.
• Affected by range of stress, frequency, stress reversal, rest periods, moisture, strength of concrete and air entrainment. Important in bridges, pavements, airfield slabs, machine foundations and offshore structures.

Impact strength

Impact strength is the capacity of concrete to resist a sudden, high-rate load, such as a falling weight, pile driving or vehicle collision. It is measured by a drop-weight test (a mass dropped repeatedly on a specimen; number of blows to first crack and to failure are recorded) or by the Charpy/Izod pendulum.

• Under rapid loading, concrete shows higher strength and higher modulus than under static loading (strength may increase by 20 to 40% for very fast rates), but it behaves in a more brittle way, with less strain at failure.
• The impact resistance rises with the strength and toughness of aggregate and with fibres; it falls with higher W/C and lack of reinforcement.
• Fibre reinforcement (steel, polypropylene), polymer modification and strong aggregates greatly increase impact resistance.

Fatigue effect is the loss of strength and eventual failure of concrete under repeated or fluctuating loads, even though the maximum stress is lower than the static strength. Microcracks at the aggregate-paste interface grow with each cycle until they join and cause rupture.

The fatigue strength of concrete for 10 million cycles is about 50 to 55% of its static strength (compressive or flexural). It is shown by an S-N curve and is important in bridges, pavements, crane girders and machine foundations.

Four types of behaviour

 (i) Linear elastic    (ii) Non-linear elastic
  s |    /               s |     _.-
    |   /                  |   .'
    |  /  load=unload      |  /  load = unload
    | /   same path        | /   same curved path
    |/____ e               |/______ e

 (iii) Linear inelastic (iv) Non-linear inelastic
  s |    /|                s |     _.-'
    |   / |                  |   .'  \
    |  /  | residual        |  /  .'   \ residual
    | /  ./  strain          | / .'      strain
    |/__/___ e               |/_/______ e

• (i) Linear elastic: stress proportional to strain, and on unloading the strain returns to zero along the same line (e.g. steel below yield).
• (ii) Non-linear elastic: the curve is curved but unloading follows the loading path with no residual strain (e.g. rubber).
• (iii) Linear inelastic: straight loading line, but on unloading a permanent strain remains (idealised plastic behaviour).
• (iv) Non-linear inelastic: curved loading path and permanent strain on unloading; this is the actual behaviour of concrete.

Reasons for non-linearity of concrete

1. Heterogeneous, three-phase structure. Aggregate is stiff and elastic, paste is softer and the interface is weak. Their different stiffness and strain capacity give a curved composite response.
2. Microcracks. Bond cracks already exist at the interface before load (from bleeding, shrinkage, temperature). Above 30% of $f_c$ they grow, and above 50 to 75% mortar cracks join them. Each new crack reduces stiffness, so the slope falls continuously.
3. Creep and time-dependent flow of gel. Part of the strain is viscous and not recovered, so unloading leaves residual strain.
4. Porosity and moisture. Water in the gel pores and capillaries moves under stress, adding to non-linear deformation.
5. Stress concentration at pores and voids leads to local yielding at stresses lower than the average strength.

Concrete is a three-phase system: aggregate, hydrated cement paste and the interfacial transition zone (ITZ) between them. The stress-strain behaviour of the composite is different from that of its parts.

 stress
   |   aggregate
   |  /
   | /     paste
   |/    /
   |   /   ___ concrete
   |  /  .'
   | / .'
   |/.'
   +------------------ strain

Aggregate

Linear elastic up to brittle failure; high strength and high modulus (typically 50 to 100 GPa), small strain at failure.

Hydrated cement paste

Also nearly linear up to failure and brittle, but of lower modulus (10 to 30 GPa) and strength; large strain capacity compared to aggregate.

Concrete

Although both components behave almost linearly, concrete gives a curved, non-linear curve with lower stiffness than aggregate and lower strength than aggregate:

• The ITZ is porous and contains oriented Ca(OH)2 crystals and microcracks formed by bleeding water and shrinkage. It is the weakest link.
• Under load the elastic mismatch between the stiff aggregate and the softer paste causes stress concentration at the interface, so bond cracks grow from about 30% of $f_c$ and matrix cracks from about 50 to 75%.
• This progressive cracking gives the curved shape; the strain at the peak is about 0.002, and the descending branch shows that the failure is governed by the matrix and the ITZ, not by the strength of aggregate.

So the modulus of concrete lies between that of aggregate and paste, and the strength depends on the paste and ITZ quality rather than on the aggregate.

Effect of w/c ratio on porosity

Porosity of the paste is the volume of pores (gel pores plus capillary pores). Mixing water which is more than needed for hydration remains as capillary pores after it evaporates or is consumed. About 0.23 of the cement mass is chemically combined and about 0.15 is held in gel pores, so full hydration needs $w/c \approx 0.38$. For higher $w/c$, the extra water makes capillary pores. With higher $w/c$:

• Capillary porosity is higher and pores are larger and interconnected.
• Strength falls and permeability rises sharply; a pore volume of about 25% and above makes capillaries continuous.
• Thus durability is worse (carbonation, chloride and sulphate ingress, freeze-thaw).

Powers' concept

Powers and Brownyard treated the paste volume as cement + water, and found that hydrated cement (gel including gel pores) occupies about 2.06 times the volume of the cement. For 1 g cement, specific volume 0.319 cm$^3$:

$$\begin{aligned} \text{Gel volume} &= 0.657\alpha, \qquad \text{Total paste} = 0.319 + w/c \\ \text{Capillary porosity } P_c &= \frac{w/c - 0.36\alpha}{w/c + 0.32} \end{aligned}$$

Here $\alpha$ is degree of hydration. Capillary porosity is the space left in the paste after subtracting unhydrated cement and gel. Strength follows the gel/space ratio, $f_c = 240x^3$.

Proper w/c ratio for durability (IS 456, Table 5)

Thus a lower w/c (0.40 to 0.45) is required in the marine, cold or aggressive environments and 0.50 to 0.55 for mild weather.

Basis: 1 cm$^3$ of cement, $w/c = 0.5$, specific gravity 3.15, so water = $0.5\times3.15 = 1.575$ cm$^3$ and total paste = 2.575 cm$^3$. 1 cm$^3$ of cement forms 2 cm$^3$ of hydrated products (standard condition). Capillary porosity is taken as the volume not filled by products or unhydrated cement.

$$\text{Capillary pores} = 2.575 - 2\alpha - (1-\alpha) = 1.575 - \alpha$$ $$P = \frac{1.575 - \alpha}{2.575}\times100$$

Answer: porosity is 41.7% at 50% hydration, 32.0% at 75% hydration and 26.2% at 90% hydration. Porosity falls as hydration proceeds, so strength and impermeability improve with good curing.

The gel/space ratio $x$ is the ratio of the volume of hydrated cement (gel) to the sum of the volume of gel and capillary pores (the space available to it):

$$x = \frac{0.657\,\alpha}{0.319\,\alpha + w/c}$$

Powers found by tests on mortar cubes that strength depends on $x$ rather than on age, as:

$$f_c = 240\,x^3\ \text{MPa}$$

Effect

• When $x$ increases, the pores are filled by gel, the solid fraction rises and strength increases. The relation is a cubic, so a small increase in $x$ makes a large strength gain.
• $x$ increases with the degree of hydration $\alpha$ (better curing, more age, finer cement, higher temperature) and falls with higher $w/c$.
• Examples at $w/c = 0.5$: $\alpha = 0.5$ gives $x = 0.498$ and $f_c = 29.7$ MPa; $\alpha = 1.0$ gives $x = 0.802$ and $f_c = 123.9$ MPa.

• Maximum $x$ is 1.0 (when all space is filled), which gives the 240 MPa intrinsic strength.
• The values are for paste; in concrete the actual strength is lower because of air voids, ITZ and aggregate, but the trend is the same and it explains why strength falls with w/c and rises with curing.

Powers' model gives the strength of concrete in terms of the gel/space ratio $x$:

$$x = \frac{\text{volume of gel}}{\text{volume of gel} + \text{capillary pores}} = \frac{0.657\,\alpha}{0.319\,\alpha + w/c}$$

where $\alpha$ is the degree of hydration and $w/c$ the water-cement ratio by mass. The numbers come from cement of specific gravity 3.15: 1 g of cement has 0.319 cm$^3$ volume and produces gel of $2.06\times0.319 = 0.657$ cm$^3$ (including gel pores). Compressive strength:

$$f_c = 240\,x^3\ \text{MPa}$$

(240 MPa is the intrinsic strength of the gel; it varies from about 234 to 255 MPa for different cements.) The ratio $x$ does not depend on the mass of cement; the mass is only needed to give volumes.

Given

Cement = 300 g; water = 150 g; so $w/c = 150/300 = 0.5$.

Case I: fully hydrated ($\alpha = 1$)

$$\begin{aligned} x &= \frac{0.657}{0.319+0.5} = \frac{0.657}{0.819} = 0.802 \\ f_c &= 240\times0.802^3 = 123.9\ \text{MPa} \end{aligned}$$

Case II: 60% hydrated ($\alpha = 0.6$)

$$\begin{aligned} x &= \frac{0.657\times0.6}{0.319\times0.6+0.5} = \frac{0.3942}{0.6914} = 0.570 \\ f_c &= 240\times0.570^3 = 44.5\ \text{MPa} \end{aligned}$$

Answer: Case I gives gel/space ratio 0.802 and strength 123.9 MPa; Case II gives 0.570 and 44.5 MPa.

Powers' model gives the strength of concrete in terms of the gel/space ratio $x$:

$$x = \frac{\text{volume of gel}}{\text{volume of gel} + \text{capillary pores}} = \frac{0.657\,\alpha}{0.319\,\alpha + w/c}$$

where $\alpha$ is the degree of hydration and $w/c$ the water-cement ratio by mass. The numbers come from cement of specific gravity 3.15: 1 g of cement has 0.319 cm$^3$ volume and produces gel of $2.06\times0.319 = 0.657$ cm$^3$ (including gel pores). Compressive strength:

$$f_c = 240\,x^3\ \text{MPa}$$

(240 MPa is the intrinsic strength of the gel; it varies from about 234 to 255 MPa for different cements.) The ratio $x$ does not depend on the mass of cement; the mass is only needed to give volumes.

Given

Cement = 1 kg; $w/c = 0.5$ (water = 0.5 kg); hydration at 28 days $\alpha = 0.9$.

$$\begin{aligned} x &= \frac{0.657\times0.9}{0.319\times0.9+0.5} = \frac{0.5913}{0.7871} = 0.751 \\ f_c &= 240\times0.751^3 = 101.7\ \text{MPa} \end{aligned}$$

(Volumes for 1 kg: gel $= 1000\times0.657\times0.9 = 591.3$ cm$^3$; space $= 1000\times0.319\times0.9 + 500 = 787.1$ cm$^3$; ratio 0.751.)

Answer: gel/space ratio = 0.751 and theoretical strength at 28 days = 101.7 MPa.

Powers' model gives the strength of concrete in terms of the gel/space ratio $x$:

$$x = \frac{\text{volume of gel}}{\text{volume of gel} + \text{capillary pores}} = \frac{0.657\,\alpha}{0.319\,\alpha + w/c}$$

where $\alpha$ is the degree of hydration and $w/c$ the water-cement ratio by mass. The numbers come from cement of specific gravity 3.15: 1 g of cement has 0.319 cm$^3$ volume and produces gel of $2.06\times0.319 = 0.657$ cm$^3$ (including gel pores). Compressive strength:

$$f_c = 240\,x^3\ \text{MPa}$$

(240 MPa is the intrinsic strength of the gel; it varies from about 234 to 255 MPa for different cements.) The ratio $x$ does not depend on the mass of cement; the mass is only needed to give volumes.

Reading of the question

The hydration at 28 days is taken as complete ($\alpha_{28} = 1.0$) and at 14 days $\alpha_{14} = 0.9$ (90% of that at 28 days). Cement = 500 g, $w/c = 0.45$ (the mass cancels in the ratio). "Percent strength gain" is the strength at 14 days as a percentage of the 28-day strength.

28 days ($\alpha = 1$)

$$x_{28} = \frac{0.657}{0.319+0.45} = 0.854, \qquad f_{28} = 240\times0.854^3 = 149.7\ \text{MPa}$$

14 days ($\alpha = 0.9$)

$$x_{14} = \frac{0.657\times0.9}{0.319\times0.9+0.45} = \frac{0.5913}{0.7371} = 0.802, \qquad f_{14} = 240\times0.802^3 = 123.9\ \text{MPa}$$ $$\frac{f_{14}}{f_{28}} = \frac{123.9}{149.7}\times100 = 82.8\%$$

Answer: the concrete gains about 82.8% of its 28-day strength at 14 days (123.9 MPa against 149.7 MPa). Note that 90% hydration gives only about 83% strength because of the cubic relation.

Temperature affects both the rate of hydration and the structure of the hydrated cement paste, and so affects strength.

Effect of casting (initial) temperature

• Higher temperature gives faster hydration and higher early strength (1 to 7 days).
• But the hydration products form quickly and are non-uniform, with a coarser pore structure, so the 28-day and later strength is lower (a concrete cast at 40 °C may have 10 to 20% less 28-day strength than the one cast at 20 °C).
• Low casting temperature (5 to 10 °C) slows the early strength but results in a higher long-term strength, since the gel is more dense and uniform.

Effect of curing temperature

• Cured at higher temperature (steam curing) the early strength rises, but the long-term strength falls (crossover effect).
• Low curing temperature slows the strength gain; hydration almost stops near 0 °C to -10 °C, and if concrete freezes before it has 3.5 MPa strength, it suffers permanent damage.
• Temperature fluctuations and rapid drying add thermal and shrinkage cracks.

Maturity

Strength is a function of time and temperature, expressed by maturity $M=\sum (T+10)\Delta t$.

Hence a moderate temperature (15 to 25 °C) with continuous moist curing is best.

Casting and curing temperature change the speed of hydration and the final structure of the gel. A high temperature gives high early strength but a lower 28-day and long-term strength (non-uniform gel, coarse pores, drying cracks). A low temperature slows early strength but improves later strength, as the gel forms more densely; near freezing, hydration almost stops and fresh concrete can be damaged by ice.

Concreting in Kathmandu

Kathmandu valley has a temperate climate: mild to warm summers (up to about 30 °C) and cool winters (night temperature 0 to 5 °C in December to January) with fog and little rain in winter. Mostly the temperature is favourable; the special care is needed in winter and in the pre-monsoon dry period.

1. Season: prefer casting in spring and autumn, between 10 and 30 °C; avoid pouring in the cold early morning of Poush to Magh, or heavy monsoon rain unless protected.
2. Cement: OPC 43 or PPC (Nepal Standard), with fresh stock; PPC is acceptable with extended curing. For durable work follow the min cement and max w/c of IS 456 Table 5 for the exposure (mild or moderate: w/c 0.50 to 0.55).
3. Winter: keep the concrete at 5 °C or higher at placing (IS 456, Cl. 14.1); use warm mixing water, cover with plastic and straw/jute at night, delay formwork stripping, avoid casting if frost is expected.
4. Summer (dry spells): cast in early morning or evening, wet the forms, shade, and do not allow plastic shrinkage (IS 456, Cl. 14.2).
5. Curing: keep moist for at least 7 days for OPC and 10 days for PPC (IS 456, Cl. 13.5.1) with ponding or wet hessian, and use a curing compound where water is scarce.
6. Mix: use a water-reducing admixture, keep w/c low, compact properly, and cast cubes cured under site conditions for stripping decisions.

Thus normal procedures with attention to cold-night protection and proper curing are enough for Kathmandu.