Chapter

1Materials

1.1 Notations Used in This Chapter

A Area of concrete cross-section

Cs Constant depending on the type of curing

Ct Creep coefficient (Ct = εsp/εi)

Cu Ultimate creep coefficient (on average Cu = 2.35)

D Diameter of cylinder for split test (Brazilian test)

Ec Modulus of elasticity of concrete

Es Modulus of elasticity of non-prestressed reinforcement

I Moment of inertia of section about centroidal axis

L Length of cylinder for split test (Brazilian test)

M Applied moment

P Applied concentrated load

Psh Correction factor for shrinkage strain

2 Chapter 1

Qcr Correction factor for creep strain

T Tensile force

a Shear span, distance from application point of concentrated load to support

b Width of member

fc Compressive stress in concrete

fc′ Specified compressive strength of concrete

fr Modulus of rupture of concrete

fs Calculated tensile stress in reinforcement at specified loads

fsp Splitting tensile strength of concrete

ft Concrete tensile stress due to applied loads

fy Specified yield strength of non-prestressed reinforcement

h Overall thickness or height of member

t Time

∆T Temperature variation

αT Coefficient of thermal expansion

γc Density of concrete

ε Normal strain

εc Strain at the extreme concrete compression fibre

εcp Creep strain in concrete

εcpic Strain in concrete corresponding to fc′

εcu Maximum strain at the extreme concrete compression fibre at ultimate (εcu = 0.0035)

εi Instantaneous elastic strain

εsh Shrinkage strain

εshu Ultimate shrinkage strain

εth Thermal expansion strain

λ Factor to account for low-density concrete (λ = 1 for normal-density concrete)

ν Poisson’s ratio

σ Effective normal stress

Materials 3

1.2 Concrete

Concrete is a material obtained by hardening a mixture of aggregates (sand, gravel), hydraulic lime (cement), water, and additives (such as entrained air) in pre-determined proportions.

Concretes are classified according to their density γc as follows:

➟ low-density concrete with γc ≤ 1850 kg/m3

➟ semi-low-density concrete with 1850 kg/m3 < γc ≤ 2150 kg/m3

➟ normal-density concrete with 2150 kg/m3 < γc ≤ 2500 kg/m3

➟ high-density concrete with 2500 kg/m3 < γc

In addition to its density, concrete is characterized by:

➟ its mechanical properties: compressive strength fc′ and tensile strength ft,

➟ its elastic properties: modulus of elasticity Ec, ultimate strain εcu, and Poisson’s ratio ν,

➟ its volumetric change properties: thermal expansion αT, creep strain εcp, and shrinkage strain εsh.

Five basic types of Portland cement are produced according to their applications (Table 1.1).

Table 1.1 – Cement Classifications

Cement Qualification Application

GU General useGeneral purpose, used in ordinary construction where special properties are not required

MSModerate sulphate resistant

Moderate exposure of concrete to sulphate attack

Used when less heat of hydration than GU cement is required

HEHigh early strength

Rapid achievement of a given level of strength

LHLow heat of hydration

Used when a low heat of hydration is desired

HSHigh sulphate resistant

Concrete exposed to severe sulphate action

4 Chapter 1

■■ Compressive Strength

The compressive strength of concrete, denoted by fc′, is obtained from crushing tests on 150 × 300 mm concrete cylinder samples at 28 days of aging. (If the concrete cylinder samples are 100 × 200 mm, use 0.95 fc′.) Typical stress-strain curves for concrete in compression are shown in Figure 1.1.

A normal-density concrete of structural quality has a compressive strength fc′ ranging between 20 MPa (minimum) and 40 MPa. High-strength concrete (fc′ > 40 MPa) can also be used for special projects.

Figure 1.1 – Concrete under Compressive Load

■■ Tensile Strength

The tensile strength may be obtained using three types of tests (Figure 1.2): a) direct tension, b) flexure test, c) split or Brazilian test.

For guidance:

fsp = 1.2 to 1.6 ft; fr = 1.4 to 2 ft (1.1)

Moreover, there is a strong relationship between λ ′fc and fr. Clause 8.6.4 of the CSA A23.3-04 Standard provides the following relationship for fr:

f fr c= ′0 6. λ (1.2)

Materials 5

where λ = 1.0 for normal-density concrete and λ = 0.75 for low-density concrete.

Figure 1.2a

a) Direct tension

σ = ft = T/A

directly provides the tensile strength but is difficult to achieve in laboratory

Figure 1.2b

b) Flexure

σ =

M

I

h

2

σ = fr = modulus of rupture = 6

2

Pa

bh

Figure 1.2c

c) Split or Brazilian test

σ

π= =f

P

LDsp

2

Figure 1.2 – Tensile Strength of Concrete

■■ Modulus of Elasticity

According to CSA A23.3-04 Standard (Clause 8.6.2), the modulus of elasticity, the secant modulus between σc = 0 and σc = 0.4fc′, may be estimated by:

E fc cc= ′ +

3300 69002300

1 5γ .

; 1500 ≤ γc ≤ 2500 kg/m3 (1.3)

In addition, for concrete of normal density and compressive strength, 20 MPa ≤ fc′ ≤ 40 MPa, Ec may be estimated using the following simplified equation:

E fc c= ′4500

(1.4)

6 Chapter 1

■■ Strain

The strain in concrete, εcpic, corresponding to fc′ increases with fc′. The approximate

value of εcpic is 0.002. It may also be estimated as a function of fc′ by:

εcpic cf=

+ ′≥

140

80 0000 002

,.

(1.5)

The ultimate concrete strain in compression generally varies between 0.003 and 0.004. However, the CSA A23.3-04 Standard limits the value of εcu to:

εcu = 0.0035 (1.6)

■■ Poisson’s Ratio

For uncracked concrete, Poisson’s ratio varies between 0.15 and 0.20 for a concrete compressive stress fc less than 0.7fc′.

■■ Creep

Creep is a phenomenon by which, under sustained loads and stresses, concrete undergoes strain. The strain increases with time, but at a progressively decreasing rate (Figure 1.3). According to the American Concrete Institute (ACI) Committee 209-1982, the creep strain in concrete, εcp, may be estimated in terms of the instantaneous elastic strain, εi, by:

ε εcp t i t u crC C

t

tC Q= =

+where

0 6

0 610

.

.

(1.7)

where

εcp = creep strain

εi = instantaneous elastic strain

Ct = creep coefficient = εcp/εi

Cu = ultimate creep coefficient, which varies between 1.30 and 4.15, with an average value of 2.35

Qcr = correction factor that takes into consideration the conditions of use (relative humidity, percentage of air, aggregate content, thickness of the element, type of curing) [see Table 1.2]

t = time in days.

Materials 7

Instantaneous recovery

Progressive recovery

Residual creep strain

Time since the application of compressive stress

Tot

al s

trai

n

Ult

imat

e cr

eep

stra

in

Unl

oadi

ngFigure 1.3 – Typical Strain-Time Curve for Concrete under Axial Compression

■■ Shrinkage

Shrinkage is a phenomenon by which the concrete undergoes strain caused by the decrease in the volume of concrete due to drying at constant temperature. Shrinkage strain generally develops during the first two to three years after casting of concrete (Figure 1.4).

Figure 1.4 – Shrinkage-Time Curve for Concrete after 7 Days of Curing

8 Chapter 1

Table 1.2 – Creep and Shrinkage Modification Factors (Adapted from Table 1.2 of CSA A23.3-04 Standard)

Creep: Qcr = Qa Qh Qf Qr Qs Qv Shrinkage: Psh = Pc Ph Pf Pr Ps Pv

Qa: to account for curing

Age at loading (days)

Qa

Moist curing

Steam curing

1 72060

1.251.000.870.77

1.000.940.850.76

Pc: to account for cement content

Cement content (kg/m3)

225 300 410

Pc 0.89 0.93 1.00

Qh: to account for humidity

Relative humidity (%) Qh

40 60 80100

1.000.870.730.60

Ph: to account for humidity

Relative humidity (%) Ph

40 60 80100

1.000.800.600.00

Qf : to account for fine aggregates

Ratio of fine to total aggregates

Qf

0.300.400.500.70

0.950.981.001.05

Pf : to account for fine aggregates

Ratio of fine to total aggregates

Pf

0.300.400.500.70

0.720.861.001.04

Qr: to account for volume/surface ratio

Volume/Surface ratio (mm)

Qr

38 75150250

1.000.820.700.67

Pr: to account for volume/surface ratio

Volume/Surface ratio (mm)

Pr

38 75150250

1.000.840.590.37

Materials 9

Creep: Qcr = Qa Qh Qf Qr Qs Qv Shrinkage: Psh = Pc Ph Pf Pr Ps Pv

Qs: to account for slump

Slump (mm) Qs

50 70125

0.951.001.15

Ps: to account for slump

Slump (mm) Ps

50 70125

0.971.001.09

Qv: to account for air content

Air (%) Qv

≤ 6 8 10

1.001.181.36

Pv: to account for air content

Air (%) Pv

≤ 6 8 10

1.001.011.03

According to ACI Committee 209-1982, the shrinkage strain may be estimated using the following formula (Figure 1.4):

ε εsh

s

shu sh

t

C tP=

+

(1.8)

where

εsh = shrinkage strain

εshu = ultimate shrinkage strain, 0.0002 ≤ εshu ≤ 0.0008. In the absence of a specific value, it is recommended to use εshu = 0.00078.

Cs = constant; Cs = 35 for seven-day moist curing of concrete and Cs = 55 for one- to three-day steam curing

Psh = correction factor taking into account the conditions of use (relative humidity, air content, aggregate and cement contents, thickness of the element) [see Table 1.2]

t = time in days.

■■ Thermal Expansion of Concrete

The coefficient of thermal expansion of concrete is αT = 10 × 10–6 mm/mm/°C. The thermal expansion strain, εth, can therefore be represented as follows:

ε αth T T= ∆

(1.9)

where ∆T is the temperature variation assumed.

10 Chapter 1

1.3 Steel Reinforcement

Steel reinforcement for concrete can be achieved by using: a) deformed bars and wires, b) welded wire fabric, or c) smooth wires. Smooth wires are allowed to be used for wire fabric, spirals, stirrups, and ties with diameters of 10 mm or less.

■■ Grades

The CSA G30.18 Standard defines five grades of steel reinforcement in concrete: 300R, 400R, 500R, 400W and 500W. The W grade indicates that a ductile and weldable steel is required. The number of each grade indicates the minimum guaranteed specified yield strength in MPa. Grade 400R is the most frequently used for reinforcement, with a specified yield strength fy = 400 MPa. Table 1.3 presents the geometric and physical characteristics of steel bars commonly used in practice.

■■ Stress-Strain Curves

Figure 1.5 shows actual and idealized stress-strain curves for steel reinforcement. The modulus of elasticity of steel reinforcement is Es = 200,000 MPa.

Table 1.3 – Characteristics of Reinforcing Bars

Bar Designation No.

Nominal dimensions

Area(mm2)

Diameter(mm)

Perimeter(mm)

Mass(kg/m)

10M 100 11.3 35.5 0.785

15M 200 16.0 50.1 1.570

20M 300 19.5 61.3 2.355

25M 500 25.2 79.2 3.925

30M 700 29.9 93.9 5.495

35M 1000 35.7 112.2 7.850

45M 1500 43.7 137.3 11.775

55M 2500 56.4 177.2 19.625

Materials 11

Figure 1.5 – Actual and Idealized Stress-Strain Curves for Steel Reinforcement

■■ Thermal Expansion of Steel

The coefficient of thermal expansion of steel is αT = 12 × 10–6 mm/mm/°C.

1.4 Examples

Example 1.1 – Stress, Creep, and Shrinkage

■■ Problem Statement

Consider a 3-m-high reinforced concrete column with a cross-section of 400 mm × 400 mm. It is reinforced with 4 No. 30M steel bars. The column is subjected to an axial compression load of 1600 kN after one week of moist curing.

a) Calculate the instantaneous compressive and tensile stresses in concrete and steel and the corresponding instantaneous strain.

b) What is the shortening of the column after 180 days of loading?

Use: fc′(at 7 days) = 20 MPa; Type GU cement (300 kg/m3); relative humidity = 60%; air content = 5%; slump of fresh concrete = 125 mm; sand = 670 kg/m3; coarse aggregate = 1000 kg/m3.

12 Chapter 1

■■ Solution

a) Instantaneous Stresses and Strain

Stress in concrete, fci

E fc c= ′4500

Ec = =4500 20 20 120, MPa

n

E

Es

c

=

n = =200 000

20 1209 9

,

,.

Ac = net concrete area = Ag – As Ac = − =160 000 2800 157 200 2, , mm

Ace = equivalent concrete area = Ac + nAs Ace = + × =157 200 9 9 2800 184 920 2, . , mm

f

P

Aci

ce

=

fci =×

=1600 10

184 9208 65

3

,. MPa

Stress in steel reinforcement, fsi

f nfsi ci= fsi = × =9 9 8 65 85 6. . . MPa

Instantaneous strain, εi

εi

ci

c

f

E=

εi = = × −8 65

20 120430 10 6.

,mm/mm

The instantaneous reduction is:

∆l li i= ε ∆li = × × =−430 10 3000 1 296 . mm

b) Shortening of the Column at t = 180 Days

Shortening due to creep

Cu = 2 35. (average value)

Q Q Q Q Q Q Qcr a h f r s v= (see Table 1.3)

Qcr = × × × × × =1 00 0 87 0 98 0 78 1 15 1 00 0 76. . . . . . .

Note: Ratio (volume/surface) = 400 400

2 400 2 400100

×× + ×

=( )( ) ( )

Materials 13

C

t

tC Qt u cr=

+

0 6

0 610

.

.

Ct =+

× × =180

10 1802 35 0 76 1 24

0 6

0 6

.

.. . .

ε εcp t iC=

εcp = × × = × −1.24 430 10 mm/mm–6 533 10 6

∆l lcp cp= ε

∆lcp = × × =−533 10 3000 1 66 . mm

Shortening due to shrinkage

Cs = 35

εshu = 0 00078. mm/mm (suggested average value in the absence of a specific value)

P P P P P P Psh c h f r s v= Psh = × × × × × =0 93 0 80 0 86 0 76 1 09 1 00 0 53. . . . . . .

ε εsh

sshu sh

t

C tP=

+ εsh =

+× × = × −187

35 1870 00078 0 53 348 10 6. . mm/mm

∆l lsh sh= ε ∆lsh = × × =−348 10 3000 1 046 . mm

Total Shortening

∆ ∆ ∆l l lcp sh= + ∆l = + =1 60 1 04 2 64. . . mm

1.5 Problems

Problem 1.1

By analyzing the creep and shrinkage strain equations (Equations 1.7 and 1.8) and the modification factors Qcr and Psh (Table 1.2), determine the three factors that have the most influence on creep and shrinkage.

Problem 1.2

Consider a rectangular section of a prestressed concrete column with dimensions 700 mm × 700 mm × 4 m. The section is subjected to a prestressed force of 2500 kN acting at the centroid of the section. The force is applied after seven days of moist curing.

a) Calculate the instantaneous stress and the instantaneous strain in concrete.

b) Determine the shortening of the column one year after the prestressed force was applied.

14 Chapter 1

Use: fc′(at seven days) = 25 MPa; Type GU cement (300 kg/m3); relative humidity = 70%; air content = 5%; slump of fresh concrete = 120 mm; sand = 660 kg/m3; coarse aggregate = 1050 kg/m3.

Problem 1.3

Consider a 4-m-high concrete column having a 500 mm × 500 mm square section. The longitudinal steel reinforcement consists of 4 No. 25M bars, that is, one No. 25M bar in each corner. The beam is subjected to a specified dead load of 1000 kN (unfactored) and a specified live load of 900 kN (unfactored). The dead load is applied 14 days after concrete casting.

a) What are the stresses in concrete and steel reinforcement, assuming an elastic behav-iour and perfect compatibility between the concrete and steel strains, for the following load cases:

• specified dead load (unfactored)?

• total factored load?

b) What is the total strain experienced by the column due to creep and shrinkage, 365 days after concrete casting?

Use: fc′(at 14 days) = 25 MPa; seven-day moist curing; Type GU cement: 300 kg/m3; sand: 700 kg/m3; coarse aggregate: 1000 kg/m3; slump: 100 mm; air content: 6%; relative humidity: 60%; unit weight of concrete = 24 kN/m3; Cu = 2.35.