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Plain & Reinforced Concrete Theory

Concrete

  • Concrete is a mixture of cement, fine and coarse aggregate.
  • Concrete mainly consists of a binding material and filler material. If filler material size is < 5mm it is fine aggregate and > 5mm is coarse aggregate.

Plain Cement Concrete (PCC)

  • Mixture of cement , sand and coarse aggregate without any reinforcement is known as PCC.
  • PCC is strong in compression and week in tension. Its tensile strength is so small that it can be neglected in design.

Reinforced Cement Concrete (RCC)

  • Mixture of cement , sand and coarse aggregate with reinforcement is known as RCC. (Tensile strength is improved)
  • Mix Proportion
Cement : Sand : Crush
1 : 1.5 : 3
1 : 2 : 4
1 : 4 : 8
  • Water Cement Ratio (W/C) W/C = 0.5 – 0.6
  • For a mix proportion of 1:2:4  and W/C = 0.5, if cement is 50 kg

Sand       = 2 x 50 = 100 Kg
Crush     = 4 x 50 = 200 Kg        Batching By Weight
Water      = 50 x 0.5 = 25 Kg

Mechanism of Load Transfer

Function of structure is to transfer all the loads safely to ground. A particular structural member transfers load to other structural member. Untitled-1

Merits of Concrete Construction

1.Good Control over cross sectional dimensions and Shape One of the major advantage of concrete structures is the full control over the dimensions and structural shape. Any size and shape can be obtained by preparing the formwork accordingly.
2.Availability of Materials All the constituent materials are earthen materials (cement, sand, crush) and easily available in abundance.
3.Economic Structures All the materials are easily available so structures are economical.

4.Good Insulation Concrete is a good insulator of Noise &  heat and does not allow them to transmit completely.

5.Good Binding Between Steel and Concrete There is a very good development of bond between steel and concrete.
6.Stable Structure Concrete is strong in compression but week in tension and steel as strong in tension so their combination give a strong stable structure.
7.Less Chances of Buckling Concrete members are not  slim like steel members so chances of buckling are much less.
8.Aesthetics concrete structures  are aesthetically good and cladding is not required
9.Lesser Chances of Rusting steel reinforcement is enclosed in concrete so chances of rusting are reduced.

Demerits of Concrete Construction

1.Week in tension Concrete is week in tension so large amount of steel is required.
2.Increased Self Weight Concrete structures have more self weight compared with steel structures so large cross-section is required only to resist self weight, making structure costly.

3.Cracking Unlike steel structures concrete structures can have cracks. More cracks with smaller width are better than one crack of larger width

4.Unpredictable Behavior If same conditions are provided for mixing, placing and curing even then properties can differ for the concrete prepared at two different times.
5.Inelastic Behavior concrete is an inelastic material, its stress-strains curve is not straight so its behavior is more difficult to understand.
6.Shrinkage and Creep Shrinkage is reduction in volume. It takes place due to loss of water even when no load is acting over it. Creep is reduction in volume due to sustained loading when it acts for long duration. This problem is not in steel structures.
7.Limited Industrial Behavior Most of the time concrete is cast-in-situ so it has limited industrial behavior.

Specification & Codes

These are rules given by various organizations in order to guide the designers for safe and economical design of structures. Various Codes of Practices are

  1. ACI 318-05 By American Concrete Institute. For general concrete constructions (buildings)
  2. AASHTO Specifications for Concrete Bridges. By American Association of State Highway  and Transportation Officials.
  3. ASTM (American Standards for Testing and Materials)  for testing of materials.

No code or design specification can be construed as substitute for sound engineering judgment in the design of concrete structures. In the structural practice, special circumstances are frequently encountered where code provisions can only serve as a guide, and engineer must rely upon a firm understanding of the basic principles of structural mechanics applied to reinforced or pre-stressed concrete, and the intimate knowledge of nature of materials

Design Loads

Dead Load The loads which do not change their magnitude and position w.r.t. time within the life of structure Dead load mainly consist of superimposed loads and self load of structure.
Self Load It is the load of structural member due to its own weight.
Superimposed Load It is the load supported by a structural member. For instance self weight of column is self load and load of beam and slab over it is superimposed load.
Live Load Live loads consist chiefly  of occupancy loads in buildings and traffic loads on bridges
  • They may be either fully or partially in place or not present at all, and may also change in location.
  • Their magnitude and distribution at any given time are uncertain, and even their maximum intensities throughout the life time of the structure are not known with precision.
  • The minimum live loads for which the floor and roof of a building should be designed are usually specified in the building codes that governs at the site construction.

Densities of Important Materials

Material Density (Kg/m3)
PCC 2300
RCC 2400
Brick masonry 1900-1930
Earth/Sand/Brick ballast 1600-1800

Intensities of Live Loads

(Table 1.1, Design of concrete structures by Nilson)

Occupancy / Use Live Load(Kg/m2)
Residential/House/Class Room 200
Offices 250-500
Library Reading Room 300
Library Stack Room 750
Warehouse/Heavy storage 1250

Basic Design Equation

Applied Action x F.O.S = Max. Internal Resistance

Factor of Safety

F.O.S. = Max. Failure load/Max. Service LoadFollowing points are relevant to F.O.S1.It is used to cover uncertainties due to

  • Applied loads
  • Material strength
  • Poor workmanship
  • Unexpected behavior of structure
  • Thermal stresses
  • Fabrication
  • Residual stresses
2.If F.O.S is provided then at service loads deflection and cracks are within limits. 3.It covers the natural disasters.

Ultimate Strength Design (USD)/LRFD Method

Strength design method is based on the philosophy of dividing F.O.S. in such a way that Bigger part is applied on loads and smaller part is applied on material strength. Material Strength ≥ Applied Load  x  F.O.S.(1) x F.O.S.(2) {1 / F.O.S.(2)} Material Strength ≥ Applied Load  x  F.O.S.1 F.O.S.(1)  = Overload factor or Load Factor {greater than 1} 1/F.O.S.(2) = Strength Reduction factor or Resistance Factor {less than 1}
                         ΦSn ≥ U
Where
  Sn    = Nominal Strength
  ΦSn  = Design Strength
  Φ     = Strength Reduction Factor
  U = Required Strength, calculated by applying load factors
For a member subjected to moment, shear and axial load:
                            ΦMn ≥ Mu
                            ΦVn ≥ Vu
                            ΦPn ≥ Pu

Allowable Strength Design (ASD)

In allowable strength design the whole F.O.S. is applied on material strength and service loads (un-factored)  are taken as it is.
Material Strength / F.O.S. ≥ Service Loads
In both Allowable strength design and Ultimate strength design analysis carried out in elastic range

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Plastic Design

     In plastic design, plastic analysis is carried out in order to find the behavior of structure near collapse state. In this type of design material strength is taken from inelastic range. It is observed that whether the failure is sudden or ductile. Ductile failure is most favorable because it gives an warning before the failure of structures

Capacity Analysis

In capacity analysis size, shape, material strengths and cross sectional dimensions are known and maximum load carrying capacity of the structure is calculated. Capacity analysis is generally carried out for the existing structures.

Design of Structure

In design of structure load, span and material properties are known and cross sectional dimensions and amount of reinforcement are to be determined.

Objectives of Designer

There are two main objectives

1.Safety
2.Economy

Safety

The structure should be safe enough to carry all the applied throughout the life.

Economy

Structures should be economical. Lighter structures are more economical.
Economy α  1/self weight (More valid for Steel Structures)
In concrete Structures overall cost of construction decides the economy, not just the self weight.

Load Combinations

To combine various loads in such a way to get a critical situation.
Load Factor = Factor by which a load is to be increased x probability of occurrence

1.1.2D + 1.6L
2.1.4D
3.1.2D + 1.6L + 0.5Lr
4.1.2D + 1.6Lr + (1.0L or 0.8W)
Where
D = Dead load
L = Live load on intermediate floors
Lr = Live load on roof
W = Wind Load

Strength Reduction Factor / Resistance Factor, Φ

Strength Condition Strength Reduction Factor
Tension controlled section (bending or flexure) 0.9
Compression controlled section
Columns with ties 0.65
Column with spirals 0.7
Shear and Torsion 0.75

Shrinkage

“Shrinkage is reduction in volume of concrete due to loss of water”
Coefficient of shrinkage varies with time. Coefficient of shortening is:

  • 0.00025 at 28 days
  • 0.00035 at 3 months
  • 0.0005 at 12 months

Shrinkage =  Shrinkage coefficient x Length
Excessive shrinkage can be avoided by proper curing during first 28 days because half of the total shrinkage takes place during this period

Creep

“creep is the slow deformation of material over considerable lengths of time at constant stress or load”
Creep deformations for a given concrete are practically proportional to the magnitude of the applied stress; at any given stress, high strength concrete show less creep than lower strength concrete.

Compressive strength Specific Creep
(MPa) 10-6 per MPa
21 145
28 116
41 80
55 58

How to calculate shortenings due to creep?

Consider a column of 3m which is under sustained load for several years.
Compressive strength, fc’ = 28 MPa
Sustained stress due to load = 10 MPa
Specific creep for 28 MPa fc’ = 116 x 10-6 per MPa
Creep Strain = 10 x 116 x 10-6 = 116 x 10-5
Shortening due to creep =  3000 x 116 x 10-5   = 3.48 mm

Specified Compressive Strength Concrete, fc’

“28 days cylinder strength of concrete”

  • The cylinder has 150mm dia and 300mm length. 
  • According to ASTM standards at least two cylinders should be tested and their average is to be taken.

ACI 5.1.1: for concrete designed and constructed in accordance with ACI code, fc’ shall not be less than 17.5 Mpa (2500 psi)

BSS specifies the compressive strength in terms of cube strength.

  • Standard size of cube is 6”x6”x6”
  • BSS recommends testing three cubes and taking their average as the compressive strength of concrete

Cylinder Strength = (0.75 to 0.8) times Cube Strength

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Relevant ASTM Standards

  • “Methods of Sampling Freshly Mixed Concrete” (ASTM C 172)
  • Practice for Making and Curing Concrete Test Specimens in Field” (ASTM C 31)
  • “Test Methods for Compressive Strength of Cylindrical Concrete Specimen” (ASTM C 39)

Testing of Samples for Compressive Strength

Cylinders should be tested in moist condition because in dry state it gives more strength.
ACI 5.6.2.1: Samples for strength tests of each class of concrete placed each day shall be taken :

  • Not less than once a day
  • Not less than once for each 115m3 of concrete.
  • Not less than once for each 450m2 of concrete.

Code allows the site engineer to ask for casting the test sample if he regards it necessary.

Acceptance Criteria for Concrete Quality

ACI 5.6.3.3: Strength level of an individual class of concrete shall be considered satisfactory if both of the following requirements are met:

  • Every arithmetic average of any three consecutive strength tests equals or exceeds fc’.
  • No individual strength test (average of two cylinders) falls below fc’
  • by more than 3.5 MPa (500 psi) when fc’ is 35 MPa (5000 psi) or less; or
  • by more than 0.10fc’ when fc’ is more than 35 MPa

Example

For Required fc’ = 20 MPa, if following are the test results of 7 samples

19, 20, 22, 23, 19, 18, 24 MPa
Mean 1 = (19 + 20 + 22) / 3 = 20.33 MPa
Mean 2 = (20 + 22 + 23) / 3 = 21.67 MPa
Mean 3 = (22 + 23 + 19) / 3 = 21.33 MPa
Mean 4 = (23 + 19 + 18) / 3 = 20.00 MPa
Mean 5 = (19 + 18 + 24) / 3 = 20.33 MPa
1.Every arithmetic average of any three consecutive strength tests equals or exceeds fc’.
2.None of the test results fall below required fc’ by 3.5 MPa.
Considering these two point the quality of concrete is acceptable

Mix Design

  • Ingredients of concrete are mixed together in order to get a specified Required Average Strength, fcr’ .
  • If we use fc’ as target strength during mix design the average strength achieved may fall below fc’.
  • To avoid under-strength concrete fcr’ is used as target strength in-place of fc’.

fcr’ > fc’
ACI-5.3.2 Required Average Compressive Strength

Table 5.3.2.1-Required Average Compressive Strength when Data are Available to Establish a Sample Standard Deviation

Specified Compressive Strength, fc’ (MPa) Required Average Strength, fcr’ (MPa) 
fc’ ≤ 35 Larger of value computed from Eq. (5-1) & (5-2)
 fcr’  = fc’  + 1.34 Ss                        (5-1)
 fcr’ = fc’  + 2.33 Ss – 3.45        (5-2)
fc’ > 35 Larger of value computed from Eq. (5-1) & (5-3)
 fcr’ = fc’  + 1.34 Ss                         (5-1)
 fcr’ = 0.9fc’  + 2.33 Ss               (5-3)

Table 5.3.2.2-Required Average Compressive Strength when Data Are Not Available to Establish a Sample Standard Deviation

Specified Compressive Strength, fc’ (MPa) Required Average Strength, fcr’ (MPa) 
fc’ < 21 fcr’  = fc’  + 7                              
21≤ fc’ ≤ 35 fcr’ = fc’  + 8.5
fc’ > 35 fcr’ = 1.1fc’  + 5

Stress Strain Curve of Concrete

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Modulus of Elasticity

Concrete is not an elastic material therefore it does not have a fixed value of modulus of elasticity

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Secant modulus (Ec)

is the one which is being used in design.

Ec = 0.043 wc1.5√fc’
wc = density of concrete in kg/m3
fc’ = specified cylinder strength in MPa
For normal weight concrete, say wc = 2300 kg/m3
Ec = 4700√fc’

Reinforcing Steel

Steel bars are:

  • Plain
  • Deformed (currently in use)

Deformed bars have longitudinal and transverse ribs. Ribs provide a good bond between steel and concrete. If this bond fails steel becomes in effective.

The most important properties for reinforcing steel are:

  • Young’s modulus, E (200 GPa)
  • Yield strength, fy
  • Ultimate strength, fu
  • Size and diameter of bar

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