Reinforced Concrete
Table of Contents
Reinforced Concrete Constituents
Concrete is an engineering material which has been used since the time of the Romans and is comprised of cement,water and aggregate. Steel bars can be added to the concrete to enhance its flexural strength as the tensile capacity of concrete is assumed to be zero when design reinforced concrete.
Portland Cement
Portland cement is the most commonly used cement which originates from limestone and was developed in England in the 19th century. It is an energy intensive manufacturing process where limestone (calcium carbonate)with other materials is heated to 1450 degrees in a kiln, in a process known as calcination,to form calcium oxide (quicklime). This quicklime is then ground with gypsum into powder which is ordinary portland cement.
Water
Water is added to the cement which creates cement paste that coasts the aggregate and chemically reacts with cement (hydration). The volumne of water added affects the strength and workability and the lower the water-cement ratio, the stronger the mix but less workability. A balance is needed between the right mix of concrete but typical water-cement ratios are between 0.35 – 0.5.
Aggregates
Course aggregates are crushed gravel or stone, (particles greater than 5mm) and fine aggregates are sand, (particles less than 5mm). They make up the bulk of the concrete mixture (between 60% – 80%).
Reinforcement
Reinforced concrete is a composite material with steel bars added due to the weak tensile capacity of concrete. Reinforced concrete is generally designed to resist tensile stresses,cracking and failure of concrete members.
Material properties
- Density of concrete: \(24 kN/m^2\)
- Density of reinforced concrete: \(25 kN/m^2\)
- Tensile strength: Weak but generally taken as 10% of compressive strength.
- Compressive Strentgh: \(fcd = acc fck /gm = 0.85 fck /1.5 = 0.567 fck\)
- Concrete classes are expressed as C20/25, C30/37, C35/45 in EC2 where the first number is the cylinder strength and the second number is the cube strength
- Young’s modulus of elasticity: 30 GPa but depends on concrete grade
- Shrinkage: The loss of moisture from concrete during different stages in it’s life results in the concrete section to shrink. The factors affecting shrinkage are water-cement ratio, environmental condition, time, type of aggregates and admixtures
- Creep: It is the deflection over sustained loading and is a time-dependent process. The factors affecting creep are the type of aggregate used, the mix proportion of the concrete and age of the concrete when it is loaded.
Reinforced Concrete Mixes
Concrete mix ratios are a proportion of Portland cement, sand, aggregates and water. The mix ratios determine the strength of the concrete, and depending on the project or environment, some mixes may not be suitable. A “prescribed mix” is when the purchaser prescribes the exact composition of the concrete and is responsible for ensuring these proportion produce concrete with it’s required performance. A “design mix” is when the engineer specifies the required performance and the concrete producer will select the mix to achieved the specified performance. The concrete will be tested via concrete cube test to ensure compliance with the specified performance.
Cement Combination Types
Prescribed Concrete Mixes
Concrete Durability and Exposure Classes
Durability
As well as designing reinforced concrete to withstand permanent and variable load, it must also withstand environmental/chemical attacks. Durability is the ability to last design life without significant deterioration. Durability requirements are based in BS8500-1:2015 and exposure classes for different environments have been identified. The exposure classes defines the required concrete cover, which is normally the minimum cover plus a margin for any deviation when on site.
Reinforced concrete develops microscopic cracks when loaded, which can widen depending on the applied loads. Designers set a minimum crack width which is generally 0.3mm. If these cracks widen over, issues arise as chemicals can penetrate the concrete and cause the rebar to corrode. The corroded material builds up and causes internal cracking and spalling. If corrosion goes uncheck, this will lead to failure of a concrete member over time.
Aggressive Chemical Attack
When reinforced concrete is placed in the ground (i.e, foundations), it is susceptible to chemical attacks due to ground contamination, gas, sulphate attack and chemicals in groundwater. Due to the unpredictability of ground conditions, precaustion should be taken when designing foundations and sulphate-restsiing concrete can specified. Sulphate-resisting concrete is where the amount of tricalcium aluminate is restricted in the cement mix, which reduces the formation of sulphate ions. This reduces the chance of sulpahate attack on concrete. In addition, Special BRE digest 1 (concrete in aggressive grounds) should be read as it discuess modes of chemical attack and how to deal with these issues.
Chloride Attack
Water is added to the cement which creates cement paste that coasts the aggregate and chemically reacts with cement (hydration). The volumne of water added affects the strength and workability and the lower the water-cement ratio, the stronger the mix but less workability. A balance is needed between the right mix of concrete but typical water-cement ratios are between 0.35 – 0.5.
Aggregates
Course aggregates are crushed gravel or stone, (particles greater than 5mm) and fine aggregates are sand, (particles less than 5mm). They make up the bulk of the concrete mixture (between 60% – 80%).
Reinforcement
Reinforced concrete is a composite material with steel bars added due to the weak tensile capacity of concrete. Reinforced concrete is generally designed to resist tensile stresses,cracking and failure of concrete members.
What are the Exposure classes?
Exposure classes are defined in accordance to the condition of the concrete when it’s been built. (Is the concrete located indoors or outside where it is exposed to rain and freezing).
The Eurocodes have developed a table which can be found in BS 8500 which defines each of these conditions.
| Class Designation | Class Description | Examples |
|---|---|---|
| No risk or corrosion or attack (X0 Class) |
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| X0 | Very Dry | Unreinforced concrete completely buried in soil classed as AC‑1. Unreinforced concrete permanently submerged in non‑aggressive water. Unreinforced concrete surfaces in cyclic wet and dry conditions not subject to abrasion, freezing or chemical attack. Reinforced concrete surfaces exposed to very dry. |
| Corrosion induced by carbonation (XC Classes) (where concrete containing reinforcement or other embedded metal is exposed to air and moisture) |
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| XC1 | Dry or permanently wet | Reinforced and prestressed concrete surfaces inside enclosed structures except voided superstructures and areas of structures with high humidity. Reinforced and prestressed concrete surface permanently submerged in non‑aggressive water |
| XC2 | Wet, rarely dry | Reinforced and prestressed concrete surfaces permanently in contact with soil not containing chlorides. |
| XC3 and XC4 | Moderate humidity or cyclic wet and dry | External reinforced and prestressed concrete surfaces sheltered from, or exposed to, direct rain. Reinforced and prestressed concrete surfaces subject to high humidity (e.g. poorly ventilated bathrooms,kitchens) Reinforced and prestressed concrete surfaces exposed to alternate wetting and drying Interior concrete surfaces of pedestrian subways not subject to de‑icing salts, voided superstructures or cellular abutments |
| Corrosion induced by chlorides other than from sea water (XD Classes) (where concrete containing reinforcement or other embedded metal is subject to contact with water including chlorides including de-icing salts, from sources other than from seawater) |
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| XD1 | Moderate humidity | Reinforced and prestressed concrete wall and structure supports more than 10 m horizontally from a carriageway Parts of structures exposed to occasional or slight chloride conditions Bridge deck soffits more than 5 m vertically above the carriageway |
| XD2 | Wet, rarely dry | Reinforced and prestressed concrete surfaces totally immersed in water containing chlorides Buried highway structures more than 1 m below adjacent carriageway |
| XD3 | Cyclic wet and dry | Reinforced and prestressed concrete walls and structure supports within 10 m of a carriageway Bridge parapet edge beams Buried highway structures less than 1 m below carriageway level Reinforced pavements and car park slabs |
| Corrosion induced by chlorides from sea water (XS Classes) (where concrete containing reinforcement or other embedded metal is subject to contact with seawater or airborne salt origination from sea water) |
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| XS1 | Exposed to airborne salt but not in direct contact with sea water | External reinforced and prestressed concrete surfaces in coastal areas |
| XS2 | Permanently submerged | Reinforced and prestressed concrete surfaces and completely submerged or remaining saturated, e.g. concrete below mid-tide level |
| XS3 | Tidal, splash and spray zones | Reinforced and prestressed concrete surfaces in the upper tidal zones and the splash and spray zones, including exposed soffits above sea water |
| Freeze‑thaw attack (XF classes) (where concrete is exposed to significant attack from freeze‑thaw cycles whilst wet) |
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| XF1 | Moderate water saturation without de‑icing agent | Verticl concrete surfaces such as facades and columns exposed to rain and freezing Non‑vertical concrete surfaces not highly saturated, but exposed to freezing and to rain or water |
| XF2 | Moderate water saturation with de‑icing agent | Concrete surfaces such as parts of bridges, which would otherwise be classified as XF1, but which are exposed to de‑icing salts either directly or as spray or run‑off |
| XF3 | High water saturation without de‑icing agent | Horizontal or near horizontal concrete surfaces, which are exposed to freezing whilst wet Concrete surfaces subjected to frequent splashing with water and exposed to freezing |
| XF4 | High water saturation with de‑icing agent or sea water | Horizontal concrete surfaces, such as roads and pavements, exposed to freezing and to de-icing salts either directly or run off Concrete surfaces subjected to frequent splashing with water containing de-icing agnets and exposed to freezing |
Reinforced Concrete Design
Column Design
Reinforced concrete columns that are subjected to axial loads are only considered columns if \(h\leq4b\), otherwise, they are considered as walls. During the design process, it is important to identify whether the columns structure is a braced frame (shear walls and cores) or unbraced, as these are factors when calculating the effective length of columns. In a braced frame, lateral loads are not resisted by the columns and in unbraced frames, they are resisted by columns. Also, columns can be identified as either slender or stocky columns.
Types of RC columns
- Rectangular
- Cicular
- Encased Steel section
- Concrete filled steel tube
Columns mainly act in compression, but bending moments always exist (construction tolerance, imperfections…)
Short columns – concrete crushes in compression
Slender columns – buckling may happen
The effective length of a column is determined by the following:
\(\lambda=\frac{l_0}{i}=\frac{l_0}{\sqrt(I/A)}\)
L depends on the end condition of the column (i.e, is the column fixed at the bottom and top or pinned?) The table below shows the effective length factors for arious conditions.
Slenderness Ratio
The slenderness ratio is a non-dimensional value that can be calculated below:
\(\lambda=\frac{l_0}{i}=\frac{l_0}{\sqrt(I/A)}\)
- Where
- \(l_0\) = effective height
- i = radius of gyration
- I = second moment of area
- A = cross-sectional area
Limiting slenderness ratio – short or slender?
When \(\lambda\leq\lambda_{lim}\) The column is short and \(2^{nd}\) order moments can be ignored.
When \(\lambda\geq\lambda_{lim}\) The column is slender and \(2^{nd}\) must be considered.
- \(\lambda=20\times A\times B\times C/ \sqrt(n)\) (Eq. 5.13N)
- A = 0.7
- B = 1.1
- C = 1.7 – \(r_m\), where \(r_m = M_{01}/M_{02}\) (moment ratio bewteen the moment at the top and bottom of the column)
- In the following cases, \(r_m\) should be taken as 1.0 (i.e. C = 0.7)
- For braced members in which the first order moments arise only from or predominantly due to imperfections or transverse loading
- For unbraced members in general
Failure modes of columns
The possible failure mechanism are either crushing or buckling
The crushing load is \(N_{ud}=0.567f_{ck}A_c+0.87f_{yk}A_s\)
Critical buckling load of pin-ended column: \(N_{crit}=\frac{\pi^2 EI}{l^2}\)
- For typical columns:
- For \(l/i<50\), \(N_{crit}>>N_{ud},\), short column, concrete crushing
- For \(l/i>110\), \(N_{crit}>>N_{ud},\), slender column, buckling
- For \(l/i\) between 50 and 110, intermediate column, crushing with possible buckling
Column Design - Reinforcement Detailing
- At least 4 bars for rectangular section and 6 for circular section. Bar diameter \(\phi \geq 12mm\)
- Links to form steel cage and confine concrete
- Minimum size =1/4 x size of compression bar & \(\geq 6mm\)
- Maximum spacing =min (20x size of the smallest compression bar, the least lateral dimension, 400mm)
- Every longitudinal bar placed in a corner should be held transver reinforcement
- No compression bar should be further than 150 mm from a restrained bar.
Concrete Bending Capacity
Bending of concrete beams
Concrete is strong in compression and weak in tension. When a beam supported at 2 ends, cracking will occur along the bottom of the concrete. Reinforcement is added to minimise cracking and the top flange/section of concrete will be in compression.
Material properties
- EC2 uses the characteristic cylinder strength fck unlike BS 8110 which uses the characteristic cube strength fcu
- Concrete classes are expressed as C20/25, C30/37, C35/45 in EC2 where the first number is the cylinder strength and the second number is the cube strength
- The design compressive strength of concrete is given by
- \(f_{cd} = \alpha f_{ck} \gamma_m = 0.85 f_{ck} /1.5 = 0.567 f_{ck}\)
- (where \(\alpha\) = 0.85 from UK National Annex for flexure and axial loading)
- The density of concrete is given as 25 kN/m3″> kN/m3 in EN 1991-1-1
Steel Reinforcement Material Properties
- The design strength of reinforcement in tension and compression fyd is given by:
- \(f_{yd}/ \gamma_m = f_{yk}/1.15\)
- fyk is the characteristic yield strength (5%) and gm is the material factor of safety for reinforcement
- The characteristic strength of reinforcement \(f_{yk}\) = 500 MPa
- The elastic modulus of reinforcement is 200 GPa
ULS Section Analysis - Assumptions
- 1. Plane sections remain plane.
- 2. Stresses in the flexural compressive zone may be derived from a design curve relating stress and strain.
- 3. The strain in the extreme compressive fibre εcu is defined at failure (εcu = 0.0035 for flexure in EC2).
- 4. The tensile strength of the concrete is neglected.
- 5. The stress in the reinforcement is calculated using an idealised bi-linear stress-strain diagram.
- Design Stress = \(\frac{ \alpha f_{ck} }{\gamma_c} = \frac{0.85f_{ck}}{1.5} = 0.567f_{ck}\)
- \(f_{cd}=0.85f_{ck}/1.5 = 0.567f_{ck}\)
Concrete Shear
When concrete is subjected loading from it’s self-weight, permanent and variable loads, diagonal cracks forms in beams and additional reinforcement is required to minimise cracking. In slabs and foundations, large loading can result in high localised “punching shear” occuring around supports(columns), which may also require additional reinforcement in those regions. See the image below which shows the types of cracks forming in a typical beam.
- Shear is carried out at ULS only (i.e, SLS checks not required).
- Shear failure is much more complex than flexural failure.
- It is still a subject of research.
- However, current theories are adequate for design.
Concrete Sections that do no require Shear reinforcement
- Lightly loaded floor slabs, pad foundations etc
- Where shear forces are small the concrete section on its own may have sufficient shear capacity (VRd,c) to resist the ultimate shear force (Ved)
- In beams a minimum amount of shear reinforcement will usually be provided.
- The shear capacity of the concrete, VRd,c in such situations is given by an empirical expression: \(V_{RD,c} = (0.12k(100\rho f_{ck})^{1/3})b_{w}d\)
- Where \(k = (1 + \sqrt{\frac{200}{d}})\leq2.0\)
- and \(\rho = \frac{A_{sl}}{b_wd} \leq 2.0 \)
- with a minimum value of: \(V_{Rd,c} = (0.035k^{3/2}f_{ck}^{1/2}) b_{w}d \)
Variable Strut Inclination method
When designing for shear in beams, the applied loading/actions are represented by an analogous truss. The concrete acts as the top compression member and at an angle \(\theta\). The bottom chord is the tension steel (main steel bars) and the steel links (stirrups) will act as tension members. In subsequent calculations when determining the shear capacity using shear links, the concrete does not contribute to the shear capacity of the beam (i.e, capacity is determined from steel links).
Analysis of the beam to find the ULS Shear resistance is carried out in three stages:
- 1. Check the compressive strength of the diagonal concrete strut and its angle \(\theta\)
- 2. Calculate required shear reinforcement \(A_{sw}/s\)
- 3. Calculate additional area of tension steel \(A_{sl}\) required in the bottom chord.
1. Diagonal compressive strut
Excessive compressive stresses must not occur in the diagonal strut:
- The effective cross sectional area of concrete acting as the diagonal strut is taken as: \(b_w \times z\cos\theta\)
- and the design concrete stress: \(f_{cd} = f_{ck}/1.5\)
- The ultimate strength of the strut = ultimate design stress × area = \((f_{ck}/1.5) \times (b_w \times z\cos\theta)\)
- and its vertical component = \([(f_{ck}/1.5) \times (b_w \times z\cos\theta)] \times \sin\theta\)
- so that \(V_{Rd,max} = (f_{ck} b_{w} z \cos\theta sin\theta)/1.5 \)
- By conversion of the trigometric functions this can be expressed as: \(V_{Rd,max} = \frac{f_{ck}b_wz}{1.5(\cot\theta + \tan\theta)}\)
- In EC2 this equation is modified b ythe inclusion of a strength reduction factor \(v_1\) for concrete cracked in shear
- Thus \(V_{Rd,max} = \frac{f_{ck}b_wzv_1}{1.5(\cot\theta + \tan\theta)}\)
- Where the strength reduction factor takes the value \(v_1=0.6(1-f_{ck}/250)\)
- and z = 0.9d then the above becomes:
- \(V_{Rd,max} = \frac{0.9dxb_w\times0.6(1-f_{ck}/250)f_{ck}}{1.5(\cot\theta + \tan \theta)}\)
- \(=(0.36b_wd(1-f{ck}/250)f_{ck})/(\cot\theta+\tan\theta)\) …. Eqn 1
EC2 Limits
- EC2 limits \theta to a value between \(22^{\circ}\) and \(45^\circ\)
- (i) With \(\theta = 22^\circ\) (this is the usual case for udl loads) from Eqn 1: \(V_{Rd,max(22)} = 0.124b_wd(1-f_{ck}/250)f{ck}\)
- If \(V_{Rd,max.(22)}\) \(<\) \(V_{Ed}\) then a larger \(\theta\) must be used so that the diagonal concrete strut has a larger vertical component to balance \(V_{Ed}\).
- (ii) With \(\theta\) = \(45^\circ\) (the maximum value of \(\theta\) as allowed by EC2) again from Eqn 1:
- \(V_{Rd,max(45)} = 0.18b_wd(1-f_{ck}/250)f_{ck}\)
- Which is the upper limit on the compressive strength of the concrete diagonal member in the analogous truss.
- If \(V_{Rd,max.(45)} < V_{Ed}\) then a larger concrete section is required
- (iii) With = between 22 and 45 The required value of can be obtained by equating VEd to VRd,max and solving in Eqn 1 as follows:
- \(V_{ED}=V_{Rd,Max}=\frac{0.36b_wd(1-f_{ck}/250)f_{ck}}{\cot\theta + \tan\theta}\)
- and \(1/(\cot\theta + tan\theta)=\sin\theta \times \cos\theta = 0.5\sin2\theta\)
- Therefore by substitution \(\theta = 0.5\sin^{-1}[V_{ED}/0.18b_wd(1-f_{ck}/250)f{ck}]\leq 45^\circ\)
- This can also be shown as \(\theta=0.5sin^{-1}[V_{Ef}/V_{Rd,max.(45)}]\leq 45^\circ\)
- This calculated angle \(\theta\) can now be used to determine \(\cot\theta\) when calculating the required shear reinforcement.
2. Vertical Shear Reinforcement
- Shear is resisted by shear links with no contribution from the concrete.
- Using the method of sections (cut at X-X), the force in the vertical link member (\V_{wd}\) must equal the shear force \(V_{ED}\), that is
- \(V_{wd} = V_{Ed} = 0.87f_{yk}A_{sw}\)
- If the links are spaced at a distance s apart, then the force in each link is reduced proportionately and is given by
- \(V_{wd} = B_{Ed}=0.87\frac{A_{sw}}{s}zf_{yk}\cot\theta = 0.87\frac{A_{sw}}{s}0.9df_{yk}\cot\theta\)
- thus rearranging: \(\frac{A_{sw}}{s}=\frac{V_{Ed}}{0.78df_{yk}\cot\theta}\)
- EC2 specifies a minimum area of links of: \(\frac{A_{sw,min}}{s}=\frac{0.08f_{ck}^{0.5}b_w}{f_{yk}}\)
3. Additional Longitudinal Force
- Allowance must be made for the additional longitudinal force in the tension steel
- Resolving forces horizontally (cut at Y-Y), the longitudinal component of the force in the compressive strut is given by :
- Longitudinal Force = \((V_{Ed}/\sin\theta) \times \cos \theta\) = \(V_{Ed}\cot\theta\)
- It is assumed that half this force is carried by the reinforcement in the tension zone of the beam, then: \(F_{td}=0.5V_{Ed}\cot\theta\)
- This can be provided by additional longitudinal reinforcement above that required for bending reinforcement
Shear reinforcement - variable strut inclination method
Punching Shear
When looking at shear stresses in slabs, the loads will be distruted through the slab (general actions are uniformly distributed loads/pressure loads), which will have a ultimate shear force \(V_{Ed}\) less than the shear resistance of the concrete \(V_{Rd,c}\). Shear reinforcement is not necessarily required (check point loads as a seperate case). However, localised ‘punching shear’ forms around the column due to the concentrated shear loads at this area (reaction force from applied loading); these cases are critical and a basic control perimeter is 2.0d from the loaded area and will generally require shear reinforcement.
The equation for punching shear stress can be expressed as: \(\frac{\beta V_{Ed}}{U_id}\)
- Where
- \(\beta\) = 1.0 for columns with no eccentricity/continuity
- \(\beta\) = 1.15 for internal columns
- \(\beta\) = 1.4 for edge columns
- \(U_i\) = 1.5 for corner columns
- d = effective depth
When the applied shear stress is greater than the shear resistance of the concrete, shear reinforcement will be required. In general, the additional reinforcement is procured from manufactor reinforcement systems for this issue and they will undertake the calculations through their softwares.
Bar Bending Schedule Tables
Reinforcement bar bending schedule guide
Steel reinforcement within reinforced concrete has to be specified in accordance to BS8666, which contains the requirement for dimensioning, bending and cutting of the bar. Also, the Istrute guide to detailing is commonly used which gives detailed sketches and minimum requirement for specific members (i.e, beams, slabs retaining walls etc.)
Scheduling table from BS 8666
Below is a table extracted from BS 8666