This page includes some notes on reinforced concrete as used in the construction of walls and structures.    It is important to note that I have based most of the content of this page on BS 8110 -1:1997.    This standard has now been replaced by Eurocode BS EN1992.   There are a number of differences between BS 8110 and BS EN 1992 e.g the symbols are generally different (N in BS8110 = N _ED in BS EN 1992).    The notes are not intended to enable detail design to the latest codes they are simply provided to enable mechanical engineers to understand the topic and produce basic design studies.    Formal design work must be completed in accordance with the relevant codes.   

Reinforced concrete is probably the most prolific and versatile construction material .    It is composed of two distinct materials concrete and reinforcement , each of which can be varied in quality disposition and quantity to fulfill a wide range of construction requirements.

The concrete composition is an based on three constituents : aggregate, cement and water.    These are mixed together in a homogeneous mass and are then put in place and left for the chemical and physical changes to occur that result in a hard and durable material.    Information on concrete forms is provided on webpage concrete.       The strength and durability of the resulting concrete depends on the quality and quantity of each of the constituents and on any additional additives have been added to the wet mix.     Much of the mixing is now done off site by ready mix companies ..The strength of the resulting mixes is generally confirmed by a cube crushing test.   

The concrete reinforcement is generally steel although other materials are sometimes used such as glass fibre.     The main reinforcement bars are generally high yield deformed bars (  f _y >= 460 N/m² ) .    Reinforcement links are often mild steel (  f _y >= 250 N/m² ) although high yield steel is becoming more popular.    For reinforced concrete slabs and walls it is convention to use mesh reinforcement.

The reinforcement is generally located to compensate for the concrete being weak in tension e.g in spanning beams the reinforcement is primarily located in the bottom half of a section and at the midspan.    In a cantilever beam the reinforcement will generally be at the top of the beam with the maximum concentration at the support.

In the notes below the characteristic strength for concrete ( f _cu ) is the value of the cube strength and the characteristic strength for the reinforcement ( f _y ) is the designated proof/yield strength.    The referenced standard for these notes (BS 8110) also identifies a partial factor of safety γ _m which is applied to these strengths to take into account the difference between test and practical conditions.    It should be noted that the strength of concrete related to flexure is actually accepted as 0,67. f _cu.

The characteristic load regime on a structure comprises a characteristic dead load (G _k) and a characteristic imposed load ( Q _k ) and sometimes a characteristic wind load ( W _k ) these are each modified by an appropriate partial safety margin γ _f

Structures made from concrete to BS 8110 (and to the latest codes ) should be designed to transmit the design ultimate dead , wind , and imposed loads safely from the highest supported level to the foundations.    The structure and interactions between the included members should ensure a robust and stable design . The design should also be such that the structure is able to remain is service . Account should be taken of temperature, creep, shrinkage, sway, settlement and cyclic loading as appropriate.

Summary of load symbols

Summary of strength symbols

When completing analysis of cross sections to determine the ultimate resistance to bending certain assumptions are made

1)The strain distribution in tension or compression assumes plane sections remain plane
2) The compressive stresses in concrete are derived from stress strain curve shown below with γ _m = 1,5
3) The tensile strength of concrete is ignored
4) The stresses in the reinforcement are derived from the stress/strain curve as shown below γ _m = 1,15
5) When the section is resisting only flexure the lever arm should not be greater than 0,95x the effective depth

The effective depth is the depth from the compression face to the centre of the area of the main reinforcement group

The formwork is the timber , steel or plastic moulds into which the concrete is poured on site to create the various concrete components.    It is a vital part of the construction process and the formwork costs can be up to 50% of the concrete construction costs.    The reinforcement must be located in the formword prior to pouring the concrete.

The formwork must be leaktight, strong and rigid to contain and maintain dimensions of the full liquid concrete mass. It must also be designed for standardisation and reuse to reduce the construction costs to a minimum.

It is clear that the formwork should not only be designed for construction it should also include features allowing safe and convenient removal and re-use

The design notes provided on this page relate to BS 8110-1 :1997 which has been superseded by the standards referenced below.

Beams
Beams are horizontal structural items specifically design to support vertical loads .    Concrete beams are generally rectangular in section with width b and height h and length L.     Beams can also be of a variety of section including channel section, tee section, I section etc.     There are a number of support configurations for beams including cantilever, simply supported , continuous, etc each one tending to produce different stress and deflection characteristics.    As concrete is not able to withstand tensile stress , loaded concrete beams are generally reinforced in the area under tensile stress.

Concrete columns
Vertical structural elements of clear height = l and cross section = b x h where h < 4b are columns , otherwise they are walls.     A column should not have an unrestrained length greater than than 60b.

Concrete Walls
Vertical load bearing element with length exceeding 4 times the thickness.    The clear height is designated l _o and the thickness h.    Concrete walls can be plain walls which have zero to minimum reinforcement (< 0,4% of section area ) or reinforced walls with reinforcemnt > 0,4% of area .   

Flat Slabs
Flat slabs are horizontal slabs , used for floors or upper structural surface which are supported on walls or beams.

Solid Slabs
These are horizontal slabs supported on pads or columns instead of walls

L = effective span = smaller of distance between bearing centres or clear distance (between supports ) + depth of section

Typical Values of γ _m for Ultimate limit state... (Persistent and transient)

Load Combinations and values γ _f for Ultimate limit state.

The theory supporting the relationships in this section,in outline, is covered on web page Reinforced concrete beams theory

The sketch below identifies the types of simple reinforced beams that are relevant to the notes provided

Condition of strain and stress in a rectangular section at ultimate limit state of loading is shown in the figure below

Note: The factor 0,67 is not a partial safety margin it relates to the direct relationship between the cube strength as indicated by f _cu and the strength in flexure of the concrete.
The total compressive force generated within the concrete at the ultimate moment capacity is

0,9 . 0,67.f_cu . b . x / (γ _m =1,5)    =     0,4 . f_cu. b. x

The ultimate moment capacity of a unreinforced concrete beam section where there is less than 10% moment distribution is

M _u = 0,156 .F _cu.b.d²

M = K.[ f _cu.b.d² ]

If the applied moment is less than M _u then the area of tension reinforcement =

If the applied moment is greater than M _u ( K > 0,156 ) then tension  and compression reinforcement is necessary
The tension reinforcement required is.

Compression reinforcement required is

M_u is simply calculated as 0,4.f_cu.b.h_f.(d - h_f/2) if this is greater than M then the neutral axis is within the flange and the tee can be assessed using the equations above..

The design shear capactity (v) in a concrete beam at any section is calculated from

v = V / (b _v.d )

V is the Shear force at a section
v should never exceed 0,8 √ f _cu or 5 N/mm² if lower.

Values of concrete shear capacity v _c related to % reinforcement and effective depth (d) of section

If the applied shear stress is less than 0,5 v _c throughout the beam then

Minimum links should be provided, and in elements of low importance e.g lintels then no links need be included

Suggested shear area provided = A _sv > 0,2 b _v.s _v / (0,87 .f _yv)

If the applied shear stress is greater than 0,5 v _c and less than (0,4 + v _c) throughout the beam then

Minimum links should be provided for the whole length of beam to provide shear resistance of 0,4 N/mm²

Suggested shear are provided = A _sv > 0,4 b _v.s _v / (0,87 . f _yv)

If the applied shear stress is greater than (0,4 + v _c) and less than (0,8 Sqrt(F _cu or 5 N/mm²) + v _c) throughout the beam then

Links should be provided for the whole length of beam to provide shear resistance at no more than 0,75d spacing .    No tension bar should be more the 150 mm for a vertical shear link.

Suggested shear are provided = A _sv > b _v.s _v (v - v _c) / (0,87 . f _yv)

>

A table is provided below which gives the basic span/ depth ratio for beams which limit the total deflection to span/250 or 20mm (if less) , for spans up to 10m.

For b _w/b >0,3 interpolation between the rectangular and flanged values is acceptable

The allowable span/depth = F ₁.F ₂F ₃F ₄. Basic span/depth ratio

F ₁ For long spans exceeding 10m the values in the table should be multiplied by 10/span.
F ₂= A factor to allow for tension reinforcement. See chart below
F ₃= A factor to allow for compression reinforcement. See chart below
F ₄= A factor to allow for stair waists where the staircase occupies over 60% of the span

Cross section of a typical simple column.

A column should not have a clear distance between restraints which exceeds 60.b ( b being the small cross section dimension ).    If one end of a column is not restrained (a cantilever column) then its clear height must be the smaller of 60.b or 100.b² /h .

It is important early in the design process to determine if the column is a stocky design and if there is significant bracing associated with the columns.    The effective length of a column l _e = l _o.β where β is the effective length constant which is dependent on the end support conditions (see tables below).     A column is considered a stocky column if its effective length divided by b = l _e/b is less than 15.     A longer column must be assessed in the design for risk of buckling.

Some simple rules for column reinforcement.
A _sc should by more than 1% and less than 6% of gross cross section area of column (b x h)
The minimum dia of bars should be 12mm.
Lateral binders/ties should be arranged to restrain each bar from buckling and the end of the binders should be anchored.     The pitch of the binder should not exceed b or 12 times the dia of the longitudinal bars, nor 300 mm . The diameter of the binders should not be less than 25% of the diameter of the longitudinal bars

Typical column head designs. (columns and associated heads can also be circular.

Braced Columns -Table of effective length coefficients (β)

A column is considered to be braced if lateral stability is provided by walls or buttresses. The column is effectively only taking axial loads and moments resulting from eccentricity of lateral loads

Unbraced Columns -Table of effective length coefficients (β)

Rigidly Fixed
The end of the column is connected monolithically (solidly) to beams on either side which are at least as deep as the overall dimension of the column in the plane considered.    Where the column is connected to a foundation structure, this should be of a form specifically designed to carry moments.

Fixed.
The end of the column is connected monolithically to beams or slabs on either side which are shallower than the overall dimension of the column in the plane considered

Pinned with some angular restraint
The end of the column is connected to members which, while not specifically designed to provide restraint to rotation of the column will, nevertheless, provide some nominal restraint.

Free.
The end of the column is unrestrained against both lateral movement and rotation. e.g the free end of a cantilever column in an unbraced structure

When a stocky column is subject to a simple axial loads with induced moments , assuming a well balanced load scenario , it need only be design for the ultimate design axial force + a nominal allowance for an eccentricity of force (e _c ) of h/20 (with a maximum of 20mm).

When a stocky column is subject Axial forces and bending stresses it is generally necessary to use design charts .Reference Column Design charts.

Stocky beams resisting moments
Following equations include provision for γ _m.

When a stocky column cannot be subjected to significant moments,it is sufficient to design the column such that the design ultimate load is less than

N = 0,4 f _cu.A _c + 0,75 A _sc.f _y

When a stocky column is supporting to a reasonably symmetrical arrangement of beams of similar spans and which are for uniformly distributed loads , it is sufficient to design the column such that the design ultimate load is less than

N = 0,35 f _cu.A _c + 0,67 A _sc.f _y

Biaxial bending in columns
When it is necessary to consider biaxial moments, the design moment about one axis is enhanced to allow for the biaxial loading condition and the column is designed around the enhanced axis.    Consider the column as loaded below.

The design axial forces in a reinforced wall may be calculated on the assumption that the beam and floor slabs being supported are simply supported.

The effective length of a wall l _e should be obtained as if the wall was a column which is subject to moments in the plane normal to the wall.    The determination if a wall is stocky or slender is also obtained using the same criteria as for a column.

Stocky reinforced walls

A stocky braced reinforced wall supporting reasonably symmetrical load should be designed such than..

n _w = 0,55.f _cu.A _c + f _y.0,67A _SC.

n _w = total design axial load on wall due to design ultimate loads :providing the slab loads are uniform in loading and relatively evenly distributed.

Except for short braced walls loaded symmetrically the eccentricity in the direction at right angles to a wall should not be less than h/20 or 20mm if less.

When the eccentricity results from only transverse moments the design axial load may be assumed to be evenly distributed along the length of the wall.    The cross section should be designed to resist the design ultimate load and the transverse moment .   The assumptions made for the calculation of beam sections apply.

When a wall is subject to in-plane moments and uniform axial forces the cross section of the wall should be designed to support the ultimate resulting axial loads and inplane moments.

Slender reinforced walls

The maximum slenderness ratio l _e/h should be 40 for braced walls with <1% reinforcement: 45 for braced walls with => 1% reinforcement: 30 for unbraced walls

A suitable design procedure is to first consider axial forces and in-plane moments to obtain the distribution of forces along the wall assuming the concrete does not resist tension.    The transverse moments are then calculated.    At various points along the wall the results are combined.

Walls subject to significant transverse moments additional to the ones allowed for by assuming a minimum eccentricity are considered by assuming such walls are slender columns bent about the minor axis .    If the wall is reinforced with only one central layer of reinforcement the additional moments should be doubled

Plain walls

Plain walls include less than 0,4% reinforcement.

The effective height of plain unbraced concrete walls is assessed as ( l _e = 1,5 l _o ) if the wall is supporting a roof or floor slab, otherwise it is calculated as (l _e = 2 l _o).

When a plain concrete wall is braced with lateral supports resisting both rotation and movement then ( l _e = 0,75 l _o )
Where the lateral support only resists lateral movement then ( l _e = l _o )or if relevant ( l _e = 2,5 (distance between support and a free edge )

Solid slabs can be simply one-way loaded plates or two way loaded plates depending on the support arrangements

Design Moments and shear forces in simple one way spanning continuous slabs
Uniformly Distributed Loads

F = Total Design Ultimate load on one slab (1,4 G _k + 1,6 Q _k)
l _s = is effective span of slab

G _k = Dead Load
Q _k = Imposed load)

Design Moments and shear forces in simple one way spanning continuous slabs

Design Moments and shear forces in two way spanning continuous slabs
Uniformly Distributed Loads

Design Moments and shear forces in two way spanning continuous slabs

Shear Force coefficients

The deflection can be limited by the application of the span/depth ratio as indicated in the table for beams and modified by the use of F₂ as shown in the relevant graph Design of Beams...... Only conditions at the centre of slab in the width of the should be used to influence the deflection.

For two way spanning slabs the ratio should be based on the shorter span.

Flat slabs should be designed to satisfy deflection requirements and to resist the shear load around the column supports.

BS 8110 allows for a simplified method for determining moments subject to certain provisions

1) design is based on a single load case of all spans being loaded with the maximum design ultimate load.
2) There are at least three rows of panels of approx equal span in the direction under consideration
4)The ratio of imposed to dead load does not exceed 1,25
5)The characteristic imposed load does not exceed 5kN/m2

This method involve simply using the table provided for design moments in simple one way spanning continuous slabs as provided above as copied below.
Moments at supports resulting from the table below are reduced by 0,15F.h _c
The design moments resulting should be divided between the column strips and mid-strips as shown in the figure below in proportions as shown in table below

F = Total Design Ultimate load (1,4 G _k + 1,6 Q _k)        l _s = is effective span of slab        G _k = Dead Load        Q _k = Imposed load)

The critical shear condition for flat slabs is punching shear around the column heads.    The shear load supported by a column is the basic calculated shear force (V) uprated to account for moment transfer.    For slabs with approximately equal spans the uprated shear force, designated the effective shear force V _eff ,can be simply estimated using the following rules.

For internal columns V _eff = 1,15 V
For corner columns V _eff = 1,15 V
For corner columns V _eff = 1,15 V
For edge columns with the moment parallel to the edge V _eff = 1,25 V
For edge columns with the moment normal to the edge V _eff = 1,4 V

The slab shear at the column face is calculated as

d = thickness of slab and U _o is the perimeter of the slab at the column head edge

ν _o should be less than 0,8 √ f _cu or 5 N/mm² if less

Perimeters U _i radiating out from the column edge should be checked with the first perimeter (i =1) at a distance 1,5.d from the column face and subsequent perimeters i = 2,3... with intervals of 0,75.d

successive perimeters are checked until the applied shear stressν _i is less than the allowable shear stress ν _c.    Reinforcement links are required between the perimeters at which the shear stress is greater than ν _c.   

When the gross width of drops in both directions exceed 1/3 the respective span the deflection can be limited by the application of the span/depth ratio as indicated in the table for beams Design of Beams...... Otherwise the resulting span/effective depth should be multiplied by 0,9.

The assessment should be completed for the most critical direction.

Details based on BS 8666 .Normally grade H bars used f _y = 500 N/m²