#### Design of Â Reinforced concrete wallsÂ are in many ways like columns.

They support vertical loads and are vulnerable to minor axis bending.

They often act as primary elements of a structureâ€™s lateral stability systems and by doing so are subjected to high in-plane bending forces.

A wall is a vertical or near vertical element with a breadth/length that is greater than 4 times its thickness.

While a minimum thickness of 150mm for a wall is possible, 180-250mm is recommended to aid detailing and construction.

Walls thinner than 250mm can incur concrete compaction problems due to the congestion of reinforcement, particularly near openings.

A wallâ€™s thickness is determined based on the forces it is being subjected to, required fire resistance, durability and buildability.

There are three types of action a reinforced concrete wall can be subjected to: axial forces, minor axis bending and shear, and major axis bending and shear.

The action to which walls are most vulnerable, is bending, and this guidance note principally addresses this form of stress. How walls resist the effects of these actions is dependent on the wallâ€™s geometry.

Other aspects that may impact the design of a wall are its design life and fi re protection. Both set parameters for the concrete strength of the wall, its thickness and cover to reinforcement.

The stages of design for a reinforced concrete wall can be summarised as follows:

- Establish design life of the structure to which the wall is to form a part
- Determine actions on the wall and their combinations
- Select a concrete strength based on durability requirements
- Determine cover to reinforcement and thickness based on fi re protection requirements
- Calculate shear force and bending moments acting on the wall
- Check for slenderness
- Calculate required amount of steel reinforcement based on applied forces
- Review minimum and maximum areas of reinforcement and bar spacing.

The thickness of reinforced concrete walls is affected by the fire rating of the structure to which they are to form a part.

* Axis distance is measured from the surface of the concrete to the centreline of the main reinforcing bars; minimum cover to reinforcement to be kept at 20mm.

The slenderness of a wall is defined by the ratio of the effective height to the thickness of the wall. When a wall is slender its resistance to bending moments and axial forces is reduced.

It is possible to simplify the process of determining the slenderness of concrete walls by satisfying the following expression:

M_{1z} Â andÂ M_{2z}Â are the smaller and larger minor axis bending moments on the considered span.

is a function of the applied design axial load (N_{Ed}) per metre length of wall.

Where

Â Â Â Â Â

Â Â Â Â Â or 0.003 f_{yk}/f_{ck}

f_{yk} tensile or characteristic yield strength of reinforcement, which is typically 500 N/mm2.

f_{ck }the characteristic cylinder compressive strength of the concrete.

h thickness of the wall.

A_{s} the area of vertical reinforcement in the wall.

l_{0z }the effective height of the wall along its minor axis.

Factors applied to height of wall in its minor axis to determine its effective height

*Condition 1**: *Wall is connected monolithically to slabs that are at least as thick as the wall. In instances where a wall is fixed into a foundation, that

sub-structure must be designed to support bending moments as well as axial and lateral forces.

*Condition 2**: *Wall is connected monolithically to slabs that are at least half the thickness of the wall, but not shallower than the overall wall thickness.

*Condition 3**: *Wall is connected to elements that provide nominal restraint to rotation.

If the wall proves to be slender, it needs to be designed to withstand increased minor axis bending moments – the applied moments that have been made greater to allow for the slender state of the wall. These modified moments.

M_{2} or M_{zi}+N_{Ed}(e_{2z}+e_{a}), whichever is the greater.

The estimated mid height minor axis bending moment of the wall is the greater of either:

0.6M_{2}+0.4M_{1} or 0.4M_{2}

Where M_{1} is the smaller end moment in the minor axis of the wall derived from analysis

M_{2} is the larger end moment in the minor axis of the wall derived from analysis

M_{zi} is the applied minor axis bending moment that is derived from analysis

E_{2z }is the deflection of the wall in its minor axis due to axial factored design actions

E_{2z }= f_{yk}(l_{0z}/d)^{2} 10^{-6} in mm

d is the effective depth of reinforcement (mm) within the wall in its minor axis

e_{a} is the eccentricity of the applied design axial force due to â€˜out of plumbâ€™ state of the wall during construction = where a notional drift of the wall as stipulated (in mm).

Notional inclination of walls

Care should be taken with respect to sign convention of bending moments within walls. Depending on the location of the eccentricity of the applied axial force on the wall, the N_{Ed} (e_{2z} + e_{a}) expression may prove to be the lesser of the two.

When calculating the forces being applied to the wall, certain assumptions need to be made concerning how they are modelled. When considering bending in the major axis, all elements that are framing into it need to be taken to be simply supported.

Bending moments that are applied due to lateral forces such as wind, must assume that the wall is acting as a cantilever from its foundations. With respect to minor axis bending, all elements framing into the wall should be considered monolithic with the appropriate level of full fixity, e.g. if these elements are also concrete, then the full bending moment is transmitted from the horizontal element to the wall.

Before establishing the required amount of reinforcement, the magnitude of stresses due to the applied actions must be calculated.

It is common practice to split the wall into 1m strips, which simplifies the design of the reinforcement within the wall.

These can each be considered subject to axial compression and minor axis bending only, with the design of reinforcement based on the extreme fibre stresses (*f _{t}*) in each idealised 1m segment of the wall.

All relevant axial forces and bending moments are applied to each segment under consideration and each unit length is treated like a column for design purposes.

The extreme fibre stressesÂ

N is the design axial force in the 1 m strip of the wall

L is the overall length of the wall

h is the overall thickness of the wall

M is the applied design major axis bending moment in the segment of wall.

The resulting stress is then multiplied by the thickness of the wall (h) to give a stress per metre length. The reinforcement can then be determined using a column design methodology where minor axis bending, as well as the axial stresses due to both major axis bending and axial forces, is applied.

A simplified approach to determining the amount of compression and tension reinforcement required.

It can only be applied in instances where there are no significant minor axis bending moments being applied to a wall e.g. when supporting slabs of the same span and depth are either side of the wall, then a simplified expression of minor axis bending stress.

For compressive forces: f_{t }h equal or less than 0.43f_{ck }h + 0.67 f_{yk}A_{sc}

f_{t} is the compressive or tensile force applied to the wall.

A_{sc} is the area of reinforcement in mm2 per metre length of wall.

For tension forces: A_{s} = f_{t }h L_{t}/0.87f_{yk}

L_{t} is the length of wall subject to tension.

All tension reinforcement should be installed at least within a zone that is measured as 0.5L_{t} from the end of the wall. However, it is common practice to place this amount of reinforcement along the entire length of the wall for ease of construction.

The area of reinforcement requirements is can be summarised as:

- Minimum of 0.002A
_{c}split between both layers for vertical reinforcement, near face and far face. - Maximum area of vertical reinforcement is set at 0.04A
_{c} - For horizontal reinforcement, the minimum is either 25% of the provided vertical reinforcement or 0.001 A
_{c}, whichever is the greater. - The minimum size of bar for horizontal reinforcement is 0.25 times the diameter of the vertical reinforcement.

A_{c} is the overall cross section area of the wall

The spacing of vertical bars should not be greater than 300mm or 3 times the thickness of the wall, whichever is the lesser, though more prescriptive requirements may be dictated by the construction method, e.g. for slip forming.

It is considered best practice that horizontal bars should not be spaced further than 300mm apart, in order to limit early age thermal cracking, although up to 400mm is permitted.

In terms of size, reinforcement bars should be at least 0.25 times the diameter of the vertical bars.

Where the vertical reinforcement exceeds 2% of the cross-sectional area within any 1m wide wall segment, containment links are to be provided.

**Worked example**

A 20m high, 3.5m long shear wall is acting as both a lateral and vertical support to a 4-storey building.

There are 6 columns between it and the next shear wall.

Design the reinforcement in the wall at its base and mid-height. Floor slabs frame into it at 3.2m centres and are 200mm thick.

The foundations to the wall have not been designed to withstand bending moments in the minor axis.

The building has a fire rating of 60 minutes, a 50-year design life and the reinforcement is to have tension strength of 500N/mm2.

The wall is exposed to the external environment on one face. The design actions (factored loads) applied to the wall are:

Axial design force at base 3,000kN

Axial design force at mid-height 1,800kN

Design bending moments 2,200kNm (major axis at base)

1,200kNm (major axis at mid-height)

80kNm/m (minor axis) per floor

Geometry of wall L=3500mm H=3200mm b=1m

TryÂ h=200 mm thick wall made from C35/40 concrete

f_{ck}=35N/mm^{2}

f_{yk}=500N/mm^{2}

Exposed to fire, rain and moisture

Fire minimum thickness h_{min}=130 mmÂ less than h=**200.000**mm

Durability Exposure class XC4 50 year Design life S4 Structure

Div=10mm

Vertical bar=20mm

Horizontal bar =10mm

Therefore C_{min} =30mm;

Therefore d=h-(C_{min}+Div+(V_{bar}/2)+ h_{bar})=**140.000**mm

At Base

N=3000kN

M_{y}=2200kNm

@ Base of wall; f_{t1}=(N/(L*h)) +(6*M_{y}/(h*L*L))=**9.673**N/mm^{2 }

& f_{t2}=(N/(L*h))-(6*M_{y}/(h*L*L))=**-1.102**N/mm^{2}

f_{t}=max(f_{t1},f_{t2})=**9.673**N/mm^{2}

Therefore force/mm @ base f_{t}*h=**1934.694**N/mm;

Slenderness Check

@ Base of wall M_{2z}=80kNm

Â M_{1z}=0 kNm

=0.003* f_{yk}/ f_{ck}=**0.043**

Therefore @ base =0.69*Sqrt((1+(2*))*(1000*h*f_{ck})/( f_{t}*h*1000)) =**1.368** equal or greater than 1

Â Slenderness Ratio Limit 4.38*(1.7- M_{1z}/M_{2z}) =**10.183**

Condition 1 @ the top & 3 @ the bottom

Therefore coefficient C_{ef} = 0.9

Therefore L_{0z}=C_{ef}*H=**2880.000**mm

Slenderness Ratio S_{r}= L_{0z}/h=**14.400**; Greater than Slenderness Ratio Limit 4.38*(1.7- M_{1z}/M_{2z})=**10.183**

Therefore, Wall is slender @ Base 1^{st} level.

Due to slenderness status of wall @ Base, additional Bending moments need to be accounted for

Modified Bending moment due to slenderness:

M_{2z}=**80.000**kNmÂ force/mm @ base f_{t}*h=**1934.694**N/mm

e_{z2}=f_{yk}*(L_{0z}/d)^2*1 mm/MPa/1000000 =**0.212** mm

Â T_{hita} =1/400 =**0.003** from the table

e_{a}= T_{hita} *L_{0z}/2=**3.600**mm

M_{base}= M_{2z} + f_{t}*h *(e_{z2}+e_{a})*1m=**87.374**kNm

Reinforcement Design

Using the column design charts to determine the amount of reinforcement in the wall

@ Base of the wall

d/h=**0.700**

M_{base}/(b*h*h*f_{ck})=**0.062**

f_{t}/f_{ck}=**0.276**

At mid height

N=1800kN

M_{y}=1200kNm

@ Mid-Height of wall; f_{t1}=(N/(L*h)) +(6*M_{y}/(h*L*L))=**5.510**N/mm^{2 }& f_{t2}=(N/(L*h)) -(6*M_{y}/(h*L*L))=**-0.367**N/mm^{2}

f_{t}=max(f_{t1},f_{t2})=**5.510**N/mm^{2}

Therefore force/mm @ Mid-height f_{t}*h=**1102.041**N/mm

@ Mid height of wall; M_{2z}=80kNm

Â M_{1z}=-80kNm

=0.003* f_{yk}/ f_{ck}=**0.043**

Therefore @ base =0.69*Sqrt((1+(2*))*(1000*h*f_{ck})/( f_{t}*h*1000)) =**1.812** equal or greater than 1

Slenderness Ratio Limit 4.38*(1.7- M_{1z}/M_{2z})* =**21.429**

Condition 1 @ the top & 1 @ the bottom

Therefore coefficient C_{ef} = 0.75

Therefore L_{0z}=C_{ef}*H=**2400.000**mm

Slenderness Ratio S_{r}= L_{0z}/h=**12.000** Not Greater than Slenderness Ratio Limit 4.38*(1.7- M_{1z}/M_{2z}) =**21.429**

Therefore, Wall is not slender @ Mid Height.

Reinforcement Design

Using the column design charts to determine the amount of reinforcement in the wall

@ Mid height of the wall

d/h=**0.700**

M_{2z} /(b*h*h*f_{ck})=**0.057**

f_{t}/f_{ck}=**0.15**