Lecture 18 Shear Walls, Deep Beams and Corbels (b&w)

Department of Civil Engineering, University of Engineering and Technology Peshawar, Pakistan

Lecture-18

Shear Walls and Coupling beams By: Prof Dr. Qaisar Ali Civil Engineering Department UET Peshawar [email protected]

Prof. Dr. Qaisar Ali

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Advance Design of Reinforced Concrete Structures

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Topics Addressed  Shear Wall  Introduction  Behavior  ACI Recommendations  Design Examples


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Topics Addressed  Coupling Beam  Introduction  Behavior  ACI Recommendations  Design Examples


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SHEAR WALLS


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Introduction  Shear Walls 

The term “shear wall” is used to describe a wall that resists lateral (wind or earthquake) loads acting parallel to the plane of the wall in addition to the gravity loads from the floors and roof adjacent to the wall.



Such walls are also referred to as “structural walls”.



Non structural walls and partitions, whether directly considered or not also add to the total lateral stiffness of the structure.


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Introduction  Difference between Wall and Column 

The differentiation between columns and walls in the code is based on the principal use rather than on arbitrary relationships of height and crosssectional dimensions, ACI 318-02, Chapter 2 Definitions. 

While a wall always encloses or separates spaces, it may also be used to resist horizontal or vertical forces or bending.



A column is normally used as a main vertical member carrying axial loads combined with bending and shear. It may, however, form a small part of an enclosure or separation.



The code permits walls to be designed using the principles stated for column design .


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Introduction  Difference between Wall, Column and Pier 

For the sake of terminology, however, following difference is recognized by the code.



Column: Member with a ratio of height-to-least lateral dimension exceeding 3 used primarily to support axial compressive load.



Wall: Though not specifically mentioned in the code, members of heightto-least lateral dimension NOT exceeding 3 are considered as WALLS.



Pier: This is a wall segment and refers to a part of a wall bounded by openings or by an opening and an edge. 

Traditionally, a vertical wall segment bounded by two window openings has been referred to as a pier, ACI 318 -02, R21.7.4.2


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Introduction 

Other Definitions 

DIAPHRAGM is a horizontal or nearly horizontal system acting to transmit lateral forces to the vertical-resisting elements. The term “diaphragm” includes horizontal bracing systems.



DIAPHRAGM or SHEAR WALL CHORD is the boundary element of a diaphragm or shear wall that is assumed to take axial stresses analogous to the flanges of a beam.



BOUNDARY ELEMENT is an element at edges of openings or at perimeters of shear walls or diaphragms.



COLLECTOR is a member or element provided to transfer lateral forces from a portion of a structure to vertical elements of the lateral-forceresisting system.



STRUCTURAL DIAPHRAGMS are structural members, such as floor and roof slabs, which transmit inertial forces to lateral- force-resisting members.


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Introduction  Other Definitions 

1629.6.4 Moment-resisting frame system. A structural system with an essentially complete space frame providing support for gravity loads. Moment-resisting frames provide resistance to lateral load primarily by flexural action of members.



1629.6.5 Dual system. A structural system with the following features: 

1. An essentially complete space frame that provides support



for gravity loads.



2. Resistance to lateral load is provided by shear walls or braced frames and moment-resisting frames (SMRF, IMRF, MMRWF or steel OMRF). The moment-resisting frames shall be designed to independently resist at least 25 percent of the design base shear.



3. The two systems shall be designed to resist the total design base shear in proportion to their relative rigidities considering the interaction of the dual system at all levels.


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Introduction  Types of Shear Walls Shape Length to height ratio Seismic demand


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Introduction  Importance of Shear Walls 

Shear walls are extremely important in high-rise buildings. If unaided by walls, high rise frames could not be efficiently designed to satisfy strength requirements or to be within acceptable lateral drift limits.



Since frame buildings depend primarily on the rigidity of connections for their resistance to lateral loads, they tend to be uneconomical beyond a certain height range.





11 to 14 stories, in regions of high to moderate seismicity



15 to 20 stories, elsewhere.

Many times, however, shear walls are also provided in low rise (1 to 5) or medium rise frame buildings (6 to 10) in order to reduce sizes of columns.


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Introduction  Locations of Shear Walls 

It should be located such that the center of mass and center of rigidity of the structure coincide.



If there is eccentricity as illustrated in the fig, the building will undergo torsional distortions. Though the structure can be designed for such effects, it would be relatively uneconomical.

Center of mass

Center of resistance

Shear wall

Eccentricity Prof. Dr. Qaisar Ali

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Introduction  Locations of Shear Walls 

Most multi-story buildings are constructed with a central core area.



The core usually contains, among other things, elevator, plumbing and HVAC shafts etc.



Walls provided for such core can be used as Shear Walls.



Additional walls can be provided at other appropriate locations.


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Introduction  Frame-Wall Interaction 

In a RC frame structure, the floor systems (RC slabs) distribute the lateral loads to the vertical framing elements in proportion to their rigidities.



Though the actual distribution of lateral loads will depend on the relative rigidities of walls and columns, the structural walls usually being substantially stiffer than the columns attract major portion of the lateral loads, leaving only small portion for the frame members.



With adequate wall bracing, the frame can be considered as non-sway for column design.


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The analysis and design of the structural system for a building frame of moderate height can be simplified if the structural walls are sized to carry the entire lateral load.



Members of the frame (columns and beams or slabs) can be proportioned to resist the gravity loads only.



Neglecting frame-wall interaction for buildings of moderate size and height will result in reasonable member sizes and overall costs.



When the walls stiffness is much higher than the stiffness of the columns in a given direction within a story, the frame takes only a small portion of the lateral loads.



Thus, for low-rise buildings, neglecting the contribution of frame action in resisting lateral loads and assigning the total lateral load resistance to walls is an entirely reasonable assumption.


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In contrast, frame-wall interaction must be considered for high-rise structures where the walls have a significant effect on the frame: in the upper stories, the frame must resist more than 100 % of the story shears caused by the wind loads.



Thus, neglecting frame-wall interaction would not be conservative at these levels. Clearly, a more economical high-rise structure will be obtained when frame-wall interaction is considered.


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Behavior of Shear Walls  A typical shear wall, which is part of a lateral

load

resisting

system,

is

subjected to following actions. 

In-plane

shear

and

bending

moment

(along major axis) 

Out-of-plane shear and bending moment (along minor axis)



Axial Load


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Behavior of Shear Walls  In-plane

shear

and

bending

moment (along major axis)  In-plane shear 

A variable shear, which reaches a maximum at the base.



Both

horizontal

and

vertical

reinforcement are provided for shear.


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Behavior of Shear Walls  In-plane shear and bending moment (along major axis)  In-plane bending moment 

A variable bending moment which reaches a maximum at the base and tends to cause vertical tension near the loaded edge and compression at the far edge.



Vertical distributed reinforcement (fig a) or reinforcement at the edges in boundary zones (fig b) will be required against this action

Fig a

Fig b Prof. Dr. Qaisar Ali

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Behavior of Shear Walls  Out-of-plane shear and bending moment (along minor axis)  Out-of-plane bending moment 

Depending on a number of parameters, the wall may bend in an out-of-plane mode either 

as a whole from top to bottom called global bending or



as Individual wall segments in a story called local bending



In both cases vertical reinforcement distributed all along the length of the wall shall be provided.


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Behavior of Shear Walls  Out-of-plane shear and bending moment (along minor axis)  Out-of-plane shear 

Out-of-plane shear is not usually a problem in shear walls.


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ACI Code Recommendations  Types of Walls according to Seismic Hazard (Definitions, Chapter 21) 

Walls located in regions of low to moderate seismic hazard (zones 1, 2a and 2b UBC 97), shall comply with the requirements of ordinary reinforced concrete structural walls of the chapter 14 of ACI 318-02. 

There are no special requirements for structural walls located in regions of low to moderate seismic hazard, except for the connection requirements.



Walls located in regions of high seismic hazard (zones 3 and 4 of UBC 97), shall comply with the requirements of Special reinforced concrete structural wall of chapter 21 of the ACI 318-02,, in addition to the requirements for ordinary reinforced concrete structural walls.


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ACI Code Recommendations  Types of Walls according to Seismic Hazard (Definitions, Chapter 21) The provisions for the design of Ordinary reinforced concrete structural wall from chapter 14 will be presented first. Special provisions for Special reinforced concrete structural wall from chapter 21 will be presented next.


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ACI Code Recommendations  Ordinary reinforced concrete structural wall (Chapter 14) 

According to section 14.2.3, walls subjected to shear forces shall be designed in accordance with the provisions of chapter 11, section 11.10 on provisions of shear reinforcement for structural walls.



According to section 14.4, Walls subjected to flexure load, axial load or combined flexure and axial load shall be designed in accordance with the provisions for flexure and axial loads of chapter 10. (like column design)



Walls shall be properly anchored into all intersecting elements, such as floors, columns, other walls, and footings.


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Wall sizing 

A minimum of 6 in thickness will be required for a wall with a single layer of reinforcement and 10 in for a wall with double layer.(ACI 14.3.4)



Moreover, according to (ACI 318-89) the shear wall must have a total stiffness of at least six times the sum of stiffness of all columns in a given direction within the story I(walls) > 6I(columns)


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Shear: (Section 11.10, ACI 318-02) 

Shear Wall Capacity contributed by concrete alone is given as



ФVc =0.75 x 2 x √fc′ x h x d



where d = 0.8 lw

(ACI 11.10.4) ( ACI 11.10.4)

Vu

hw

lw Prof. Dr. Qaisar Ali

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h


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Shear: (Section 11.10, ACI 318-02) Minimum reinforcement for shear



Both horizontal and vertical shear reinforcement shall be provided as per following criterion.



 

Conditions

Horizontal Shear Reinforcement

Vertical Shear Reinforcement

Vu ≤ ФVc /2 (11.10.8)

ρh = 0.0020 for #5 and smaller ρh = 0.0025 for other bars (14.3)

ρn = 0.0012 for #5 and smaller with fy>60ksi ρn= 0.0015 for other bars (14.3)

ФVc/2 ≤ Vu ≤ ФVc (11.10.8)

ρh = 0.0025

ρn = 0.0025 (11.10.9.4)

Vu > ФVc (11.10.8)

s =0.75 x Av x fy x d / (Vu – ФVc)

(11.10.9.2)

ρn = 0.0025 +0.5(2.5-hw / lw )(ρh - 0.0025) (11.10.9.4)

ρh = ratio of horizontal shear reinforcement area to gross concrete area of vertical section ρn = ratio of vertical shear reinforcement area to gross concrete area of horizontal section


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Maximum Spacing of Shear reinforcement 

Horizontal Shear reinforcement 

Spacing of horizontal shear reinforcement shall not exceed 



Vertical Shear reinforcement 

Spacing of vertical shear reinforcement shall not exceed 


Iw/5 , 3h nor 18 inch, (whichever is less)

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Iw/3 , 3h nor 18 inch, (whichever is less )


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The ACI code additionally requires that 

Vu < Ф 10√f c h(0.8lw ) ( ACI 318-02,11.10.3)



Increase thickness of wall, if this happens


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Flexure (14.2, 14.3) 

Walls must be designed as compression members by the strength design provisions in Chapter 10 for flexure and axial loads.



Vertical reinforcement, however, need not be enclosed by lateral ties if vertical reinforcement area is not greater than 0.01 times gross concrete area.



Minimum ratio of vertical reinforcement area to gross concrete area shall be 

(a) 0.0012 for deformed bars not larger than No. 5 with a specified yield strength not less than 60,000 psi; or




(b) 0.0015 for other deformed bars

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Placement of Reinforcement (14.3.4) 

Walls more than 10 in. thick, except basement walls, shall have reinforcement for each direction placed in two layers parallel with faces of wall in accordance with the following: 

(a) One layer consisting of not less than one-half and not more than twothirds of total reinforcement required for each direction shall be placed not less than 2 in. nor more than one-third the thickness of wall from the exterior surface;



(b) The other layer, consisting of the balance of required reinforcement in that direction, shall be placed not less than 3/4 in. nor more than one-third the thickness of wall from the interior surface.


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Reinforcement around openings (14.3.7) 

In addition to the minimum reinforcement, not less than two No. 5 bars shall be provided around all window and door openings. Such bars shall be extended to develop the bar beyond the corners of the openings but not less than 24 in.


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Design of Ordinary Reinforced Concrete Structural Wall  General 

In the case of low-rise walls, shear requirements usually govern, so a preliminary thickness can be determined based on shear.



In high-rise structures, a preliminary wall thickness is not as obvious. In such structures, the wall thickness can vary a number of times over the height of the structure, and a thickness is usually determined from experience.



While fire resistance requirements will seldom govern wall thickness, the governing building code requirements should not be overlooked.


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The size of openings required for stairwells and elevators will usually dictate minimum wall plan layouts. Thus, the lengths of walls are usually dictated by architectural considerations.



Therefore, the first step in the design procedure is to determine a preliminary thickness of the wall.



From a practical standpoint, a minimum thickness of 6 inches will be required for a wall with a single layer of reinforcement, and 10 inches for a wall with a double layer.


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In low-rise walls,

which are typically governed by shear

requirements, it is common practice to determine the amount of vertical and horizontal reinforcement based on the shear provisions of Section 11.10. 

The flexural and axial force requirements of the appropriate design method are then checked based on the reinforcement for shear.



It is not uncommon for low-rise walls to have minimum amounts of reinforcement over their entire height.


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In the case of high-rise walls, wall sections at the base of the structure will usually, but not always, be governed by the requirements for flexure and axial load. Once the required amount of reinforcement is established for those requirements, the shear requirements of Section 11.10 are checked.



The amounts of reinforcement are typically varied over the height of high-rise walls.



In no case shall the provided areas of reinforcement be less than the minimum values prescribed in the code.


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Design of Ordinary Reinforced Concrete Structural Wall  Shear 

ФVn = ФVc + ФVs



ФVs =Vu – ФVc = 0.75 x Av x fy x d/ s



Therefore s = 0.75 x Av x fy x d /(Vu – ФVc)



“s” is center to center to center spacing of horizontal reinforcement in inches



Av is single bar area for one curtain and two times bar area for two curtains of reinforcement. Vu hw

h

lw Prof. Dr. Qaisar Ali

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Design of Ordinary Reinforced Concrete Structural Wall  Shear 

Horizontal and vertical shear reinforcement Ash & Asv from minimum reinforcement ratio “ρ” can be calculated as follows



Ash or Asv = (inch2 per foot ) = ρ x 12 x h ; h is thickness of wall



Spacing “s” (inch c/c) = (Av /Ash ) x 12



s = Av /(ρ x 12 x h ) x 12 (substituting Ash )



s = Av /(ρ x h )



ρ = Av /(s x h )

hw

h h lw


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Design of Ordinary Reinforced Concrete Structural Wall  Flexure 

In general, when designing a wall as a compression member, an interaction diagram needs to be constructed for sections subjected to combined flexure and axial load, and the applied factored moments must be magnified to account for slenderness effects.



Details on how to construct such a diagram have been discussed earlier.


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Approximate procedure for design of In-plane bending 

For buildings of moderate height, walls with uniform cross-sections and uniformly distributed vertical and horizontal reinforcement are usually the most economical.



Concentration of the reinforcement at the extreme ends of a wall or small segment (boundary zones) is usually not required except in high and moderate seismic zones (special walls).



Uniform distribution of the vertical wall reinforcement required for shear wall usually provides adequate moment strength as well.



Minimum amounts of reinforcement will usually be sufficient for both shear and moment requirements.


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Approximate procedure for design of In-plane bending 

In general, walls that are subjected to axial load or combined axial and flexure load need to be designed as compression members according to the provisions given in ACI Chapter 10.



For rectangular shear walls containing uniformly distributed vertical reinforcement and subjected to an axial load smaller than that producing balance failure, the following approximate equation can be used to determine the nominal moment capacity of the wall. ( Cardens A.E et. al, Design Provisions for Shear walls,” Journal of the ACI, Vol 70, No. 3 March 1973, pp 221-230)


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•

•

Approximate procedure for design of In-plane bending

Where

0.5

1

1

= total area of vertical reinforcement, in.2 = horizontal length of wall, in. = factored axial compressive load, kips = yield strength of reinforcement = 60 ksi


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Out-of-plane bending 

As wall is mostly slender along its minor axis, moment magnification shall be done before wall is designed for out-of-plane bending



Once moment is magnified, wall shall be designed for this moment either using interaction diagram or approximate procedure.


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Moment Magnification


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Moment Magnification 

Cracked moment of inertia, Icr

Es Icr =


Ec

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As +

Pu fy

(d –

c)2

+

 w c3 3


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out-of-plan deflection requirements

s =

5Mc2 48EcIe

1–

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c 150

Msa

M=




5Psc2 48EcIe


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ACI Provisions for Special reinforced concrete structural walls


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Provisions for Special Boundary Elements 

The minimum reinforcement ratio for both the longitudinal and transverse reinforcement is 0.0025, unless the design shear force does not exceed Acv fc , where Acv is the net area of concrete bounded by the web thickness and the length of the wall in the direction of analysis; in this case, the minimum reinforcement must not be less than that given in 14.3. The reinforcement provided for shear strength must be continuous and distributed uniformly across the shear plane with a maximum spacing of 18 in. At least two curtains of reinforcement are required if the in-plane factored shear force assigned to the wall exceeds Acv fc



n = ratio of area of distributed reinforcement parallel to the plane of Acv to gross concrete area perpendicular to that reinforcement.(horizontal, denoted by h in chapter 14)



v = ratio of area of distributed reinforcement perpendicular to the plane of Acv to gross concrete area Acv.(vertical, denoted by n in chapter 14)



Acp = area of concrete section, resisting shear, of an individual pier or horizontal wall segment, in.2



Acv = gross area of concrete section bounded by web thickness and length of section in the direction of shear force considered, in.2


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Provisions for Special Boundary Elements 21.7.2.1:1. ordinary

If Vu ≤ Acv

′, provide minimum reinforcement as given for

reinforced structural walls, where Acv is the net area of concrete

bounded by the web thickness and the length of the wall in the direction of analysis Acv = h × lw

h

h

lw

’


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Provisions for Special Boundary Elements 21.7.2.2: 2. If Vu> Acv ′ , both the longitudinal (v)and transverse reinforcement (n ) must not be less than 0.0025 21.7.2.2: 3. If Vu>2 Acv both directions.

′ ,

Two curtains of reinforcement are required in

21.7.2.3:4. Anchoring or splicing of reinforcement as per 21.5.4 21.7.4: Shear Strength. Vn = Acv (c

′ + n fy)

c = 3 (for hw/lw ≤ 1.5) &c= 2 (for hw/lw ≥ 2.0) & varies linearly for other values.

(c = 2.0 conservatively)

21.7.4.3: Walls shall have distributed reinforcement providing resistance in two orthogonal directions in the plane of the wall. If the ratio hw/lw does not exceed 2.0 then reinforcement v shall not be less than n Prof. Dr. Qaisar Ali

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Provisions for Special Boundary Elements R 21.7.4: The ratio hw/lw may refer to overall dimensions of a wall, or of a segment of the wall bounded by two openings or an opening and an edge. To restrain the inclined cracks effectively, reinforcement included in n and v should be appropriately distributed along the length and height of the wall. Chord reinforcement provided near wall edges in concentrated amounts for resisting bending moments is not to be included in determining n and V. 21.7.5:

Design of flexure and axial loads.

21.7.5.1: 21.7.5.2:


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21.7.6 Boundary Elements of Special Reinforced Concrete Structural Walls 21.6.3. Compression zones shall includes special boundary elements where the maximum extreme fiber stress corresponding to the factored forces, including earthquake effects, exceeds 0.2 fc’ (see Fig. 29-21).


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21.7.6 Boundary Elements of Special Reinforced Concrete Structural Walls

Fig 29-21: Special Boundary Element Requirements per 21.7.6.3


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21.7.6 Boundary Elements of Special Reinforced Concrete Structural Walls When special boundary elements are required, they must extend horizontally from the extreme compression fiber a distance not less than the larger of c – 0.1 lw and c/2 (21.7.6.4(a); see Fig. 29-20). In the vertical direction, the special boundary elements must extend from the critical section a distance greater than or equal to the larger of lw or Mu/4Vu (21.7.6.2). This distance is based on upper bound estimates of plastic hinge lengths, and is beyond the zero over which concrete spalling is likely to occur. From earlier codes, it is 0.15 to 0.25 lw See chapter 6 simplifired approach.


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Fig 29-20: Special Boundary Element Requirements per 21.7.6.2


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21.7.6 Boundary Elements of Special Reinforced Concrete Structural Walls Section 21.7.6.4 contains the details of he reinforcement when special boundary elements are required by 21.7.6.2 or 21.7.6.3. The transverse reinforcement must satisfy the same requirements as per special moment frame members subjected to bending and axial load (21.4.4.1 through 21.4.4.3), excluding Eq. (21-3) (21.7.6.4(c); see Fig. 29-22). Also, the transverse reinforcement shall extend in the support a distance not less than the development length of the largest longitudinal bar in the special boundary element; for footing or mats, the transverse reinforcement shall extend at least 12 in. into the footing or mat (21.7.6.4(d)). Horizontal reinforcement in the wall web shall be anchored within the confined core of the boundary element within the confined core of the boundary element to develop its specified yield strength (21.7.6.4(c)). To achieve this anchorage, 90-deg hooks or mechanical anchorages are recommended. Mechanical splices and welded splices of the longitudinal reinforcement in the boundary elements shall conform to 21.2.6 and 21.2.7, respectively (21.7.6.4(d) )


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21.7.6 Boundary Elements of Special Reinforced Concrete Structural Walls when special boundary elements are not required, the provisions of 21.7.6.5must be satisfied. For the cases when the longitudinal reinforcement ratio at the wall boundary is greater than 400/fy, transverse reinforcement, spaced not more than 8 in. on center, shall be provided that satisfies 21.4.4.1(c), 21.4.4.3, and 21.7.6.4(c)(21.7.6.5(a)). This requirement helps in preventing bucking of the longitudinal reinforcement that can be caused by cyclic load reversals. The longitudinal reinforcement ratio to be used includes only the reinforcement at the end of the wall as indicated in Fig. R21.7.6.5. Horizontal reinforcement terminating at the edges of structural walls must be properly anchored per 21.7.6.5(b)in order for the reinforcement to be effective in resisting shear and to help in preventing buckling of the vertical edge reinforcement. The provisions of 21.7.6.5(b)are not required to be satisfied when the factored shear force Vu ′ is less than Acv


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Fig 29-22: Reinforcement Details for Special Boundary Elements


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21.7.7 Coupling Beams When adequately proportioned and detailed, coupling beams between structural wall can provide an efficient means of energy dissipation under seismic forces, and can provide a higher degree of overall stiffness to the structure. Due to their relatively large depth to clear span ratio, ends of coupling beams are usually subjected to large inelastic rotations. Adequate detailing and shear reinforcement are necessary to prevent shear failure and to ensure ductility and energy dissipation. coupling beams with ln/h ≥ 4 shall satisfy the requirement of 21.3for flexure members of special moment frames, excluding 21.3.1.3and 21.3.1.4(a)if it can be shown that the beam has adequate lateral stability (21.7.7.1). When ln/h < 4, coupling beams with two intersecting groups of diagonally-placed bars symmetrical about the midspan is permitted (21.7.7.2). The diagonal bars are required for deep coupling beams (ln/h < 2) with a factored shear force Vu greater than 4 ′ Acp, unless it can be shown otherwise that safety and stability are not compromised (21.7.7.3). Experiments have shown that diagonally oriented reinforcement is effective only if the bars can be placed at large inclination. Prof. Dr. Qaisar Ali

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21.7.7 Coupling Beams Note that in 2002 code, h replaces d in the definition of the aspect ratio (clear span/depth) and Acp replaces bwd in the shear equations. The first change simplifies the code requirements, since d is not always readily known for beams with multiple layers of reinforcement. The second change removes an inconsistency between 21.6.4.5 and 21.6.7.4 of the 1999 code; Acp is now consistently used in 21.7.4.5and 21.7.7.4. Section 21.7.7.4 contains the reinforcement details for the two intersecting groups of diagonally placed bars. Figure 29-23 provides a summary of these requirements. The requirement on side dimensions of the cage and its core is to provide adequate toughness and stability when the bars are stressed beyond yielding. The nominal shear strength of a coupling beam is computed from the following (21.7.7.4(b)): Vn = 2Avdfysin ≤ 10


′ AcpEq. (21-9)

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21.7.7 Coupling Beams The additional reinforcement specified in 21.7.7.4(f) is used to confine the concrete outside of the diagonal cores.

Fig 29-23: Coupling Beam with Diagonally Oriented Reinforcement


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References  ACI 318-02  Design of Concrete Structures (13th Ed.) by Nilson, Darwin and Dolan  PCA Notes on ACI 318-02


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The End


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Design of Shear Wall  Ordinary reinforced concrete structural wall 

Flexure 

Wall design is further complicated by the fact that slenderness is a consideration in practically all cases of out-of-plane bending.



The approximate evaluation of slenderness effects prescribed in Section 10.11 may be used


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Ash Asv


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Lecture 18 Shear Walls, Deep Beams and Corbels (b&w)

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