3.6 shear walls · a single cantilever shear wall, such as the one s hown in figure 3.6-7, can be...

Sabah Shawkat Cabinet of Structural Engineering 2017

3.6 Shear walls

Walls carrying vertical loads should be designed as columns. Basically walls are designed in

the same manner as columns, but there are a few differences. A wall is distinguished from a

column by having a length that is more than five times the thickness.

Plain concrete walls should have a minimum thickness of 120 mm. Where the load on the wall

is eccentric, the wall must have centrally placed reinforcement of at least 0.2 percent of the

cross-section area if the eccentricity ratio exceeds 0.20. This reinforcement may not be included

in the load-carrying capacity of the wall.

Shear walls should be designed as vertical cantilevers, and the reinforcement arrangement should be

checked as for a beam. Where the shear walls have returns at the compression end, they should be treated

as flanged beams.

If the walls contains openings, the assumption for beams that plane sections remain

plane is no longer valid. Shear walls connected by beams or floor slabs. The stability of shear-

wall structure is often provided by several walls connected together by beams or floors.

Where the walls are of uniform section throughout the height and are connected by regularly

spaced uniform beams. Many shear walls contain one or more rows of openings.

Figure 3.6-1: Building with shear walls

When walls are used to brace a framed structure, it may be acceptable to disregard the lateral

stiffness of the frame and assume the horizontal load carried entirely by the walls.

The equilibrium and compatibility equations at each level produces a set of simultaneous

equations which are solved to give the lateral deflection and rotation at each level.


Figure 3.6-2: Shear walls subjected to bending moment and vertical load

If a tall building has an asymmetrical structural plan and is subjected to horizontal loading,

torsional as well as bending displacements will occur, and hence a full three-dimensional

analysis is required. In many tall building shear wall provide most, if not all, of the required strength

for lateral loading resulting from gravity, wind, and earthquake effects.

Figure 3.6-3


The system (Hull - Core Structures) has been used for very tall buildings in both steel and

concrete. Lateral loads are resisted by both the hull and the core, their mode of interaction

depending on the design of the floor system.

Figure 3.6-4: Shear walls subjected to horizontal load and vertical load

A floor slabs of multi-story buildings, when effectively connected to the wall, acting as

stiffeners, provide adequate lateral strength. As essential prerequisites, adequate foundations

and sufficient connection to all floors, to transmit horizontal loads, must be assured.

Figure 3.6-5: side view of shearing wall shows the thickness of bearing wall in accordance

with boundary conditions of the members

Normally, for wind loading, the governing design criterion or limit state will be

deflection. Shear walls, when carefully designed and detailed, hold the promise of giving the


greatest degree of protection against non-structural damage in moderate earthquakes, while

assuring survival in case of catastrophic seismic disturbances, on account of their ductility.

Yielding of the flexural bars will also affect the width of diagonal cracks. The shear

strength of tall shear walls may also be controlled by combined moment and shear failure at the

base of the wall. Door and service openings in shear walls introduce weaknesses that are not

confined merely to the consequential reduction in cross-section. Stress concentrations are

developed at the corners, and adequate reinforcement needs to be provided to cater for these

concentrations.

This reinforcement should take the form of diagonal bars positioned at the corners of the

openings. The reinforcement will generally be adequate if it is designed to resist a tensile force

equal to twice the shear force in the vertical components of the wall as shown, but should not

be less than two 16mm diameter bars across each corner of the opening.

Figure 3.6-6: Diagonal reinforcement in coupling beams, beam cross-section and possible

mechanisms involving openings

A single cantilever shear wall, such as the one shown in figure 3.6-7, can be expected to behave

in the same way as a reinforced concrete beam. The shear walls will be subjected to bending

moments and shear forces originating from lateral loads, and to axial compression induced by

gravity.

At the base of the wall, where yielding of the flexural reinforcement in both faces of the

section can occur, the contribution of the concrete towards shear strength should be disregarded

where the axial compression on the gross section is less than 12% of the cylinder crushing


strength of the concrete. Sectional area of the concrete and should be equally divided between

the two faces of the wall. The maximum area of vertical reinforcement should not exceed 4%

of the gross cross-sectional area of the concrete. Horizontal reinforcement equal to not less than

half the area of vertical reinforcement should be provided between the vertical reinforcement

and the wall surface on both faces. The spacing of the vertical bars should not exceed the lesser

of 300mm or twice the wall thickness. The spacing of horizontal bars should not exceed 300mm

and the diameter should not be less than one-quarter of the vertical bars.

Figure 3.6-7: Geometry and reinforcement of typical shear wall

The prime function of the vertical reinforcement, passing across a construction joint, is to

supply the necessary clamping force and to enable friction forces to be transferred.

Figure 3.6-8: Geometry and reinforcement of shear wall in tall building


Figure 3.6-9: Precast reinforced concrete walls

Figure 3.6-10: Shear subjected to lateral load

Figure 3.6-11: Shear walls with flexible coupling beams


Figure 3.6-12


Figure 3.6-13


Calculation of sectional forces and moments of structures

Example 3.6-1: Reinforced concrete wall subjected to horizontal load Ho or Wo

Construction height H = 27.5 m

Storey height l = 2.75 m

Sectional area of the first pillar 1 and A1 or 1 = 2 m2

Sectional area of the second pillar 2 and A2 or 2 = 1.6 m2

Moment of inertia of the first pillar I1 = 4 m4

Moment of inertia of the first pillar I2 = 2 m4

Moment of inertia of the cross-sectional area

Structures weakened openings I = 39 M4

Moment of inertia of girders IPR = 0.006 m4

Modulus of elasticity of pillars E = 10 GPa

Modulus of elasticity of girder E´ = 20 GPa

Static moment of sectional area

Walls weakened with openings S = 5.42 m3

Shear force applied in base Construction Ho = 354 kN

The distance between the center of gravity of the pillars 2c = 6.10 m

Width of the window type openings 2a = 2m


Figure 3.6.1-1: Shear walls contains openings

Figure 3.6.1-2: Geometry calculated reinforcing walls subjected to horizontal loading Ho

Data:

Determination the value of

E 10000 MPa E´ 20000 MPa 1 2 m2

I1 4 m4

2 1.6 m2

I2 2 m4

S 5.42 m3

I 39 m4

i 0.006 m4

l 2.75 m

c

1

1

1

2

S

2 c 3.049m

Z 10 l Ho 354 kN a 1 mHo l S

I135.292kN

3 E´ i

E I1 I2

I

S

c

a3

l

20.048m

2 0.219m

1

Z 6.016

6.0162

T ( )d

d

2

2T ( )

2 1 ( ) T 0( ) 0 T' 1( ) 0 Odesolve 1( )


Diagram () vs

where the value of the function can read from the graf, based on the coefficient

for paying the relationship H

Diagram () vs

Solving the values of K and J

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.06

0.12

0.18

0.24

0.3

0.36

0.42

0.48

0.54

0.6

( )

( )

1

( )

d 0 0.01 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.04

0.08

0.12

0.16

0.2

0.24

0.28

0.32

0.36

0.4

( )

v1 30tonne

l v2 20

tonne

l e1 0.5 m e2 1 m

KS

Iv2 e2

I1 I2

2 c

1

2

v1 e1

I1 I2

2 c

1

1

K 1.893 103

m1

kg


Determination the values of M1 and M2

JS

Iv2 e2

I1 I2

2 c

1

2

v1 e1

I1 I2

2 c

1

1

J 1.893 103

m1

kg

i 2 12 j 1 11 1

0 i

i 1 0.1 ( )

Ho l S

I ( )

M1 ( )I1

I1 I2Ho Z

1 ( )2

22 c

S

I ( )

M2 ( )I2

I1 I2Ho Z

1 ( )2

22 c

S

I ( )

j 22

0-35·10

0.02

0.045

0.08

0.125

0.18

0.245

0.32

0.405

0.5

1 j

21.828

25.068

32.611

42.226

52.451

62.055

69.575

72.767

67.778

47.701

-147.526·10

kN

1 j

0

0.017

0.038

0.066

0.101

0.143

0.192

0.245

0.297

0.341

0.361

1 ( )j

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

1 j

0.161

0.185

0.241

0.312

0.388

0.459

0.514

0.538

0.501

0.353

0


External load w acting on the structure induces in the individual pillars bending moments.

2 cS

I 1

j

0

0.014

0.032

0.056

0.085

0.121

0.163

0.208

0.252

0.289

0.306

j 22

2 cS

I 1

j

0-3-9.383·10

-0.012

-0.011

-3-5.313·10

-33.761·10

0.017

0.037

0.068

0.116

0.194

q j

21.828

25.068

32.611

42.226

52.451

62.055

69.575

72.767

67.778

47.701

-147.526·10

kN

1 ( )2

2

2 c m

I ( )

Ho l S

I ( ) q ( ) 1 ( )( )

2 c m

I

0 0.2 0.4 0.6 0.8 15 10

5

0

5 105

1 106

1.5 106

M 1 j

j

0 0.2 0.4 0.6 0.8 12 10

5

0

2 105

4 105

6 105

8 105

M 2 j

j

M1 1 j

0

-60.893

-79.765

-69.196

-34.479

24.407

112.802

243.358

440.758

750.922

31.258·10

kN m

M2 1 j

0

-30.447

-39.883

-34.598

-17.239

12.204

56.401

121.679

220.379

375.461

629.068

kN m


Example 3.6-2: Solution of reinforcing concrete walls with openings subjected to vertical load

The reinforcing walls, as mentioned in the introduction, other than the horizontal load and the

vertical load are transmitted as well.

This chapter is about solving the stiffening of reinforced walls in terms of a vertical load defined

the basic assumptions that in dealing with all three types of reinforcing walls.

L - floor height

H - total height of the wall

A1A2 - cross-sectional area of each pillar

2c - distance between pillars

2 - width of openings

N - normal force acting in the pillar

shear force applied in the girders

E - modulus of elasticity of the walls

E'- modulus of girders

V1 - vertical loads on pillar 1 at level each floor

v2 - vertical loads on the Pillar 2 at the level of each floor

E1 - eccentricity at which the load acts v1

e2 - eccentricity at which the load applied v2


Figure: 3.6.2-1

Data:

Diagram () vs

E 10000 MPa E´ 20000 MPa 1 2 m2

I1 4 m4

2 1.6 m2

I2 2 m4

S 5.42 m3

I 39 m4

i 0.006 m4

l 2.75 m Z 10 l Ho 354 kN a 1 m

c

1

1

1

2

S

2 c 3.049m v1 300 kN v2 200 kN e1 0.5 m e2 1 m

Ho l S

I135.292kN K

S

Iv2 e2

I1 I2

2 c

1

2

v1 e1

I1 I2

2 c

1

1

3 E´ i

E I1 I2

I

S

c

a3

l

20.048m

2 K 52.06 kN 0.219m

1

Z 6.016 i 2 12 j 1 11 1

0 i

i 1 0.1

( ) K ( )

6.0162

T ( )d

d

2

2T ( )

2 1 ( ) T 0( ) 0 T' 1( ) 0

Odesolve 1( )


Figure: 3.6.2-2

Diagram () vs

Figure: 3.6.2-3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.06

0.12

0.18

0.24

0.3

0.36

0.42

0.48

0.54

0.6

( )

( )

1

( )

d

0 0.01 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.04

0.08

0.12

0.16

0.2

0.24

0.28

0.32

0.36

0.4

( )

M1 ( )I1

I1 I2

Z

l 1 ( ) v1 e1 v2 e2 2 c K ( )

M2 ( )I2

I1 I2

Z

l 1 ( ) v1 e1 v2 e2 2 c K ( )


Diagram () vs

Figure: 3.6.2-4

Diagram () vs

N1 ( )Z

lv1 1 ( ) K ( )

N2 ( )Z

lv2 1 ( ) K ( )

( )

1

( )

d

0 0.01 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.09

0.18

0.27

0.36

0.45

0.54

0.63

0.72

0.81

0.9

( )

K 12

T ( )d

d

2

2

T ( ) K T 0( ) 0 T' 1( ) 0 Odesolve 1( )


Figure: 3.6.2-5

Figure 3.6.2-6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.50.6

0.7

0.8

0.9

1

( )

0.4( ) 0.91 0.5( ) 0.95

1 j

0

-0.099

-0.199

-0.298

-0.396

-0.492

-0.585

-0.673

-0.75

-0.809

-0.834

1 j

39.606

39.57

39.447

39.194

38.714

37.829

36.21

33.251

27.85

17.992

-142.654·10

kN

1 ( )j

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

1 j

-0.995

-0.994

-0.991

-0.985

-0.973

-0.95

-0.91

-0.835

-0.7

-0.452

0

0 0.2 0.4 0.6 0.8 10

2 105

4 105

6 105

8 105

1 106

M1 j

j

0 0.2 0.4 0.6 0.8 10

1 105

2 105

3 105

4 105

5 105

M2 j

j


Diagram vs

M1 1 j

0

72.384

145.081

218.519

293.404

370.982

453.501

545.055

653.105

791.268

984.388

kN m

M2 1 j

0

36.192

72.54

109.259

146.702

185.491

226.75

272.527

326.553

395.634

492.194

kN m

N1 1 j

0

260.406

520.889

781.554

31.043·10

31.304·10

31.567·10

31.832·10

32.101·10

32.378·10

32.668·10

kN

N2 1 j

0

239.594

479.111

718.446

957.424

31.196·10

31.433·10

31.668·10

31.899·10

32.122·10

32.332·10

kN


Figure: 3.6.2-7


Diagram vs

Figure: 3.6.2-8

Diagram vs


Figure: 3.6.2-9

Diagram vs


Figure: 3.6.2-10

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,1 0,2 0,3 0,4

3.6 shear walls · a single cantilever shear wall, such as the one s hown in figure 3.6-7, can be...

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