Sabah Shawkat Cabinet of Structural Engineering 2017 3.2 Reinforced Concrete Slabs Slabs are divided into suspended slabs. Suspended slabs may be divided into two groups: (1) slabs supported on edges of beams and walls (2) slabs supported directly on columns without beams and known as flat slabs. Supported slabs may be one-way slabs (slabs supported on two sides and with main reinforcement in one direction only) and two-way slabs (slabs supported on four sides and reinforced in two directions). In one-way slabs the main reinforcement is provided along the shorter span. In order to distribute the load, a distribution steel is necessary and it is placed on the longer side. One- way slabs generally consist of a series of shallow beams of unit width and depth equal to the slab thickness, placed side by side. Such simple slabs can be supported on brick walls and can be supported on reinforced concrete beams in which case laced bars are used to connect slabs to beams. Figure 3.2-1: One –way slab, two-way slab, ribbed slab, flat slab, solid flat slab with drop panel, waffle slab In R.C. Building construction, every floor generally has a beam/slab arrangement and consists of fixed or continuous one-way slabs supported by main and secondary beams.
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3.2 Reinforced Concrete Slabs - vsvu.sk · Figure 3.2-1: One –way slab, two-way slab, ribbed slab, flat slab, solid flat slab with drop panel, waffle slab In R.C. Building construction,
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Sabah Shawkat Cabinet of Structural Engineering 2017
3.2 Reinforced Concrete Slabs
Slabs are divided into suspended slabs. Suspended slabs may be divided into two
groups:
(1) slabs supported on edges of beams and walls
(2) slabs supported directly on columns without beams and known as flat slabs. Supported
slabs may be one-way slabs (slabs supported on two sides and with main reinforcement in one
direction only) and two-way slabs (slabs supported on four sides and reinforced in two
directions). In one-way slabs the main reinforcement is provided along the shorter span. In order
to distribute the load, a distribution steel is necessary and it is placed on the longer side. One-
way slabs generally consist of a series of shallow beams of unit width and depth equal to the
slab thickness, placed side by side. Such simple slabs can be supported on brick walls and can
be supported on reinforced concrete beams in which case laced bars are used to connect slabs
to beams.
Figure 3.2-1: One –way slab, two-way slab, ribbed slab, flat slab, solid flat slab with drop
panel, waffle slab
In R.C. Building construction, every floor generally has a beam/slab arrangement and
consists of fixed or continuous one-way slabs supported by main and secondary beams.
Sabah Shawkat Cabinet of Structural Engineering 2017
Figure 3.2-2: Solid flat slab, solid flat slab with drop panels
The usual arrangement of a slab and beam floor consists of slabs supported on cross-
beams or secondary beams parallel to the longer side and with main reinforcement parallel to
the shorter side. The secondary beams in turn are supported on main beams or girders extending
from column to column. Part of the reinforcement in the continuous is bent up over the support,
or straight bars with bond lengths are placed over the support to give negative bending
moments.
Figure 3.2-3: Types of the reinforced concrete slab systems
Sabah Shawkat Cabinet of Structural Engineering 2017
3.2.1 Flat Slabs
Flat plate is defined as a two-way slab of uniform thickness supported by any
combination of columns, without any beams, drop panels, and column capitals. Flat plates are
most economical for spans from 4,5 to 7,5m, for relatively light loads, as experienced in
apartments or similar building.
-A flat slab is a reinforced concrete slab supported directly on and built monolithically with the
columns, the flat slab is divided into middle strips and column strips. The size of each strip is
defined using specific rules. The slab may be in uniform thickness supported on simple
columns. These flat slabs may be designed as continuous frames. However, they are normally
designed using an empirical method governed by specified coefficients for bending moments
and other requirements which include the following:
1. There should be not less than three rectangular bays in both longitudinal and transverse
directions.
2. The length of the adjacent bays should not vary by more than 10 %.
Figure 3.2.1-1: Post punching behaviour of slab- critical section
The general layout of the reinforcement is based on the both bending moments (in spans) and
bending moments in addition to direct loads (on columns).
Sabah Shawkat Cabinet of Structural Engineering 2017
Figure 3.2.1-2: Combined punching shear and transfer of moments
Figure 3.2.1-3
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Figure 3.2.1-4
Sabah Shawkat Cabinet of Structural Engineering 2017
3.2.1-1 Analysis and Design of Flat Plate
To obtain the load effects on the elements of the floor system and its supporting
members using an elastic analysis, the structure may be considered as a series of equivalent
plane frames, each consisting of vertical members – columns, horizontal members - slab.
Such plane frames must be taken both longitudinally (in x-direction) and transversely (in y
direction) in the building, to assure load transfer in both directions.
For gravity load effects, these equivalent plane frames can be further simplified into
continuous beams or partial frames consisting of each floor may be analysed separately together
with the columns immediately above and below, the columns being assumed fixed at their far
ends. Such a procedure is described in the “Equivalent Frame Method”. When frame geometry
and loadings meet certain limitations, the positive and negative factored moments at critical
sections of the slab may be calculated using moment coefficients, termed “Direct Design
Method”. These two methods differ essentially in the manner of determining the longitudinal
distribution of bending moments in the horizontal member between the negative and positive
moment sections. However, the procedure for the lateral distribution of the moments is the same
for both design methods.
Figure 3.2.1.1-1: Steel shear –heads, steel plats joined by welding
Sabah Shawkat Cabinet of Structural Engineering 2017
Since the outer portions of horizontal members (slab) are less stiff than the part along the
support lines, the lateral distribution of the moment along the width of the member is not
Uniform. The procedure generally adopted is to divide the slab into column strips (along the
column lines) and middle strips and then apportion the moment between these strips and the
distribution of the moment within the width of each strip being assumed uniform.
Figure 3.2.1.1-2: Moments and frames
Sabah Shawkat Cabinet of Structural Engineering 2017
Figure: 3.2.1.1-3
Example: 3.2-1 Design and calculation of Flat Plate
Geometric Shapes
Slab thickness
The geometry of the building floor plans:
Construction height of object:
Dimensions columns:
The peripheral dimensions of the beam:
Figure: 3.2.1-1
Load calculation
Load per area
Reinforced concrete slab thickness of 300 mm
hd 300 mm
l1 7.7 m lk 2.3 m l2 3.6 m ly 7.7 m
kv 2.850 m
bs 400 mm hs bs
ho 0.5 m bo 0.30 m
qdo hd 25kN
m3
1.35 qdo 10.125kN
m2
Sabah Shawkat Cabinet of Structural Engineering 2017
floor layer:
Live load (apartments):
Total load on 1 m 2 of slab:
Force load Peripheral masonry thickness of 400 mm YTONG:
Total load acting on the console:
Investigation replacement frame in the X-axis Frame 1:
Calculation model
Figure: 3.2.1-2
load calculation
q1d 3kN
m2
1.4 q 1d 4.2kN
m2
vd 2.0kN
m2
1.5 vd 3kN
m2
qd qdo q1d vd qd 17.325kN
m2
F1 10kN
m3
kv ly 400 mm 1.35 F1 118.503kN
F1d F1 F1d 118.503kN
Sabah Shawkat Cabinet of Structural Engineering 2017
Load width in a direction perpendicular to the x:
Load in the x-direction:
Calculation of internal forces
Moment on a console:
Moment of inertia:
Transverse replacement frame:
Central girders replacement of frame:
column:
Bending stiffness:
Transverse replacement frame:
Central girders replacement frame:
Column
zsx ly
qdx qd zsx qdx 133.403kN
m
Mk F1d lk qdxlk
2
2
Mk 625.407kN m
Iply hd
312
Ip 0.017m4
Ist Ip
Is1
12bs hs
3 Is 5.208 10 3 m4
KpIp
l11000
kN
rad m2
Kp 2.25 kNm
rad
KstIst
l2 1000kN
rad m2
Kst 4.813 kNm
rad
KsIs
kv1000
kN
rad m2
Ks 1.827 kNm
rad
Sabah Shawkat Cabinet of Structural Engineering 2017
GivenM97 kN m M97o Ks 3 9 rad M911 kN m M911o Ks 2 9 rad M910 kN m M910o Kp 2 9 rad 10 rad M108 kN m M108o Ks 3 10 rad M109 kN m M109o Kp 2 10 rad 9 rad M1012 kN m M1012o Ks 2 10 rad M1010' kN m M1010ò Kst 10 rad M10'10 kN m M10'10o Kst 10 rad
Mk M97 kN m M910 kN m M911 kN m 0 kN m
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The calculated moments of individual members of equilibrium conditions:
The computation of shear forces in the individual members:
Maximum moment between 9-10 Mmax
Maximum moment between 10-10 Mstr
M109 kN m M1012 kN m M1010' kN m M108 kN m 0 kN mv Find M97 M910 M911 M109 M1012 M1010' M108 9 10 M10'10
M910 v 1.0( ) kN m M910 691.408 kN mM10'10 v 9 0( ) kN m M10'10 282.659kN mM1010' v 5.0( ) kN m M1010' 282.659 kN mM911 v 2.0( ) kN m M911 26.401 kN mM109 v 3.0( ) kN m M109 545.787kN m
V910o qdxl12
V109o V910o
V910 V910oM910 M109
l1
V910 532.511kN
V109 V109oM910 M109
l1
V109 494.688 kN
V1010ò qdxl14
V1010' V1010òV1010' 256.8 kN
v
0
01
2
3
4
5
6
7
8
9
39.601-691.408
26.401
545.787
-105.251
-282.659
-157.877
7.223
-28.797
282.659
a V910l1
V910 V109 a 3.992m
Mmax V910 a M910 qdxa
2
2 Mmax 371.423kN m
Mstr V1010'l14 M1010' qdx
l22
2
2 Mstr 4.431 kN m
Sabah Shawkat Cabinet of Structural Engineering 2017
Figure: 3.2.1-4
Transformation moments for the part columned strip and between the columns
Moments over support:
Positively support moments:
Ma M910 Mb M109 Ma 691.408 m kN Mb 545.787m kNMc' Mmax 1.25 Mc Mstr 1.25 Mc' 464.278kN m Mc 5.539 kN m
p 0.75M1a p Ma M1a 518.556 kN m M2a 1 p MaM2a 172.852 kN m M1b p Mb M1b 409.34 kN mM2b 1 p Mb M2b 136.447kN m
m 0.60M3c Mc' m M3c 278.567kN m M4c Mc' 1 m M4c 185.711kN m
Sabah Shawkat Cabinet of Structural Engineering 2017
Figure: 3.2.1-5
Dimensioning of the reinforcement:
Material characteristic of concrete f ckcyl and steel fyk
The top reinforcement for moments:
effective height:
width, which act the moment
Column strip M 1a:
fyd 375 MPa fcd 12 MPa
d hd 3 cm
bly2
b 3.85m
M1a 518.556 kN m M1a 0.518 MN m
d 0.27m fcd 12 MPa b 3.85m
M1a
b d2 fcd
0.0475 0.154
Ast b d fcd Ast b d fcd
MN100 cm
2 Ast 59.252cm2
Ast
3.8515.39 cm
2
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Among the columned strip M 2a:
Column strip M 1b:
Among the columned strip M 2b:
The lower reinforcement for moments:
M2a 172.852 kN m M2a 0.172 MN m
d 0.27m fcd 12 MPa b 3.85m
M2a
b d2 fcd
0.01439 0.051
Ast b d fcd
MN100 cm
2 Ast 17.95 cm2
Ast
3.854.662 cm
2
M1b 409.34 kN m M1b 0.409 MN m
d 0.27m fcd 12 MPa b 3.85m
M1b
b d2 fcd
0.03596 0.121
Ast b d fcd
MN100 cm
2 Ast 44.857cm2
Ast
3.8511.651 cm
2
M2b 136.447kN m M2b 0.136 MN m
fcd 12 MPa b 3.85m d 0.27m
M2b
b d2 fcd
0.01145 0.04
Ast b d fcd
MN100 cm
2 Ast 14.283cm2
Ast
3.853.71 cm
2
Sabah Shawkat Cabinet of Structural Engineering 2017
Column strip M 3c:
Among the columned strip M 4c:
Investigation replacement frame in y Frame 2
Calculation Model
M3c 278.567kN m M3c 0.278 MN m
b 3.85m d 0.27m fcd 12 MPa
M3c
b d2 fcd
0.0249 0.083
Ast b d fcd
MN100 cm
2 Ast 31.06cm2
Ast
3.858.068 cm
2
M4c 185.711kN m M4c 0.185 MN md 0.27m fcd 12 MPa b 3.85m
M4c
b d2 fcd
0.055 0.01588
Ast b d fcd
MN100 cm
2 Ast 19.809 cm2
Sabah Shawkat Cabinet of Structural Engineering 2017
Figure: 3.2.1-6
Load calculation
Calculation internal forces
Support part:
Among the supports:
a magnification between support:
Transformation moments for the part columned strip and among columned
support t of Ma2
Between the support of M c
q2d qd1
2 l1 l2 q2d 97.886
kN
m
Ma1
12
q2d ly
2 Ma 483.64 kN m
Mc'1
16q2d ly
2 Mc' 362.73 kN m
Mc Mc' 1.25 Mc 453.412kN m
p 0.75 Ma1 p Ma Ma1 362.73 kN m
Ma2 1 p Ma Ma2 120.91 kN m
m 0.6 Mc1 m Mc Mc1 272.047kN m
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Dimensioning of reinforcement
Upper reinforcement of moment:
Effective depth:
Column strip M 1a:
The width on which acting the moment:
Column strip M 1a:
Between the column strip M 2a:
Mc2 1 m Mc Mc2 181.365kN m
d hd 3 cm
bl14
l24
b 2.825m
M1a 0.518 MN m d 0.27m fcd 12 MPa b 2.825m
M1a
b d2 fcd
0.06687 0.21
Ast b d fcd
MN100 cm
2 Ast 61.206cm2
M2a 0.172MN m d 0.27m fcd 12 MPa b 2.825m
M2a
b d2 fcd
0.02037 0.07
Ast b d fcd
MN100 cm
2 Ast 18.645cm2
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Column strip M 1a:
Between column strip M 2a:
Investigation extreme frame replacement
Calculation Model:
Mc1 0.272MN m d 0.27m fcd 12 MPa b 2.825m
Mc1
b d2 fcd
0.03277 0.11
Ast b d fcd
MN100 cm
2 Ast 29.994cm2
Mc2 0.181MN m d 0.27m fcd 12 MPa b 2.825m
Mc2
b d2 fcd
0.0216 0.073
Ast b d fcd
MN100 cm
2 Ast 19.77cm2
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Figure: 3.2.1-7
Calculation of load:
From the slab:
Peripheral masonry thickness of 400 mm YTONG:
Total load replacement frame:
Calculation of internal forces
Moment of the end strip:
q3do qd lkl12
q3do 106.549
kN
m
F1 10kN
m3
kv 400 mm 1.4 F1 15.96kN
m
qkd q3do F1 q kd 122.509kN
m
Sabah Shawkat Cabinet of Structural Engineering 2017
Support bending moment:
Between the column bending moment:
Transformation moments for the part columned bands and among columned
columned strip width:
Moments over support:
Between the column moments:
Design the reinforcement to the reinforced concrete slab
The top reinforcement for moments:
Mka1
12 qkd ly
2 Mka 605.295 kN m
Mkc1
16qkd ly
2 Mkc 453.971kN m
bp3 lkl12
bp3 6.15m
MextaMka
41 2
lk
bp3
Mexta 264.509 kN m
Minta Mka Mexta Mk4a p Minta Mk4a 255.59 kN m
Mk3a 1 p Minta Mk3a 85.197 kN m
MextcMkc
41 2
lk
bp3
Mextc 198.382kN m
Mintc Mkc Mextc Mk4c m Mintc Mk4c 153.354kN m
Mk3c 1 m Mintc Mk3c 102.236kN m
Sabah Shawkat Cabinet of Structural Engineering 2017
effective height:
Column extreme strip Mexta:
width which act moment
Column extreme strip Mexta: see diagram B3-B3.3
Column strip inside M k4a:
width, which acts moment, see diagram B3-B3.3
Among the columned strip M k3a:
width, which acts moment
d d 3 cm
b lk b 2.3m
Mexta 0.265 MN m d 0.24m fcd 12 MPa b 2.3m
Mexta
b d2 fcd
0.0509 0.166
Ast b d fcd
MN100 cm
2 Ast 33.716cm2
bl14
b 1.925m
Mk4a 0.256 MN m d 0.24m fcd 12 MPa b 1.925m
Mk4a
b d2 fcd
0.05965 0.192
Ast b d fcd
MN100 cm
2 Ast 33.07cm2
Ast
1.92517.179 cm
2
Sabah Shawkat Cabinet of Structural Engineering 2017
The lower reinforcement for moments:
Column extreme strip Mextc:
width, which acts moment, see diagram B3-B3.3
Column strip inside M k4c:
width, which acts moment
bl14
b 1.925m
Mk3a 0.085 MN m d 0.24m fcd 12 MPa b 1.925m
Mk3a
b d2 fcd
0.01739 0.064
Ast b d fcd
MN100 cm
2 Ast 9.641cm2
Ast
1.9255.008 cm
2
b lk b 2.3m
Mextc 0.198MN m d 0.24m fcd 12 MPa b 2.3m
Mextc
b d2 fcd
0.03758 0.125
Ast b d fcd
MN100 cm
2 Ast 24.893cm2
Ast
2.310.823 cm
2
bl14
b 1.925m
Mk4c 0.153MN m d 0.24m fcd 12 MPa b 1.925m
Sabah Shawkat Cabinet of Structural Engineering 2017
Among the columned strip M k3c:
width, which acts moment
See diagram B3-B3.3
Mk4c
b d2 fcd
0.03436 0.115
Ast b d fcd
MN100 cm
2 Ast 19.049cm2
Ast
1.9259.896 cm
2
bl14
b 1.925m
Mk3c 0.102MN m d 0.24m fcd 12 MPa b 1.925m
Mk3c
b d2 fcd
0.02189 0.077
Ast b d fcd
MN100 cm
2 Ast 12.136cm2
Ast
1.9256.304 cm
2
Sabah Shawkat Cabinet of Structural Engineering 2017
Example 3.2-2: In the example we are considering reinforced concrete slab flat, floor slab
thickness is hd = 0.3m, Column diameter (round column) d =0.50 m, the maximum force applied
one column at Nd= 1800 kN.
Material characteristics:
Figure: 2.3.2‐1
Coefficient of shear strength
in both directions
On 1m plate
d 0.5 m hd 0.3 m b 1 m Nd 1800 kN
fcd 17 MPa fctm 1.2 MPa fyd 375 MPa
1 18 mm As1 1
2
4 n 5 Ast n As1 Ast 0.00127 m
2
stw
Ast
b hd stw 0.004241 m
stmin1
3
fctm
fyd
stmin 0.001067
b 1 s 1.159 n 1.0 f 1.25 b 1
h 1.42
3
hd
m h 1.2
g s n h f g 1.74
Sabah Shawkat Cabinet of Structural Engineering 2017
Carrying capacity of the concrete section
Assess the resistance of the concrete section
Maximum force per columns
Basic critical perimeter
Shear force on the critical perimeter
Shear resistance of concrete
We suggest shear reinforcement
Incorrect design, head to be designed so that they apply condition:
correct proposal
Proposal of hidden head
Maximum critical perimeter with hidden head
qbu 0.42 hd g b fctm qbu 262.86 m1
kN
Vcd Nd Vcd 1800 kN
ucr 2.51m
qd
Vcd
ucr
qd 716.2 m1
kN
qbu 262.86m1
kN qd 716.2m1
kN qd qbu
qd 2 qbu 2 qbu 525.72 m1
kN
qd 2 qbu
Ucrmax 1.9 ucr Ucrmax 4.78m
qda
Vcd
Ucrmax
qda 376.95m1
kN qbu 262.86m1
kN
qda qbu 2 qbu 525.72m1
kN
Sabah Shawkat Cabinet of Structural Engineering 2017
If we want to make a proposal without head, subject to the following parameters:
Carrying capacity of the concrete section
Proposal visible head
Geometry head
Figure: 2.3.2‐2
We suggest shear reinforcement
Proposal shear reinforcement - reinforced by bins
n is the number of bins reinforced, Ass area of reinforcement to a bin
Given
d 1.0 m fctm40 1.40 MPa ucr40 d 2hd
2
ucr40 4.08m
qbu40 0.42 hd g b fctm40 ucr40 qbu40 1252.47kN Vcd 1800 kN
2 qbu40 2504.94 kN 2qbu40 Vcd
45 deg sin ( ) 0.71 cos ( ) 0.71
hh 0.6 m dh 2.0 m Ucr2 dh Ucr2 6.28m
qd2
Vcd
Ucr2
qd2 286.48m1
kN qbu 262.86m1
kN qd2 qbu
qd2 2 qbu
qd2
qbu
qsu qsu n A ss ss s fyd qsu qd2 qbu qsu 23.62 m1
kN
fyd 190 MPa ss 1 s 1 Ass 1 m2
n 1
qsu n Ass ss s fyd m1
Ass Find Ass Ass 0.000124 m2
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number of bars in one bin / m 'ss = 0.25m
diameter of one profile
Assessment of the punching according to EC 2
design value of shear resistance of plate without shear reinforcement (per unit length of critical perimeter)
shear resistance
average width tension section
concrete area
The maximum degree of reinforcement
The average degree of reinforcement
n1 5
1 8 mm
Asssku n1 1
2
4 Asssku 0.00025133 m
2 Asssku Ass
vRd1 Rd k 1.2 40 1 d
Rd 0.3MN
m2
hd 0.3 m k 1.6hd
m k 1.3
bt 1 m
fyk 410MN
m2
min1 0.6 bthd
fyk
kN
m4
min1 0.00000044
min2 0.0015 bthd
m2
min2 0.00045
min
min1
min2
min max min min 0.00045
Ac hd bt Ac 0.3m2
max 0.04Ac
m2
max 0.01
1
min max
2 1 0.01
Sabah Shawkat Cabinet of Structural Engineering 2017
The maximum design value of shear resistance of plate with shear reinforcement (per unit
length of critical perimeter)
Design value of shear resistance of plate with shear reinforcement (per unit length of critical
perimeter)
Column diameter
Diameter of critical perimeter
Critical perimeter
concrete shear area
Assumption degree of shear reinforcement
carrying capacity
vRd1 Rd k 1.2 40 1 hd vRd1 169.53m1
kN
vRd2 1.6 vRd1 vRd2 271.25m1
kN
vRd3 vRd1i
As fyd sin ( ) u
Ps 0.5 m
Pu 2 1.5 hd Ps Pu 1.4m
u Pu u 4.4m
Acw Pu
2
4
Ps2
4 Acw 1.34m
2
´w 0.0013 0.6
As ´w Acw As 0 m2
2 fyd 360
MN
m2
vRd3 vRd1
As fyd sin ( ) u
vRd3 255.28m1
kN
Sabah Shawkat Cabinet of Structural Engineering 2017
The load effects
Computing shear force internal columns
Figure: 2.3.2‐3
incorrect design, design head
as being applicable condition
Geometry head
Critical perimeter with head
Concrete shear area
The expected level of reinforcement by shear reinforcement
Slab with shear reinforcement to a void punching.
Vsd 1800 kN 1.15
vsd
Vsd
u vsd 470.64m
1kN vsd vRd3
vsd vRd3
lh 0.9 m hh 0.6 m d2crit 3.11 m
ucrit d2crit ucrit 9.77m
Acwh d2crit
2
4
Ps2
4 Acwh 7.4m
2
Ash ´w Acwh Ash 0.01m2
vsd
Vsd
ucrit
vsd 211.87m1
kN
vRd3 vRd1
Ash fyd sin ( ) ucrit
vRd3 382.21m1
kN vsd vRd3
Sabah Shawkat Cabinet of Structural Engineering 2017
Space inside the critical perimeter less the contact surface
For dimensioning elements requiring shear reinforcement
Smallest section width in the range of effective height
Height of the floor slab
Maximum distance of stirrups
Maximum diameter of reinforcement stirrups with a smooth surface
Sectional area of shear reinforcement in the length range