aci - shear in beam
TRANSCRIPT
-
7/26/2019 Aci - Shear in Beam
1/15
A. J. Clark School of Engineering Department of Civil and Environmental Engineering
Fifth Edition
CHAPTER
4b
Reinforced Concrete Design
ENCE 355 - Introduction to Structural DesignDepartment of Civil and Environmental Engineering
University of Maryland, College Park
SHEAR IN BEAMS
Part I Concrete Design and Analysis
FALL 2002By
Dr Ibrahim Assakkaf
CHAPTER 4b. SHEAR IN BEAMS Slide No. 1ENCE 355 Assakkaf
Shear Analysis Procedure
The shear analysis procedure involves
the following:
Checking the shear strength in an existing
member
Verifying that the various ACI code
requirements have been satisfied and met.
Note that the member may reinforced or
plain.
-
7/26/2019 Aci - Shear in Beam
2/15
CHAPTER 4b. SHEAR IN BEAMS Slide No. 2ENCE 355 Assakkaf
Shear Analysis Procedure
Example 1
A reinforced concrete beam of rectangular
cross section shown is reinforced with
seven No. 6 bars in a single layer. Beam
width b = 18 in., d= 33 in., single-loop No.
3 stirrups are placed 12 in. on center, and
typical cover is 1 in. Find Vc, Vs, and the
maximum factored shear force permitted
on this member. Use = 4,000 psi andfy= 60,000 psi.
CHAPTER 4b. SHEAR IN BEAMS Slide No. 3ENCE 355 Assakkaf
Shear Analysis Procedure
Example 1 (contd)
33
bars6#7
COV.2
11
stirrup21@3#
81
-
7/26/2019 Aci - Shear in Beam
3/15
CHAPTER 4b. SHEAR IN BEAMS Slide No. 4ENCE 355 Assakkaf
Shear Analysis Procedure
Example 1 (contd) The force that can be resisted by concrete
alone is
The nominal shear force provided by the
steel is
The maximum factored shear force is
( )( )kips1.75
1000
3318000,422 === dbfV wcc
( )( )( )kips3.36
12
336011.02=
==
s
dfAV
yv
s
( )
kips7.94
3.361.7585.0maximum
=
+=+= scu VVV
CHAPTER 4b. SHEAR IN BEAMS Slide No. 5ENCE 355 Assakkaf
ACI Code Provisions for Shear Design
According to the ACI Code, the design of
beams for shear is based on the following
relation:
Shear Reinforcement DesignRequirements
un VV Where
= strength reduction factor (= 0.85 for shear)
Vn = Vc + VsVs = nominal shear strength provided by reinforcement
stirrupsinclinedfors
dfA yv=
(1)
(2)
-
7/26/2019 Aci - Shear in Beam
4/15
CHAPTER 4b. SHEAR IN BEAMS Slide No. 6ENCE 355 Assakkaf
ACI Code Provisions for Shear Design
Symbols
Av= total cross-sectional area of web reinforcement within
a distances, for single loop stirrups,Av = 2AsAs = cross-sectional area of the stirrup bar (in
2)
bw = web width = b for rectangular section (in.)
s = center-to-center spacing of shear reinforcement in a
direction parallel to the longitudinal reinforcement (in.)
fy = yield strength of web reinforcement steel (psi)
Shear Reinforcement Design
Requirements
CHAPTER 4b. SHEAR IN BEAMS Slide No. 7ENCE 355 Assakkaf
ACI Code Provisions for Shear Design
For inclined stirrups, the expression for
nominal shear strength provided by
reinforcement is
For = 450, the expression takes the form
( )
s
dfAV
yv
s
cossin +=
s
dfAV
yv
s
414.1=
(3)
(4)
Shear Reinforcement DesignRequirements
-
7/26/2019 Aci - Shear in Beam
5/15
CHAPTER 4b. SHEAR IN BEAMS Slide No. 8ENCE 355 Assakkaf
ACI Code Provisions for Shear Design
The design for stirrup spacing can be
determined from
Shear Reinforcement Design
Requirements
stirrups)45(for414.1
required
and
stirrups)cal(for vertirequired
0
s
yv
s
yv
V
dfAs
V
dfAs
=
=
where
cus
VVV
=
(5)
(6)
(7)
CHAPTER 4b. SHEAR IN BEAMS Slide No. 9ENCE 355 Assakkaf
ACI Code Provisions for Shear Design
According to the ACI Code, the maximum
spacing of stirrups is the smallest value of
Shear Reinforcement DesignRequirements
in.24
2
50
max
max
max
=
=
=
s
ds
b
fAs
w
yv
(8)
If Vs exceeds , the maximum spacing must
not exceed d/4 or 12 in.
dbf wc4
-
7/26/2019 Aci - Shear in Beam
6/15
CHAPTER 4b. SHEAR IN BEAMS Slide No. 10ENCE 355 Assakkaf
ACI Code Provisions for Shear Design
It is not usually good practice to space
vertical stirrups closer than 4 in.
It is generally economical and practical to
compute spacing required at several
sections and to place stirrups accordingly
in groups of varying spacing. Spacing
values should be made to not less than 1-in. increments.
Shear Reinforcement Design
Requirements
CHAPTER 4b. SHEAR IN BEAMS Slide No. 11ENCE 355 Assakkaf
ACI Code Provisions for Shear Design
Critical Section
The maximum shear usually occurs in this
section near the support.
For stirrup design, the section located a
distance dfrom the face of the support is called
the critical section Sections located less than a distance dfrom
the face of the support may be designed for the
same Vu as that of the critical section.
Shear Reinforcement DesignRequirements
-
7/26/2019 Aci - Shear in Beam
7/15
CHAPTER 4b. SHEAR IN BEAMS Slide No. 12ENCE 355 Assakkaf
ACI Code Provisions for Shear Design
Critical Section
Shear Reinforcement Design
Requirements
d
d
Critical Section
Figure 1
L
CHAPTER 4b. SHEAR IN BEAMS Slide No. 13ENCE 355 Assakkaf
ACI Code Provisions for Shear Design
Critical Section (contd)
The stirrup spacing should be constant from the
critical section back to the face of the support
based on the spacing requirements at the
critical section.
The first stirrup should be placed at a maximumdistance ofs/2 from the face of the support,
wheres equals the immediately adjacent
required spacing (a distance of 2 in. is
commonly used.
Shear Reinforcement DesignRequirements
-
7/26/2019 Aci - Shear in Beam
8/15
CHAPTER 4b. SHEAR IN BEAMS Slide No. 14ENCE 355 Assakkaf
The design of stirrups for shearreinforcement involves the
determination of stirrup size and
spacing pattern.
A general procedure is as follows:
1. Determine the shear values based on
clear span and draw a shear diagram for
Vu.
2. Determine if stirrups are required.
Stirrup Design Procedure
CHAPTER 4b. SHEAR IN BEAMS Slide No. 15ENCE 355 Assakkaf
3. Determine the length of span over which
stirrups are required.
4. On the Vu diagram, determine the area
representing required Vs. This will
display the required strength of the
stirrups to be provided.
5. Select the size of the stirrups. Find thespacing required at the critical section ( a
distance dfrom the face of the support.
Stirrup Design Procedure
-
7/26/2019 Aci - Shear in Beam
9/15
CHAPTER 4b. SHEAR IN BEAMS Slide No. 16ENCE 355 Assakkaf
6. Establish the ACI Code maximum
spacing requirements.
7. Determine the spacing requirements
based on shear strength to be furnished
by web reinforcing.
8. Establish the spacing pattern and show
detailed sketches.
Stirrup Design Procedure
CHAPTER 4b. SHEAR IN BEAMS Slide No. 17ENCE 355 Assakkaf
Example 2
A continuous reinforced concrete beam
shown in the figure is 15 in. wide and has
an effective depth of 31 in. The factored
loads are shown, and the factored uniform
load includes the weight of the beam.
Design the web reinforcement if = 4000psi andfy = 60,000 psi.
Stirrup Design Procedure
cf
-
7/26/2019 Aci - Shear in Beam
10/15
CHAPTER 4b. SHEAR IN BEAMS Slide No. 18ENCE 355 Assakkaf
Example 2 (contd)
Stirrup Design Procedure
51
13
As
spanclear051
100 k100 k
0-5 0-5 0-5
wu = 1.0 k/ftA
A
Section A-A
CHAPTER 4b. SHEAR IN BEAMS Slide No. 19ENCE 355 Assakkaf
Example 2 (contd)
Establish the shear force diagram for Vu:
Stirrup Design Procedure
100 k100 k
0-5 0-5 0-5
wu = 1.0 k/ft
107.5 k 107.5 k
( ) ( )k5.107
2
151100221 =
+==RR
-
7/26/2019 Aci - Shear in Beam
11/15
CHAPTER 4b. SHEAR IN BEAMS Slide No. 20ENCE 355 Assakkaf
100 k100 k
0-5 0-5 0-5
wu = 1.0 k/ft
107.5 k 107.5 k
107.5102.5
Vu (kips)2.5
107.5
2.5
+
-
Example 2 (contd)
See Fig. 2 for enlargement
CHAPTER 4b. SHEAR IN BEAMS Slide No. 21ENCE 355 Assakkaf
100 k100 k
0-5 0-5 0-5
wu = 1.0 k/ft
107.5 k 107.5 k
Example 2 (contd)
M
V
107.5 k
50for5.107 = xxVu
M
V
100 k
105for1005.107 = xxVu107.5 k
( ) k9.10458.25.10758.2
85.213
*
===
==
uu VV
d
x
x
-
7/26/2019 Aci - Shear in Beam
12/15
CHAPTER 4b. SHEAR IN BEAMS Slide No. 22ENCE 355 Assakkaf
Example 2 (contd)
Because of the symmetry, we will focus on
the left half of the shear diagram as shown
in Fig. 2.
Determine if stirrups required:
Stirrup Design Procedure
( ) ( )( )
( ) kips25502
1
2
1
kips50100
3115000,4285.02
==
===
c
wcc
V
dbfV
Since ( =104.9 k) > (1/2 Vc = 25 k), stirrups are required.*
uV
CHAPTER 4b. SHEAR IN BEAMS Slide No. 23ENCE 355 Assakkaf
Example 2 (contd)
Stirrup Design Procedure
9.104
5.107*
uV
cV
5.102
50
25cV2
1
*
*
85.213 ==d
0.5 5.2
Sym.CL
sVrequired
Vu(kips) 0
Figure 2
-
7/26/2019 Aci - Shear in Beam
13/15
CHAPTER 4b. SHEAR IN BEAMS Slide No. 24ENCE 355 Assakkaf
Example 2 (contd) Stirrups are required to the point where
From Fig. 2, this point is located at the first
concentrated load and it is at distance 5 ft from
the face of the support.
Determine the required Vs on the Vu diagram:
Stirrup Design Procedure
kips252
1== cu VV
52.58for5.57required
505.107maxrequired
=
==
xxV
xwxVVV
s
cus
CHAPTER 4b. SHEAR IN BEAMS Slide No. 25ENCE 355 Assakkaf
Example 2 (contd)
Assume No. 3 vertical stirrups (Av = 0.22 in2):
Establish ACI Code maximum spacing
requirements:
Stirrup Design Procedure
( )( )( )
in.6use
in.3.6509.104
316022.085.0
requiredrequired
*
* =
==s
yy
V
dfAs
( )( )
kips6.6485.0
509.104
kips6.1171000
3115400044
** =
==
==
ss
wc
VV
dbf
-
7/26/2019 Aci - Shear in Beam
14/15
CHAPTER 4b. SHEAR IN BEAMS Slide No. 26ENCE 355 Assakkaf
Example 2 (contd) Since 64.6 kips < 117.6 kips, the maximum
spacing shall be the smallest of the
following values (see Eq. 8):
Stirrup Design Procedure
( )( )
in.24
in.5.152
31
2
in.6.171550
000,6022.0
50
max
max
max
=
===
===
s
ds
b
fAs
w
yv
Therefore, use a maximum spacing of 15 in.
controls
CHAPTER 4b. SHEAR IN BEAMS Slide No. 27ENCE 355 Assakkaf
Example 2 (contd)
Determine the spacing requirementsbetween the critical section and the firstconcentrated load:
The results of applying above equation for
values ofx
range from 3 to 5 are tabulatedas shown
Stirrup Design Procedure
( )( )( )xV
dfAs
s
yy
==
5.57
316022.085.0
requiredrequired
6.65
6.54
6.43
Requireds (in)x (ft)
-
7/26/2019 Aci - Shear in Beam
15/15
CHAPTER 4b. SHEAR IN BEAMS Slide No. 28ENCE 355 Assakkaf
Example 2 (contd) Since no stirrups are required in the
distance between the first concentrated
load , it is clear that the maximum spacing
of 15 in. need not be used in that distance.
A spacing of 6 in. will be used between the
face of the support and the concentrated
load.
The center part of the beam will be
reinforced with stirrups at a spacing slightly
less than the maximum spacing of 15 in.
Stirrup Design Procedure
CHAPTER 4b. SHEAR IN BEAMS Slide No. 29ENCE 355 Assakkaf
Example 2 (contd)
Final Sketch for Shear Reinforcement:
Stirrup Design Procedure
67
56@spaces10 =
3
2
111
2
111
LCSym.