This document summarizes the process for designing shear reinforcement in reinforced concrete beams. It first describes calculating the shear strength provided by concrete (Vc) and steel (Vs), and checking that the nominal shear strength (Vn) exceeds the factored shear force (Vu). It then outlines ACI code provisions for shear design including requirements for stirrup spacing, area, and yield strength. An example problem is worked through step-by-step to demonstrate how to determine required stirrup spacing and design a reinforcement pattern.

1. 1
• A. J. Clark School of Engineering •Department of Civil and Environmental Engineering
Fifth Edition
CHAPTER
4b
Reinforced Concrete Design
ENCE 355 - Introduction to Structural Design
Department of Civil and Environmental Engineering
University of Maryland, College Park
SHEAR IN BEAMS
Part I – Concrete Design and Analysis
FALL 2002
By
Dr . Ibrahim. Assakkaf
CHAPTER 4b. SHEAR IN BEAMS Slide No. 1
ENCE 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.

2. 2
CHAPTER 4b. SHEAR IN BEAMS Slide No. 2
ENCE 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 ½ in. Find Vc, Vs, and the
maximum factored shear force permitted
on this member. Use = 4,000 psi and fy
= 60,000 psi.
CHAPTER 4b. SHEAR IN BEAMS Slide No. 3
ENCE 355 ©Assakkaf
Shear Analysis Procedure
Example 1 (cont’d)
33 ′′
bars6#7−
COV.
2
1
1
″
stirrup21@3# ′′
81 ′′

3. 3
CHAPTER 4b. SHEAR IN BEAMS Slide No. 4
ENCE 355 ©Assakkaf
Shear Analysis Procedure
Example 1 (cont’d)
– 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
3318000,42
2 ==′= dbfV wcc
( )( )( ) kips3.36
12
336011.02
=
×
==
s
dfA
V yv
s
( )
kips7.94
3.361.7585.0maximum
=
+=+= scu VVV φφ
CHAPTER 4b. SHEAR IN BEAMS Slide No. 5
ENCE 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 Design
Requirements
un VV ≥φ
Where
φ = strength reduction factor (= 0.85 for shear)
Vn = Vc + Vs
Vs = nominal shear strength provided by reinforcement
stirrupsinclinedfor
s
dfA yv
=
(1)
(2)

4. 4
CHAPTER 4b. SHEAR IN BEAMS Slide No. 6
ENCE 355 ©Assakkaf
ACI Code Provisions for Shear Design
– Symbols
Av = total cross-sectional area of web reinforcement within
a distance s, for single loop stirrups, Av = 2As
As = cross-sectional area of the stirrup bar (in2)
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. 7
ENCE 355 ©Assakkaf
ACI Code Provisions for Shear Design
– For inclined stirrups, the expression for
nominal shear strength provided by
reinforcement is
– For α = 450, the expression takes the form
( )
s
dfA
V
yv
s
αα cossin +
=
s
dfA
V
yv
s
414.1
=
(3)
(4)
Shear Reinforcement Design
Requirements

5. 5
CHAPTER 4b. SHEAR IN BEAMS Slide No. 8
ENCE 355 ©Assakkaf
ACI Code Provisions for Shear Design
– The design for stirrup spacing can be
determined from
Shear Reinforcement Design
Requirements
stirrups)45(for
414.1
required
and
stirrups)cal(for vertirequired
0
s
yv
s
yv
V
dfA
s
V
dfA
s
=
=
where
φ
φ cu
s
VV
V
−
=
(5)
(6)
(7)
CHAPTER 4b. SHEAR IN BEAMS Slide No. 9
ENCE 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 Design
Requirements
in.24
2
50
max
max
max
=
=
=
s
d
s
b
fA
s
w
yv
(8)
If Vs exceeds , the maximum spacing must
not exceed d/4 or 12 in.
dbf wc
′4

6. 6
CHAPTER 4b. SHEAR IN BEAMS Slide No. 10
ENCE 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. 11
ENCE 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 d from the face of the support is called
the “critical section”
• Sections located less than a distance d from
the face of the support may be designed for the
same Vu as that of the critical section.
Shear Reinforcement Design
Requirements

7. 7
CHAPTER 4b. SHEAR IN BEAMS Slide No. 12
ENCE 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. 13
ENCE 355 ©Assakkaf
ACI Code Provisions for Shear Design
– Critical Section (cont’d)
• 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 maximum
distance of s/2 from the face of the support,
where s equals the immediately adjacent
required spacing (a distance of 2 in. is
commonly used.
Shear Reinforcement Design
Requirements

8. 8
CHAPTER 4b. SHEAR IN BEAMS Slide No. 14
ENCE 355 ©Assakkaf
The design of stirrups for shear
reinforcement 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. 15
ENCE 355 ©Assakkaf
3. Determine the length of span over which
stirrups are required.
4. On the Vu diagram, determine the area
representing “required φVs.” This will
display the required strength of the
stirrups to be provided.
5. Select the size of the stirrups. Find the
spacing required at the critical section ( a
distance d from the face of the support.
Stirrup Design Procedure

9. 9
CHAPTER 4b. SHEAR IN BEAMS Slide No. 16
ENCE 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. 17
ENCE 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 = 4000
psi and fy = 60,000 psi.
Stirrup Design Procedure
cf ′

10. 10
CHAPTER 4b. SHEAR IN BEAMS Slide No. 18
ENCE 355 ©Assakkaf
Example 2 (cont’d)
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. 19
ENCE 355 ©Assakkaf
Example 2 (cont’d)
– 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
1511002
21 =
+
== RR

11. 11
CHAPTER 4b. SHEAR IN BEAMS Slide No. 20
ENCE 355 ©Assakkaf
100 k100 k
0-5 ′′′0-5 ′′′0-5 ′′′
wu = 1.0 k/ft
107.5 k 107.5 k
107.5
102.5
Vu (kips)
2.5
107.5
2.5
+
-
Example 2 (cont’d)
See Fig. 2 for enlargement
CHAPTER 4b. SHEAR IN BEAMS Slide No. 21
ENCE 355 ©Assakkaf
100 k100 k
0-5 ′′′0-5 ′′′0-5 ′′′
wu = 1.0 k/ft
107.5 k 107.5 k
Example 2 (cont’d)
M
V
107.5 k
50for5.107 ≤≤−= xxVu
M
V
100 k
105for1005.107 ≤≤−−= xxVu
107.5 k
( ) k9.10458.25.10758.2
85.213
*
=−==
′=′′=
uu VV
d
x
x

12. 12
CHAPTER 4b. SHEAR IN BEAMS Slide No. 22
ENCE 355 ©Assakkaf
Example 2 (cont’d)
– 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
( ) ( )( )
( ) kips2550
2
1
2
1
kips50
100
3115000,4285.0
2
==
==′=
c
wcc
V
dbfV
φ
φφ
Since ( =104.9 k) > (1/2 φVc = 25 k), stirrups are required.*
uV
CHAPTER 4b. SHEAR IN BEAMS Slide No. 23
ENCE 355 ©Assakkaf
Example 2 (cont’d)
Stirrup Design Procedure
9.104
5.107
*
uV
cVφ
5.102
50
25cVφ
2
1
*
*
85.213 ′=′′=d
0.5 ′ 5.2 ′
Sym.
CL
sVφrequired
Vu
(kips) 0
Figure 2

13. 13
CHAPTER 4b. SHEAR IN BEAMS Slide No. 24
ENCE 355 ©Assakkaf
Example 2 (cont’d)
– 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 φVs” on the Vu diagram:
Stirrup Design Procedure
kips25
2
1
== cu VV φ
52.58for5.57required
505.107
maxrequired
≤≤−=
−−=
−−=
xxV
x
wxVVV
s
cus
φ
φφ
CHAPTER 4b. SHEAR IN BEAMS Slide No. 25
ENCE 355 ©Assakkaf
Example 2 (cont’d)
– Assume No. 3 vertical stirrups (Av = 0.22 in2):
– Establish ACI Code maximum spacing
requirements:
Stirrup Design Procedure
( )( )( )
in.6use
in.3.6
509.104
316022.085.0
required
required *
*
=
−
==
s
yy
V
dfA
s
φ
φ
( )( )
kips6.64
85.0
509.104
kips6.117
1000
311540004
4
*
*
=
−
==
==′
φ
φ s
s
wc
V
V
dbf

14. 14
CHAPTER 4b. SHEAR IN BEAMS Slide No. 26
ENCE 355 ©Assakkaf
Example 2 (cont’d)
– 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.5.15
2
31
2
in.6.17
1550
000,6022.0
50
max
max
max
=
===
===
s
d
s
b
fA
s
w
yv
Therefore, use a maximum spacing of 15 in.
controls
CHAPTER 4b. SHEAR IN BEAMS Slide No. 27
ENCE 355 ©Assakkaf
Example 2 (cont’d)
– Determine the spacing requirements
between the critical section and the first
concentrated load:
– The results of applying above equation for
values of x range from 3 to 5 are tabulated
as shown
Stirrup Design Procedure
( )( )( )
xV
dfA
s
s
yy
−
==
5.57
316022.085.0
required
required
φ
φ
6.65
6.54
6.43
Required s (in)x (ft)

15. 15
CHAPTER 4b. SHEAR IN BEAMS Slide No. 28
ENCE 355 ©Assakkaf
Example 2 (cont’d)
– 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. 29
ENCE 355 ©Assakkaf
Example 2 (cont’d)
Final Sketch for Shear Reinforcement:
Stirrup Design Procedure
67 ′′−′
56@spaces10 ′=′′
3′′ ″
−′
2
1
11
″
−′
2
1
11
LC
Sym.