CHAPTER CHAPTE R 3: Reinf Reinforced orced Concre Concrete te Slabs Slabs and Beams Beams
3.4 Rei Reinfo nforc rced ed Conc Concre rete te Beams Beams - Siz Sizee Selec Selectio tion n
Description
This application calculates the spacing for shear reinforcement o f a concrete beam supporting a uniformly distributed load. The application uses the strength design method of ACI 318-89, and calculates shear strength of the concrete con crete using the simplified method of ACI Section 11-3, Eq. (11-3). The required input includes strength of concrete, strength of reinforcement, unit weight of concrete, length of beam between the face of support and the point where shear passes through zero, uniformly distributed factored load, effective beam depth, beam web width, flange thickness and cross sectional area of the ties or stirrups. A summary summary of input and calculated c alculated values is shown on pages pag es 13 and 14.
Reference: ACI 318-89 "Building Code Requirements for Reinforced Concrete." (Revised 1992)
Input Notation
Input Variables Enter the following variables.
Clear span length:
Ln ≔ 24
Factor k, 1.15 for 1st interior support or 1 for all other supports:
k≔ 1.15
Uniformly distributed factored load:
wu ≔ 6.5
Effective depth:
d≔ 17.5
Beam web width:
bw ≔ 12
Beam flange thickness:
h f ≔ 4
Stirrup spacing factor:
SpF ≔ 1
Area of shear reinforcement within distance s:
Av ≔ 0.22
2
Final stirrup spacing is calculated to a multiple of the specified stirrup spacing factor SpF. Since stirrup spacing may be governed by minimum spacing requirements, the smallest stirrup size should be tried first and increased only if the spacing is too small. Computed Variables
The following variables are calculated in this doc ument. Vuf
factored shear force at face of support
Vud
critical factored shear force, at distance d from face of support
Vc
nominal shear strength provided by concrete (ACI 318, 9.3.2.3, 11.3.1.1, Eq. (11-3), 11.5.4.3)
Vs
nominal shear provided by shear reinforcement (ACI 318, 9.3.2.3, 11.5.6, Eq. (11-17))
Xo
distance from face of support to the point where shear passes through zero
Xs
distance from face of support to the point where shear is equal to 1/2 the useable concrete shear strength
Xc
distance from face of support to the point where shear is equal to the useable concrete shear strength
s
spacing of shear reinforcement in direction parallel to longitudinal reinforcement
sp
spacing of shear reinforcement in direction parallel to longitudinal reinforcement from face of support to center of e ach stirrup
Material Properties
Enter values for f'c, f y, wc, and k v if different from that shown. Specified compressive strength of concrete:
f' c ≔ 4
Specified yield strength of reinforcement (f y may not exceed 60 ksi, ACI 318 11.5.2):
f y ≔ 60
Unit weight of concrete:
wc ≔ 145
Shear strength reduction factor for lightweight concrete k v = 1 for normal weight, 0.75 for alllightweight and 0.85 for sand-lightweight concrete (ACI 318, 11.2.1.2.): Strength reduction factor for shear (ACI 318, 9.3.2.3):
kv ≔ 1
ϕv ≔ 0.85
Limit the value of f'c for computing shear strength to 10 ksi (ACI 3 18, 11.1.2): f' c ≔
f' c > 10 ⋅
, 10 ⋅
, f' c
f' c = 4 ksi
Limit the effective yield strength of shear reinforcement to 60 ksi as required by ACI 318, Section 11.5.2: f y ≔
f y > 60 ⋅
, 60 ⋅
, f y
f y = 60 ksi
The following values are computed from the entered material properties. Nominal concrete shear strength per unit area:
vc ≔ kv ⋅ 2 ⋅
‾‾‾ f' c ― ⋅
vc = 126.491 psi
If the member is subject to significant axial tension the concrete shear strength may be defined as 0. Shear reinforcement will then support the total shear (ACI 318 11.3.1.3). Nominal shear strength per unit area for shear reinforcement spaced at d/4 or less:
‾‾‾ f' c
vs_max ≔ 8 ⋅
⋅
vs_max = 505.964 psi
Nominal shear strength per unit area for shear reinforcement spaced at greater than d/4 and less than d/2: vs_max2 ≔ 4 ⋅
f' c
⋅
vs_max2 = 252.982 psi
Calculations
Factored shear force at face of support: V uf ≔ k ⋅
wu ⋅ Ln
2
= 89.7
Critical factored shear force, at distance d from face of support: V ud ≔ V uf − wu ⋅ d = 80.221
Maximum useable shear strength with shear reinforcement: ϕV n_max ≔ ϕv ⋅ vc + vs_max
⋅ bw ⋅ d = 112.893
Note: If Vn_max is less than Vu_d the beam size must be increased.
If you do not increase the beam size the following expression will increase the e ntered value of the beam width to the minimum value necessary to support the shear force using the maximum permissible shear reinforcement. bw ≔
⎛
ϕV n_max > V ud , bw ,
⎝
ϕV n_max ≔ ϕv ⋅ vc + vs_max
⎞ ⋅ bw = 12 ϕV n_max ⎠ V ud
⋅ bw ⋅ d = 112.9
Useable shear strength provided by concrete (ACI 318, 11.3): ϕV c ≔ ϕv ⋅ vc ⋅ bw ⋅ d = 22.579
Defined factor k c for determining maximum permissible factored shear force before shear reinforcement is required (ACI 318, 11.5.5.1 (c)): kc ≔
d ≤ 10
+ d ≤ bw + d ≤ 2.5 ⋅ h f , 1 , 0.5 = 0.5
Note: For wide shallow beams the full concrete shear strength of the beam is used in determining the maximum permissible factored shear force before shear reinforcement is required. For all other beams half the concrete shear strength is used. If the member is a slab or footing, or a joist as defined in ACI 318, Section 8.11, the user may define the value of k c as equal to 1 at this point:
Maximum permissible factored shear force before shear reinforcement is required: kc ⋅ ϕV c = 11.289
Maximum spacing of shear reinforcement determined by ma ximum permissible spacing of d/2 (ACI 318, 11.5.4) or by minimum required shear reinforcement area (ACI 318, 11.5.5):
⎛ s' max ≔ ⎜V ud ≤ kc ⋅ ϕV c , 0 ⋅ ⎝
⎛⎡ , min ⎜⎢
0.5 ⋅ d ⎤⎞⎞ Av ⋅ f y ⎥⎟⎟ = 8.75 ――― ⎝⎣ 50 ⋅ ⋅ bw ⎦⎠⎠
Maximum spacing rounded off to a multiple of Sp F: smax ≔ floor
⎛ s' max ⎞ ⋅ SpF = 8 ⎝ SpF ⎠
Minimum shear capacity: ϕV n_min ≔ ϕV c +
⎛ ⎝
smax ＝ 0 ⋅
,0⋅
ϕv ⋅ Av ⋅ f y ⋅ d ⎞ , ――― = 47.1 smax ⎠
Defined variable "Case": Case = 0: no shear reinforcement required Case = 1: minimum shear reinforcement required Case = 2: greater than minimum shear reinforcement is required Case ≔ Case = 2
V ud ≤ kc ⋅ ϕV c , 0 ,
V ud ≤ ϕV n_min , 1 , 2
Minimum stirrup spacing: s' min ≔ ‖ if Case ＝ 0
| | = 3.406 ‖ || 0 ‖ || Case else if ＝ 1 ‖ || ‖ s' max || ‖ || else ‖ || ‖ ⎛ ⎡ ⎛ ⎞ ⎤ ⎞ d d ‖ || V ud ≥ ϕV n_max , , ‖ ⎜ ⎟ ⎢ ⎥ ‖ 4 2 ⎠ || ⎝ ‖ ‖ min ⎜⎢ ⎥⎟ | | ϕv ⋅ Av ⋅ f y ⋅ d ‖ ‖ ⎜⎢ ⎥⎟ | | V ϕV − ⎦⎠ | | ⎝⎣ ud c ‖ ‖
Minimum spacing rounded off to a multiple of SpF: smin ≔ floor
⎛ s' min ⎞ ⋅ SpF = 3 ⎝ SpF ⎠
If the minimum spacing is too small or too large the shea r reinforcement area should be revised. First possible spacing larger than d/4 rounded off to multiple of SpF: s1 ≔ floor
⎛ d ⎞ ―― ⋅ SpF + SpF = 5 ⎝ 4 ⋅ SpF ⎠
Minimum useable spacing for stirrup spacings greater than d/4:
⎛ ⎛⎡ ⎤⎞ ⎛⎡ ⎤⎞⎞ s1 s1 d d ⎜ ⎜ ⎟ ⎜ ⎥ > , , max ⎢ ⎥⎟⎟ = 5.115 Av ⋅ f y ⋅ d Av ⋅ f y ⋅ d s' min2 ≔ max ⎢ 2 2 ⎝ ⎝⎣ ϕv ⋅ vs_max2 ⋅ bw ⋅ d ⎦⎠ ⎝⎣ ϕv ⋅ vs_max2 ⋅ bw ⋅ d ⎦⎠⎠ Spacings larger than d/4 and less than the minimum spacing required to develop half the maximum shear reinforcement stress provide no useable additional shear strength. Minimum useable spacing for stirrup spacings greater than d/4 rounded to a multiple of SpF: smin2 ≔ floor
⎛ s' min2 ⎞ ⋅ SpF = 5 ⎝ SpF ⎠
Number of stirrups at a spacing less than or equal to d/4, greater than smin and less than smax: n≔
⎛⎛
smin >
d⎞
+ smin ＝ 0 ⋅
+ smin ＝ smax , 0 ,
s1 − SpF − smin ⎞
=1
4⎠
⎝⎝
SpF
⎠
Minimum stirrup spacing smin plus any additional stirrup spacings less than or equal to d/4: j1 ≔ 0 ‥ n
≔ smin + j1 ⋅ SpF j1
s=
s
⎡3⎤ ⎣4⎦
Stirrup spacings from smin2 or smin to smax: s2 ≔ n1 ≔
smin2 ≤ smax
⎛smin2 ＝ s , n , n + 1⎞ = 2 n
j2 ≔ n1 ‥ n1 + s
j2
≔
⋅ smin2 > smin , smin2 , smin = 5
⎛ smax − s2 ⎞ V ud ≥ kc ⋅ ϕV c , ―― , 0 SpF ⎝ ⎠
V ud ≤ kc ⋅ ϕV c , 0 ⋅
, s2 +
s2 ＝ smax , 0 ⋅
, j2 − n − 1 ⋅ SpF
Stirrup spacings from smin to smax: j ≔ 0 ‥ last s s
T
= 3 4 5 6 7 8
Useable strength of shear reinforcement at each stirrup spacing:
⎛ ≔ ⎜V ud ≤ kc ⋅ ϕV c , 0 ⋅ j ⎝
ϕV s
ϕV s
T
,
ϕv ⋅ Av ⋅ f y ⋅ d ⎞ s
j
⎟ ⎠
= 65.5 49.1 39.3 32.7 28.1 24.5
Total useable shear strength at each stirrup spacing: ϕV n ≔ ϕV c + ϕV s ϕV n
T
= 88 71.7 61.8 55.3 50.6 47.1
Total useable shear capacity from face of support to the point of zero shear:
u ≔ last s ϕV n
ϕV n
u−1
T
+2
u=7
≔ kc ⋅ ϕV c
ϕV n
u
≔ kc ⋅ ϕV c
= 88 71.7 61.8 55.3 50.6 47.1 11.3 11.3
j3 ≔ 0 ‥ u
Distance from the left reaction to the point where shear force is less than the limiting concrete shear strength and shear reinforcement is not required (ACI 318, 11.5.1): X s ≔
⎛ ⎝
V ud ≤ kc ⋅ ϕV c , 0 ⋅
,
V uf − kc ⋅ ϕV c ⎞ wu
⎠
= 12.063
Distance from the left reaction to the point where shear is less than or equal to the useable concrete shear capacity: X c ≔
⎛
V uf ≤ ϕV c , 0 ⋅
⎝
V uf − ϕV c ⎞ , ――― = 10.326 wu ⎠
Distance from face of support to the point whe re shear passes through 0: V uf X o ≔ ― = 13.8 wu
Distances from the face of support to each point where stirrup spacing changes, and to the point of zero shear rounded up to the nearest foot: xs
0
≔0
⎛ V uf − ⎛s ＝ smax , kc ⋅ ϕV c , ϕV n ⎞ xs ⎞ j j + 1 j ⎜ xs ≔ xs j + s j ⋅ ceil − ⎟ j + 1 wu ⋅ s s j j ⎠ ⎝ xs
xs
⎛ X o ⎞ ≔ ceil ― ⋅ u ⎝ ⎠ T
xs
= 0 36 52 67 73 80 152 168
u
= 14
Length covered by each stirrup spacing: Ls
j
Ls
≔ xs j + 1 − xs j
T
= 36 16 15 6 7 72
Number of stirrups at each spacing:
⎛ ⎞ Ls j N1 ≔ ⎜V ud > kc ⋅ ϕV c , , 0⎟ j s j ⎝ ⎠ N1 s
T
T
= 12 4 3 1 1 9
= 3 4 5 6 7 8
Spacing of shear reinforcement from face of suppo rt: s sp
0
sp
≔
T
0
sp ≔ augment sp , s
2
T
T
= 1.5 3 4 5 6 7 8
Number of stirrups or ties at each spacing from face of support: N
0
≔
V ud ≤ kc ⋅ ϕV c , 0 , 1
N ≔ augment N , N1 T
N
T
T
N
1
≔
N
1
≤ 1 , 0 , N − 1 1
= 1 11 4 3 1 1 9
Factored shear force as a function of distance x from face of support: V u x
≔ V uf − wu ⋅ x
Range variable i and distances x0 and x1 for plotting shear diagram: i≔0‥1
x
0
≔0
x
1
≔ X o
Factored Shear Force Vu and Shear capacity Vn in kips versus Distance from the Left Reaction to the Point of Zero Shear (in ft)
130 120 110 100 90
V u ⎛x ⎞ i
80 70 60
ϕV n
j3
50 40 30 20 10 0
0
1.5
3
4.5
6
x
i
7.5
9
10.5
12
13.5
15
16.5
xs
j3
This chart is formatted for values of Xo less than 16 feet, and values of shear force up to 120 kips. The chart may be reformatted for any v alues outside these limits by changing the values of Xo and maximum shear force, which serve as "markers" defining the limits of the chart.
Summary Input
Specified compressive strength of concrete:
f' c = 4
Unit weight of concrete:
wc = 145
Clear span length:
Ln = 24
Factor k, 1.15 for 1st interior support or 1 for all other supports:
k= 1.15
Effective depth:
d= 17.5
Area of shear reinforcement within distance s:
Av = 0.22
Specified yield strength of reinforcement:
f y = 60
Shear strength reduction factor for lightweight concrete:
kv = 1
Uniformly distributed factored load:
wu = 6.5 ― ft
Strength reduction factor for shear:
ϕv = 0.85
Beam web width:
bw = 12
Beam flange thickness:
h f = 4
Stirrup spacing factor:
SpF = 1
2
Computed Variables
Critical shear force:
V ud = 80.2
Maximum useable shear strength with shear reinforcement:
ϕV n_max = 112.9
Number of ties or stirrups at each spacing:
N
Stirrup or tie spacing from face of support:
sp
Total number of ties:
∑ N = 30
Maximum useable shear strength without shear reinforcement:
ϕV c = 22.6
T
= 1 11 4 3 1 1 9
T
= 1.5 3 4 5 6 7 8
Notes
1) For differing input variables some N values may be 0, which means that the particular spacing is not useable. 2) A minimum of 3 spaces will always be listed. The 1st spacing is half the minimum spacing, the 2nd is the minimum spacing, and the last spacing is the maximum. If the spacing is uniform throughout, the second and third spacing will be equal.
