Transcript Lean Construction
Reinforced Concrete Design-8
Design of 2 way Slabs
By
Dr. Attaullah Shah
Swedish College of Engineering and Technology Wah Cantt.
One way vs Two way slab system
L>2S
As ≥ Temp. steel Min. Spacing ≥
∅
main steel ≥ 4/3 max agg. ≥ 2.5 cm (1in) Max. Spacing ≤ 3 t ≤ 45 cm (17in)
Min slab thickness =
Shear Strength of Slabs
In two-way floor systems, the slab must have adequate thickness to resist both bending moments and shear forces at critical section. There are three cases to look at for shear.
1.
Two-way Slabs supported on beams 2.
Two-Way Slabs without beams 3.
Shear Reinforcement in two-way slabs without beams.
Shear Strength of Slabs Two-way slabs supported on beams
The critical location is found at d distance from the column, where
V
c 2
f
c
bd
The supporting beams are stiff and are capable of transmitting floor loads to the columns.
Shear Strength of Slabs
The shear force is calculated using the triangular and trapezoidal areas. If no shear reinforcement is provided, the shear force at a distance d from the beam must equal
V
ud
V
c 2
f
c
bd
where,
V
ud
w
u
l
2 2
d
Shear Strength of Slabs Two-Way Slabs without beams
There are two types of shear that need to be addressed 1.
One-way shear or beam shear at distance d from the column 2.
Two-way or punch out shear which occurs along a truncated cone.
Shear Strength of Slabs
1.
One-way shear or beam shear at distance d from the column 2.
Two-way or punch out shear which occurs along a truncated cone.
Shear Strength of Slabs
One-way shear considers critical section a distance d from the column and the slab is considered as a wide beam spanning between supports.
V
ud
V
c 2
f
c
bd
Shear Strength of Slabs
Two-way shear fails along a a truncated cone or pyramid around the column. The critical section is located d/2 from the column face, column capital, or drop panel.
Shear Strength of Slabs
If shear reinforcement is not provided, the shear strength of concrete is the smaller of:
V
c 2 b 4 c
f
c
b
o
d
4
f
c
b
o
d
b o b c = = perimeter of the critical section ratio of long side of column to short side
Shear Strength of Slabs
If shear reinforcement is not provided, the shear strength of concrete is the smaller of:
V
c a s
d b
o
2
f
c
b
o
d
a s is 40 for interior columns, 30 for edge columns, and 20 for corner columns.
Shear Strength of Slabs Shear Reinforcement in two-way slabs without beams
.
For plates and flat slabs, which do not meet the condition for shear, one can either - Increase slab thickness - Add reinforcement Reinforcement can be done by shearheads, anchor bars, conventional stirrup cages and studded steel strips.
Shear Strength of Slabs Shearhead
consists of steel I-beams or channel welded into four cross arms to be placed in slab above a column. Does not apply to external columns due to lateral loads and torsion.
Shear Strength of Slabs Anchor bars
consists of steel reinforcement rods or bent bar reinforcement
Shear Strength of Slabs Conventional stirrup cages
Shear Strength of Slabs Studded steel strips
Shear Strength of Slabs
The reinforced slab follows section 11.12.4 in the ACI Code, where V n can not
V
c 4
f
c
b
o
d V
n
V
c
V
s 6
f
c
b
o
d V
s
A
v
f
y
d
The spacing, s, can not exceed d/2.
s
If a shearhead reinforcement is provided
V
n 7
f
c
b
o
d
Example Problem
Determine the shear reinforcement required for an interior flat panel considering the following: V u = 195k, slab thickness = 9 in., d = 7.5 in., f c = 3 ksi, f y = 60 ksi, and column is 20 x 20 in.
Example Problem
Compute the shear terms find b 0 for
V
c
4
f b d
c 0
b
0 4 column width 110 in.
d
Example Problem
Compute the maximum allowable shear
V
c 4
f b d
c 0 1 k 1000 lbs 135.6 k V u =195 k > 135.6 k Shear reinforcement is need!
Example Problem
Compute the maximum allowable shear
V
c 6
f b d
c 0 1 k 1000 lbs 203.3 k So V n >V u Can use shear reinforcement
Example Problem
Use a shear head or studs as in inexpensive spacing. Determine the a for
V
c 2
f b d
c 0
b
0 4 column width 2
a
Example Problem
Determine the a for
V
u 2 19500 lb
f b d
c 0 41.8 in.
2
a
7.5 in.
The depth = a+d = 41.8 in. +7.5 in. = 49.3 in. 50 in.
Example Problem
Determine shear reinforcement
V
s
V
u
V
c 59.4 k The V s per side is V s / 4 = 14.85 k
Example Problem
Determine shear reinforcement
V
s 14.85 k 0.75
19.8 k Use a #3 stirrup A v = 2(0.11 in 2 ) = 0.22 in 2
V
s
A f d
v y
s A f d
v y
V
s
Example Problem
Determine shear reinforcement spacing
s
A f d
v y
V
s 0.22 in 2 19.8 k 5.0 in.
Maximum allowable spacing is
d
2 7.5 in.
3.75 in.
2
Example Problem
Use s = 3.5 in.
# of stirrups 50 in.
3.5 in.
14.3 Use 15 stirrups The total distance is 15(3.5 in.)= 52.5 in.
Example Problem
The final result: 15 stirrups at total distance of 52.5 in. So that a = 45 in. and c = 20 in.
Direct Design Method for Two-way Slab
Method of dividing total static moment M o positive and negative moments.
into Limitations on use of Direct Design method 1.
Minimum of 3 continuous spans in each direction. (3 x 3 panel) 2.
Direct Design Method for Two-way Slab
Limitations on use of Direct Design method 3.
Successive span in each direction shall not differ by more than 1/3 the longer span.
4.
Columns may be offset from the basic rectangular grid of the building by up to 0.1 times the span parallel to the offset.
Direct Design Method for Two-way Slab
Limitations on use of Direct Design method 5.
All loads must be due to gravity only (N/A to unbraced laterally loaded frames, from mats or pre-stressed slabs) 6.
load
Direct Design Method for Two-way Slab
Limitations on use of Direct Design method 7.
For panels with beams between supports on all sides, relative stiffness of the beams in the 2 perpendicular directions.
a 1
l
2 2 a 2
l
1 2 Shall not be less than 0.2 nor greater than 5.0
Definition of Beam-to-Slab Stiffness Ratio,
a Accounts for stiffness effect of beams located along slab edge reduces deflections of panel adjacent to beams.
a
flexural stiffness flexural stiffness of beam of slab
Definition of Beam-to-Slab Stiffness Ratio,
a a 4E cb
I
b /
l
4E cs
I
s /
l
4E cb
I
b 4E cs
I
s E cb I I b s Modulus of elasticity E sb Modulus of elasticity of slab concrete Moment of inertia Moment of inertia of of of beam concrete uncracked uncracked beam slab With width bounded laterally by centerline of adjacent panels on each side of the beam.
Two-Way Slab Design
Static Equilibrium of Two-Way Slabs
Analogy of two-way slab to plank and beam floor Section A-A: Moment per ft width in planks Total Moment
M
f
M
wl
1 2 k ft/ft 8 2
l
1 2 8 k ft
Two-Way Slab Design
Static Equilibrium of Two-Way Slabs
Analogy of two-way slab to plank and beam floor Uniform load on each beam Moment in one beam (Sec: B-B)
wl
1 k/ft 2
M
lb
wl
1 2
l
2 2 k ft 8
Two-Way Slab Design
Static Equilibrium of Two-Way Slabs
M
1
l
2 2 k ft Total Moment in both beams 8 Full load was transferred east-west by the planks and then was transferred north-south by the beams; The same is true for a two-way slab or any other floor system.
Basic Steps in Two-way Slab Design
1.
Choose layout and type of slab.
2.
Choose slab thickness to control deflection. Also, check if thickness is adequate for shear.
3.
Choose Design method − Equivalent Frame Method- use elastic frame analysis to compute positive and negative moments − Direct Design Method - uses coefficients to compute positive and negative slab moments
Basic Steps in Two-way Slab Design
4.
Calculate positive and negative moments in the slab.
5.
Determine distribution of moments across the width of the slab. - Based on geometry and beam stiffness.
6.
Assign a portion of moment to beams, if present.
7.
8.
Design reinforcement for moments from steps 5 and 6.
Check shear strengths at the columns
Minimum Slab Thickness for two-way construction
Maximum Spacing of Reinforcement At points of max. +/- M:
s
and
s
2
t
ACI 13.3.2
18 in.
ACI 7.12.3
Min Reinforcement Requirements
A
A
Distribution of Moments
Slab is considered to be a series of frames in two directions:
Distribution of Moments
Slab is considered to be a series of frames in two directions:
Distribution of Moments
Total static Moment, M o where
M
0
w
u
l
2
l
n 2 ACI 13 3 8
w
u factored load per unit area
l
2 transvers e width of the strip
l
n clear span between columns for circular columns, calc.
l
n using h 0.886d
c
Column Strips and Middle Strips
Moments vary across width of slab panel Design moments are averaged over the width of column strips over the columns & middle strips between column strips.
Column Strips and Middle Strips
Column strips Design w/width on either side of a column centerline equal to smaller of 0 .
25
l
2 0 .
25
l
1 l 1 = length of span in direction moments are being determined.
l 2 = length of span transverse to l 1
Column Strips and Middle Strips
Middle strips: Design strip bounded by two column strips.
Positive and Negative Moments in Panels
M 0 is divided into + M and -M Rules given in ACI sec. 13.6.3
Moment Distribution
Positive and Negative Moments in Panels
M 0 is divided into + M and -M Rules given in ACI sec. 13.6.3
M
u
M
u
M
0
w
u
l
2
l
n 2 8
Longitudinal Distribution of Moments in Slabs
For a typical interior panel, the total static moment is divided into positive moment 0.35 M o moment of 0.65 M o . and negative For an exterior panel, the total static moment is dependent on the type of reinforcement at the outside edge.
Distribution of M 0
Moment Distribution
The factored components of the moment for the beam.
Transverse Distribution of Moments
The longitudinal moment values mentioned are for the entire width of the equivalent building frame. The width of two half column strips and two half-middle stripes of adjacent panels.
Transverse Distribution of Moments
Transverse distribution of the longitudinal moments to middle and column strips is a function of the ratio of length l 2 /l 1 , a 1 , and b t .
Transverse Distribution of Moments
Transverse distribution of the longitudinal moments to middle and column strips is a function of the ratio of length l 2 /l 1 , a 1 , and b t .
a 1
E
cb
I
b b t
E
cs
I
s
E
cb
C
2
E
cs
I
s
C
1 0 .
63
x y
x
3
y
3 torsional constant
Distribution of M 0
ACI Sec 13.6.3.4
For spans framing into a common support negative moment sections shall be designed to resist the larger of the 2 interior M u ’s ACI Sec. 13.6.3.5
Edge beams or edges of slab shall be proportioned to resist in torsion their share of exterior negative factored moments
Factored Moment in Column Strip
a 1 Ratio of flexural stiffness of beam to stiffness of slab in direction l 1 .
b t Ratio of torsional stiffness of edge beam to flexural stiffness of slab(width= to beam length)
Factored Moment in an Interior Strip
Factored Moment in an Exterior Panel
Factored Moment in an Exterior Panel
Factored Moment in Column Strip
a 1 Ratio of flexural stiffness of beam to stiffness of slab in direction l 1 .
b t Ratio of torsional stiffness of edge beam to flexural stiffness of slab(width= to beam length)
Factored Moment in Column Strip
a 1 Ratio of flexural stiffness of beam to stiffness of slab in direction l 1 .
b t Ratio of torsional stiffness of edge beam to flexural stiffness of slab(width= to beam length)
Factored Moment in Column Strip
a 1 Ratio of flexural stiffness of beam to stiffness of slab in direction l 1 .
b t Ratio of torsional stiffness of edge beam to flexural stiffness of slab(width= to beam length)
Factored Moments
Factored Moments in beams (ACI Sec. 13.6.3) Resist a percentage of column strip moment plus moments due to loads applied directly to beams.
Factored Moments
Factored Moments in Middle strips (ACI Sec. 13.6.3) The portion of the + M u and - M u not resisted by column strips shall be proportionately assigned to corresponding half middle strips.
Each middle strip shall be proportioned to resist the sum of the moments assigned to its 2 half middle strips.
ACI Provisions for Effects of Pattern Loads
The maximum and minimum bending moments at the critical sections are obtained by placing the live load in specific patterns to produce the extreme values. Placing the live load on all spans will not produce either the maximum positive or negative bending moments.
ACI Provisions for Effects of Pattern Loads
1.
The ratio of live to dead load. A high ratio will increase the effect of pattern loadings.
2.
The ratio of column to beam stiffness. A low ratio will increase the effect of pattern loadings.
3.
Pattern loadings. Maximum positive moments within the spans are less affected by pattern loadings.
Reinforcement Details Loads
After all percentages of the static moments in the column and middle strip are determined, the steel reinforcement can be calculated for negative and positive moments in each strip.
M
u
A
s
f
y
d
a
2
R
u
bd
2
Reinforcement Details Loads
Calculate R =0.9. A s u and determine the steel ratio r , where = r bd. Calculate the minimum A s from ACI codes. Figure 13.3.8 is used to determine the minimum development length of the bars.
R
u
w
u
f
c 1 0 .
59
w
u
w
u r
f
y
f
c
Minimum extension for reinforcement in slabs without beams(Fig. 13.3.8)
Moment Distribution
The factored components of the moment for the beam.
Transverse Distribution of Moments
The longitudinal moment values mentioned are for the entire width of the equivalent building frame. The width of two half column strips and two half-middle stripes of adjacent panels.