MATH: Reinforced Concrete Design 02

Chapter 10
Two-way Slab
When a rectangular reinforced-concrete slab is supported on all four sides, reinforcement placed perpendicular to the side may be assumed effective in the two directions. These slabs are known as two-way slabs. The bending on these slabs occurs in both directions. However, if a rectangular slab is supported in all four sides but the long side is two or more times the short side, the slab will, for all practical purposes, act as a one way slab, with bending occurring in the short direction.

The code specifies two methods of designing two-way slabs. These are the direct design method (Section 5.13.6) and equivalent frame method (Section 5.13.7). However, there are other methods that can be used. These include the strip method and moment coefficients method (Method 2).
COLUMN AND MIDDLE STRIPS
When the design moments have been determined by either the direct design method or equivalent frame method, the moments are distributed across each panel. The panel is divided into column and middle strips. Column strip is a design strip with a width on each side of a column centerline equal to 0.25 x L2 or 0.25 x L1, whichever is less. Column strip includes beams, if any. The middle strip is a design strip bounded by two column strips.


MINIMUM SLAB THICKNESS (SECTION 5.9.5.3.2)
The minimum thickness of slabs without interior beams spanning between the supports shall be in accordance with the provisions of Table 10.1 and shall not be less the following values:
Slabs without drop panels---------------------------------------------------------125 mm
Slabs with drop panels-------------------------------------------------------------100 mm
Table 10.1: Minimum Thickness of Slabs without Interior beams



Yield Stress f MPa Note (1)
Without drop panels
Note (2)
With drop panels
Note (2)

Exterior panels
Interior panels
Exterior panels
Interior panels

Without edge beams
With edge beams
Note (3)

Without edge beams
With edge beams
Note (3)

275
Ln/33
Ln/36
Ln/36
Ln/36
Ln/40
Ln/40
415
Ln/30
Ln/33
Ln/33
Ln/33
Ln/36
Ln/36

For values of reinforcement yield stress between 275 and 415 MPa minimum thickness shall be obtained by linear interpolation.
Drop panel is defined in Sections 5.13.4.7.1 and 5.13.4.7.2.
Slabs with beams between columns along exterior edges. The value of α for the edge beam shall not be less than 0.8.

The minimum thickness of slabs with or without beams spanning between the supports on all sides and having a ratio of long to short span not exceeding 2 shall be

h=Ln(800+0.73fy)36,000+5,000β[αm=0.12(1+1β)]
Eq. 10-1
but not less than
h=Ln(800+0.73fy)36,000+9,000β
Eq. 10-2
and need not be more than
h=Ln(800+0.73fy)36,000
Eq. 10-3

the values obtained from Eq. 10-1, Eq. 10-2, or Eq. 10-3 shall be modified as required by Sec. 5.9.5.3.4 and Sec. 5.9.5.3.5 but in no case shall the thickness be less than
For αm< 2.0 ………………………………………………………………………………………………………….125 mm
For αm 2.0 ……………………………………………………………………………………………………………90 mm
Section 5.9.5.3.4: For slabs without beams, but with drop panels extending in each direction from centerline of support a distance not less than one-sixth the span length in that direction measured center-to-center of supports, and projection below the slab at least one-quarter the slab thickness beyond the drop, thickness required by Eq. 10-1, Eq. 10-2, or Eq. 10-3 may be reduced by 10%.
Section 5.9.5.3.5: At discontinuous edges, an edge beam shall be provided with a stiffness ratio α not less than 0.80; or the minimum thickness required by Eq. 10-1, Eq. 10-2, or Eq. 10-3, shall be increased by at least 10% in the panel with a discontinuous edge.
where:
Ln= length of clear span in long direction of two-way construction, measured face-to-face of supports in slabs without beams and face-to-face of beams or other supports in other cases.
αm= average value of α for all beams on edges of a panel.


α = ratio of flexural stiffness of beam section to flexural stiffness a width of slab bounded laterally by centerline of adjacent panel (if any) in each side of beam.
α=EcbIbEcsIs
Eq. 10-4

β = ratio of clear spans in long to short direction of two-way slabs.
Ecb= modulus of elasticity of beam concrete
Ecs= modulus of elasticity of slab concrete
Ib= moment of inertia about centroidal axis of gross section of beams as defines in Sec.5.13.2.4.

Section 5.13.2.4: For monolithic or fully composite construction, a beam includes that portion of slab on each side of the beam extending a distance equal to the projection of the beam above or below the slab, whichever is greater, but not greater than four times the slab thickness.

Is= moment of inertia about centroidal axis of gross section of slab
Is= h3/12 times width of slab defined in notations α and β.

The student may find this Code procedure in solving the minimum slab thickness very tedious, but for slabs up to 8 m in length and with beams in all four sides having almost the same size, the minimum slab thickness is usually not governed by Eq. 10-1.
DIRECT DESIGN METHOD (SECTION 5.13.6)
Limitations of Direct Design Method
There shall be a minimum of three continuous spans in each direction.
Panels shall be rectangular with a ratio of longer to shorter span center-to-center of supports within a panel not greater than 2.
Successive span lengths center-to-center of supports in each direction shall not differ by more than one-third the longer span.
Columns be offset a maximum of 10 percent of the span (in direction of offset) from either axis between centerlines of successive columns.
All loads shall be due to gravity only and uniformly distributed over an entire panel. Live load shall not exceed three times dead load.
For a panel with beams between supports on all sides, the relative stiffness of beams in two perpendicular directions.




α1L22α2L12
Eq. 10-5
and shall not be less than 0.2 nor greater than 5.0.
where: L1= length of span in direction that moments are being determined, measured center-to-center of supports.
L2= length of span transverse to L1, measured center-to-center of supports. See also Figure 10.2.
Moments in Slabs (Section 5.13.6.2)
The total moment that is resisted by the slab equals absolute sum of positive and average negative factored moments in each direction shall not be less than
Mo=(WuL2)Ln28
Eq. 10-6
where Wuis the factored load in Pa or kPa.
If the transverse span of panels on either side of the centerline of supports varies, L2 in Eq. 10-6 shall be taken as the average of adjacent transverse spans. When the span adjacent and parallel to an edge is being considered, the distance from edge to panel centerline shall be substituted for L2 in Eq. 10-6.
Clear span Ln shall extend from face to face columns, capitals, brackets, or walls. Value of Lnused in Eq. 10-6 shall not be less than 0.65L1. Circular or regular polygon shaped supports shall be treated as square supports with the same area.
Negative and Positive Factored Moments (5.13.6.3)
Negative factored moments shall be located at face of rectangular supports. Circular or regular polygon shaped supports shall be treated as square supports with the same area.
In an interior span, total static moment Mo shall be distributed as follows:
Negative factored moment ………………………………………………………………………………… 0.65
Positive factored moment …………………………………………………………………………………. 0.35



In an end span, total factored static, moment Mo shall be distributed as given in Table 10.2.

Table 10.2: Distribution of total span moment in an end span

1
2
3
4
5



Exterior edge unrestrained

Slab with beams between all supports
Slab without beams between interior supports


Exterior edge fully restrained



Without edge beam*
With edge beam

Interior negative factored moment
0.75
0.70
0.70
0.70
0.65
Positive factored moment
0.63
0.57
0.52
0.50
0.35
Exterior negative factored moment
0
0.16
0.26
0.30
0.65
*See Sec. 5.13.6.3.6
Negative moment sections shall be designed to resist the larger of the two interior negative factored moments determined for spans framing into a common support unless an analysis is made to distribute the unbalanced moment in accordance with stiffness of adjoining elements.
Edge beams or edged of slab shall be proportioned to resist in torsion their share of exterior negative factored moments.
For moment transfer between slab and an edge column, column strip nominal moment strength provided shall be used as the transfer moment for gravity load.
Factored Moments in Column Strips
Column strips shall be proportioned to resist the following portions in percent of interior negative factored moments:
Table 10.3
I2/Is
0.5
1.0
2.0
(α1I2/I1)=0
(α1I2/I1) 1.0
75
90
75
75
75
45

Linear interpolation shall be made between values shown.


Table 10.4: Interpolated values of Table 10.3
Column strips shall be proportioned to resist the following portions in percent of exterior negative factored moments:
Table 10.5
L2/Ls

0.5
1.0
2.0
(α1L2/L1)=0
β1= 0
100
100
100

β1 2.5
75
75
75
(α1L2/L1) 1.0
β1= 0
100
100
100

β1 2.5
90
75
45

Linear interpolation shall be made between values shown.
Where supports consist of columns or walls extending for a distance equal to or greater than three-quarters the span length L2 used to compute Mo, negative moments shall be considered to be uniformly distributed across L2.
Column strips shall be proportioned to resist the following portions in percent of positive factored moments:
Table 10.6
L2/L1
0.5
1.0
2.0
(α1L2/L1)=0
(α1L2/L1) 1.0
60
90
60
75
60
45
Linear interpolations shall be made between values shown.
Table 10.7: Interpolated values of Table 10.6
For slabs with beams between supports, the slab portion of column strips shall be proportioned to resist that proportion of column strip moments not resisted by beams.
Factored Moments in Beams (Section 5.13.6.5)
Beams between supports shall be proportioned to resist 85 percent of column strip moments if (α1L2/L1) is equal to or greater than 1.0. for values of (α1L2/L1) between 1.0 and zero, proportion of column strip moments resisted by beams shall be obtained by linear interpolation between 85 and zero percent.
Factored Moments in Middle Strips
That portion of negative and positive factored moments 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 two half middle strips. A middle strip adjacent to and parallel with an edge supported by a wall shall be proportioned to resist twice the moment assigned to the half middle strip corresponding to the first row of interior supports.
Modification of Factored Moments
Negative and positive factored moments may be modified by 10 percent provided the total static moment for a panel in the direction considered is not less than that required by Eq. 10-6.




Factored Shear in Slab System with Beams
Beams with (α1L2/L1) equal to or greater than 1.0 shall be proportioned to resist shear caused by factored loads on tributary areas bounded by 45° lines drawn from the corners of the panels and the center-lines of the adjacent panels parallel to the long sides. Beams with (α1L2/L1) less than 1.0 may be proportioned to resist shear obtained by linear interpolation, assuming beams carry no load at α = 0.






















ILLUSTRATIVE PROBLEMS
DESIGN OF SQUARE FOOTING
Problem 9.1
A square column footing is to support a 400-mm square tied column that carries a dead load of 880 kN and a live load of 710 kN, the column is reinforced with 8-25mm bars. The base of the footing is 1.50 m below the natural grade where the allowable soil pressure is 235 kPa. The soil above the footing has a weight of 15.6 kN/m3. Assuming fy = 27.5 MPa, fc = 27.5 MPa, and unit weight of concrete as 23.50 kN/m3, design the footing. Use 25 mm main bars.
Solution
Our first task in the design of footing is the determination of its depth. This requires several cycles of trial and error procedure because its value affects the effective soil bearing capacity. There are several rules of thumb used by designers for making initial thickness estimates, such as 20% of the footing width plus 75mm. However, with the aid of computer (available at GERTC), this will become easier.
Initial estimate of footing depth:
Aftg= L2= (880+710)/235
L = 2.6 m
L = 2600 mm
Depth = 20% (2600) + 75 = 595 mm say 600 mm

Effective soil bearing capacity:
qc = qa-γh
= 235-23.5 (0.6) – 15.6 (1.5 – 0.6)
qc = 206.86 kPa

Dimension of the footing:
Aftg=Unfactored loadqc
= 880+710206.86
Aftg = 7.69 m2 = L x L
L= 2.77 m say 2.8 m
Footing dimension = 2.8 m x 2.8 m







qu=Factored loadArea of Footing
=1.4880+1.7(710)2.8(2.8)=311.1 kPa
qu=0.3111 MPa

Based on wide-beam shear:
Vu=quAshaded
= (0.3111)[2800(1200-d)]
Vu=871.081200-dN
Vc=16f'cbwd
=1627.5(2800)d
Vc=2447.2d N

Vu=ϕvc
871.08(1200- d) = 0.85(2447.2d)
1200 – d = 2.388d
d= 354.2 mm

Based on the two-way or punching shear:
Vu=quAshaded
= (0.3111)[(2800)² - (400 + d)²]
= 0.3111 (2800² - 160000 – 800d - d²)
Vu=0.3111(7,680,000-800d-d2)
Vc=13f'cbod
bo=4400+d
Vc=1327.54400+dd
Vc=6.99 (400d+d2)

[Vu=ϕvc]
0.3111(7,680,000 – 800d - d²) – 0.85[6.99(400d + d²)]
7,680,000 – 800d - d² - 7639d + 19d²
d=-8439±84392-420(-7,680,000)2(20)
d= 443.6 mm say 445 mm







Total depth of footing = 445 + 1.5(25) + 75
Total depth = 557.5 mm < 600 mm (OK)

Required Steel Area:
d= 445 mm
Mu=311.11.22.81.22
= 627.18 kN-m
Mu=ϕRubd2

627.18 x 106-0.9Ru28004452
Ru=1.26 MPa

ρ=0.85f'cfy1-1-2Ru0.85f'c
=0.85(27.5)2751-2(1.26)0.85(27.5)
ρ = 0.00471

ρmin=1.4fy=1.4275
ρmin=0.005091

Use ρ = 0.005091
As=ρbd
= 0.005091(2800)(445)
As=6,343 mm2

Number of 25-mm bars:
π4252N=6,343
N = 12.9 say 13

Development Length:
ldb=0.02Abfyf'c
ldb= 0.02 x π4252(275)/ 27.5 = 515 mm
or ldb=0.06dbfy=0.0625275 = 412.5 mm
Furnished Ld - 1200 – 75 – 1125 mm > 515 mm (OK)






Verify if dowels or column bars extension are necessary:
Actual bearing strength = Pu=1.4880+ 1.7710
Pu=2439 kN

Permissible bearing stress:
Φ 0.85fc.A1=0.70.8527.54002
= 2,618,000 N
Φ 0.85fc.A1=2,618 kN

But this may be multiplied by A2A1 2
A1=0.4 x 0.4=0.16 m²
A2=2.8 x 2.8=7.84 m²
A2A1 =7 use 2
Permissible bearing stress = 2,618(2) = 5,236 kN > 2,439 kN (no need)

Minimum area of dowel or extension bar required by the Code:
Area = 0.005(400 x 400) = 800 mm²
At least two column bars (25-mm) must be extended into the footing.

Use 2.8 m x 2.8 m footing with an effective depth to top bars of 445 mm (total depth = 560 m), with 13-25 mm bars on each side of the footing, and at least two column bars (25-mm as given) must be extended into the footing.