1. Theory of Reinforced Concrete and Lab. II
11. Footings and Foundations
by Prof. Jae-Yeol Cho
INTRODUCTION
SPREAD FOOTINGS
DESIGN FACTORS
LOADS, BEARING PRESSURES, AND FOOTING SIZE
WALL FOOTINGS
COLUMN FOOTINGS
COMBINED FOOTINGS
TWO-COLUMN FOOTINGS
STRIP, GRID, AND MAT FOUNDATIONS
PILE CAPS
Theory of Reinforced Concrete and Lab. II
Fall 2009

2. 11. Footings and Foundations
INTRODUCTION
Functions
As a part of structure, transmits the load to the underlying soil or rock, and spread the load over a sufficiently large area.
Requirements
The total settlement of the structure should be limited to a tolerably small amount.
The differential settlement of the various parts of the structure should be eliminated as nearly as possible.
 limiting the differential settlement is more IMPORTANT.
Theory of Reinforced Concrete and Lab II Fall 2009

3. 11. Footings and Foundations
INTRODUCTION
Types
- Spread foundation (footing) :
If satisfactory soil directly underlies the structure, it is merely necessary to spread the load.
Deep foundation (piles or caissons) :
If adequate soil is not found immediately below the structure, it becomes necessary to use deep foundation.
- Mat foundation
Spread foundation: 확대기초
Deep foundation: 깊은기초
Mat foundation: 전면기초
Theory of Reinforced Concrete and Lab II Fall 2009

4. 11. Footings and Foundations
SPREAD FOOTINGS
Most frequently used types of foundations on soils of reasonable bearing capacity.
Spread footings
- wall footings
- single-column footings
- combined / strap footings
If the soil is weak and/or column loads are great, the required footing areas become so large as to be uneconomical.
Wall
Property line
W 줄기초
S 독립확대기초
C 연결확대기초/복합확대기초
Theory of Reinforced Concrete and Lab II Fall 2009

5. 11. Footings and Foundations
DESIGN FACTORS
(a)
(b)
(c)
The bearing pressure is assumed to be uniformly distributed.
Coarse-grained soils – individual grains moves outward.
Clay soils – shear resistance around the perimeter.
Theory of Reinforced Concrete and Lab II Fall 2009

6. 11. Footings and Foundations
DESIGN FACTORS
(b)
(c)
It is customary to disregard these nonuniformities,
Because their numerical amount is uncertain and highly variable depending on types of soils
and their influence on bending moment and shearing forces in the footings is relatively small.
Theory of Reinforced Concrete and Lab II Fall 2009

7. 11. Footings and Foundations
Note
Layout of structural supports vary widely and soil conditions differ from site to site and within a site.
As a result, the type of foundation to be selected has to be governed by these factors and by optimal cost considerations.
Basic knowledge of soil mechanics and foundation engineering is assumed when designing footings.
Background knowledge of the methodology determining the resistance of soils is necessary to select the appropriate bearing capacity value for the particular site and the particular foundation system under consideration.
Theory of Reinforced Concrete and Lab II Fall 2009

8. 11. Footings and Foundations
LOADS, BEARING PRESSURES, AND FOOTING SIZE
Allowable Bearing Pressures
under service loads
to keep settlements within tolerable limits.
(1)
Footing size
For concentrically loaded footings, the required area
In the presence of wind and earthquake effects
하중계수가 적용되지 않음에 주의
(2) or
(3)
Theory of Reinforced Concrete and Lab II Fall 2009

9. 11. Footings and Foundations
LOADS, BEARING PRESSURES, AND FOOTING SIZE
Footing Size
Note
Footing sizes are determined for unfactored service loads and soil pressures. This is because, for footing design, safety is provided by the overall safety factors (2.5~3.0)
The load used in determination of footing size must be calculated at the level of the base of the footing
; footing weight and fill pressure should be included.
Theory of Reinforced Concrete and Lab II Fall 2009

10. 11. Footings and Foundations
LOADS, BEARING PRESSURES, AND FOOTING SIZE
Eccentricity Load Effect
When the eccentric moment is very large, tensile stress on one side of the footing can result, since the bending stress distribution depends on the magnitude of load eccentricity.
It is always advisable to proportion the area of these footings such that the loads fall within the middle kern.
In such a case, the location of load is in the middle third of the footing dimension in each direction, thereby avoiding tension in the soil that can theoretically occur prior to stress redistribution.
Theory of Reinforced Concrete and Lab II Fall 2009

11. 11. Footings and Foundations
Theory of Reinforced Concrete and Lab II Fall 2009

12. 11. Footings and Foundations
LOADS, BEARING PRESSURES, AND FOOTING SIZE
Eccentricity Load Effect
Case 1 : e1 < L/6 (Fig.(a))
The direct stress P/Af is larger than the bending stress Mc/I
(4)
Theory of Reinforced Concrete and Lab II Fall 2009

13. 11. Footings and Foundations
LOADS, BEARING PRESSURES, AND FOOTING SIZE
Eccentricity Load Effect
Case 2 : e2 = L/6 (Fig.(b))
direct stress =
bending stress =
where
s and L are the width and length of the footing, respectively
(5)
(6)
Theory of Reinforced Concrete and Lab II Fall 2009

14. 11. Footings and Foundations
LOADS, BEARING PRESSURES, AND FOOTING SIZE
Eccentricity Load Effect
Case 2 : e2 = L/6 (Fig.(b))
In order to find the limiting case where no tension exists on the footing, the direct stress P/Af has to equivalent to the bending stress so that
(7)  
(8)
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15. 11. Footings and Foundations
Case 3 : e3 > L/6 (Fig.(c))
If the maximum bearing pressure qmax due to load P does not exceed the qa of the soil, no uplift is expected at the left end of the footing.
The center of gravity of the triangular bearing stress distribution coincides with the point of action of load P
Therefore, the width of the triangle is
Hence, the maximum comprehensive bearing stress is
(9)
(10)
Theory of Reinforced Concrete and Lab II Fall 2009

16. 11. Footings and Foundations
WALL FOOTING
The simple principle of beam action apply to wall footings
with only minor modifications.
For concrete wall, the moment
is computed at the face of the wall.
The vertical shear force is computed
at a distance d from the wall face
(11)
(12)
Theory of Reinforced Concrete and Lab II Fall 2009

17. 11. Footings and Foundations
WALL FOOTING
Note
For footings supporting masonry walls, the maximum moment
is supported midway between the middle and the face of the
wall.
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18. 11. Footings and Foundations
Example 11.1 Design of Wall Footing
A 400mm concrete wall supports a dead load D=180 kN/m and a live load L=140 kN/m. The allowable bearing pressure is qa = 200 kN/m2 at the level of the bottom of the footing, which is 1.2 m below grade.
Design a footing for this wall using fck=21 MPa and fy=300MPa.
400 mm
1.2 m
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19. 11. Footings and Foundations
Solution
Assume the thickness of footing 300 mm.
footing concrete unit weight 24 kN/m3
 Footing weight per square meter is
2) The weight of the ( ) meter fill on the top of the footing is
unit weight of soil
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20. 11. Footings and Foundations
3) The portion of the allowable bearing pressure that is available or effective for carrying the wall load is
4) The required width of the footing is
5) The bearing pressure for strength design of the footing is
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21. 11. Footings and Foundations
6) The factored moment for strength design is
7) Assume d=220m , the shear at section 2-2 is
Shear usually governs the depth of footings, particularly since the use of shear reinforcements in footing is generally avoided as UNECONOMICAL .
Theory of Reinforced Concrete and Lab II Fall 2009

22. 11. Footings and Foundations
8) The design shear strength per meter is
Homework #5
Complete the design and sketch the reinforcements and
discuss it !
Theory of Reinforced Concrete and Lab II Fall 2009

23. (a) Single slab footing
11. Footings and Foundations
COLUMN FOOTINGS
An individual reinforced concrete footing for a column, also termed an isolated spread footings, is the most common and most economical of the various types of footings used for structures.
Column footings are usually square.
; rectangular for space restriction or
column with large length/width
(a) Single slab footing
(b) Stepped footing
(c) Sloped footing
Theory of Reinforced Concrete and Lab II Fall 2009

24. 11. Footings and Foundations
COLUMN FOOTINGS
The single column footing behavior under concentric load is that of TWO-WAY cantilever action.
- The footing is loaded upward by the soil pressure
- Tensile stresses are induced in both of directions at the bottom of the footing
- The footing should be reinforced by two layers perpendicular to each other and parallel to the edges.
Theory of Reinforced Concrete and Lab II Fall 2009

25. allowable soil bearing pressure, qa
11. Footings and Foundations
The required bearing area Areq (footing-soil contact area)
Total loads (including the weight of footing)
In computing bending moment and shears, only the upward pressure qu caused by the factored column loads is considered, that is, the weight of column is not included. Why? =
allowable soil bearing pressure, qa
Note
Weight of footing must be estimated and usually amount to 4 to 8 percent of the column load.
편평한 책상위에 있는 책에 휨이나 전단이 발생하지 않는 것과 같은 원리.
Theory of Reinforced Concrete and Lab II Fall 2009

26. 11. Footings and Foundations
COLUMN FOOTINGS
Shear Design
In single footings, the effective depth d is mostly governed by SHEAR
- It is generally not economical in footings to use shear reinforcement.
 All shear is carried by the concrete in footing design
TWO types of shear behaviors are distinguished in footings
- one-way shear : beam shear
- two-way shear : punching shear
Theory of Reinforced Concrete and Lab II Fall 2009

27. 11. Footings and Foundations
COLUMN FOOTINGS
Shear Design
Punching shear
The two-way shear is termed punching shear, since the column or pedestal tends to punch through the footing, induces stresses around the perimeter of the column
- This average shear stress acts on vertical planes through the critical section abcd which is at a distance of d/2
Punching shear failure
in single footing
Critical sections for shear
Theory of Reinforced Concrete and Lab II Fall 2009

28. 11. Footings and Foundations
COLUMN FOOTINGS
Shear Design
Punching shear
punching shear stress
+ vertical compression
+ horizontal compression in both direction
 This triaxiality of stresses increases the shear strength of the concrete
shear stress at the critical perimeter ≥ beam action shear
 due to the stresses spreading out from the column
 due to the biaxial bending moments
Theory of Reinforced Concrete and Lab II Fall 2009

29. 11. Footings and Foundations
Punching shear
KCI Code provisions (KCI ) for nominal punching-shear strength, whichever is smaller.
for square column
(13)
for column of
elongated cross section
(14)
for very large bo/d
(15)
Beta_c = 무한대? -- 결국 oneway shear 와 같은 값이 된다.
where βc=a/b is the ratio of the long to short sides of the column cross section
αs=40, 30, and 20 for interior, edge, and corner loading, respectively.
Theory of Reinforced Concrete and Lab II Fall 2009

30. 11. Footings and Foundations
Shear Design
One-way shear (beam action)
Shear failures can also occur, as in beams or one-way slabs, at a section a distance d from the column.
Therefore, the nominal shear strength is
(16) or
(17)
Theory of Reinforced Concrete and Lab II Fall 2009

31. 11. Footings and Foundations
Shear Design
The required depth of footing d is calculated from
But should be separately applied to punching shear and one-way shear.
여기서 파이는 0.75
Theory of Reinforced Concrete and Lab II Fall 2009

32. 11. Footings and Foundations
COLUMN FOOTINGS
Bearing Design
; Transfer of load from column into footing
All loads applied to a column must be transferred to the top of the footing (through a pedestal, if there is one) by compression in the concrete, by reinforcement, or by both.
The allowable bearing stress on the actual area A1 of the column base or footing top area of contact is
where,
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33. 11. Footings and Foundations
COLUMN FOOTINGS
Bearing Design
When the supporting area is wider than the loaded area on ALL sides, the design bearing strength is (KCI 6.8)
Loaded
area
(18)
A2 measured
on this plane
where φ=0.65
fck is the cylinder strength of the footing concrete
Theory of Reinforced Concrete and Lab II Fall 2009

34. 11. Footings and Foundations
Bearing Design
Note
A minimum area of reinforcement of 0.005Ag has to be provided across the interface of the column and the footing EVEN when the concrete bearing strength is not exceeded, Ag being the gross area of the column cross section.
Theory of Reinforced Concrete and Lab II Fall 2009

35. 11. Footings and Foundations
Flexure Design
Critical sections
Theory of Reinforced Concrete and Lab II Fall 2009

36. 11. Footings and Foundations
Flexure Design
Square footings
The reinforcement should be uniformly distributed over the width of the footing in each direction.
Since the bending moment is the same in each direction, the reinforcing bar size and spacing should be the same in each direction.
However in reality, the effective depth is not the same in both directions.
 Determine As based on the same average depth of two layers.
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37. 11. Footings and Foundations
Flexure Design
Rectangular footings
Rectangular footings are generally used where space limitations require it.
In two-way rectangular footing supporting one column, the bending moment in the short direction is taken as EQUIVALENT to that in the long directions.
The distribution of reinforcement differs in the long and short directions.
The effective depth is assumed to be equal in both the long and short directions.
Theory of Reinforced Concrete and Lab II Fall 2009

38. 11. Footings and Foundations
Rectangular footings
Code recommendations for reinforcement distribution
(KCI )
- Reinforcement in the long direction is to be uniformly distributed over the short width of the footing.
- For reinforcement in the short direction, a central band of width equal to the width of footing in the short direction shall contain a major portion of the reinforcement total areas.
Theory of Reinforced Concrete and Lab II Fall 2009

39. 11. Footings and Foundations
Flexure Design
Rectangular footings
Code recommendation for reinforcement distribution
(KCI )
- Equation uniformly distributed along the band width
where β is the ratio of the long to the short side of the footing.
- The remainder of the reinforcement is uniformly distributed outside the center band of footing.
Reinforcement in band width
Total reinforcement in short direction β+1 =
(19)
Theory of Reinforced Concrete and Lab II Fall 2009

40. 11. Footings and Foundations
Flexure Design
Minimum reinforcement requirement (KCI 6.3.2)
The usual minimum flexural reinforcement ratios for beams need NOT be applied to either slabs or footings.
Instead, the min. requirements for shrinkage and temperature crack control for structural slab are to be applied. (KCI 5.7.2)
(a) Slab where fy≤400MPa deformed bars are used
(b) Slab where fy≥400MPa measured at a yield strain of 0.35% is used x400/fy
Theory of Reinforced Concrete and Lab II Fall 2009

41. 11. Footings and Foundations
Flexure Design
Minimum reinforcement ratio (KCI 6.3.2)
The max. spacing of bars in the direction of the span is reduced to
the lesser 3h and 450mm
rather than 5h as is normal for shrinkage/temperature steel.
These requirements are to be applied to mat foundations as well as individual footings.
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42. 11. Footings and Foundations
Example 11.2 Design of square footing
A column 450mm square, with fck=24MPa supports a dead load of 1,000kN, and a live load of 800kN. The allowable soil pressure qa is 250kN/m2.
Design a square footing with base 1.5m below grade, using fck=24MPa and fy=300MPa.
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43. 11. Footings and Foundations
Solution
1) Average unit weight of material between the bottom of footing and the surface.
That space is occupied partly by concrete and soil.
Assume the unit weight is 20kN/m3
2) The pressure of this material at 1.5m depth
 qe=(250-30)=220kN/m3 is available to carry the column service load.
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44. 11. Footings and Foundations
3) The required footing area
 A base 2.85m square is selected
4) For strength design calculate the upward pressure.
5) Determine the depth of footing
(usually determined based on the punching shear)
Trial calculation suggest d= 500mm
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45. 11. Footings and Foundations
The length of critical perimeter is
6) The shear force acting on this perimeter is
7) The corresponding nominal shear strength is
250mm
250mm
2,850mm
column
450x450mm
700mm
500mm
1,200mm
2,850mm 
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46. 11. Footings and Foundations
8) Check the selected d=500mm for one-way (beam) shear on critical section ef.
The factored shear force acting on that section is
250mm
250mm
2,850mm
column
450x450mm
9) The corresponding nominal shear strength is
700mm
500mm
1,200mm
2,850mm
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47. 11. Footings and Foundations
Therefore, the assumed d=500mm is adequate for one-way shear as well as punching shear.
10) The bending moment on critical section gh is
250mm
250mm
2,850mm
column
450x450mm
700mm
500mm
1,200mm
Note
Because the depth required for shear is GREATLY in excess of that required for bending, the reinforcement ratio will be LOW and the corresponding depth of the rectangular stress block is small.
2,850mm
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48. 11. Footings and Foundations
Homework #6
Check the previous design for flexure and bearing pressure and complete the design.
Should illustration and discussion be included!!
Theory of Reinforced Concrete and Lab II Fall 2009