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This portion of reinforced concrete design of spreading footing follows the requirement of ACI code 318-99. Factored loads should be used instead of service load. Factored footing pressure is used to determine footing depth and reinforcement.
The topics include:
Example 4: Check footing depth for punching shear and direct shear for footing subjected to axial column load only
Example 5: Check footing depth for punching shear and direct shear for footing subjected to axial column load and moment
The depth of footing usually governs by punching shear and direct shear because shear reinforcement is normally not used. For a square footing, punching shear usually governs the design. For a rectangular footing, direct shear may be more critical.
The critical section of punching shear is located at one half effective distance from the faces of column. Shear strength of concrete should be larger than factored shear stress at critical section as
The punching shear strength f v_{c} for normal weight concrete in ACI 318-05, -08, -11 is
[2.8]
where f = 0.75 for shear, c is the ratio of long to short sides of column, and f'_{c} is the compressive strength of concrete.
For sand light weight concrete, f v_{c} shall be multiplied by 0.85. For all light weight concrete, it shall be multiplied by 0.7.
The punching shear stress is factored shear force at the critical section divided by the perimeter of the critical section and the effective depth of the footing. The factored shear force at the critical section is factored column load minus factored footing pressure under the critical section. It can be calculated as
[2.9]
where
v_{u} is punching shear stress,
Q_{u} is factor footing pressure,
B, L are width and length of footing,
b, c are width and length of square column
d is effective depth of the footing.
Figure 2.3 Critical sections of punching shear
The critical section of direct shear is at one effective distance from the face of column. The direct shear strength of normal weight concrete specified in ACI 318-05, -08, -11 is
[2.10]
The factored shear stress at the critical section is the factored shear force divided by the width and depth of the footing at the critical section. The factored shear force at the critical section is the area from the critical section to the edge of the footing multiply average factored footing pressure in the area.
Given:
Column loads:
Live load: 25 kips
Dead load: 25 kips
Footing and column information:
Footing sizes = 4 ft x 4 ft x 1ft
Column size: 1 ft x 1 ft
Concrete strength at 28 day = 3000 psi
Design code: ACI 318-05
Requirement: Check if the footing depth is adequate for punching shear and direct shear.
Solution:
Factored column load P_{u}= 1.2*25+1.6*25= 70 kips
Factored footing pressure Q_{u}= 70/16=4.38 ksf
Effective footing depth d = 12”-3” (cover) – 0.5” (assumed diameter of #4 bar) = 8.5”
Check punching shear:
Area under critical section A = [(12”+8.5”)/12]2=2.92 ft2.
Factored shear force at critical section V_{u}= 70-4.38*2.92=57.2 kips
Perimeter of critical section p= 4 (12” + 8.5”) = 82”
Factor shear stress at critical section v_{u}= 57200/(82*8.5) = 82.1 psi
Punching shear strength of concrete fv_{c}= 0.75*(4Ö3000) = 164 psi > 82.1 psi O.K.
Check direct shear:
Distance from critical section to edge of footing l = 4/2-1/2-8.5/12=0.83’
Factored shear force V_{u}= 4.38*0.83*4=14.5 kips
Factored shear stress v_{u}= 14500/(4*12*8.5) = 35.6 psi
Director shear stress of concrete fv_{c}= 0.75*(2Ö3000) = 82 psi >35.6 psi O.K.
Given:
Column loads:
Live load: 25 kips
Dead load: 25 kips
Live load moment = 25 ft-kips
Dead load moment = 25 ft-kips
Footing and column information:
Footing sizes = 6 ft x 4 ft x 1ft
Column size: 1 ft x 1 ft
Concrete strength at 28 day = 3000 psi
Design code: ACI 318-05
Requirement: Check if the footing depth is adequate for punching shear and direct shear.
Solution:
Factored column load = 1.2*25+1.6*25= 70 kips
Factored column moment = 1.2*25+1.7*25=70 ft-kips
Eccentricity of factored column load = 70/70 = 1 ft
Maximum footing pressure = [70/(4x6)][1+1*6/6]=5.83 ksf
Minimum footing pressure = [70/(4x6)][1-1*6/6]=0 ksf
Effective footing depth = 12”-3” (cover) – 0.5” (assumed diameter of #4 bar) = 8.5”
Check punching shear:
Average factored footing pressure = (5.83+0)/2=2.92 ksf
Area under critical section = [(12”+8.5”)/12]2=2.92 ft^{2}.
Factored shear force at critical section = 70-2.92*2.92= 61.5 kips
Perimeter of critical section = 4 (12” + 8.5”) = 82”
Factor shear stress at critical section = 61500/(82*8.5) = 88.2 psi
Punching shear strength of concrete = 0.75*(4Ö3000) = 164 psi > 88.2 psi O.K.
Check direct shear:
Distance from critical section to edge of footing with maximum footing pressure
= 6/2-1/2-8.5/12=1.83’
Distance from critical section to edge of footing with maximum footing pressure
= 6-1.83=4.17’
Factor footing pressure at the location of critical section=5.83*4.17/6=4.05 ksf
Factored shear force = [(5.83+4.05)/2]*1.83*4=36.1 kips
Factored shear stress = 36100/(4*12*8.5) = 88.5 psi
Director shear stress of concrete = 0.75*(2Ö3000) = 82 psi < 88.5 psi N.G. Need to increase footing depth.