Determine depth of spread footings for shear

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:

• Punching shear (Two-way shear)

• Director shear (One-way shear)

• 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

Determine depth of footing for punching shear and direct shear

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.

Punching shear (Two-way shear)

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  in ACI 318-05 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.

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

Director shear (One-way shear)

The critical section of direct shear is at one effective distance from the face of column.  The direct shear strength of specified in ACI 318-05 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.

Figure 2.4. Critical section of direct shear

Example 4: Check footing depth for punching shear and direct shear for footing subjected to axial column load only

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.

Example 5: Check footing depth for punching shear and direct shear for footing subjected to axial column load and moment

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 ft².

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.