📘 Lecture: Column Footing Stresses
1. Introduction
Column footings (also called isolated footings or pad foundations) are structural elements that
transfer loads from a column into the supporting soil. The stress distribution beneath the
footing is a critical design consideration because:
If soil stresses exceed its bearing capacity → shear failure of soil.
If stresses are uneven → differential settlement and tilting of the structure.
2. Load Transfer Mechanism
Column load (P) → transferred to the footing.
Footing spreads the load over a larger area (A) so that the soil pressure does not exceed
the safe bearing capacity (qₐ).
Stress in the soil is assumed uniform if the load is concentric and the footing is rigid
relative to soil.
3. Basic Stress Distribution
a) Normal Bearing Stress
For an axially loaded footing:
q=PAq = \frac{P}{A}q=AP
Where:
qqq = uniform soil pressure (kN/m² or kPa)
PPP = applied column load (kN)
A=B×LA = B \times LA=B×L = area of footing (m²)
Design Requirement:
q≤qallowableq \leq q_{allowable}q≤qallowable
b) Eccentric Loading
In reality, footings may be subjected to axial load + moment due to wind, seismic action, or
column eccentricity.
Soil pressure is then non-uniform:
q(x,y)=PA±MxIxy±MyIyxq(x,y) = \frac{P}{A} \pm \frac{M_x}{I_x}y \pm \frac{M_y}{I_y}xq(x,y)=AP
±IxMxy±IyMyx
Where:
Mx,MyM_x, M_yMx,My = moments about x and y axes
Ix,IyI_x, I_yIx,Iy = footing section moments of inertia
x,yx, yx,y = distances from centroid
📌 Key Effects:
Pressure distribution becomes trapezoidal or triangular.
Maximum soil pressure occurs at the edge nearest to the load.
If eccentricity eee is too large → part of the footing lifts off → soil stress becomes
triangular (no tension in soil).
c) One-Way Shear (Beam Shear)
Critical section at d (effective depth) from the column face.
τv=Vb dwith V=shear force at section\tau_v = \frac{V}{b \, d} \quad \text{with } V = \text{shear
force at section}τv=bdVwith V=shear force at section
If τv≤τc\tau_v \leq \tau_cτv≤τc (shear strength of concrete) → safe.
d) Two-Way (Punching) Shear
Critical section around column perimeter at distance d/2d/2d/2.
Punching shear stress:
τp=Vub0d\tau_p = \frac{V_u}{b_0 d}τp=b0dVu
Where:
Vu=V_u =Vu= ultimate column load minus soil reaction inside the critical perimeter
b0=b_0 =b0= critical perimeter length at d/2d/2d/2 from column
If τp≤τc,p\tau_p \leq \tau_{c,p}τp≤τc,p (permissible punching shear stress) → safe.
e) Bending (Flexural) Stresses in Footing
The footing acts like a slab-on-grade spanning between soil reactions.
Maximum bending moment under column face:
M=q⋅l22M = q \cdot \frac{l^2}{2}M=q⋅2l2
Where lll = projection of footing beyond column face.
Design reinforcement must resist this moment.
4. Stress Distribution Cases in Soil
Case 1: Concentric Load, Rigid Footing
Uniform soil stress.
Safe if q≤qaq \leq q_{a}q≤qa.
Case 2: Eccentric Load, Small Eccentricity
Trapezoidal soil stress distribution.
All soil in compression.
Case 3: Large Eccentricity (Beyond Middle Third Rule)
Triangular soil stress distribution.
Soil under one side in compression, the other side lifts off (no tension allowed in soil).
5. Design Considerations
Soil Bearing Pressure Check:
qmax=PA(1+6eL)≤qaq_{max} = \frac{P}{A} \left(1 + \frac{6e}{L}\right) \leq q_{a}qmax=AP(1+L6e
)≤qa
Shear Check: one-way shear and punching shear.
Flexure Check: design reinforcement for bending moments.
Settlement Check: ensure total and differential settlement within permissible limits.
6. Practical Example
Column load = 1000 kN
Footing = 2.5 m × 2.5 m = 6.25 m²
Soil allowable = 200 kN/m²
Soil pressure:
q=10006.25=160 kN/m²<200 kN/m² (safe)q = \frac{1000}{6.25} = 160 \,\text{kN/m²} < 200 \,\
text{kN/m² (safe)}q=6.251000=160kN/m²<200kN/m² (safe)
If moment = 100 kNm, footing length L = 2.5 m,
qmax=PA+6ML⋅B2q_{max} = \frac{P}{A} + \frac{6M}{L \cdot B^2}qmax=AP+L⋅B26M
Check that qmaxq_{max}qmax ≤ qaq_aqa.
7. Summary
Footing stresses arise from the interaction of column load, footing rigidity, and soil
bearing capacity.
Key stress checks:
o Soil bearing pressure (uniform or non-uniform).
o One-way shear.
o Two-way punching shear.
o Flexure (bending stress).
Ensure stresses in concrete, reinforcement, and soil all remain within permissible limits
to avoid soil failure, structural cracking, or settlement issues.
