A free standing brick wall 20 cm thick is subjected to a wind pressure of 75 kg/m². The maximum height of wall from stability consideration is:
🔬 Understanding the Stability of a Freestanding Wall
A freestanding wall is essentially a vertical cantilever fixed at its base. When wind blows against it, it creates an overturning moment that tries to topple the wall about its base. This is counteracted by the resisting moment generated by the wall's own weight.
Condition for Stability: For the wall to be stable, the resisting moment must be greater than or equal to the overturning moment. The maximum height is reached at the limit state where these two moments are equal.
Resisting Moment (Mᵣ) ≥ Overturning Moment (Mₒ)
⚖️ Detailed Calculation
Let's analyze the forces and moments for a 1-meter length of the wall.
- Wind Pressure (P): 75 kg/m²
- Wall Thickness (t): 20 cm = 0.2 m
- Wall Height (h): To be calculated
- Density of Brick Masonry (ρ): A standard value is approx. 1800 kg/m³
1. Calculate the Overturning Moment (Mₒ)
The total wind force acts at the center of the wall's height (h/2).
- Total Wind Force (F) = Pressure × Area = P × (h × 1) = 75h kg
- Overturning Moment (Mₒ) = Force × Lever Arm = F × (h/2) = 75h × (h/2) = 37.5h² kg-m
2. Calculate the Resisting Moment (Mᵣ)
The wall's self-weight acts at the center of its thickness (t/2), which is the lever arm from the toe (pivot point).
- Weight of Wall (W) = Volume × Density = (t × h × 1) × ρ = (0.2 × h) × 1800 = 360h kg
- Resisting Moment (Mᵣ) = Weight × Lever Arm = W × (t/2) = 360h × (0.2 / 2) = 360h × 0.1 = 36h kg-m
3. Equate Moments to Find Maximum Height (h)
At the point of tipping, Mᵣ = Mₒ.
36h = 37.5h²
Divide both sides by h (since h is not zero):
36 = 37.5h
h = 36 / 37.5
h = 0.96 m
💡 Study Tips for Stability Problems
- Identify the Moments: Always start by identifying the overturning force (wind, water pressure, etc.) and the resisting force (self-weight).
- Find the Lever Arms: The overturning force on a rectangle acts at half the height (h/2). The resisting weight of a rectangular wall acts at half the thickness (t/2).
- Memorize Key Densities: Knowing a standard density for brick masonry (1800-2000 kg/m³) is often necessary to solve these problems.
- Set Resisting = Overturning: The maximum stable condition is always when the resisting moment equals the overturning moment.