Overview

This document runs through the calculations that we have used to design our pyramid. We attempt to estimate wind loading, and how to distribute these loads to the various parts of the structure. We assume that the pyramid’s base is 70 ft square, and that it is 30 ft high. We will consider the wind loading for 60mph sustained wind with 80mph gusts.

Wind Loading

We will attempt to estimate the wind load using two techniques. First, we use the traditional drag-force calculation for non-laminar flow. [Need a reference for this equation.]

Where

Fdrag = Drag force

A = cross-sectional area = ½ * 70 ft * 30 ft = 97 m^2

Cd = Drag coefficient = 0.5 (typical for structures of this shape)

= density of air = 1.2 kg/m^3

v = wind velocity = 60 mph = 27 m/s

(80 mph = 36 m/s)

This gives a total drag force of 21 kN = 4.7 kip. (1 kip = 1000 lb)

With wind speed of 80 mph, we get a drag of 8.4 kip of total drag.

Second, we will calculate wind drag using a momentum-transfer technique. We will figure out how much instantaneous force is applied to the pyramid if all the momentum from a thin section of air moving at high speed is transferred to the pyramid. This represents the absolute worst case aerodynamic situation. The drag force calculated in this manner will necessarily be much higher than any actual sustained drag, but is a reasonable upper bound for the drag force resulting from wind gusts.

The basic equation for momentum is

P = m * v

P = momentum

M = mass

V = velocity

We will consider a thin sheet of air with a cross sectional area A and a thickness of Δw. Thus

M = A * Δw * rho

Newton tells us that force is the time rate of change of momentum

At velocity v, the amount of time it will take for the column of air to completely hit the structure is

Δt = Δw / v

The final momentum of the air column is 0 since the air must completely stop in order to transfer all of its momentum to the structure. The initial momentum is

Pinitial = m * v = (A * Δw * rho) * v

Thus the total force on the structure is

F = (P final – P initial) / Δt

= (0 – A * Δw * rho * v) / (Δw / v)

= - A * rho * v^2

From this we can see that total momentum transfer results in the same force as an object with a drag coefficient of 2.0. The negative sign on the force is merely indicative of the direction in which the force will be applied.

So, we can estimate gust wind loads as being 4 times that of the sustained wind drag we calculated earlier. Specifically, 34 kip at 80 mph.

We will design the structure to take the maximum wind load of 34 kip. This calculation of wind drag gives us a reasonable upper limit on the amount of force that will be applied to the pyramid, but really doesn’t tell us anything about how this load will be applied to the structure.