Billboard Analysis - Part 1

Billboard Problem Solving

1)    Problem Analysis
A pipe is to be designed to support a billboard. Figure below illustrates the case.



Tha available data are:
-    Dimension of billboard:  height = 2 m and  thickness = 10 cm
-    Free stream velocity = 5 m/s (constant)
-    The wind loading is in +x direction and determined by using CFDSOF.

a.    Comments
Choosing a pipe design for this problem is an open ended problem. There will be many solutions we can choose. In order to obtain a good design, it is required to obtain additional constraints to the problem which lie outside of the problem definition.  Depending on the application,  it may be reasonable to apply one or more of the following goals:
    •  Minimize the cost of the pipe
    •  Minimize the weight of the pipe
    •  Maximize availability of stock by choosing standard sizes

There are many size and material of the pipe we can choose. So. It’s good enough for us to use the pipe that are available in the market.

Some simple steps according to Fundamentals of Mechanical Design:
- Our purpose
- Loading that works
- Stress determination
- Mohr’s circle
- Effective stress allowable
              

b.    Finding Critical Point



There are about 3 loadings that work on a billboard as shown in figure above. Let’s assume that:
    Fw = loading of wind
    Fb = loading of billboard
    Fp = loading of pipe

From the illustration, it’s clear that the point which’s given the most loading will be the bottom of the pipe. But if the cross section is round, where will the most critical point? Just go to the analysis details section.

c.    Analysis Details
Note that the loads result in the following stresses:
1.  Fb & Fp results in a uniform compressive axial stress in the pipe wall.  The magnitude equal to the weight of the billboard and pipe divided by the cross sectional area at the bottom.  This means that the bottom of the pipe is most important.
2.  Fw results in an axial momen which are independent of height,  and is maximum at the bottom of the pipe. 

Therefore, the critical section is at the bottom of the pipe. Now we wish to find the critical location in this section.  On the critical cross section,  all of the stresses are constant except for the moment stresses.  Since the pipe is so long,  it is predicted that the pipe will fail by bending and not shear. Because the span of the billboard is not given, there will be no shear stress due to the wind loading twist. Thus, only locations where bending stresses are large will be considered. 

Moments that caused by Fw is more critical.  It is has a much large moment arm.  This moment causes compression in the +x direction,  and since the other normal stresses are compressive, this is the more critical location. (In the -x direction,  the tensile bending stresses will be offset by the compressive stresses.)  Finally, the critical location is at the right side of the cross section as shown in Fig. 3. with the blue rectangular section.



d. Stress Determination
Now, we will determine the stress at critical section as a function of pipe area A, outer diameter d0,  and moment of inertia I.  Draw a stress element showing the stress state. Let’s assume there’s a very small segment in the bottom of the pipe as shown in Figure below.



The stress components will be:
1) The compressive stress due to the weight of the billboard and pipe
2) The stress due to the moment of wind loading
3) The shear stress due to twist of wind loading (if we consider about the span of the billboard)

- Wind Loading Determination
We can find wind loading using CFDSOF. And solve it using numeric integration or just some common calculation. But here Mr Ahmad Indra asked us to find the wind loading using two methods. They are:
1. Using perssure distribution that works on the billboard
2. Using momentum difference of the air before and after pass throurh the billboard

It's easy to calculate using pressure distirbution. Just see the color of point in the billboard simulation and match it with the value shown by CFDSOF. The pressure that we should see is the front and back side of the billboard. We should consider the possibility of the pressure on the back side of the billboard. It's possible there is some points which its pressure direction is opposite to the pressure direction on the front side of the billboard. Once we find the value of each point, we can use numeric integration or common calculation to determine the amount of pressure. Once we get the pressure, we can use this equation to determine the loading: F = P x A.

It's the same if we use the second method. Just see the color of point in the billboard simulation and match it with the value shown by CFDSOF. The velocity that we should see is in some distance from the front and back side of the billboard. Sum it and the integrate it.

We want to compare the result of those two mehtods.

Simulation

From the case above, a simulation is needed to determine the force acting on the billboard using pressure difference or momentum difference. There are 5 cases will be discussed here.

1.       Case One

The first case is by adjusting the domain like shown in the following figure. It’s a domain with 20, 6, and 1 meter for the dimension of length, height and width (default). The amount of cell is 200 and 60 for length and height respectively. We know that the billboard is usually placed in outdoor. From this the domain is:

-        -   The upper cells and the right cells are outlet,

-         - The left cells are inlet where the wind will come, and

-         - The lower cells are wall because of it’s ground.

To simulate this domain, first we should enter the condition, which are:

-          The inlet air velocity is 5 m/s

-          Density of air is 1 kg/m3 and viscosity is 10-3 Pa.s

-          Activate the turbulent, this simulation is using K-epsilon

Result for Case 1

This is the result for case 1. The iteration is convergence after the 2555^th iteration like shown in figure below.

 

Here is the “grafik residu” for this iteration result.

The graph of stream function for Case 1 is shown by the following figure.


We can see from the figure above that the upper vortex is bigger than the lower vortex.

For the velocity vector is shown by the following figure.

 

We can see that the velocity right on the top of the ground is the highest. We want to see the velocity vector around the billboard. The following figure is shown this.

We can see that the velocity vector in the left-middle of the billboard moves up and down. The Velocity factor of the center of the billboard is about zero. We want to see the velocity vector in another place. The following picture is the velocity vector in the top-left of the billboard, top-middle near the billboard, and the top-middle  of the domain.