Engineering Computations For The Design Structure

Translation

Here three problems have been asked which is based on the problem of truss design and weight or forces acting on the members of truss as well as design analysis of water waste pipeline which cross the ravine using a trough slung below the bridge. The challenging part is the design of bridge with slung of waste water pipe where weight of bridge, weight of vehicle and the weight of pipeline addition may cause catastrophe. Truss are generally made of combination of triangles; some truss might have different combination of design structure[1].

The total mass of water in the trough at any given time is given by the expression:

Where m [kg] is the mass of water in the trough, r [m] is the trough radius, L [m] is the length of the trough, and h [m] is the height of water from the top of the trough. 

The MATLAB code for the give problem and equation is presented in appendix at the last of the report. From the given equation of mass which is having terms, radius of half cylindrical open pipe, length and height. For the given value of length, radius and mass, the height from the top of the trough is derived using MATLAB with the help of for loop.

When bridge or roof top structure made with truss subjected to various reaction forces acting in horizontal and vertical directions. As well each member of truss may subject to tension and compression. In bridge design truss are very common if truss is not used in bridge construction than it may be made of simple beam connection. Very common truss designs are warren truss[2], pratt truss[3], howe truss[4], K truss[5] etc. In the design procedure of the truss along with the newtons law of motion many other things are taken into consideration. The joint also plays important role in truss design. Very common joints are pin joint at where member of truss meet at that point[6]. Also, the forces of compression and the tension acts upon the member of the truss[7]

The total mass of water in the trough at any given time is given by the expression: 

Where m [kg] is the mass of water in the trough, r [m] is the trough radius, L [m] is the length of the trough, and h [m] is the height of water from the top of the trough.

Derive the equation in terms of h. 

The methodology opted for truss design calculation is the equilibrium equations. Which is The summation of the forces at a assumed point is always zero. 

With the simple concept of vectors, So according to the equilibrium equations

Similarly, the sum of the forces acting upon the Y direction also zero. 

Two common methods are methods of joins and method of sections are use for truss analysis. Both the methods are explained in the next section, here in our example we opted method of section for the solution of numerical[8]. And the A force that causes a counterclockwise moment is positive moment. Stress that the only forces used when summing the moments are the external forces. Members’ internal forces are not used[9].  Note: Choose the point that we will use as our “pivot” when summing the moments[10].  

Methodology

Using the concept of trigonometry cosine and sine which helps in finding the vectors x and y. 

Lets assume that all elements of the truss are subjected to the tension, if answer is positive it means all the element is in tension and if the answer is negative it means element is subjected to compression[11]. 

As from the previous assumption if sing of answer is different need to update the free body diagram with the right magnitude and direction. If the direction of a member force changes, there will be two free body diagrams that need to be updated[12].

For example, here by making free body diagram for elements. Some random forces are shown with some magnitude.

Each and every element of the truss assumed to be subjected to tension[13]. If the sign of answer is positive it means the element is subjected to tension and the minus sign means it is subjected to compression. The selected joint must have at lease two unknowns. Here in below diagram reaction forces at A and C are better to select to start the calculations.

Method of joints explained in the above section. In our assignment method of section is followed. As the name suggest, truss design is divided into various section and each section are analysed for the forces action upon the members. For example, here in below figure whole truss section is divided into and section is shown by dashed line. From this section model we will be able to find the forces acting upon the member 2-4, 2-3, and 1-3. In the next step moving towards the opposite direction and next section is made and the forces are determined. If truss structure is symmetrical it is easy to find the forces in other members due to symmetry of truss structure.

Make cut or reduce the truss structure design split it with sectional line put it where the forces are needed to be determined. Make cut at least in two section because at least two equilibrium forces are needed to be find. At the last step apply equilibrium condition and find forces in all the members. 

The Length of the bridge = 50m

Length/member =50/6=8.33 m

Assuming Height of member =8.33 m

The total load carried is

Reactions are given as  in upward direction

As the truss is symmetrical the forces in all other members are same as calculated above

Weight of water =50,000 kg=50,000X10 N

       =500 kN=R

Let

So the forces in the members are given as below  

Solution 3:

Introduction

Hilltown bridge designed here was originally use for heavy vehicles and it was subjected to risk of overloading. And the total limit over loading and forces on truss element was kept 550 kN. So the use of bridge as transportation way limited to only 5 vehicles at a time.  The force applied on reaction points subjected to type of vehicle passing over the bridge. Where truck considered to be applying 50 kN and car 15 kN during three time slot intervals  

Translation

To solve the problem poisson distribution given and solution based on probabilistic 

And for the three condition of vehicles passing equation can be formed as below, 

Methodology

Given the bridge is considered to be overloaded with forces in any truss element exceeds 550 kN.

At any time, there may be up to 5 vehicles on the bridge are present

Mean=5 

Solution 3 (a): The question is a poisson distribution mean=5 (as per the given data)

So, if number of vehicles exceeds more than 5 then bridge will be overloaded

Take number of vehicles n=6 for overloading bridge at any time

So

Solution 3 (b): Now we have the probability of traffic condition throughout the day. So we can find the expected value of car or truck occupied a reaction point throughout the day

So expected value of car or truck occupied a reaction point early in a day is  

Solution 3 (c): So we can expected value of car or truck occupied a reaction point is high is case of late in the day. So in between 26:00 to 23:59 the bridge is max likely to be overloaded. 

Most of diagonal elements which are run outward are subjected to the compression stress. And the other member with inner diagonal subjected to tensile stress.  The truss design used here is somewhat related to Baltimore truss bridge. The mass of the water slung of waste water line and vehicle affect the forces acting upon the bridge structure. The reaction forces of horizontal and vertical direction sum are equal to zero it means at equilibrium state. And from the probability analysis we can expected value of car or truck occupied a reaction point is high is case of late in the day. So in between 26:00 to 23:59 the bridge is max likely to be overloaded   

References:

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[6] K. Shea, J. Cagan, and S. J. Fenves, “A shape annealing approach to optimal truss design with dynamic grouping of members,” Journal of Mechanical Design, vol. 119, no. 3, pp. 388-394, 1997.

[7] K. Svanberg, “Optimization of geometry in truss design,” Computer Methods in Applied Mechanics and Engineering, vol. 28, no. 1, pp. 63-80, 1981.

[8] M. N. Priestley, F. Seible, G. M. Calvi, and G. M. Calvi, Seismic design and retrofit of bridges. John Wiley & Sons, 1996.

[9] H. T. Kang and C. J. Yoon, “Neural network approaches to aid simple truss design problems,” Computer?Aided Civil and Infrastructure Engineering, vol. 9, no. 3, pp. 211-218, 1994.

[10] E. Bayo and J. Stubbe, “Six?axis force sensor evaluation and a new type of optimal frame truss design for robotic applications,” Journal of Robotic Systems, vol. 6, no. 2, pp. 191-208, 1989.

[11] M. J. Ryall, G. A. Parke, N. Hewson, and J. Harding, The manual of bridge engineering. Thomas Telford, 2000.

[12] F. Jarre, M. Kocvara, and J. Zowe, “Optimal truss design by interior-point methods,” SIAM Journal on Optimization, vol. 8, no. 4, pp. 1084-1107, 1998.

[13] R. L. Brockenbrough and F. S. Merritt, Structural steel designer’s handbook. McGraw-Hill New York, 1999.