Tension in the Bridge development

This Blog post has some relations with the earlier post. I advise to read that post first.

Different elements in the bridge design
In the design some steps have to be prescribed. This is not about how a model works but how I think it should be build. It is done in this way because the equations are very large and so mistakes are made very soon.
In this prescription only a simple rectangular beam is used but also other shapes would be possible. This would only mean that different I-values are needed.

1) A certain area / floor plan is made. This floor plan has a load.
2) One central line, the basic line of the primary beam, is created out of the lines that create the area.
3) Some points on the central line are made. First they are used in making the basic line for secondary beams. Also they got used for the calculation of moments in this point.

The next steps can be done independently because they both only depend on the load.

A) The primary beam get its shape from a maximum committed tension. (earlier mentioned)
B) The secondary beam get its shape from maximum committed tension.

In both situations first the line of moments has to be found. This is easy in phase B but not so easy in phase A.
After that with maximum stress the cross sections of a beam can be calculated. This can only been done if decisions about the height and width relations are made.

A) The shaping of the primary beam
The primary beam is the support for the secondary beams. In this they all create a point load on the primary beam. Because the length of the secondary beam can change, also the point load on the primary beam can change.
Because the loads are known and the distances between the secondary beams can be calculated also the support points for the primary beam can be calculated. Then also the moments on every point on the basic line, where the support for the secondary beams take place, can be calculated.
From the moments in combination with the maximum tension and information about the width and the height relation the cross section on a certain point can be shaped. When these cross sections got connected the beam gets it shape.


In This picture the next values are showed:
y…. = length of the secondary beam
Fn = First support point of the primary beam
Fn+1 = Second support point of the primary beam
x…. = distance between secondary beams

The next equations the different values that are calculated:


By using these equations no cross section would exist in the support points of the primary beam. Of course this is not possible. A solution for this has to be found. A minimum cross section on this point has to be calculated. This can be done by introducing a maximum shear force that comes forward out of the support points in Point a and b.

B) The shaping of the secondary beam
From the points on the primary beam and the area / floor plan the basic lines for the secondary beams are created. The beam that is related to this line is thought of being fixed to the secondary beam. (for now the torsion that this creates is not used in the calculations but this will be important in shaping the construction and so it might be needed to use more beams than only one or to make other decisions)
As was told in the shaping of the primary beam also the secondary beam gets it shape from the maximum committed tension in the material. This is also done by using the line of moments and some information about the cross section height and width relations.
This construction is made with two different width and height relations in the cross section. In the first the relation gives that the width is half the height. In the second a constant width is given.

The same pictures are possible for a simply supported beam with an equally divided load. This gives the next view.

(is inserted in this blog in a later phase)

From lines of moment to tension and cross section

Lines of moment in GC
In the next text many formulas are introduced. These formulas are the result of many calculations.

File: Feat_qload01
Equally divided loads and a certain length are variables. On this length some points are created and the variable x is the distance of this point from the start point of the line. The equation of the line of moment then is:

parabola fromand in this

File: Feat_qload01
For an overhang with an equally divided load the next equation can be written:

parabola from

and in this

and because of parabola
The tension in a construction.
Because the tension in a material has its maximum value the construction can be adapted to this phenomenon. In this the tension depends on the cross section of the construction. This means that a bigger cross section is needed where a higher tension is found. This comes forward in the next equations:

(where M is the moment on a point and z is the distance from the outside of the construction to the center of the construction)

For a simple rectangular beam this means:
because

and

This would mean the following for the tension in the earlier situations:

File: Feat_qload01_2
The formula for the line of moment is inserted in the formula for the tension.

File: Feat_qload02_2

In these last equations the width and the height of the beam can be influenced because of the maximum tension in a material.

What does this mean for the bridge?
For the bridge this can mean the following: every certain distance, about a meter, a beam for carrying the floor plan is introduced. On this beam the load is equally divided. If the length of these beams is different then the primary beam would have a load that is not equally divided Which would mean a variable q-load.

The load on the primary beam gets build up out of point loads that come from the support points of the secondary beams.

A last point is about the torsion. Because a higher moment from the secondary beam means a higher torsion moment the deformation because of this could be high. Than it would sometimes be better to use more than one beam in the construction. As a research point this could be something to work out in a later stage.

The moments that come out of the q-load have create tension in the construction. In this the Izz-value has its influence. The width and the height of the beam get influenced because only a certain maximum tension is allowed.

Equation of Clapeyron - three moment equation

After midterm presentation I will work further on the project. I now will try to make a bridge with a longer beam but with more support points. This will be harder and it needs some attention of the tutors. In this I will introduce the equation of Émile Clapeyron.

Eliza Guse has send me a pdf with information about this equation. In the next text I will try to tell how I interpret the formula

The equation of Émile Clapeyron.
This equation gives the moments that are introduced in the support points. In that case the load can be seen as just a load on a beam from point to point. Every part can in this way been seen as a loose part.

ln Line with a certain length
An Area under the line of moment
xn Distance from Point Pn on the line ln


Every beam length now has influence from a moment on every side and as well a load on the beam. This means the following for line ln:

Moment in point Pn
Moment in point Pn+1
Moment because of the q load

As I see it now I can calculate of give a formula for a prescription of the moment on the side points of a line. Then also the moment because of the q load can be calculated of prescribed. The total line of moment is the sum of all the graphs of the moments.
From the total moment graph it would be possible to calculate the total displacement per point. Further it would also be possible to use the same method as used in the bridge where the height and the width become variables.

For a simple situation, with equally divided load, the same EI everywhere, where a beam is supported in three places and where both the end points are supported this means the following:


(simplified because An = qln and because EI is the same everywhere.)


(in the middle point of the line and 3/4 Mln,max on the quarter parts of the line, because of the parabolic form)

The bridge: possible further developments

Mechanical elements of a construction can depend on many parameters. In earlier calculations only the displacement under one condition is taken care of. Of course many other things are possible:

Other conditions, beams that contain more support points. It would also be possible to make a model where a beam splits in two separate beams if a load or a distance to the side of the floor plan is larger than a certain value.

Other possibilities would be making a bridge form depend on the line of moment and stresses in a construction. The area of the construction can be changed by the value of tension. The maximum tension on the lower parts of the construction. This would mean that the construction on the construction would be larger where a higher moment can be found.

Torsion could also be a mechanical thing that crates the construction. The torsion creates tension in the material and in the material a maximum tension is allowed. Where the tension is higher, the construction must be bigger.

In the first presentation is told about a bridge which is designed. This bridge was the starting point for the research. Also in this presentation was told about things that I think that are important. Shortly I told about structuring and optimizing and this is what comes back every time, in the development of the models.

In the early stage of this course is told that a bridge construction or a general construction has its influence on the load, which depends on the area that rests on the construction. In this was concluded that changes in the program (changes in the floor plan) should be able to change the construction. In this the displacement was the point that is worked out under a certain condition.
Further in the course also the possibility of introducing lines of moment and beams on more than two support points. The summation of lines of moments was important to find out.
The possible models for the summation of lines of moments can be worked out with tension. In the next weeks I will try to develop a way of making the construction depend on this tension. The maximum tension in a material should be the creator of the constructional form.