Bay Bridge Tutorial

by Tony Alfrey   (tonyalfrey at earthlink dot net)
(updated Jan. 19, 2010, under construction, some images missing)

Contents

Motivation
Introduction
Tension and Compression
Basic Truss Bridge
The Forth Rail Bridge
Computers and Bridge Models
A Basic Truss Bridge Model
Modeling the Bay Bridge
Calculate the Eyebar Stress
Onward To Repair 1.0
Conclusion
More Obsessive Details

Motivation

The motivation for our earlier Bay Bridge investigation was to figure out what broke on Labor Day, 2009.  So we figured out the "what", but now we'd like to know a little more about the "why".  We want to know why the Bay Bridge repair failed, but most importantly, we'd like to know what science we'll need to find the answer so that you, too, could find this out for yourself.  Ultimately, we want to understand how important those eyebars really are.

Introduction

We'll study a little bit about bridges so that we can understand what is all the fuss about broken eyebars on the San Francisco-Oakland Bay Bridge.
If you've not been following the Bay Bridge story, check this out.  Then come back to learn some real bridge engineering.

Here is a picture of the entire San Francisco-Oakland Bay Bridge.




(Image from here and here.  Also, see other fine aerial photography in LA, my hometown,  here)


The part that we'll be most concerned about is the long span just east (the upper half of this photo) of Yerba Buena Island that is unsupported in the middle.  Long spans are what bridge designers have been trying to perfect for 3000 years (maybe more).  This is what the long span looked like (from the southeast) very near to its completion in 1936.


Image from here

The key thing to see in this picture is that the bridge itself rests on two supports that are separated by a long distance.   But equally important are the two supports immediately adjacent to each of these (the one on the far right is just out of the frame).  To understand how this works, we need to study simple bridges first. 

Tension and Compression

Here is the simplest possible bridge, consisting of a beam resting between two supports and loaded with some weight.  Our beam is nothing more than a stick, but it will help us make a valuable point in just a minute.


(Image: The beam is sagging)


Our simple bridge is close to failure.  Let's see why.  Let's take a beam of clay and bend it;  clay is a terrible bridge material but it will tell us what forces the bridge feels.

(Image: A beam of clay)


Look closely at the clay beam.  The top side of the "beam" is wrinkled and squeezed together, the bottom side is stretched and ripped apart.
We say that the top side is in compression and the bottom side is in tension.  In this model, the more intense the color, the greater the compression or tension.

(Graphic: beam in tension and compression)


OK, so we intuitively know that the easiest thing to do to improve our bridge is to get a taller beam.  Here is a board that is just as wide as before, but now it is much taller.  The beam does not bend as much and there is less compression and tension.

(Image: Tall beam under load)

The problem is that bigger beams are expensive.  What to do? 
First we think "if the top of the beam is in compression and the bottom of the beam is in tension, what happens in the middle of the beam?"
The answer is easy:  nothing much!  The middle of the beam isn't helping our bridge too much, it just adds material we don't need and extra weight we don't want.  So let's get rid of some of it.

(Image: Tall beam with symmetric holes)

In fact, here is an important example of a "beam" that must withstand considerable load; the inside of an airplane wing.  It doesn't look like the middle of the beam is too important, otherwise those big holes wouldn't be there!  Let's get rid of that extra weight;  punch big holes in those beams (called "ribs").





Basic Truss Bridge

This punching-of-holes can be taken to extremes to create a truss bridge. 
Now it's mostly holes.		 There are many different flavors of truss bridge, but the common feature of all of them is that the horizontal member on the top is in compression and the member on the bottom is in tension.  In fact, trusses are found everywhere.  Here they are holding up a roof.

Cascade, CO through-truss bridge image from here.
Image from here.


If we're clever, we can make truss bridges with longer spans that don't need much material and are not very heavy.  But there are limits.  Eventually the compression on the top members is too much for the material to tolerate without making the member prohibitively expensive.  How to solve the problem?  One solution is to build a scheme called a double cantilever.  In one historic picture, the principles of the double cantilever can be seen.



The Forth Rail Bridge Double Cantilever
From Imperial College, London

"A historical demonstration at Imperial College in 1887 showing the weight of the central span of a bridge being transmitted to the banks through diamond shaped supports. The central "weight" is Kaichi Watanabe, one of the first Japanese engineers who came to study in the UK. Sir John Fowler and Benjamin Baker of Imperial College provide the supports."

Demonstration of Forth Bridge principles 1887

"To commemorate the centenary of the construction of the Forth Rail Bridge, undergraduate students at Imperial College re-enacted the demonstration in 1988."

Demonstration of Forth Bridge principles re-enactment 1988



The distance to be spanned is represented by the space between the two chairs.
The load (roadway and traffic) is represented by the person in the middle.
The two towers are represented by the people on each side.
The arms of the people represent top members that are being stretched in tension, and the poles represent two bottom members that are being squeezed in compression.  Finally, on either side are two heavy counterweights that balance the weight of the roadway in the middle, much like two children sitting on opposite sides of a seesaw.

Let's see how this works on a real bridge, the Firth of Forth Rail Bridge in Scotland.  The cantilever on each side supports a small truss bridge in the center.




Image from here.


Enjoy this excellent video about the design of the Forth Bridge


Computers and Bridge Models

Now let's apply this idea to the Bay Bridge.
We'll simplify the structure of the Bay Bridge to make it easy to understand how each piece works.  Then we'll use a computer to calculate the tension and compression on the important bridge components.



Complete Drawing of the Double Cantilever Span of the San Francisco-Oakland Bay Bridge



Simplified Model


In our simplified model, the dark green parts are the double cantilevers and the light green part is the truss bridge, considered to be the "load".  The counterweights are orange and the support towers are grey.  The very heavy pier on the left side (on Yerba Buena Island) serves not only as the counterweight for the double cantilever, but it serves to anchor the entire double cantilever firmly to bedrock, because the bottom of the San Francisco Bay is an inherently squishy place to anchor a bridge.  Notice that we've removed many components from the actual bridge to construct our model; there are two reasons for this.  The first is that this helps us recognize the shape of the basic double cantilever design "hiding" inside the elaborate drawing of the bridge.  And the second, more important, reason is that it gives us a reasonable number of components to work with when we use a simple computer modeling application that you, too, can experiment with on-line.

The double cantilever in the Bay Bridge model looks a little different than the one in the Forth Bridge model.  But the only real difference is that the lower diagonal parts in compression in the Forth Bridge have been brought up to be horizontal, and form the roadway itself, in the Bay Bridge.  And this photo of the construction process of the Bay Bridge, displays the crux of the double cantilever principle;  counterweight on one side supports load on the other.  Can you find the additional counterweight support temporarily added to the right cantilever?


Image from here


A Basic Truss Bridge Model

Now let's get more detailed about the compression and tension in the bridge members.  To do that, we'll use a very nice web-based application that can be found here at the Johns Hopkins University Engineering Department that you can use on your own computer to design a bridge.  Read the instructions here.  We'll start by building a simple truss bridge, a little bit like the Basic Truss Bridge example seen above.  This is what the application window will look like on your computer after you have added some truss parts.




The key components of this application are nodes, members and loads.
Members = long beams, sticks or cables.
Nodes = connections at the ends of the long members.
Loads = pushs or pulls, like the weight of a car or the weight of a bridge member itself.

Rules: 
a)  Loads are always applied to nodes.
b)  Nodes are not "glued" connections, but they are pins that would be free to pivot.  But when one builds a truss bridge, one always makes the members connect in such a way that the structure has "shear strength", that is, it won't simply fold up by itself.
c)  We need a foundation for the model.  We rigidly attach the left side of the bridge to a fixed support, and we allow the right side to sit on rollers.  This means that, in the model, the bridge is free to stretch a bit as it heats up, or as the load changes.  Forcing rigid supports on both ends can distort the bridge, and the computer application will not allow you to do it, anyway!

Limitations (and a Homework Problem for the Advanced Student)

(Skip directly to A Basic Truss Model: Results if you are just starting your modeling study.  Come back to this after you have worked with the examples below).
There are limitations to this modeling program.  The program is limited to designs that are "statically determinate".  This is similar to saying that if any one member is pulled out of the truss, the truss collapses.  The railroad bridge in the first picture is not statically determinate;  it is actually "overconstrained".  This is why we did not use it as the example for our first computer modeling attempt.  Three of the diagonal elements under any loading condition are unnecessary, but which ones are unnecessary depends on the loading.  Here's a hint:  get rid of one each of the diagonals on the end "boxes" and one of the center diagonals.  Then load up the bridge as if the train were on the left, center and right.  As the train chugs along, use only diagonals that will be under tension.  Because this bridge design uses simple rods or cables for the diagonals (which only work in tension, not in compression) you will see that both of the center diagonals are necessary, just not all the time.  Therefore, you need to remove one when doing the modeling.



A Basic Truss Model : Results

Here is the result of building a simple truss bridge similar to the picture of the railroad bridge above.  Although the load is applied to specific nodes, one can imagine the load as uniformly distributed over the length of a member, but this makes the calculation more complex and is really unnecessary.




Just like what we saw for a solid beam of clay, the bottom side (called the lower chord) of the truss bridge is in tension and the top (the upper chord) is in compression.  We also see that the diagonal members can be either in compression or tension depending on which way they connect.  This is important, because it means that the parts that are only in tension can be replaced with things that only need to tolerate being pulled, like a rope, a cable or a chain.  For just a second, return to the picture of the Basic Truss Bridge and think about which pieces must surely be in tension.

In the truss model results above, the numbers alongside each beam show how much tension or compression there is in each member.  We've chosen a load of "50" and placed it on the center node.  The results show that the foundation (the yellow and red nodes) is pushing up on the ends of the bridge with "25" each, since the load of "50" is right in the center of the bridge.


Making the Truss Longer

Now we'll make the truss bridge longer by duplicating all of the members into a longer span.  We'll also keep the same load per node:  if the train covers the whole bridge, then each car on the bridge can be thought of as the same load per node.  See how the amount of compression and tension increases as the span gets longer.  Also note how the diagonal members switch back and forth as compressive or tensile members.





Modeling the Bay Bridge

Now we'll see how the principles of the double cantilever apply to the Bay Bridge.
We'll forget that we know something about the double cantilever and, instead, we'll just try to make a truss bridge with only the members that exist in the Bay Bridge between the support towers.

Setting Up the Simple Truss Model

Here is how we'll assemble the model.  First, it is very important to scale the model properly.  In other words, the lengths of the members on the model must match the lengths on the bridge itself.  The working window on the computer app is not very large and it turns out that, if we select each unit of measure on the computer app to equal 100 feet, we have just enough space for the whole cantilever section.  You could make your model smaller, but then it is difficult to place the nodes just where you need them (the computer app has a hidden grid that restricts the placement of the nodes).  Anyway, you'll need to properly position the nodes vertically as well as horizontally.  The very top of the two towers is about 300 feet off of the lower roadbed, the second-highest nodes on either side of the towers are at about 190 feet off of the roadbed and finally, the height of the center truss is about 145 feet.




If the Bay Bridge Were Just a Truss Bridge




Computer Model if the Bay Bridge Were Just a Truss Bridge

Now, how shall we load the bridge?  For our model, we will consider only static loads;  the weight of the bridge members, and the traffic, not any sort of variable loads like earthquakes or wind.  The weight of the static loads is distributed continuously out along each beam and chunk of roadway.  But the modeling program only allows loads to be placed at nodes (where members connect).  For a very large structure like the Bay Bridge with so many individual pieces, this is perfectly OK.  So then we need to decide where to place these loads.  Certainly all of the bridge members have some weight, and this weight is exerted at every node.  But eventually, all of this weight can be considered to be exerted on the lowest nodes of the bridge, the nodes along the roadbed.  We've performed a few calculations with this modeling program in which we've distributed the weight in a logical way over the whole bridge, but the results really do not change that much, and the modeling windows gets very cluttered this way.  So a reasonable approximation is to simply assume that the load applied to each portion of the roadway is uniform.  And, because it is caused by gravity, it points down.

We've selected an arbitrary amount of load ("25")  to be placed at each node on the roadway itself.  If we know the weight per each foot of roadway, we can convert the number "25" into an actual number of pounds.  But for now, we'll just use this number to compare the load on the roadway to the stresses in the bridge members.  So for example, the distance between the end supports in this model is the distance between the inner support towers at some 427 meters, or 1400 feet.  The distance between the diagonals is 1/7 of that, or 200 feet.  So "25" represents the weight of 200 feet of the double deck roadway and all of the superstructure steel components above the roadways within that 200 foot section.  The weight of the members in the upper portions of the bridge structure might be more accurately placed at the upper nodes in this model, but these loads are considerably smaller than the roadbed and "active" loads, and the model results do not change substantially if all loads are concentrated at the roadbed nodes.  It also keeps the model from becoming hopelessly cluttered.

Results : Simple Truss Model








We see that the top of the truss is in compression and the bottom of the truss is in tension as we would expect for a simple beam lying between two supports.  Note the numbers for the amount of tension and compression top and bottom, at bridge center, are huge, nearly 10 times the load applied to each node.  This is why a simple truss doesn't work for a long span.  Go back up to this picture;  look further east of the long span we're studying (upper right side of the photo).  There you will see shorter bridge sections, each one of which is a separate truss bridge.


Setting Up the Double Cantilever Model


Now let's add the rest of the components needed to make the double cantilever.





Complete Bridge Structure




Complete Bridge Computer Model


Results: Double Cantilever Model







What has changed?
1.  The compression and tension is reduced in the central portion of the span.
2.  The roles of the upper and lower chords switch:  the top is now in tension and the bottom is in compression.
3.  Stress has been moved from the long central span to the short sections on either end (where the broken eyebar was found).

Also, how did we choose "40" for the load on the ends?  By running the model several times and seeing what load minimizes the stress on the central truss.  It also just about balances the "see-saw" formed by the weight on either side of the support towers.*

Finally, we look at a drawing of the Bay Bridge and we highlight the important eyebar that recently cracked.  We see that not only is this eyebar (and its partners) under tension, they are under lots of tension.  That is why this assembly consists of a total of 8 eyebars side-by-side on each side of the bridge!  If even a few of them were to break, it is possible that the entire bridge would fail.  While only one of the sixteen eyebars is cracked, Caltrans engineers are justified in being worried about this repair and certainly want to get this repair, of any on the bridge, working correctly.  Just a note about this model:  as shown in the instructions that come with the modeling program,  this simplified bridge design which incorporates only the principal elements of the bridge is statically determinate.  That means that if we lose any of these principal pieces, the bridge goes down.  So each of these components must be well-designed.  Also note that for each of these components, there is a twin on each side of the bridge, plus a collection of "companion" pieces, and a whole range of interconnecting beams that couple the two sides of the bridge.  So the bridge is more robust than just these key components would imply.




Bay Bridge Cantilever Model:  Broken Eyebar Highlighted


Calculate the Eyebar Stress

Is it possible to calculate real numbers for the eyebar stress?  To do this, we need to find the weight per foot of all of the components that are in each slice of a length of bridge.  It's been almost impossible to find this information on-line, but certainly anyone intimately involved in the construction or various redesigns of the Bay Bridge must know this information.  We've found this reference; it gives the weight of the 300 foot-long replacement section of the double-deck roadway as 3600 tons.



The white thing is the piece we'll use to calibrate our model.  We know how long it is and how much it weighs, and we assume that it has the same weight as any other equivalent length section of the bridge.  Then we make a guess as to how much additional weight to add for superstructure and traffic.


This replacement section appears to be roughly equivalent to the double-deck roadway section of the existing cantilever portion of the bridge and we'll multiply this number by a "weight-to-load conversion" of 1.5 to account for the truss structure above the roadbed itself and the traffic load.  Therefore this yields a total load of 5400 tons per 300 ft of bridge length.  This means that each 100 ft of bridge experiences a load of 1800 tons.  In all of our models, we've applied a load of "25" along each 200 foot section of the cantilever, so this means that "25" must be the same as 2 x 1800 tons = 3600 tons.  This conversion ("25" = 3600 tons) means that we can convert all of the other numbers that appear on the model output.  For example, the eyebar assembly that we've been considering has a "59" attached to it in our model results.  So to convert this into a real number, we would perform the following calculation:

                                                                                         (59/25) x 3600 tons  = 8,496 tons

This tension is spread out over a total of 16 eyebars on either side of the bridge. 
Some of this stress might distributed over some of the other minor elements that we've left out of the model.  For example, there are three additional vertical members surrounding this eyebar set running from the roadbed to the top of the bridge superstructure.  But an analysis of the stress on these components shows that they are largely in compression and do little to change the load on the eyebars from the results of the simple model.  So we will estimate that the force on the whole 16 eyebar set is about 8,400 tons so the force on one eyebar is 525 tons. 

How Strong is an Eyebar?

So how much stress should an eyebar be able to take before it breaks?  We need to know something called the yield strength.  This is a number that will tell us how strong a steel stick that measures only 1" by 1" would be.  The type of steel used in the Bay Bridge, the type of steel used for virtually all structural engineering projects of the mid-20th century, is A36 steel.  You can look this up in Wikipedia.  There, A36 steel is listed as having a minimum yield strength of 36,000 pounds per square inch.  This means that you could take a long stick of A36 steel that measured 1" x 1" and hang a weight of 36,000 pounds from the end and expect the stick to never break (but to just begin to stretch beyond its elastic limit).  As the stick gets thicker, it can hold more weight;  it's just like having more sticks side-by-side, each individually carrying the same weight.  The eyebars measure 12.375" wide x 1.8125" thick, so they have a cross sectional area of 22.4 square inches.  That's like having 22 1/2 individual sticks of 1" x 1" steel, side-by-side.  So the eyebars should have a minimum tensile strength of 36,000 pounds x 22.4 = 806,400 pounds or 403 tons.

However, the original material was heat-treated to push the minimum yield strength up to 50,000 pounds per square inch (abbreviated "ksi"), which pushes the minimum tensile strength up to 1,120,000 pounds or 560 tons


Our model showed that the applied stress would be 525 tons, and that 560 tons is the minimum yield strength.  This seems like it should be an adequate safety margin, but the structural building codes usually require that the components be loaded to no more than 60% of the minimum yield strength, considerably lower than the 525 tons calculated.   See more obsessive details that complicate the issue.



Onward to Repair 1.0

Now you, too, can use your new-found power to validate Repair 1.0, the famous kludge that was used on the Bay Bridge to fix the broken eyebar.  Let's review:

a)  Eyebar cracked found on Labor Day.
b)  Repair 1.0 was installed, consisting of some brackets (Saddles and Crossbars) and four long, approximately 2" diameter Tie Rods surrounding the cracked eyebar.  Because of the crack in the eyebar, all of the stress on the eyebar is now applied to the unbroken half of the eyebar that wraps around the pin, and this portion has only about half the area of the full eyebar.  So engineers may have wanted to relieve all of the stress on the eyebar with Repair 1.0, or maybe just assist, or "take up" the load that used to be applied to the now-cracked portion of the eyebar.

When we first started these calculations, we assumed that the steel used in the Tie Rods was the conventional A36 steel which is common to virtually all large structural steel projects.  When we performed calculations on this material, we concluded that the Tie Rods would have been totally insufficient to support the expected loads.  While the use of A36 steel was a good assumption, it turns out that we discovered on Jan 12, 2010 that special high-tensile steel was used in the Tie Rods.  There are at least two grades of material that could have been selected, and it is not clear which material was available on short notice for the Labor Day repair.  We've selected the lower of the two grades (based on looking at the photographs), the so-called Williams Grade 75 All-Thread Rebar, with a minimum yield strength of 75,000 pounds per square inch.  It is possible that the higher grade material was used, the so-called Williams Grade 150 KSI All-Thread-Bar, with a minimum yield strength of 150,000 pounds per square inch.

Let's calculate!

1.  Q:  What is the cross-sectional area of a 2" diameter rod?
     A:  The radius of the rod is 1".  The area of the rod is Pi x the square of the radius.  3.14 x 1" x 1" = 3.14 square inches.

2.  Q:  What is the minimum yield strength of Williams Grade 75 steel?
     A:  Approximately 75,000 pounds per square inch.

3.  Q:  What is the strength of a 2" diameter rod if made from this steel?
     A:  Multiply the (cross sectional) area of the rod times the minimum yield strength.  3.14 square inches x 75,000 pounds per square inch  =  235,000 pounds.  This number is the minimum yield strength; the point at which the material just begins to deform.  While not a limit on the material, this is the stress one does not want to exceed.

4.  Q:  What is the minimum yield strength of four rods?
     A:  Multiply the strength of one rod times 4

              4  x  235,000 pounds = 942,000 pounds

5.  Q:  How many tons is that?
     A:  Divide by 2000.

                  942,000 pounds / 2000 pounds per ton  =  471 tons

6.   Q:  How does that compare to the stress we calculated on the eyebar?
      A:  We calculate 525 tons.  We estimate that the stress required for the rods is half of this (there is still some unbroken eyebar that helps) or 262 tons.  The 471 tons that the rods can conservatively provide (Step 5) is adequate for assisting the broken eyebar, but probably not really enough for taking up the load of the entire eyebar.  The Williams Grade 150 KSI All-Thread-Bar should be used in that case.

7.  Q:  How does the strength of an unbroken eyebar compare to that of four rods?
     A:  The minimum yield strength of an unbroken eyebar is about 525 tons.
           The minimum yield strength of four Williams Grade 75 rods is 471 tons, twice that for the Williams Grade 150 rods.

8.  Q:  Why aren't we doing all this in the metric units you learned in school?
     A:  Because U.S. structural engineers aren't physicists.  "When in Rome . . . . ."


Conclusion:

It appears that the strength of the lower-grade Williams Form Engineering material would be close to adequate for the job and that the higher grade material will be adequate.  We do not know for sure which material was actually used.  Based on the pictures, the lower grade material may be the only material originally available on short notice.  

Why did the Tie Rod break?  It is possible that metal fatigue due to repeated bending at a point where the rod passes through the nut, or because of fatigue where it passes through the crossbar, caused the failure.  The manufacturer lists a variety of cautions to be taken when using this material.  It is possible that this indicates that, while the material is strong, that it may be brittle.  We will see what additional information we obtain as time passes.

Is it easy to replace the eyebars?  This would probably require unloading all eight eyebars attached to the pin on the south side of the bridge with some tool similar to the present Repair 2.0 so that eyebars could be replaced as a set.  This would require something capable of temporarily tolerating a stress of some 8 x 525 tons or 4200 tons.  This is greater that the weight of the entire roadway section that was plugged into the S-curve detour.  So the answer is NO, it would not be easy.

Will Repair 2.0 tolerate a substantial earthquake?  NO.

Would an assembly of uncracked eyebars (or the footings or most anything else about the east span) tolerate a substantial earthquake?  NO.

Does it make sense to make the Repair 2.0 capable of withstanding a substantial earthquake while the rest of the old span falls around it?  Not in my opinion.

Does Caltrans know all of this?  YES.  The technology exists to create a computer model of every bolt, rivet, weld or potato chip thrown out by a passing motorist onto the roadway surface.  The fact that Caltrans is monitoring the bridge constantly and is making announcements that they may want to do a Repair 3.0 indicates to me that they are monitoring the strain on the Tie Rods and can see some stretching.

Does Repair 2.0 need improvement?  Probably yes.  Caltrans needs some clever engineers:  they need a solid, temporary, reliable fix that will get them through the next five years, albeit with no earthquake.  Repair 2.0 probably does not qualify.  And early in December of 2009, we see that Caltrans had prepared a far superior solution in the form of Repair 3.0, which is already installed as of our most recent update.


More Obsessive Details (another advanced homework problem)

Here are more obsessive details that most readers will not care about, but they are interesting to me because I'm obsessive!  Scroll back up and look at the results of the model calculation again.  The assembly with the broken eyebar is not the one under the most tension:  according to the model, the next eyebar set further east (and its symmetric mate on the Yerba Buena end of the cantilever) are under even more tension ("72" instead of "59").  So to resolve this problem, one would need to assume that the "72" set was working closer to minimum yield strength, not the "59" set (the "72" set and "59" set are fabricated in exactly the same way).  If this is the case, then one needs to further change the "weight-to-load conversion" of 1.5 (that we used to convert the weight of the double roadway into bridge load) to something smaller .

It is a little more accurate to assume that the load is distributed preferentially near the support towers where the superstructure is heavier.  In other words, maybe the model needs lighter loads of "20" along the center truss, and heavier loads of "30" nearer the support towers.  We've also scaled the lengths in the model a little more accurately.  You can experiment with the model and see how this changes the numbers.



 
Here are the results for a more optimized model;  it decreases the load on the central truss, and swaps the values for the tension in the two sets of eyebars we're considering.  It puts our "weight-to-load-conversion" very close to 1.2 for the central portion of the span, and pushes it up to 1.8 for the portion near the support towers.  The model could be "tuned" even better if the application allowed a little more flexibility about setting the loading and node positions.  If you're really obsessive, look at this more detailed photo .  It helps show all of the important pieces in the section of the cantilever between the tower and the counterweight.  Look at the size of the really highly-stressed eyebars along the upper chord of the cantilever.  Can you model these parts, too?


  *.  In engineering jargon, "the 'moment' about the support tower is balanced".