Arch Truss Bridge, Static Determinacy

[engineer248]

Homework Statement

This is for a bridge design project.

We went for an arch bridge over the roughly parabolic valley. The arch is parabolic and rises 1.3 meters above the deck of the bridge. The deck is divided in 20 sections and supported by both vertical and diagonal members (hollow structural steel sections) connecting arch and deck.

It is no problem to figure out angles, load forces and reaction forces. The bridge is completely supported by the arch, which thrusts into the wall of the river (concrete footings for our purpose). The arch, too, is made out of discrete members (so that a true parabola is approximated). All joints are assumed to be pin joints (hinges).

These are my assumptions for my calculations:

• The arch is hinged at the footings as well
• All forces in members have a horizontal and vertical component, and therefore give two values (sin and cos) to be used in equations
• The end of the deck of the bridge is supported by nothing but the arch. The ends of the bridge don't touch the riverbank.

I have 88 equations now, and 87 members. I created a huge matrix in Matlab to solve them for me, but I get ridiculous answers. Supposed I did iron out all the typos, does this bridge actually have any chance of being statically determinate? Can I work with 88 equations (44 joints, horizontal and vertical forces for each) and 87 unknowns? Can I just add a random member somewhere to make up for the missing member?

It's really hard to find easily information on the static determinacy of arch bridges for very simple model calculations like this. It seems like there's a lot of stuff on basic trusses, but whoever's doing arch bridges needs to really know what they're doing.

So should I scrap the arch bridge and go for something simple, or how do I fix my design?

 

[haruspex]

Since there is only one unknown for each member, more equations than members implies an equation is redundant.

Static determinacy can be established by considering the consequences of removing a member; a member can be removed without the structure flopping if and only if the system is indeterminate.

You can halve the problem using symmetry, getting you down to 44 members. But I'm puzzled about the numbers. I imagine the structure consists of non-overlapping triangles, each made of three struts or two and a road section…. except, that would mean the road sections are under tension, so maybe you put struts in parallel with them. 20 road sections should then mean 20 struts along the road, 20 struts along the arch, 19 verticals and 18 diagonals; 77 in total.

 

[Halc]

I am certainly no expert, but have designed (6 months) and built (1 month) a model arch bridge, which was simplified and didn't have to deal with say thermodynamics due to its simplicity. It had no diagonal members, and the entire load hung from the arch (mostly) overhead, which was a series of trusses with diagonal braces, so a thick arch.

You seem to be looking only for static stability, a set of angles where the whole thing is in balanced equilibrium.
As such, it seems that you can assume the deck has zero force acting on it, and thus their contribution is only their weight. Ditto for the risers/diagonals, which bear their own weight plus that of the deck. Sure, if a load (a truck say) is applied somewhere, that balance is lost and the load changes, but you seem to be after the simple solution at first.
Only the arch elements do any work, so they bear their own weight and the angle that they bend at each intersection balances the weights of the components acting on that intersection. That's about 11 meaningful equations in all, the rest of them being fairly trivial.

There's a bit of instability at the points where the arch crosses the deck, dividing the bridge into three rigid sections. A point load placed say at the left one will displace the rigid left section downward and the other to sections upward. There's little to resist that if the whole thing is hinged. The arch does need more rigidity than just what it gains by all the tinker-toys attached to it.