Dear Colleagues! My Name Is NN. I Will Present the Russianteam with the Problem Invent Yourself

Dear colleagues! My name is NN. I will present the Russianteam with the problem Invent yourself.

The problem says: It is more difficult to bend a paper sheet, if it is folded “accordion style” or rolled into a tube. Using a single A4 sheet and a small amount of glue, if required, construct a bridge spanning a gap of 280 mm. We are proposed to introduce parameters to describe the strength of your bridge, and optimize some or all of them.

The first construction we studied was a foldedor accordion style bridge.

To measure the strength of a bridge we put it on a test stand and seta loading platform on its middle.

We tested bridges with different fold sizes. For each size of the fold was made a series of 10 tests. Red segments show the statistical spread of the tests. We found that the strength of the bridge reaches its maximum when the fold sizeis about 20 mm.

It is important to notice how thin-walled constructions, as folded bridges, deform and break. Under a load such a construction first deforms globally. When the load reaches its critical value, a local buckling occurs and the bridge breaks suddenly.

The next construction we testedwas a tube bridge.

For example, this construction is used in a tube overhead crane.

The tube usually breaks in the midspan by wrinkling at the top due to a local buckling.

Let medescribe a role of torques in this breaking. The force F applied to the arm L produces the torque F·L. This torque is balanced by elastic forces of tension and compression, which act in the tube walls. So we can write the balance of torques:F·L = F*·a, where F* is the force acting in the wall and a is the tube diameter. When compression F* at the top of the tube reaches its critical value, leads to a local buckling occurs.

From this model we get that the breaking load must beindependent on the diameter of the tube. In fact, halving the diameter, the number of layers is doubled. So the critical torque remains the same.

This graph shows how the breaking load depends on the inner diameter of the tube. Our expectation generally confirmed in a broad diapason of tube diameters. When the diameter is small the critical load decreases because the inner layers of the tube are underloaded. When the diameter is big the critical load decreases because the ends of the tube are jammed.

To check that explanation we didan experiment with twin tubes, dividing the sheet of paper on two halves. You can see that twin tubes of small diameters sustain more load, then single ones.

The next thing we consideredwas the distribution of torques. If the midspan load is F, the support reaction is F/2. On the arm L the torque is F/2·L. As a result, the area of torques has this form, and the most dangerous place is in the midspan of the tube. So it is desirable to strengthen the midspan.

We tried to do it changing the layout. Paper width in the middle increased 1.6 times, so we expect the breaking load increasing 1.6 times.

This diagram shows the result of the test. We expected to increase the strength, and we really got it.

Next type of a bridge we tested was a beam with triangular cross-section.

We know that the most dangerous part of the tube section is its upper part, for buckling occurs here. Let me consider the cross-section of a triangular tube. In this orientationthe upper part of the section become wider, so its compression become weaker, and we expect some increasing of the strength.

This diagram shows test results of triangular tubes in comparison with round tubes. It is seen that triangular tubes really are stronger that round ones. Note that the scatter of results in the case of triangular tubes was larger than in the case of round ones. The matter is that the strength of a triangular tube depends strongly on the details of its manufacture.

Now I will discuss bridges of truss construction, for which weobtained our best results.

This is a truss bridge over the river Ob in Novosibirsk.

A truss construction consists of straight members connected at nodes. Trusses usually are composed of triangles because of the structural stability of that shape.

External forces and reactions in trusses act such a way that truss members are only in tension or in compression. Torques in truss constructions are usually excluded.

We know that paper is very strong in tension, and short paper tubes are sufficiently strong in compression. So a truss structure may be enough good for paper bridges.

Our first truss bridge had a construction called “inverted king-post truss”. There is a horizontal beam with a vertical rod. The lower end of the rod is connected with the ends of the beam by diagonal braces. Such bridges break under the load more than 30 N. Here you can see the distribution of forces in this bridge.

This is thecutting of the sheet.

Also we built an arch truss bridge. It consists of four inclined tubes connected with horizontal braces. The first prototype of this bridge broke under the load of more than 40 N. In the series this construction brokeunder the loadof more than 30 N. In all cases inclined beams initially lost their global stability, and then one of them broke in the middle.

This is thecutting of this construction.

To strengthened this bridge we add paper layers in the middle of the tubes. The maximum load for this constructionreaches 44 N.

This is the cutting of the sheet.

So far we have only measured the strength of the paper bridges. But we would like to give for it some theoretical estimates. To do this, we need to know the strength characteristics of the paper. Also we beganwith the paper tensile test.

To measure the elastic modulus we took a twin paper bend and loaded it.

The test showed that the paper extension is proportional to the load due to Hooke’s law.

So we can calculate the Young’smodulus of a paper as a ratio of the stress to the strain.

Knowing Young’s modulus of a paper, we can make estimates for the Euler rods instability.

Consider a rod hinged to the ends, loaded with an axial compressive force. When this force exceeds some critical value, the loss of stability becomes energetically favourable.

Forthetube we used in an arch truss bridgeits calculating critical loadis about 100 N.

On this test stand 8 tubes were tested. The global buckling occurs with a load 18–22 N, and local buckling occurs with a load 22–26 N.These loads are 5–6 times lower then the calculating critical load.

Let me summarize our investigations.

The members of paper bridges have thin walls, so those bridges are broken due to local buckling of these walls.

In a truss structure all elements are working on the compression or tension, and do not work on a bending. As a result, truss paper bridges can withstand the great load.

The best design of our paper bridges withstand the load is about 60 N.

There are our references.

Thank you for your attention!

Our next thought was to replace a tube on a triangular beam. Six bridges of this type were tested. Breaking loads hadea large spread. But stillthere was some strengthening.

Bridges with twin triangular beam buckles at 48 N.