Practice — theory

SIMPLE PROBLEMS WITH FAR-REACHING CONSEQUENCIES: Load on two strings | Simplest rod construction

LOAD ON TWO STRINGS

A. Yu. Kositcyn

One of the simplest problems, which the author had to explain to his students, is the problem of a load, hanging on two strings. Knowing the load weight Ð and inclination angle of the strings , it is necessary to determine forces in the strings.

This is quite a typical but wrong answer: both strings are stretched by force, equal to a half of the load weight.
The calculation scheme of this problem looks as follows:

The equilibrium of coplanar concurrent force system is described by two equations:

Solving them we obtain the following result:
,

Equality of magnitudes of forces in the strings Ò1 and Ò2 looks evident even without these “boring” calculations, but these magnitudes essentially depend on inclination angle of strings (or their sag) and they are bigger the smaller the angle . When the angle is inclined to zero, reactions of the strings are inclined to infinity!
This simple problem has a lot of interesting applications. We mention only a few of them, which authors watched with their own eyes.

Once, while making my way to work (for sure to explain the mentioned above problem to my students!) I noticed a crowd standing around a car. Joining the crowd, I saw that the left fender of the car after an accident was so crumpled, that there wasn't any possibility to turn the wheels.

For reasons, clear only to the inhabitants of the former Soviet Union, the driver didn't call the car-care service, but solved the problem with his own powers. He winded one end of a cable around the fender, and the other around a tree trunk. Then he began jumping on the cable in the middle of it. The force in the cable, which could be estimated by formula (A), at the small sag several times as much as the driver's weight. When the fender somewhat straitened (and the sag somewhat increased), he stretched the cable and proceeded the “repair” again. Engineer keenness let him reach the garage without anybody else's help.

Another episode, which I would like to tell, is connected with freezing, causing in December 1988 great damage to the city in which the authors live and work. On one of the intersections one could see a ferroconcrete pillar, which earlier supported wires. During the freeze it broke down (!) and fell upon a balcony of the second floor flat of a nearby building.

A plan of this intersection looks as follows:

Thoroughly studying this plan, one can find, that the pillar was under action of the unbalanced force system, the resultant of which increased very much because of freezing ice on the wires. Curiously the weight of the wires with ice wasn't too big but the forces on the wires at small sag turned out to be sufficient to break the ferroconcrete pillar! Because of numerous breaks of current networks the city was without electricity for several days. Imagine the terrible nights for the apartment owners, when in pitch darkness a stone guest broke into the window of the second floor! How could one not recall the immortal Pushkin's tragedies!

The natural disaster showed that the engineers' designing current networks, too much relied on the strength of reinforced concrete.

3. Don't harm gas pipes

The instructions for using gas cookers forbid attaching ropes (for instance, clothes-lines) to gas pipes. Forces, arising at that, can lead to sizable deformations of screw joints, seal failure and, consequently, lead to dangerous gas leakages.

Once students told the author an interesting story. During some sporting event a steel cable was stretched across a gym hall. The ends of the cable were tied to water-heating radiators. When one of the participants stumbled over the cable, the competitions had to be terminated because of flooding!

4. Something about vessel tow

When towing a vessel or a barge, one should watch the sag of a trailing line. If the sag is small the line can break.

From formulae (A) and (B) we get, that when the sag h is decreasing to zero, the force in the line is increasing to infinity, but nevertheless the lines break far from always . Fortunately, the not-stretched materials don't exist and their deformations (tensions) guarantee the necessary sag. Ship cables with big compliances, e.g. made of vegetable fibers, are more reliable.

5. Along the monkey bridge

S. Ya. Bekshaev

Who doesn't remember the fairy tale about the doctor Aybolit (“Oh! Hurts!”)? We will remind you of one of the episodes. The doctor, breaking out of the malicious Barmaley prison, hastens to the ill monkey's help. “… But suddenly he saw a river before him. The river was wide; he wouldn't be able to swim across it. The Barmaley prison guards were just about to catch him! O, if there were a bridge across the river, the doctor would run along it and get at once into Monkeys' Country… Here one of the monkeys cried:
-   Bridge! Bridge! Make a bridge!
The doctor looked around him. The monkeys had neither iron nor stones. What would they make the bridge of?
But monkeys built the bridge neither of iron nor of stones, but of live monkeys. A tree grew on the bank of the river. One of the monkeys grasped it, and the other caught her tail. So all the monkeys stretched as a long chain between two high banks of the river.
- Here is the bridge for you, run!
– They cried to the doctor.

It was difficult to walk along the live monkey bridge… But it was strong, the monkeys held fast to each other and the doctor quickly ran to the other bank with all his beasts.”
It is especially important for those learning mechanics that this story has a happy ending owing to the special strength of the live bridge. Let us see what force the live links of this bridge had to possess.

To begin with let us suppose for the sake of simplicity, that the bridge is weightless. Then, in the moment when moving along it the load is in the middle of it, the shape of it will be as follows:

The size h (the sag) in comparison with the width of the river L should not be very big, because otherwise it will be impossible to go along it. Simple calculations show that forces stretching the bridge in this case are equal to:

Assuming that the kind doctor isn't too heavy, and accept P = 600 Í ( 60 Kg). We accept too, that the sag constitutes the tenth part of the bridge length, i.e.=10 . Then the poor monkeys have to endure a force, equal to

which seems absolutely unreal.
In reality the situation is much worse. The dead weight of a bridge, thrown across a wide river must be sizable. Not adducing exact calculations, we present an approximate value of the horizontal component of the force, stretching the bridge (the real value is much bigger but when the sags are small, this is a good approximation),

where - is a linear weight, i.e. the weight of the bridge length unit.

To get numerical evaluation let us assume plausible values

Then

As you can see, the little monkeys must be fantastically strong!

At last we have to recognize that it is absolutely unclear for us how the grasping of the tree on the one bank of the river, the monkeys could “reach” anything on the other. In other words, not only the strength calculations, but “erection technology” of the described above bridge as well bring forth a lot of questions and gives food for thought to an engineer.