Vector addition: may the force be with you.

Vector addition, specially applied to calculation of forces on an object (body) has a vast application in practice.

Every human made structure – be it a building, a telegraph pole, a locker or a table – needs to be calculated and the forces that are applied to that structure observed – in order to make the structure robust and strong enough to withstand the demands of it’s application (ie: a locker needs to be able to bear the mass of stuff you put in it, a building has to withstand wind and earthquakes, and a table really shouldn’t break when you stand on it. If you’re reasonably heavy).

Since every force that applies to some man made structure is a vector – has it’s own direction, orientation and intensity – we need to apply vector addition and subtraction.

Every man-made item in this picture was calculated in respect to FORCES that apply to it’s structure: the tall buildings, the huge cranes, the newly built structures, the small houses that house the workers; even the road itself was calculated in order to not give out and get messy under the wheels of heavy machinery that needs to travel there. Awesome, no?!

Observe a common man made structure: a bridge. Each bridge needs to bear many forces, including

the weight of the traffic on the bridge
the force of wind, even “bura”!
the force of snow
the force of earthquakes, should they happen!
the weight of the bridge itself, of course!

To calculate the forces that “work” on a bridge, we represent the known outside forces by vectors, and apply them to the construction designs of the bridge (before we actually MAKE one, of course!). We then calculate the forces (vectors) that are created inside the bridge construction, and see how much load (force) will each component of the structure have to bear. This allows us to further calculate how thick should each component be, and also what material should we use for our bridge to be strong&safe.

An example of a simplified bridge structure loaded only by it’s weight (m*g) and forces inside the bridge that result from it (red arrows in the schematic).

In the past, these calculations were actually done manually, by graphing all of these vectors on huge pieces of paper, using the actual triangles and protractors!

Of course, today this is all done by computer software which apply vector addition for us. The calculations in simpler software are simple, using the vector components to reach algebraic solutions.

Simple vector calculation software – the black “100” vector represents the force on the bridge, while all the other vectors are calculated by the software using simple vector addition and Newton’s laws. The little numbers represent force in kN.

More advanced software uses FEM and other more advanced mathematical models to calculate structural forces. These methods are beyond the scope of our program, though they are common engineering tools.

Example of advanced FEM calculation of forces and resulting structure deflections on a bridge.

Based on these calculations, our bridge structure is developed and built so to withstand all foreseeable forces that might occur and burden it! And… a bit more: the foreseen forces are multiplied by a “safety coefficient” which makes our structures a bit more robust and strong. Just to be on the safe side!

Of course, nobody’s perfect, so sometimes we screw up…

so now that you know how ultimately important vector addition is to the creation of civilization as we know it, here is an awesome little tool to play with your vectors:

And for those who really got into the subject of bridges, here’s an interesting documentary on how the tallest bridge on Earth was built:

1 Comment

with the simplified bridge structure, can you help me with this question: Provide them a brief explanation of some of the different forces involved and how a vector analysis of these forces can help them with bridge design. Show them an example of a solved problem and a possible design concept.

Varun

11th May 2022