After years of hard work as a student of
engineering, you’ve landed a role as an engineer
in a big company.
Congratulations!
Your new job even gives you a fancy office
in a skyscraper. It’s a sweet deal!
Until one day, a terrible storm hits the city.
That doesn’t bother you too much at first;
a bit of wind doesn’t seem like a big deal when
you’re in an office.
But as you’re working at your desk, you
notice the entire skyscraper begins to sway
in the wind.
In fact, the building tilts so much, your pencil
rolls right off your desk and the view outside your
window starts to angle toward the sky.
In 1978, the Citicorp building in Manhattan
came dangerously close to that scenario.
If a powerful enough storm had happened
before anyone realized, key parts of the structure
could have failed.
And all because the engineers who designed
it didn’t account for how all the forces acting on
the structure could affect its stability.
But by considering those forces and how to counteract them, an industrious team of engineers managed to save the Citicorp building before any disasters happened.
[Theme Music]
A force, as you’ll recall, is any interaction
with an object that, if unopposed, would cause
it to accelerate, or change its momentum.
Like, with the skyscraper, you don’t want forces
like the wind seriously deforming the structures you
build or even causing them to collapse entirely.
So it’s important to design things to be
able to withstand the forces they may encounter.
When the Citicorp tower was first built, it had
a slightly unusual structure to accommodate a
church on the corner of the block it was built on.
The whole building was raised on nine-story-high stilts,
but because of the church, the stilts needed to be in the
middle of each side rather than in the corners.
Which meant quartering winds, which blow
diagonally to the faces of a building rather than
straight on, became a problem.
Ordinarily, these winds aren’t a huge issue
since they blow past the structure without
applying much force to it.
Unfortunately, having those columns on the building’s
sides made it more vulnerable to quartering winds than
the engineers originally thought!
And they didn’t notice until after the
building had been built,
when a student writing a paper on the Citicorp
building realized there was a problem and brought it
to the attention of the lead structural engineer.
Essentially, they’d made a critical mistake
when they calculated the building’s statics
and dynamics.
Both of these are branches of mechanics,
a field that considers how forces affect a
structure’s behavior.
Statics, as the name implies, is about what
happens to objects that don’t accelerate
when a force is applied.
All the forces acting on the object balance
out, keeping it rigid – it doesn’t rotate or move
from its original position.
When this happens, we’d say the object is
in static equilibrium with its environment.
Dynamics on the other hand, deals with what
happens when the various forces don’t cancel out.
If the structure isn’t held in place by friction or attached
to its surroundings some other way, a large enough
force will cause it to move, rotate, or deform.
That’s when dynamics comes into play, and
you’d have to consider the object’s motion.
The good news about that is that the same
basic idea governs both statics and dynamics:
Newton’s second law of motion, which says
that the force acting on an object is equal to
its mass times its acceleration.
But an object like a building or bridge is
made of many connecting parts.
If the mass is distributed throughout the structure,
a force applied at one part of that structure won’t
affect the entire thing in the same way.
It’s also important to note that Newton’s
second law assumes the force acts in a
given direction.
So in reality, it’s not quite as simple
as “F” equals “m” “a”.
Instead, you have to consider the direction
the force is applied in, which also determines
the direction of the acceleration.
Quantities that have both a magnitude and direction
are called vectors, and you have to take the direction
into account when you add them together.
Both the “F” and “a” in Newton’s
second law are really vectors.
To know the final motion or overall force acting
on a structure, you need to add all the forces acting
in different directions on each part of it.
That might sound like a lot to consider, but
there’s a clever way to keep track of all of it.
As we’ve mentioned, in statics, we’re
looking at a situation where all the forces
are balanced to zero – static equilibrium.
These calculations are important for making
sure your structure is never on the verge of failure.
Objects can only handle a certain amount
of stress before they deform or break.
Statics helps you work out the force that
parts of the structure are experiencing under
a load.
Since by definition, all the forces on the
structure have to cancel out to zero,
if you know some of the forces acting on it,
you can usually figure out the others.
To make sense of all this, engineers use what’s
called a free body diagram – a sketch of the
structure and all the known forces acting on it.
Consider, for example, a bridge.
The material the bridge is made of contributes
to its weight.
Rather than show the force of gravity
acting on all the little bits of the bridge,
you can average all the contributions of its mass
and draw a single force on the diagram acting on
what’s called the center of mass.
Now, say there was a goat trying to cross
the bridge, maybe to get to some greener grass
on a hillside.
Luckily for the goat, there’s
no troll blocking its way.
Since the bridge is in equilibrium and both
the goat’s weight and the bridge’s own weight
are pulling it down,
there must be some force counteracting this to
stop the bridge from dropping into the river.
In this case, you would use static equilibrium
to calculate the force each of the supports have
to apply to the bridge to hold it up.
The details require a bit of math and
careful thinking.
But the basic idea is, if you add the contributions
of all the forces in each direction,
being careful to give opposing forces different
signs in your equations, everything should balance
out to zero.
So at a glance, you already know that the sum of all the
forces acting upward, from the supports of the bridge,
has to cancel out the downward forces of the weights.
The full treatment would require a bit more
work, of course.
If it was a suspension bridge, the weight of the
bridge and the goat would be counteracted by the
vertical component of the tension in the cables.
If those cables can only handle a certain
amount of tension before breaking,
a free body diagram could help you work out
the total weight the bridge could handle before
the cables snap.
It’s also important to consider the torque –
any force that could cause a rotation of an
object around a point.
It’s also sometimes called a moment.
Torque is usually a factor when part of a
structure is fixed to some point, about which
it’s free to rotate, called a pivot.
If you’ve ever been to a playground, you
might have seen torques in action on a seesaw.
Let’s say there are two twins with the same
weight, Parvati and Padma.
Both twins sit on a seesaw, one and a half
meters away from the pivot.
They both weigh six hundred newtons – the
equivalent of about 61 kilograms of mass being
pulled down by gravity.
They’re exerting torque because on their
own, each twin would cause the seesaw to rotate.
The torque comes from the force of the twin’s
weight pushing down on the seesaw, perpendicular
to the line connecting the seat to the pivot.
The heavier the twin, the more force they’ll
exert and the stronger the torque will be.
Their distance from the pivot matters too.
If Parvati moved closer, she’d exert less
torque than Padma and the seesaw would
swing her upward.
Those are the two ingredients you need for
calculating torques –
the force perpendicular to a line connecting to
the pivot, and the distance from the force to the pivot.
Multiply those together, and you get torque,
which is measured in units of Newton-meters since
you’re multiplying a force by a distance.
As with forces, in a statics scenario – like the twins
balanced on the seesaw – you can use the fact that
everything balances out to zero to help you.
If you add up the torques, defining either
clockwise or counterclockwise as the positive
torque direction, they’ll equal zero.
In this case, Parvati and Padma each create
torques of nine hundred Newton-meters in opposite
directions, which cancel out.
Of course, if Padma swaps places with a heavier
friend of hers, the torques will no longer add up to zero.
If the seesaw was long enough, she could balance
out the torque generated by her heavier friend
by sitting further away from the pivot.
But if they’re sitting the same distance
away on opposite sides, we have a dynamics
problem, rather than statics.
By adding up torques or forces, we could work out the total torque and Parvati’s change in angular momentum – or the momentum of her rotation – as she begins to swing upward.
So dynamics describes when the net force or
torques on the system don’t cancel out.
That was the problem for the Citicorp building
– the force of the wind had a significant
chance of overwhelming it.
Big skyscrapers ordinarily resist the wind
through a system of braces inside the building’s
structure.
As we’ve seen for the other engineering materials,
the braces can withstand a certain amount of stress
before they begin to undergo failure.
In the case of the Citicorp building, the
student’s discovery prompted the lead structural
engineer to consider the quartering winds.
He realized that the positioning of the building’s
support columns meant the total wind force on the
braces would be way more than they’d planned for.
By looking at how the force of weight was distributed
throughout the building and calculating how quartering
winds would distribute a force to the braces,
he found that some of the braces would
encounter a tension of 160% more than
they’d previously calculated!
It would only take a once-in-16-year storm
for the braces to fail!
Thankfully, he and a team of engineers worked
through the full mechanics of the situation
and realized that they could reinforce the
braces by welding 5-centimeter-thick steel
plates throughout the structure.
To get the job done without causing too much panic to the inhabitants of the building – or alerting the media – welders worked at night, and in secret to reinforce the braces.
After a few months of welding, they transformed the Citicorp building from having a 1-in-16 chance of failure in any given year, to one of the safest buildings ever built.
Today, it can handle pretty much anything
the weather throws at it.
In this episode, we looked at statics and
dynamics, and what they mean for the
structures we design.
We explored the idea of static equilibrium, forces,
and torques, and how free-body diagrams can help
us make sense of it all.
In our next episode, we switch from looking
at the math engineers use to solve problems,
to the decisions they have to make to acknowledge
and admit problems in the first place.
Crash Course Engineering is produced in association
with PBS Digital Studios.
To learn more about physics in the real world,
check out Physics Girl.
Dianna Cowern demonstrates the physics behind
puzzling phenomenon and everyday mysteries.
Check out Physics Girl and subscribe at the
link below.
Crash Course is a Complexly production and this
episode was filmed in the Doctor Cheryl C. Kinney
Studio with the help of these wonderful people.
And our amazing graphics team is Thought Cafe.
