trusses are everywhere they are used in bridges. antenna tower. cranes. even in parts of the international space station. and for good reason, they allow us to create strong structures while using materials
in a very efficient and cost effective way. so what exactly is a trust. it is essentially a rigid structure made up of a collection of straight members, but
that's not a complete definition. there are two important assumptions we need to be
able to make for a structure to be considered as a trust. first, we need to be able to assume that all of the joints in the structure can be
represented by a pin connection. meaning that members are free to rotate at the
joint. the members of a trust are often rigidly connected using what is known as a gossip
plate. but if the center lines of all the members at a joint intersect at the same point like
they do here, it's reasonable to assume that the joint behaves like a pin connection. the second assumption we need to be able to make is that loads are only ever
applied at the joints of the trust. we never have loads acting in the middle of a
member, for example. because all joints are pinned, the members cannot carry bending moments. they can
only carry axial loads. this simplifies the analysis of a trust significantly. each member
has to be an equilibrium. so the forces acting at each end of a member must be
equal and opposite. each member is either intention or compression. these assumptions are what differentiate a trust from a frame. unlike trusses, frames don't necessarily have pin joints, and so members can carry
bending moments. a frame can also have loads applied directly to its members. the base shape of a trust is three members connected to form a triangle. if a load is applied, the ankles of the triangle won't be able to change if the length of
each of the members stays the same. this means that the triangle is a very stable
shape, which won't deform when loads are applied to it. and so it is a great base
from which to build a larger structure. joining foreign members together does not form a stable structure. the angles
between members can change without any change in the length of the members. and
so using a four sided shape as the base for building a trust would be a terrible. an easy way to stabilize this configuration is to add a diagonal bracing member to
split it into triangles. we can start with our triangle and build it out to form structure. there are a lot of different ways to build a trust. but there are some particularly
popular trust designs that you will see again and again. and so they are referred to
by specific names. the one shown here is a faint roof trust, but there are many more
as you can see here. later on in this video, i'll cover how these different designs carry loads in different
ways. the members of these trusses are all located in the same plane. these are called
planar trusses, and we can analyze them as two dimensional structures. even seemingly three dimensional structures can often be analyzed as planar trusses. take a look at this bridge, for example. the loads are transmitted from the horizontal floor beams to the two vertical trusses
on each side of the bridge. each of these trusses only carries loads acting in its
plane. and so we can analyze it as a two dimensional structure. to be able to design or analyze a trust, we need to be able to determine the force in
each of its members. this allows us to check that the member can carry the loads
without failing or gives us the information we need to select the best cross section
for each of the members. there are two main methods we can use to do this. the method of joints and the
method of sections, let's look at the method of joints first using the think roof trust
we saw earlier. the method is really simple. first, you draw free body diagram showing all the external
loads acting on the trust. and you use the three equilibrium equations to calculate
the reaction forces. then you draw free body diagram for every single joint and work through them one
by one to solve the unknown forces acting and each joint, you solve the unknown
forces using the equilibrium equations. since all of the joints are pin connections,
there are no moments, and so you only need to consider equilibrium of the horizontal
and vertical forces. remember that we are calculating the forces acting at each joint. not the forces in
the member if a member is intention. the internal forces will be acting to make the
member longer for every action. there is an equal and opposite reaction, which
means that a member intention will be exerting a force on the joint, which is acting
away from the joint. for members and compression, the force will be acting towards the joint. let's work through an example for a slightly simpler trust. first, let's draw the free body diagram and determine the reaction forces using our
three equilibrium equations. taking equilibrium of the horizontal forces, the horizontal force at joint a must be
equal to zero because it is the only force in the horizontal direction. taking equilibrium of the vertical forces, the reaction forces adjoins a and e must sum
up the twenty kilometres. both joints are located at the same distance from joint sea.
so taking equilibrium of the moments acting about joint sea, we can calculate that
they both equal ten kilograms. now let's determine the forces acting on each joint, since we don't know yet which
members are intention and which are in compression. it's easiest to just assume that
all the members are intention. and so we'll draw the internal forces as pointing away
from each joint if we end up with negative values for these forces. it just means that
we guess wrong, and the member is actually in compression. now we can work through each of the joints, starting with joint a. analyzing trusses involves a lot of resolving forces to different angles. so if you want
to be good at it, you're going to need to remember your trigonometry. here's a quick
reminder. back to our joint. all we have to do is apply the equilibrium equations to determine the unknown forces
at join a taking equilibrium of the vertical forces. the ten kill newton reaction force
must bounce the vertical component of the force f a b, and so we can calculate f a b
as negative ten divided by sign of the angle of sixty degrees. taking equilibrium of the horizontal forces, we get that the force f a c must balance
the horizontal component of the force f a b, and so is equal to five point eight killing
newtons. that's all the forces acting on joint a calculated. the force in each member is constant. and so we now also know the force is acting
on the joints at the other ends of these two members. we can repeat the process for joint b. we can start by considering equilibrium of the vertical forces, which allows us to
calculate the force f b c. and then we can consider equilibrium of the horizontal forces to calculate f b d. we then need to work through all the remaining joints, but we can save a bit of time
by noticing that the trust and the applied loads have an axis of symmetry. and so the
forces on the other side of the trust must be identical. that gives us all of the forces
of the joints we can show, which members are intention and which earn compression
like this. one thing you'll notice as you analyze trusses is that some members don't carry any
loads at all. we call these zero force members. there are two main configurations
where we have zero force members. the first is where we have three members connected at a single joint and two of the
members are aligned. here, only one member has a component in the vertical direction. and so to maintain
equilibrium of forces at the joint in the vertical direction, the force in this member
must be zero. the second configuration is when we have only two members connected at a joint
and the members are not aligned. only one member has a component in the vertical direction, and so both must be
zero force members. by the way. this is true regardless of how the members are oriented because we can
rotate the orientation of the coordinate system we are using to apply the equilibrium
equations. these two configurations only contain zero force members if there are no external
loads acting at the joints. if we have external loads, there will be components in the vertical direction, and so
these will not be zero force members. let's look at an example. at this point here, we have two connected members. the members are not colonial,
and there are no external loads. so they must be zero force members. and at this joint, we have three members, of which two are colonial. the vertical
member must be a zero force member. we can remove these members and so have a much easier starting point for solving
the trust. you'll notice that we haven't removed the two members at this joint. that is
because there is an external load acting here, and so these can't be zero force
members. you might be wondering why anyone would bother, including zero fourth members in
a trust if they carry no loads, they are definitely not useless. they are usually included
to provide stability, for example, to prevent buckling of long members, which are
under compression. or they may be used to make sure that unexpected loads won't
cause the structure to fail. we've covered the method of joints. let's look at the other method we can use to
solve trusses, which is the method of sections. the first step is the same as the method of joints. we draw the free body diagram and
use the equilibrium equations to solve the reaction voices. next, we make an imaginary cut through the members of interest in our trust, and we
draw the internal forces in the cut members. the internal and external forces must be an equilibrium, and so we can apply the
equilibrium equations to solve the internal forces. when choosing how to cut your trust. remember that we only have three equilibrium
equations. if you cut through too many members. you will have too many unknowns
and not enough equations. you can choose which side of the cut you want to assess. the left side looks easier to
solve because there are less forces, but we could have chosen to solve the right side
instead. the method of sections is best use when you have a trust, which has a lot of
members, but you are only interested in the loading and a few specific members. let's
look at an example. we need to determine the internal forces in these three members. first, let's draw the
freebody diagram and apply the equilibrium equations to calculate the reaction
forces. the horizontal reaction force at joint a is the only force acting in the horizontal
direction, so it must be equal to zero. by considering equilibrium of forces in the vertical direction and equilibrium of the
moments acting about joint a, we can figure out that f a is equal to nineteen and f h
is equal to twenty one. next, let's make our imaginary cut through members f d f e and g e. and draw the
internal forces like we did earlier for the method of joints, we will assume that all
unknown forces are tensile. next, we just need to apply the equilibrium equations. the
force in member f e is the only unknown force with a component in the vertical
direction, so that's a good place to start. the diagonal members are all at forty five
degree angles, so by considering equilibrium in the vertical direction, we get that the
force and member f e is equal to twelve point seven kilometres. now let's consider
equilibrium of moments acting about joint f. this is a good joint to choose because
three of the five horses in this freebody diagram have a line of action passing
through f, and so only the force and member g e and the twenty one kilogram
newton reaction force generate a moment about this joint. both forces are located at
a distance of two meters from. and so we can conclude that the force and remember g e is equal to twenty one
killing newton's. finally, we can take equilibrium in the horizontal direction to
calculate that the force and member f d is equal to negative thirty killing newton's,
and that's it. we've calculated the internal forces in the three members we were
interested in. one member is in compression in two or. if it is possible to determine the reaction forces and the internal forces and the
members of a trust by applying the equilibrium equations, the trust is said to be
statically determinant and. real life structures sometimes contain more members that are needed for the
structure to be stable, as this makes them safer. this means we may not be able to
apply the method of joints or the method of sections because we have too many
unknowns and not enough equilibrium equations either to determine the reaction
forces or to determine the internal forces within the trust. these trusses are said to
be statically indeterminate and would need to be solved using other methods like the
force method or the displacement method, which i won't get into this video. now that we know how to calculate the loads in a trust, let's explore some of the
differences between trust designs here. we have three different bridge trusses, the
how prat and warm trusses, these trusses were all patented in the eighteen forty s at
a time when new bridge designs were being developed to accommodate the
expansion of the railroad industry. they were typically constructed from a
combination of wood and iron. we can learn a lot about trust design by figuring out which members are intention
and which are in compression. let's start with the how trust we can see that its
vertical members are intention in its diagonal members are in compression. members and compression usually need to be thicker than members intention to
reduce the risk of buckling. this means that the how trust isn't very cost effective,
since the diagonal members, which need to be thicker are quite long. the pratt trust
addresses this issue. its vertical members are mostly in compression. and its inner
diagonal members are intention. this is more cost effective than the how trust since
the longer diagonal members can be thinner, longer members are also more
susceptible to buckling under compressive loading than shorter ones. so it's a good
idea for long members to. the design of the warren trust was based on equilateral triangles. the fact that all of
the members are the same length as an advantage for construction, and it uses less
members overall than the how and practice, so it is more efficient. the diagonal
members alternate between tension and compression, so it does have some quite
long members and compressions. it can also be interesting to observe how the loading and members changes as a load
moves across a bridge in this simplified model of a load moving across the prat
bridge, we can see that some members alternate between tension and compression.
and so will need to be designed accordingly. the three dimensional bridge we looked at earlier could be assessed as a collection of
planar trusses, but this won't be possible for all structures, and sometimes a trust will
need to be assessed in three dimensions. this type of trust is called a space trust. these can be analyzed in essentially the same way as plainer trusses using the
method of joints in the method of sections. the only difference will be in the number
of equilibrium equations. we will have six equations instead of. and at each joint, we will have three equations instead of two. that's it for now. thanks for watching. and as always, please remember to subscribe.