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	<title>qh2021</title>
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        <item>
        <title><![CDATA[Understanding and Analysing Trusses]]></title>
		<link><![CDATA[https://qh2021.wodemo.com/entry/539177]]></link>
		<dc:creator><![CDATA[@qh2021]]></dc:creator>
		<pubDate><![CDATA[Thu, 21 Apr 2022 01:41:57 -0700]]></pubDate>
        <description><![CDATA[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. 
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