L7 Trusses

本站添加: 2026-05-31

[00:00:00]now let's talk about the behavior and Analysis of trusses and let's start by thinking back to our beam as we learned last week beams take perpendicular forces and convert them into a bending stress profile that looks like this with the most amount of tensile and compressive stress concentrated at top and bottom so really it's the flanges that are doing the majority of the work and a wide flange beam section the top flange provides compressive resistance and the bottom flange provides tensile resistance when we have a simply supported Beam with a load applied vertically downward together these resistive forces create a resistive moment with a truss it's also the top and bottom cord that are primarily providing resistance the top chord provides compressive resistance and the bottom cord provides tensile resistance and together those resistive forces create a resistive moment but working backwards now we did not get these forces in these chords through a bending stress profile instead it's these diagonal pieces that are doing the work to convert these perpendicular forces into horizontal forces that the chords can resist as we mentioned earlier none of the elements of a truss are typically designed for bending with the exception of the top chord but more on that later for now let's think about how I trust resists a perpendicular Force purely through axial internal forces so if I apply a load down here at this node what will happen we know that the load cannot travel into the bottom cord here because the bottom chord is perpendicular to the applied force and since we've pinned either end the load and the chord must be aligned with its axis so it must be horizontal so the load cannot get into the bottom cord directly but it can go into this diagonal member here it can also go into this vertical piece so a portion of the vertical force that we've applied will travel up this vertical piece and a portion will be resisted by this diagonal piece and as we saw with the arch demonstration along with this vertical component to resist the vertical Force there will also be a horizontal component that is generated and so the total Force is aligned with the axis of this diagonal member and as we reach the node on the other end of this diagonal member it's this horizontal component that will be transferred into the top chord as a compressive Force and back at the other end it's the horizontal component that will be transferred into the bottom cord as a tension force and so it's the top chord that goes into compression the bottom chord that goes into tension which generates our resistive moment to this perpendicular applied force and it's the diagonals that we're able to convert the vertical Force into horizontal forces for the top and bottom chord to resist without the diagonals the truss would not perform as it should and let's look at an example of what it would look like to remove those diagonals so here I have a simply supported Beam on top and a broken truss with no diagonal members on the bottom let's compare their behavior under the same loading conditions we already know that the beam will provide resistance to the applied force through bending and so it's deflected shape will look something like that and its moment diagram will look something roughly like this but what about our broken truss at the bottom we have all of these extra pieces here that we've added so will we get any extra resistance to bending from these pieces well based on our discussion so far if you think about what will happen to this applied load here this vertical piece underneath here cannot convert this vertical load that we've applied into a vertical and horizontal component and so all it can do is pass the load down to the bottom here where it meets the bottom cord which is perpendicular and since this joint here is pinned that perpendicular Force cannot travel into this bottom cord and so this bottom cord will provide no resistance meaning that the force will not travel down this vertical member and instead all of the load will be resisted by this top chord at least that's how it should work now let's see if our prediction is true so if we look at the deflected shape we see that the deflected shape of the top chord is equal to the deflected shape of the beam and if I look at the values here at this node we have 1.36 inches of deflection and we have the same deflection at our broken truss down below similarly with our moment diagram if I turn that on you can see we have the same exact moment diagram for our top chord of our broken truss as we did with our Beam with a maximum moment of 36 foot kips and so this broken truss here because it has no diagonal members is essentially just a beam with some pieces dangling off of it so trusses with pinned joints like this need diagonal members in order to work properly now let's hop back to our OneNote document so our diagonal pieces convert vertical forces into horizontal components for the top and bottom chords to resist and keep in mind that it's very important that we apply loads only at these panel points for this Behavior to hold true if we placed a load midway between panel points along this bottom cord we would be placing this bottom cord into bending which it likely was not designed for the top chord however as I mentioned earlier will often be designed for some bending since it will likely support a floor or a roof which will have some uniform load across it such as people walking around or snow load or the weight of the floor or roofing materials now let's dive deeper into the behavior of these trusses we have already established that the top chord will be in compression and the bottom cord will be in tension when we have a simply supported Truss with loads applied vertically but what about the interior pieces can we detect whether these pieces will be in tension or compression well let's start with the vertical pieces these pieces have two primary roles one is for stability they provide extra bracing to the top and bottom cords second they provide more panel points for loading to be placed for example if we apply a load here say we are hanging a large heavy Banner in an arena that load cannot go into the bottom cord directly as we learned earlier there are also no diagonal pieces at this node for the load to go into and so the entire force must go into the vertical piece and since this force is pulling down on the Node that vertical piece will be in tension similarly if we apply a load at a node up here which also has no diagonal pieces immediately available at that node but there are diagonal pieces directly below the entire force will travel through this vertical piece down to this node here and since the load is pushing down on this member it will be in compression like a column so when we have vertical pieces at a node with no diagonals at one end and diagonals at the other it's very easy to tell whether they will be in compression or tension however if we have a situation like we had up above where we have a vertical piece with diagonals framed into the nodes at either end that gets a bit more tricky and in this case we actually need to know what the forces in these diagonal numbers are before we can State what the force in this vertical piece will be so we cannot clearly state yet whether this vertical piece will be in compression as we drew or in tension without knowing what the forces in these two diagonal pieces are so let's return to our truss and see if there are any other forces we can easily predict without detailed analysis now what if we place a load at a node that has a vertical piece framing into it as well as diagonals will any of this Force go into the vertical piece well there are no diagonals at the node below and so there is no path for the load to continue if it were to travel down this vertical piece and so no none of the load will go into this vertical piece and it will be a zero Force member and lastly if you see a vertical piece with no loads applied to it at all from either end it will again be a zero Force member and let's Place one last load here at this joint which type of force will be in this vertical member it will be in compression like a column now what about the diagonal pieces this can seem difficult at first and you might think we need to resort to some math to figure out which ones are in compression and which ones are in tension but actually we don't as long as the force is applied are all vertical downward forces and we have a simply supported truss with a pin at one end and a roller at the other we can figure out the behavior of these diagonals very easily with a simple trick if we think about the deflection of this truss its deflected shape will look something like this where its maximum deflection occurs somewhere around mid-span it may not actually be at mid-span because we have unequal loading here but for now we'll simplify and say it happens somewhere around mid-span so mid span is where we have our change in curvature where it is going downward to back upward and so we draw a line cutting the truss at that point now if we look at each diagonal and its relationship to the deflected shape if the slope of the diagonal follows the slope of the deflected shape that diagonal piece will be in tension if the slope of the diagonal is against the slope of the deflected shape that diagonal will be in compression so let's move from left to right here this first diagonal follows the slope of the deflected shape and so it will be in tension the next diagonal goes against the slope of the deflected shape and so it will be in compression the next one follows the slope so it is in tension this next diagonal also follows the slope of the deflected shape on the other side of our change in curvature and so it is also in tension the next one is against the slope so it's in compression and lastly tension and finally let's recall that the top chord is in compression and the bottom chord is in tension now let's take a look at our analysis software to see how we did you can see I have the exact same configuration and loading a pin at one end and a roller at the other and now if we turn on our axial forces you can see that we were exactly correct and just to prove to you that this is a true truss if I turn on my moment diagram you can see there is no moment in any of these members and also if I turn on my Shear diagram there is no Shear in any of these members either we are only resisting these perpendicular forces through internal axial forces in the next video we will be diving deeper into how to analyze trusses and there are two tools that we have for doing so the method of joints and the method of sections