Beam Deflections with Singularity Functions (Example 2) - Mechanics of Materials
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- เผยแพร่เมื่อ 7 ก.พ. 2025
- Example problem showing how to determine slope and deflection equations of a statically determinate beam with multiple loading types using singularity functions.
Thank you for this. The whole social distancing thing forced us to figure out these topics ourselves cause the lecturers can't do much now. The material they gave was so insufficient. I was about to give up until I found this video.
p/s: 6 years later and this guy's still hearting comments. What a legend.
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You're an absolute life saver!! I could not, for the life of me, grasp this method so tq so much
I love these videos. For starters, you sound like James Franco and that just makes learning so much more enjoyable. Secondly, your teaching style just flows really well and you always use great examples. Quality videos every time.
SIMPLYxISxPRO Thank you for the very nice and detailed comment about my teaching style! Although, I suspect the only thing James Franco and I have in common is a love-hate relationship with spiderman.
structurefree Your next video should be on calculating the stress Harry Osborn went through when spiderman killed his dad
Great example!! I learned so much from this video. I did not have things in my notes such as if you have a distributed load in the center, you have to grab both bounds. Very helpful!
I love you, my prof was not doing great with his lecturing XD
This helps a lot, thank you
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Thanks so much for posting these videos.. they seriously saved me.
Great!! Finally understood how to make a moment equation using singularity functions. Thank you so much for this!!!!
Explained better than my professor which I am paying hundreds of dollars to teach me.
You make Mechanical Eng fun! lol
Been watching since I was in Foundation, currently in Year 2~
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Thanks for the videos, helps a lot. Really appreciate them!
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insane in the membrane. :)
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It comes from integrating the (x-2)^1 from the shear equation... it becomes more easily apparent when you start with the loading function W(x) and integrate to get V(x) and again to M(x)->slope->deflection
Thanks man,You really saved my day :)
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Great stuff,thank you
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ur right, but am kinda regretting not going to lectures given that i gotta learn the whole curriculum by watching this guy the day before the test hahaha
thank you so much for this information
Hi. Thanks for the video. I just tried example my own. I took different Boundary condition. I consider at roller support slope is 0. It gave me different value for C1. So it means that final answer depends on boundary condition which you apply.
hi! why dont we directly integrate the load function. I guess we would get 4 boundary conditions instead of 2.Thank you!
Anyways great video for learners! Keep it up!
how do we know what power to use
Hello, thank you so much for the video
I would like to know what would have happened for the very first singularity function if there was a roller at C? Thank you
Great video, thank you very much!
But I have a question. If EI not constant, i.e. if the some section of the beam at the middle has different EI, can I still use the singularity functions to find deflection equationof the beam?
does this method work with any type of support ?
sory why when 2kn/m uniformal distrubuted loads moment is divided to 2?
if you look at the formula sheet you'll see for a dispersed load M = (pressure/2)^2
So, anti clockwise and clockwise moment are both positive on the singularity function? Great video,anyways!!
YOur a hero!
What will happen if we place the concentrated moment somewhere along the length and not on the edge??? Will we count only 5KNm?.. 4:40
Is it fair to say that the slope of the beam is 0 at x = 0 and then you plug that into the slope equation getting C1 = 0
Thank you! Sir.
Thanks for the video!
great video. but would be better if you started the video from the 4th derivative equation( load equation) EIv= -q(x)
so what do I if the structure is a cantilever beam where the deflection is 0 at x=0 and no other supports? Would C_1 and C_2 both be 0?
hi , can u explain how can we choose which power of n we shud use , like in previous video you used for distributed load the power of 0 but here u are choosing it to be 1, why is that?
P.S. u and ur videos are awesome
+Surayans Tiwari in this video he's using the singularity function for M(x), so for UDL it would be -w/2^2, while the previous video was describing the w(x) function which is why n = 0
When we start integrating from the loading function, why isn't there any constants of integration? Because I've noticed that the constants only show up for the slope and deflection functions.
Hi thx for the videos, they helped me a lot. However, not sure if you will still answer my question now... or anyone can help... So, in 3:18, the first value of the moment function is in positive instead of negative? It is in clockwise direction so I think it should be in negative... this confusing me, and thx
Would you be able to find the moment diagram through this method? If so, is it possible if you could explain how? Thanks :)
How can i understand the powers will be either 0/1/2 over singularly brackets?
So is double integration method with a singularity function just the same as Macaulay's method? Also man your videos are so helpful it's unreal.
Owen Barnes Yes. Thank you for the kind words.
How would you set up the load equation if there is a triangular distributed load? Thanks for the videos!
+Kaan Korkmaz I think this introductory video on using singularity functions might help you figure out how to handle a triangular (or linearly distributed) load. th-cam.com/video/Fi_4nY808Q8/w-d-xo.html
I appreciate it!
+Kaan Korkmaz in case you did not want to follow the video: The singularity function for a triangularly distributed load is M = (m/6)^3. m is the slop, which is rise/run.
what if you only have 1 boundary condition. lets say its positioned in the center of the beam and you want to know the deflection on both sides. Do you just plug in your solo boundary condition and hope that both c1 and c2 are zero
using singularity function and double integration method both are same or difference between this two method
I know your videos are already quite long but I wish you took it just 1 step further to find slope and deflection at some point, any point for that matter. My confusion is what do we do when a term inside the parenthesis is negative? For example, lets say we wish to know deflection at 6m. Your 4th term would look like this: (4.625/6)KN(6m-8m)^3 Does this mean that this term is zero since according to step functions x is less than or equal to 8m ? Or does step function rules not apply at this point in the method and the actual value for that term is -6.1666667KN*m^3 Please let me know at your earliest convenience and thank you in advance
How about if udl. hinge and roller at the back. inverted version? should i include the Rb?
I understand you sample but and I have seen another exaple with a l uniform type wedge load that starts at point away from the left reaction and increases short of the right reaction. I now that the original load has to continue to the right reaction and a counter wedge shown from the end of the wedge load to the right reation. but I can't come up with a formula that gives me the proper deflections and curvature. I have it working fine when the wedge goes all the way to the right reaction
If you use Slope at x=0 is 0, then you get C1=0. Is this correct, because I am a little confused on why you chose to use y(8)=0 instead.
Why is it that you can just jump to the moment equations with no constants of integration from going to shear and then to the moment equation? You only use constants for the last two equations. Do they just always cancel out to zero?
Ginny Schilling you could start from a loading equation and integrate 4 times to get displacements. There will be constants with each integration and for shear and moment functions those constants would be the support reactions of the beam.
This was kind of asked below, but if you have a plywood beam that slopes from its lowest points from its ends to its maximum point at the center how do you handle the varying moment of inertia? In the standard example, when I is a constant, EIy''=M, EIy`=EI*slope, and EIy=EI*deflection. Is it just a straight replacement of I with I(x) [EI(x)y] or does I being a function of location complicate the integration?
Yes, for a tapered beam the MOI would be included with the integration. Ooooh, that would be a fun and challenging.
Not too challenging when you use math software. I created a Mathcad calculation that will accept 7 supports (fixed or pinned), 10 point loads, 10 distributed loads, and 10 ramp loads. Every thing gets thrown in a matrix that solves for all the reactions and c1/c2. Plot out the shear, moment, and deflection diagrams and you're the talk of the town (at least I imagine that is what is town worthy talk). Very handy for this structural engineer.
Juston Fluckey Sounds cool. I'm a big fan of Mathcad too. Definitely, "talk of the town" for a city full of engineers.
Hi. How would you go about solving a problem where there is a triangular distributed load starting at 0 and increasing to half or quarter of the beam? with the linearly distributed load you mentioned that according to the equation, the load continues to the end and has to have the non existent part cancelled off. how would you cancel off a triangular distributed load?
what if it's a triangle distribution I'm a bit confused from the solution in my text book. Can you still "turn it on and off" but instead of 2 / 2 it would be the slope / 6 ? but wouldn't the "on" position be 0 ?
As per as I know, singularity method can't be directly used in beams with internal hinge (slope discontinuity). But I have also seen only one lecture note where a slope discontinuity function was added into the equation similar to load and moment. Do you know something about this or can you suggest some literature regarding this. Thanks
What if the loading is not UDL.If triangular starting from different point of the Beam?How to write loading function using heavy side notation then?
I Wonder i think i have an explanation video on singularity functions that might be helpful.
I love you
How do you go about finding the max deflections using these equations, i try making the slope equal to 0 and solving for x on a beam with symmetrical loading, but the x value doesn't come out to be the halfway point as expected? What am i doing wrong?
Az your method is good. for this example the beam geometry and loading are not symmetrical, therefore the max deflection does not occur at midspan.
Wouldn't the integral of the moment function be the shear function? How does the integral of (d^2v/dx^2) get you the slope? shouldn't the integral of d^2v/dx^2 = d^3v/dx^3 ?
Hi,
How will you define/describe the equation if the distributed load is all over the beam?
For the distributive loading, why is it 2kNm/2? thanks!
which software can we use for lectures, I need to have the name of software that u already have it tnx :)
Does anyone know how you are meant to know what powers to use for the moment function?
you may want to watch this one...th-cam.com/video/Fi_4nY808Q8/w-d-xo.html
Why do you find v''(x) at first rather than v''''(x) and integrating 4 times?
Great explanation, but I have one question why is the last part of the equation M(x)/EI or w(x) always zero in this case 5(x-11)^0? I noticed you did this in example 1, but I'm not quite understand why. Thanks
Because X will never ge greater than 11 (which is the length of the beam) so that singularity function will not be active
how could we find the max deflection?
One way would be to identify where the slope is zero.
Can we find Max Deflection using singularity functions?
when we finding reation at A and B, during summation of moment at A. the concentrated load at C 5 k.N. you only subtract it. are we can multiply with the span upto A means -5*11
I can I use singularity equations to solve ANY deflection slope problem? I'm wondering can I use singularity on a beam that has different moments of inertia (fat part and then a small part), do I Just use 4 equations, 2 for each slope/deflection with each corresponding moment of inertia? Thanks =d
Ya. you can do that. You just need to do like in the double integration method where you had different equations for different sections of the beam except in this method you have different equations for different material properties. it would be a huge pain to do with more complicated beams
i have one question sir,y c2 is0 when u do x=10?
How to find C1 if we have only one boundary condition only one end fixed ?
° Chill out the World ° on a fixed end there are two boundary conditions. displacement = 0 and slope = 0.
structurefree Hello sir,
How to find the C1 if we have beam with three hinges supported, it is symmetric with three unknowns. The slope for hinge is not = 0, right?
Thanks in advance.
Ahmed Alshawi You are correct. One of your boundary conditions would be that the moment at the hinge location is zero.
@structurefree, I am just wondering ,how do you determine the powers of the singularity function when you start with the moment equation? I am a little confused on that
why you put +ve sign in calculation for moment 5kN.m anticlockwise ? thank you..
haidil helqi He does assume that anti-clockwise is positive, but as you know that in the bending moment diagram when there is an external moment acting clockwise on the beam the graph shoots upwards in the positive direction. Thus making it positive in his singularity equation. That is what I understand it to be. Hope that helps.
how to get the deflection of beam on 8m is zero?
Anyone know why the distributed load was divided by 2?
Watch this introductory video and you will know....th-cam.com/video/Fi_4nY808Q8/w-d-xo.html
What shall I do if my EI is not constant?
Then EI will be a function of x, when you integrate curvature to find slope and deflection. You'll have a product of two functions, M(x) and EI(x) as your initial integrand.
What program do you use to draw this all out?
info:I designed a 5 by 5 matrix by using An Integral method to solve for the unknown variables :R(a)=27/8,R(b)=37/8,Slope(a)=68/3,Slope(b)=1 and Y(c)=-51/2 end of the beam.
That was good but you sped up a little at the end...thanks a lot though!
c1 should be unitless , how u can write it . it is scalar as i know
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C2 was 181 when i put it in my calculator
khan ain't got noth on this =)
x=8 sorry
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I know your videos are already quite long but I wish you took it just 1 step further to find slope and deflection at some point, any point for that matter. My confusion is what do we do when a term inside the parenthesis is negative? For example, lets say we wish to know deflection at 6m. Your 4th term would look like this: (4.625/6)KN(6m-8m)^3 Does this mean that this term is zero since according to step functions x is less than or equal to 8m ? Or does step function rules not apply at this point in the method and the actual value for that term is -6.1666667KN*m^3 Please let me know at your earliest convenience and thank you in advance