Hi i know this is so late but other people will have this question. The positive and negative conventions for bending moment and shear force are shown in the diagram. Just a quick reminder that shear force is the internal force that keeps a segment of beam static when other forces 'removed' by imaging cutting the segment off. The bending moment is what prevents that same segment from rotating. This makes sense, because an object at rest still needs to be at rest when a segment of the whole is analyzed. Positive shear force is actually down if you are analyzing from the left end of the beam to the right (this changes if you analyze from right--> left). A good way of thinking about this is if a bending moment would make a beam bend into a smile, it's positive. Frown makes it negative. They decided to do this because they wanted the convention to be positive when resisting a supporting force (like how in region AB, the only force is going up). The reason why the moment is positive when smiling is because the convention is easier if the positive moment resists a positive shear force.
+Aggerschoen There are at least 3 ways you could go about obtaining the strain from this experiment. First, put a strain gauge on your specimen. Second, if you know the stiffness of your material, strain is stress / stiffness. However, if you are unable / unwilling to put a strain gauge on your specimen, and you are testing it with the intent of measuring it's stiffness, you are left with the third option. Use the beam deflection equations, which can relate the deflection at your supports, to the applied loads, geometry, and material properties, to back out the stiffness of the material. From the stiffness you can again calculate strain with knowledge of the stress field you saw in this video. In case you're unfamiliar with the beam deflection equations, they're in most text books, and a quick google search turned up this: www.engineersedge.com/beam_bending/beam_bending6.htm Good luck.
This TH-cam series has been helpful. Would you be able to do a more in depth example using actual numbers? Something to the complexity of the homework or tests
Thanks for the video, it is really simple and quite informative. Concerning the span to depth ratio (S/d), how to calculate the correct S/d (for ceramics) so as to avoid the effect of deflections on beam geometry?
Hi, first I wanna thank you for your useful video and secondly I have this question about the displacement formula/equation for 4-point bending. I can't find it anywhere! I found out about the 3 point bending test but what I really need is 4 point bending test displacement formula that you mentioned at the end of your video.
Moey Rad Thanks for the question. You'll find methods to go from the moment diagram to the deformation equations in most intro mechanics of materials texts (Chapter 12 of Hibbeler for instance). If we take 'v' to represent the displacement of the neutral axis, then the beam bending equations are of the form: M(x) = E*I*d^2/dx^2(v) To solve this, you should recognize that M(x) is a piecewise continuous function. Therefore, integrate the moment equations twice for each of the regions (AB, BC, CD), and then apply the proper boundary and continuity conditions for the 6 constants of integration. 2 Boundary conditions: displacement at A and D is zero. 4 Continuity conditions: displacement and slope of the beam at B and C are continuous
Hi. I am currently carrying out a project to determine the Young's Modulus of a steel beam using 3 point or 4 point bending, and after watching your videos, 4 point bending seems to be the less destructive/damaging option to use. With regards to calculating the deflection, can't that just be done using the Integration or Macaulay's Method, or is a gauge needed by force at the centre of the beam? Thanks for your time, and keep up the good work :)
+Andrew Borg I would typically recommend taking all the experimental measurements that you can, and then comparing those measurements to predictions based on the theoretical equations. If all you are trying to determine is the youngs modulus, I'd recommend using surface mounted strain gauges. You should then pretty easily be able to measure the maximum strain, and relate it to the bending stress in the beam. If you also want the deflection, you could experimentally measure the deflection of the midpoint with an extensometer, or if your sample is large enough perhaps capture the deflection on video, and then use DICM based techniques to back out the displacement field of the full beam (en.wikipedia.org/wiki/Digital_image_correlation). As long as you are well within the linear elastic range of the material, and you have a nice homogeneous material, the standard beam deflection equations should accurately match the experimental deflection. However, if you approach the yield stress, your will likely get local yielding near the supports, and this will causes deviations between the theoretical bending curves, and your experimental results. This is one of the reasons why if you need to get yield properties, you're better off using a traditional tensile test. As a final note, I wouldn't say that the 4 point method is any less destructive that the 3 point method. The destructiveness will depend on what load you apply in either case. An advantage of the 4 point test, is that the central region has a constant moment, and therefore constant bending stress. This is similar to having a uniform gauge region in a tensile test specimen. 3 point bending has no uniform stress region, making measurements slightly more challenging, and making your measurements more susceptible to flaws in the sample. Good luck
I'd recommend tracking down the ASTM specification relevant for your material. For example: : D7264/D7264M - 15 is the Standard Test Method for Flexural Properties of Polymer Matrix Composite Materials. Aside from that, I'd recommend working out what the deflection of the beam will be (see the section on Static Beam Theory: en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory). If you know the expected displacement, then you also know the surface strains. From that point you can either measure the surface strains, or use an extensometer to measure the midpoint deflection and compare those values to your predicted ones to back out what the bending stiffness is. However, if all you need is the bending stiffness, then the definition of bending stiffness as K = P/x, where K is the stiffness, P is the transverse load, and x the displacement, point to three point bending being a useful approach as well. (en.wikipedia.org/wiki/Bending_stiffness)
hi thank you so much for your great video. My samples cross section is square hollow shape. I would like to know the equaton of stress? For this cross section, the moment is the same?
The equation for calculating stress is the same, you just need to compute the moment of inertia for the shape you have. You should be able to find any examples in any common Mechanics of Materials textbook (something like Hibbeler, for example)
The direct approach would be to put a strain gauge on your beam (described near the 4.30 mark in this video). Indirectly, if you already know the stress, and you know the stiffness, you can determine the strain using hooke's law
Hi John, Thank you so much for your reply. I have the LVDT for measuring the beam deflection. the stifness of the beam is unknown. As you have mentioned the stress can be calculated using the formula in the video. I would like to calculate the strain using the beam deflection. I am using standard ISO 14125, the strain is calculated using this formula: strain=4.7sh/L^2, s is deflection, h is the thickness of the specimen. However, I do not know how this equation has derived? is this equation is applicable for any cross section?
Hi Fariborz, I'm happy to help clarify the basics, but I am unfamiliar with the details of the standard your are using. However, it appears to only be valid for a very specific geometry. At this point, assuming you are a student, you should be consulting with one of your local faculty members for assistance. If you are not at a university, it may be time to consult a practicing engineer. Your experiment sounds simple enough that anyone with a mechanical, civil, or aerospace engineering background should be able to help. Best of luck
Hi weijian, I'm not sure where in the video I say that we measure the deflection in sections. In general, for a problem like this, you want to determine the shear force, and moment throughout the beam, as they will generate shear, and normal stresses accordingly. Also, by solving for the bending moment in each of the 3 unique regions, you should eventually be able to determine the elastic curve (deflection of the neutral axis) for the beam, which can be quite helpful.
Thank you for your video. May I know why do we need to measure the deflection in sections? Is it to find out the strain value?
Thanks
Very good material for teaching purpose and practical applications.
At 2:45 why wouldn't the moment be -0.5F*x - 0.5F*(x-1/6L) aren't all forces going downwards and thus negative?
Hi i know this is so late but other people will have this question. The positive and negative conventions for bending moment and shear force are shown in the diagram. Just a quick reminder that shear force is the internal force that keeps a segment of beam static when other forces 'removed' by imaging cutting the segment off. The bending moment is what prevents that same segment from rotating. This makes sense, because an object at rest still needs to be at rest when a segment of the whole is analyzed.
Positive shear force is actually down if you are analyzing from the left end of the beam to the right (this changes if you analyze from right--> left).
A good way of thinking about this is if a bending moment would make a beam bend into a smile, it's positive. Frown makes it negative.
They decided to do this because they wanted the convention to be positive when resisting a supporting force (like how in region AB, the only force is going up). The reason why the moment is positive when smiling is because the convention is easier if the positive moment resists a positive shear force.
Hi. If you have the displacement from a test machine for this experiment, how do you convert that to the strain?
+Aggerschoen There are at least 3 ways you could go about obtaining the strain from this experiment. First, put a strain gauge on your specimen. Second, if you know the stiffness of your material, strain is stress / stiffness. However, if you are unable / unwilling to put a strain gauge on your specimen, and you are testing it with the intent of measuring it's stiffness, you are left with the third option. Use the beam deflection equations, which can relate the deflection at your supports, to the applied loads, geometry, and material properties, to back out the stiffness of the material. From the stiffness you can again calculate strain with knowledge of the stress field you saw in this video. In case you're unfamiliar with the beam deflection equations, they're in most text books, and a quick google search turned up this: www.engineersedge.com/beam_bending/beam_bending6.htm
Good luck.
Really good explanation
Can you help me in solving the same problem for a 16 ply CFRP layup Composite beam.
This TH-cam series has been helpful. Would you be able to do a more in depth example using actual numbers? Something to the complexity of the homework or tests
Thanks for the video, it is really simple and quite informative.
Concerning the span to depth ratio (S/d), how to calculate the correct S/d (for ceramics) so as to avoid the effect of deflections on beam geometry?
Hi, first I wanna thank you for your useful video and secondly I have this question about the displacement formula/equation for 4-point bending.
I can't find it anywhere! I found out about the 3 point bending test but what I really need is 4 point bending test displacement formula that you mentioned at the end of your video.
Moey Rad Thanks for the question. You'll find methods to go from the moment diagram to the deformation equations in most intro mechanics of materials texts (Chapter 12 of Hibbeler for instance).
If we take 'v' to represent the displacement of the neutral axis, then the beam bending equations are of the form:
M(x) = E*I*d^2/dx^2(v)
To solve this, you should recognize that M(x) is a piecewise continuous function. Therefore, integrate the moment equations twice for each of the regions (AB, BC, CD), and then apply the proper boundary and continuity conditions for the 6 constants of integration.
2 Boundary conditions: displacement at A and D is zero.
4 Continuity conditions: displacement and slope of the beam at B and C are continuous
Hi. I am currently carrying out a project to determine the Young's Modulus of a steel beam using 3 point or 4 point bending, and after watching your videos, 4 point bending seems to be the less destructive/damaging option to use.
With regards to calculating the deflection, can't that just be done using the Integration or Macaulay's Method, or is a gauge needed by force at the centre of the beam?
Thanks for your time, and keep up the good work :)
+Andrew Borg I would typically recommend taking all the experimental measurements that you can, and then comparing those measurements to predictions based on the theoretical equations.
If all you are trying to determine is the youngs modulus, I'd recommend using surface mounted strain gauges. You should then pretty easily be able to measure the maximum strain, and relate it to the bending stress in the beam.
If you also want the deflection, you could experimentally measure the deflection of the midpoint with an extensometer, or if your sample is large enough perhaps capture the deflection on video, and then use DICM based techniques to back out the displacement field of the full beam (en.wikipedia.org/wiki/Digital_image_correlation).
As long as you are well within the linear elastic range of the material, and you have a nice homogeneous material, the standard beam deflection equations should accurately match the experimental deflection. However, if you approach the yield stress, your will likely get local yielding near the supports, and this will causes deviations between the theoretical bending curves, and your experimental results. This is one of the reasons why if you need to get yield properties, you're better off using a traditional tensile test.
As a final note, I wouldn't say that the 4 point method is any less destructive that the 3 point method. The destructiveness will depend on what load you apply in either case. An advantage of the 4 point test, is that the central region has a constant moment, and therefore constant bending stress. This is similar to having a uniform gauge region in a tensile test specimen. 3 point bending has no uniform stress region, making measurements slightly more challenging, and making your measurements more susceptible to flaws in the sample.
Good luck
+jdomann Thanks so much for your help :)
+Dimitri Giordani you can bring out an equation relating the deflection of the beam at the centre and the Flexure Formula for stress (stress = -My/I)
how can I compute the bending stiffness using the four point bending method?
I'd recommend tracking down the ASTM specification relevant for your material. For example: : D7264/D7264M - 15 is the Standard Test Method for Flexural Properties of Polymer Matrix Composite Materials.
Aside from that, I'd recommend working out what the deflection of the beam will be (see the section on Static Beam Theory: en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory). If you know the expected displacement, then you also know the surface strains. From that point you can either measure the surface strains, or use an extensometer to measure the midpoint deflection and compare those values to your predicted ones to back out what the bending stiffness is.
However, if all you need is the bending stiffness, then the definition of bending stiffness as K = P/x, where K is the stiffness, P is the transverse load, and x the displacement, point to three point bending being a useful approach as well. (en.wikipedia.org/wiki/Bending_stiffness)
hi thank you so much for your great video. My samples cross section is square hollow shape. I would like to know the equaton of stress? For this cross section, the moment is the same?
The equation for calculating stress is the same, you just need to compute the moment of inertia for the shape you have. You should be able to find any examples in any common Mechanics of Materials textbook (something like Hibbeler, for example)
Thank you so much for your guidance. How about for calculation of the strain at maximum point from the neutral axis?
The direct approach would be to put a strain gauge on your beam (described near the 4.30 mark in this video). Indirectly, if you already know the stress, and you know the stiffness, you can determine the strain using hooke's law
Hi John,
Thank you so much for your reply.
I have the LVDT for measuring the beam deflection. the stifness of the beam is unknown. As you have mentioned the stress can be calculated using the formula in the video.
I would like to calculate the strain using the beam deflection. I am using standard ISO 14125, the strain is calculated using this formula: strain=4.7sh/L^2, s is deflection, h is the thickness of the specimen.
However, I do not know how this equation has derived? is this equation is applicable for any cross section?
Hi Fariborz,
I'm happy to help clarify the basics, but I am unfamiliar with the details of the standard your are using. However, it appears to only be valid for a very specific geometry. At this point, assuming you are a student, you should be consulting with one of your local faculty members for assistance. If you are not at a university, it may be time to consult a practicing engineer. Your experiment sounds simple enough that anyone with a mechanical, civil, or aerospace engineering background should be able to help.
Best of luck
Hi, everyone.. How can I find ''stifness [K]'' for round hollow section area and again 4 Point Bending?
Well, it would need a 4 point bending apparatus which is too complicated to set up
Thanks!
Hi weijian, I'm not sure where in the video I say that we measure the deflection in sections.
In general, for a problem like this, you want to determine the shear force, and moment throughout the beam, as they will generate shear, and normal stresses accordingly. Also, by solving for the bending moment in each of the 3 unique regions, you should eventually be able to determine the elastic curve (deflection of the neutral axis) for the beam, which can be quite helpful.
I love you II