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Consider a rectangular cross sectional solid beam, fixed at one end and a uniform shear force applied to the other end.

Below shows the side view for this beam.

enter image description here

Now, I change the support area as shown below. How should the force distribution be now?

enter image description here

In the first image, above the neutral axis we already know that the structure will be in tension while below the neutral axis, the structure will be in compression. The net upwards tensile force and net bottom compressive forces are equal, and when combined together, the resultant is a reaction moment (with no net reaction force along the longitudinal axis, and only net reaction force to counter the shear).

Now, for the second image, again the reaction moment will exist. Since no external force is applied along the longitudinal axis, there won't be any reaction force along that axis. The only reaction force that will exist is the force to counter the shear. So how would the tension and compression forces be distributed (on the cross section marked with blue line) so that they will return a total reaction moment only?

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3 Answers 3

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If your question is :

how should be the force distribution at the cross section right before the fixed support? This was my actual question, actually.

Since the structure needs to transfer internally a bending moment right before the support there will be a tension layer and a compression layer. The combination of those layers in each cross-section is that creates internally the bending moment that resists the load.

The exact size of the tension layer and the compression layer (they need to be somewhat symmetric, even for the simple case (isotropic material, constant cross-section) is not something that is usually covered in textbooks. IMHO the best way to address that is either experimentally or FEA.

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The stresses will be dense at the support and gradually fanned out per the elastic property of the material of the beam. Theory of Elastic Plate, or the finite method, is required to find the exact/approximate distribution width ($b_{EFF}$) alone the X-Axis.

enter image description here

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  • $\begingroup$ So there is going to be a reaction moment. I agree. My question is how would this reaction moment be acting on the support face in terms of forces? Will there be tension as well as compression on the support face? But the support face lies above the N.A, and we have learned that the stresses above the neutral axis should only be either tension or compression. But the moment reaction cannot exist in only tension or only compression. This paradox confuses me. $\endgroup$ Jan 12 at 19:37
  • $\begingroup$ Yes. At the support, the beam axis is meaningless but the local axis of the support plate (the red dash line). The local neutral axis remains in the midpoint of the plate at the contact plane and the stresses distribution is linear as usual (thru elastic behavior). As the effective area varies from the support toward the free end, the local neutral axis varies as well. The local neutral axis eventually will become coincient with the neutral axis of the beam at the free end. $\endgroup$
    – r13
    Jan 12 at 20:37
  • $\begingroup$ Hmmm. Interesting. So how would you anticipate the forces (tension and compression, which in turn generate the reaction moment at the small support) be distributed at this small support? The N.A should be lying exactly at the middle of the support face? What if it was an I beam, and I was only supporting the face of upper flange, would you still say it should lie exactly the middle of this flange face? $\endgroup$ Jan 12 at 20:44
  • $\begingroup$ The above explanation is from the view of the "elastic behavior" camps, which, IMO, is non-realistic for this situation. Because the b << d, the deep beam behavior and non-linear stresses distribution will be more dominant towards the free end. Computer modeling is helpful in obtaining the real stresses distribution throughout the length of the beam. $\endgroup$
    – r13
    Jan 12 at 20:46
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    $\begingroup$ I suggest starting a new question with sketches that reflect these updates. $\endgroup$
    – r13
    Jan 12 at 21:07
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We could think of the second case as a beam welded to the support only on the top part as you show say with a length H/4, with the beam length L, and its height, H, the shear substituted with a point load P.

The weld has to support the entire shear,$ \ P$, and a tensile force equal to the tensile force that is required to resist the moment of PL. $$T=C =\frac{ P*L}{2H/3} $$

There will be a complex pattern of stress distribution from the weld to the section you show. The shear will spread to the entire height of the beam, shown red, close to a uniform shear same as P/H and the tension will fill in the triangular tension I have shown with a blue line on your sketch, and the compression will be supported by the support.

If the support can not resist compression if the vertical portion under the weld can not support compression then the weld or connection must support both shear and the entire moment and transfer them to the blue section. Then there is potential for failure due to stress concentration!

Edit

After comment by OP.

The support on the left can be treated as a beam with two sections with a large stepped depth.

It will have both moment of PL and shear. Therefore it will have tension on top and compression on the bottom plus shear.

But again the huge stresses on the support combined with the fact that due to the stiffness of the rest of the beam will lead to high stress concentration in the small transition are between the support and the blue line in you drew. and will most likely buckle around the support.

I am waiting on the taxi way for my turn to take off. will elaborate later.

beam

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  • $\begingroup$ Should the support resist compression or not? The support is above the N.A and (as taught by the Solid Mechanics book) the region above the N.A should only resist tension or only resist compression. This is what perplexes me. I mean the moment reaction need to have compression and tension both on it so that it can generate a reaction moment, but the support is above the N.A so analytically it should only resist tension. What is the reality? $\endgroup$ Jan 12 at 19:41
  • $\begingroup$ i edit my answer to address it. please check it and tell me what you think. $\endgroup$
    – kamran
    Jan 12 at 19:50
  • $\begingroup$ Right. So it will have both on the support, i.e. compression as well as tension. Now, how should be the force distribution at the cross section right before the fixed support? This was my actual question, actually. $\endgroup$ Jan 12 at 20:14

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