Is Shear Stress Vertical or Horizontal in I-Beam Webs?

[UMath1]

My mechanics of materials book states that in order for an element to be in equilibrium it must have equal horizontal and vertical shear stresses. However, it also states "The shear stresses in the web of a wide-flange beam act only in the vertical direction and are larger than the stresses in the flanges" . How can the stress only act in the vertical direction if in order to maintain equilibrium they must act in the horizontal direction as well?

 

[PhanthomJay]

My mechanics of materials book states that in order for an element to be in equilibrium it must have equal horizontal and vertical shear stresses. However, it also states "The shear stresses in the web of a wide-flange beam act only in the vertical direction and are larger than the stresses in the flanges" . How can the stress only act in the vertical direction if in order to maintain equilibrium they must act in the horizontal direction as well?

Good question and insight. Yes, you are correct, there must be horizontal shear stresses in the web, but they don't act left to right, they act longitudinally into and out from the plane of the page, along the beam's long axis. If you envision a cubic element in the cross section, which has 6 sides, the front and back faces have equal and opposite vertical shears, and the top and bottom faces have equal and opposite horizontal longitudinal shears. Not much going on in the side faces. In the flanges, horizontal left to right shears vary from zero at the free edge to a max at the center of the flange, with equal and opposite longitudinal shears on the side faces of the cubic element. Shear stresses are largely carried by the web , which is why codes use web area only when determining average shear stresses.

 

[Dr.D]

How can the stress only act in the vertical direction if in order to maintain equilibrium they must act in the horizontal direction as well?

This is basically a problem of using two dimensional terminology to describe a three dimensional situation. PhanthomJay has given the correct explanation above.

 

[UMath1]

Good question and insight. Yes, you are correct, there must be horizontal shear stresses in the web, but they don't act left to right, they act longitudinally into and out from the plane of the page, along the beam's long axis. If you envision a cubic element in the cross section, which has 6 sides, the front and back faces have equal and opposite vertical shears, and the top and bottom faces have equal and opposite horizontal longitudinal shears. Not much going on in the side faces. In the flanges, horizontal left to right shears vary from zero at the free edge to a max at the center of the flange, with equal and opposite longitudinal shears on the side faces of the cubic element. Shear stresses are largely carried by the web , which is why codes use web area only when determining average shear stresses.

Where do the horizontal left to right shears arise from in the flange? I can visual transverse shears on a beam creating vertical shears on the web but I can't visualize the horizontal shears in the flange.

Similarly, I am having trouble visualizing or creating a free body diagram to understand where the horizontal shears on the nails in the box beam arise from shown in attached picture. I can see there would be longitudinal shear stresses if the box were rotated 90 degrees.

 

[PhanthomJay]

Where do the horizontal left to right shears arise from in the flange? I can visual transverse shears on a beam creating vertical shears on the web but I can't visualize the horizontal shears in the flange.

Similarly, I am having trouble visualizing or creating a free body diagram to understand where the horizontal shears on the nails in the box beam arise from shown in attached picture. I can see there would be longitudinal shear stresses if the box were rotated 90 degrees.

Actually, there are vertical shear stresses in the flange also, but they average out very small when spread out over the full flange width , b, from the shear stress formula VQ/Ib, which yields a parabolic shear stress distribution over the flange, but again, rather small, and essentially these are ignored for thin walled sections. The horizontal and complimentary longitudinal shear stresses in the flange come from shear flow concepts, where now Q is determined for the flange section which maximizes at the center of the flange using the flange thickness, not the flange width, in the shear stress formula. This horizontal shear stress distribution in the flange is linear, not parabolic, because the section of interest in the flange is always the same distance from the neutral axis. The nails in the box beam carry longitudinal shear into the plane of the page (perpendicular to the nail), using the area of the flange in between the nailed joints for determining Q.

 

[Dr.D]

This is an example of a question where we really need a good diagram.

 

[PhanthomJay]

This is an example of a question where we really need a good diagram.

Ok. Last page shows shear flow distribution (q) in flanges and web. Shear stresses are q/t.
http://www.ae.msstate.edu/tupas/exp/A14.7_ex1.html

 

[Dr.D]

That's a rather good presentation of the whole issue. As that discussion demonstrates, this is not a simple matter and requires careful and complete descripiton to get it correct.

 

[UMath1]

Thank you for this. I am still somewhat confused. I understand that the calculations prove that the max shear stress is at the center of the flange. But understanding the directions of the stress with the fluid flow analogy does not seem concrete to me. Is there a free body diagram I can make to find the directions of the stresses?

Is there a good diagram for the box beam case too?

 

[Dr.D]

The URL given by PhanthomJay in #7 is as good as you are likely to find. Study it carefully; it has all you need, I think.

 

[PhanthomJay]

A FBD shows that the horizontal shears and torques caused by those shears sum to zero, as one might expect for equlibrium, since there are no horizontal forces applied. That does not explain, however, why they are horizontal in the first place. As the Good Doctor points out, you are not likely to find a good explanation on the web. The stress distribution can be solved using the Theory of Elasticity, which is a brutal topic. An analogy might be the stress concentration at hole in a long plate subject to a uniform tensile load. That stress is 3 times the stress away from the hole. The Theory of Elasticity shows using horrendous partial differential equations that the stress concentration is exactly 3. Other than that, you are out of luck. Keep up the good work!

 

[Gunner34878]

Thank you for this. I am still somewhat confused. I understand that the calculations prove that the max shear stress is at the center of the flange. But understanding the directions of the stress with the fluid flow analogy does not seem concrete to me. Is there a free body diagram I can make to find the directions of the stresses?

Is there a good diagram for the box beam case too?

Horizontal Shear will come due to moment variation along the section. If you have dx element along the length of beam, in that you can see the unbalance horizontal force is balanced by the horizontal shear along the length of the beam and after that you can use the complementary shear concept. In box beam you cut the section at equal distance from symmetrical line and by force balance concept u find the shear stress direction along longitudinal plane and then you can use complementary shear concept to get the direction of shear flow.

Source https://www.physicsforums.com/threads/is-shear-stress-vertical-or-horizontal-in-i-beam-webs.945083/