Shear area and first moment of area to calculate shear stress

[SuG89]

Hello,
For a rectangular cross-section, the shear area is 0.833 times the cross-section area. Can this shear area be used to calculate the transverse shear stress [approach 1]?
Usually, the transverse shear stress is calculated using the formula VQ/It where Q is the first moment of area [approach 2]
Is there any link between these two approaches?
Regards,
Su

 

[rb1957]

"the shear area is 0.833 times the cross-section area" ? do you mean that max shear stress = P/(cross-section_area *0.833) ?

have you tried calculating VQ/It and derived for yourself that max shear stress is 1.2*P/A ?
then turn the section on it's side and see what your get ?

another day in paradise, or is paradise one day closer ?

 

[BAretired]

For a rectangular section, maximum shear stress is 1.5P/A, not 1.2P/A. Shear stress varies parabolically from 0 at the top and bottom to maximum at mid height.

BA

 

[rb1957]

I thought so , but just reciprocalised the 0.833.

another day in paradise, or is paradise one day closer ?

 

[Settingsun]

The term 'shear area' seems to mean different things to different people. When I was taught, it related to elastic shear deflection of beams. Shear deflection is usually negligible compared to the flexural deflection so everyone forgets it upon graduation (including me).

There's probably some relation between shear area and shear stress distribution but I couldn't say what it is.

From memory, shear deflection is something like the integral of (shear force divided by shear stiffness) over the length of the beam. Shear stiffness = shear modulus of the material multiplied by shear area of the section and is analogous to EI for flexural stiffness.

Your value of 0.833*gross area matches this definition of shear area.

 

[BAretired]

steveh49, could you kindly elaborate on the shear area of a rectangular section. I consider it to be the same as the gross area; however, in the case of an I-beam, I would consider shear area to be the area of the web.

Where does the factor 0.833 come from?

BA

 

[BAretired]

See attached for shear stress in rectangular beam and I-section:

https://www.codecogs.com/library/engineering/materials/beams/shear-stress-in-beams.php

BA

 

[JoshPlumSE]

BAretired -

I believe steveh49 is correct. I don't have the formulas in front of me, but I remember deriving the shear area reduction factors for RISA many years ago. And, I believe it was something like 0.833 for solid rectangular areas. The reference we used was "Stress, Strain, and Structural Matrices" by Walter Pilkey.

For wide flange members, there is a really complex derivation you can do for this shear deformation reduction factor. But, you're always pretty close if you just use web area divided by full area. I believe in RISA we just used this simplification for wide flanges and channels. However, for the solid shapes and pipes we derived the values explicitly.

 

[BAretired]

JoshPlum,

Thanks for the comment. I believe there is a property called shear area which, for a rectangular section is 0.8333*A. It is mentioned in a few articles on the internet. I'm not totally clear on it yet but I think it has something to do with deflection due to shear, something which is usually neglected by engineers because it is usually small compared to bending deformation. There are instances where it should not be neglected.

Shear area should not be used to determine shear stress, the question raised in the original post. Shear stress is calculated by VQ/Ib which for a rectangular section results in a maximum value of 1.5*A at mid-height.

BA

 

[Hokie93]

Roark's Formulas for Stress and Strain gives this decent treatment though it is presented as shape-dependent multiplier, not a "shear area". Mathematically, it works out to be the same. For example, the multiplier (F, in Roark's nomenclature) for a rectangle is 6/5. The multiplier is in the numerator and cross sectional area is in the denominator. Moving 6/5 to the denominator yields the 0.833 coefficient the OP cited.

 

[JoshPlumSE]

BAretired -

Yes, the 0.833 is purely related to deflection not stress. Sorry, that I didn't make that clear. However, I was just responding to steveh49 who made it really clear what he was talking about.

Note: Back in the old days of RISA (version 4.5-ish in 2002), the program used one factor (entered by the user) for both of those items. At least for any shapes that didn't come from the steel database. Since this was incorrect, and we didn't want users to have to navigate esoteric stuff like this anymore, I had to go derive the formulas (with the help of Pilkey). Now the program will set the two proper values for every shape type.

 

[Settingsun]

This website has a derivation of the 0.833 factor for a rectangular section.

http://www.roymech.co.uk/Useful_Tables/Beams/Shear_stress.html#Deflections

 

[SuG89]

Thank you all for your responses to my query
Regards,
Su