Fixed End or Zero Tolerance Load Distribution 1

[cap4000]

Can anyone explain how gravity beam loads get distributed into a fully fixed end or a zero tolerance type fitting. I have seen it assumed as an triangular distribution with the higher ends of the triangle opposing one another on each end. Any help would be appreciated.

 

[rb1957]

the triangular distribution would be the cumulative shear force ... uniform distributed load, linear increasing shear force, parabolic moment, ...

 

[cap4000]

Please check the embedment load calcs on page 3 of this link.

http://www.ncree.org.tw/iwsccc/PDF/08 - Shahrooz.pdf

 

[rb1957]

that looks like the common assumption of "plane sections remaining plane"

 

[vonlueke]

cap4000: The triangular load distribution you mentioned would be applicable if all material is linear and the stress doesn't exceed the yield strength. If this is a cantilever, and we neglect the bond horizontal shear stress (which is equivalent to assuming zero friction), and assume the stress doesn't exceed the yield strength, I'm currently getting the following rigid-body mechanics solution, where a = distance from wall surface to where triangular load distribution crosses x axis, L = beam length (measured from wall surface), Le = beam embedment length, w1 = beam self weight per unit length (positive downward), w2 = triangular load distribution ordinate at wall surface (force per unit length, positive upward), and w3 = triangular load distribution ordinate at embedment length Le (positive upward).

a = (Le/6)(3L+4Le)/(L+Le).
w2 = (w1)(L)(3L+4Le)/(Le^2).
w3 = (w2)[1-(Le/a)].

 

[cap4000]

vonlueke

Nice Job. Thanks.

 

[cap4000]

volueke

I think you meant your w3 is downwards not upwards. If you have Joseph Shelleys Volume 1 Statics book by Schaums its on page 553. Thanks.

 

[vonlueke]

cap4000: My w3 equation is written positive upward, which means if it gives a negative result (which it does), then the force is downward.

By the way, using a rigid-body mechanics solution, such as in my previous post, is reasonable only if the beam is relatively stiff compared to the embedment material. If, instead, the beam is relatively flexible compared to the embedment stiffness, the load will transfer more rapidly to the embedment material, such that embedment beyond a certain distance becomes ineffective. In other words, if the beam is relatively flexible, then one can't ignore elasticity and compliance, and the problem becomes more complicated, in which case you could use, e.g., finite element and surface contact to solve the problem.

 

[cap4000]

vonlueke

If you have the PCI Design Handbook 3rd edition pgs 6-26 and 6-68 have a lot of conditions already calculated in a nice tabular form. If you do any precast or prestressed concrete this book is a must. Thanks Again for the Tips.