Standard formula for double cantilevering beam 3

[Iasonasx]

I thought about applying double cantilevering to minimize the material for a small structure, but I don't seem to find a standard formula for the deflection of this beam (see figure - I refer to the case on top although if formulae for the maximum deflections for both cases are available I would appreciate having them both)
Thank you!

 

[BAretired]

@rb1957...M2 is the moment of a cantilever. If the cantilever has length a, then M2 = wa[sup]2[/sup]/2. In this case, a = 0.21L so M2 = w(0.21L)[sup]2[/sup]/2 or 0.022wL[sup]2[/sup]. The negative sign used in the diagram signifies tension on top. The small discrepancy in magnitude results from the fact that 0.21L is an approximation. For M2 to be equal in magnitude to M1, the cantilever length must be 0.3536*span or 0.2071*L.



BA

 

[rb1957]

I calc a downward deflection at the end (-0.000003376*wL^4/EI) ... unless I've made a math error.

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

 

[Iasonasx]

@BAretired my calculations gave me 0.207 too. There's some slight rounding in the diagram. Also, the Mu I get is 0.022wL^2.

 

[structSU10]

Are you applying this to a cantilever drop roof framing system? If so, you need to account for the point load at the ends of the double cantilevers. Or are you framing your whole floor / roof with a cantilever end?
Do you account for potential skip loading on this system?
does it have to support a facade at the cantilever end, thus adding an additional point load?
Do you have wind uplift to contend with? that loading condition is non uniform.

just some things to consider - you want to run a final design that account for all conditions and not hang your hat on a preliminary design from one formula.

 

[Iasonasx]

@structSU10, Yes I considered all of these in the specific scenario. Yes there is point load in a way, but it is not on this beam. This beam takes the point load of the end wall by the exterior wall above. All of those factors have been taken into account, but thank you for confirming.

 

[BAretired]

I calc a downward deflection at the end (-0.000003376*wL^4/EI) ... unless I've made a math error.

You may be right. One or both of us has made a math error.

BA

 

[Dinosaur]

The diagrams are also typical for field work when parts have to be lifted or rigged for unloading. The workers measure the total piece length and then compute (using the ratios in the diagram) where to locate the slings. In these cases most folks don't care what the deflection is and that may be why the formula isn't available at the primary source.

For my part, I would just use superposition or derive the deflection formula from conjugate beam for the fun of it. I was doing something very near this a few months ago while helping a colleague prepare for the PE exam.

Good luck.

 

[BAretired]

I thought about applying double cantilevering to minimize the material for a small structure, but I don't seem to find a standard formula for the deflection of this beam (see figure - I refer to the case on top although if formulae for the maximum deflections for both cases are available I would appreciate having them both)

The second case is included in the CISC handbook (and I expect also in the AISC handbook). If you don't have either handbook, you can google "Beam Diagrams and Formulas"...Diagram 24.

BA

 

[Celt83]

I get deflection results pretty close to BAretired's numbers

with E= 1 ksi, I= 1 in^4, L=20ft, w= 1 klf
@0.21L for the cantilevers an upwards deflection of 0.000121 w L^4 / EI
@0.2071L for the cantilever an upwards deflection of 0.000204 w L^4 / EI

0.21L - grey=deflection green=moment blue=load:

0.2071L:

Open Source Structural Applications:

https://github.com/buddyd16/Structural-Engineering

 

[Celt83]

If anyone is curious for the envelope deflections
Take the numbers in the attached image * 29000 * 30.8 / (1* 20^4 * 1728) = c
delta = c * w L^4 / EI

Edit:
This covers the following patterns with 1 = loaded span
1,1,1
1,1,0
0,1,0
0,0,1
0,1,1
1,0,0
0,0,1
1,0,1

Open Source Structural Applications:

https://github.com/buddyd16/Structural-Engineering

 

[IDS]

Spreadsheet results using:
ConBeamU

With L=1, EI = 1 and load = -1 in any consistent units:

With Cantilever = 0.21:
Cantilever deflection = 1.2128E-04, Mid-span deflection = -5.463E-04

With Cantilever = 0.20711 (to equalise moment magnitude at support and mid-span):
Cantilever deflection = 2.0367E-04, Mid-span deflection = -6.1328E-04

With Cantilever = 0.22315 (to equalise deflection magnitude at support and mid-span):
Cantilever deflection = -2.6973E-04, Mid-span deflection = -2.6973E-04

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BAretired]

but I don't seem to find a standard formula for the deflection of this beam

Using 'c' and 's' for the cantilever and span length respectively,
Δs = deflection at midspan; Δc = deflection at end of cantilever

Δs = 5w*s[sup]4[/sup]/384EI - M*s[sup]2[/sup]/8EI where M = cantilever moment, wc[sup]2[/sup]/2
(1) Δs*EI = 5w*s[sup]4[/sup]/384 - w*(cs)[sup]2[/sup]/16

Δc = w*c[sup]4[/sup]/8EI - θ*c where θ is the slope at the left support.
θ = w*s2/8EI * s/3 - M/EI * s/2
i.e. θ = w*s[sup]3[/sup]/24EI - w*c[sup]2[/sup]*s/4EI

(2) Δc*EI = w*c[sup]4[/sup]/8 - (w*s[sup]3[/sup]/24 - w*c[sup]2[/sup]*s/4)c

Using c = 0.21L and s = 0.58L,
from Equation (1) Δs*EI = 0.0014735w*L[sup]4[/sup] - 0.0009272L[sup]4[/sup] = 0.000546L[sup]4[/sup]

from Equation (2) Δc*EI = = 0.0002431L[sup]4[/sup] - 0.0003644L[sup]4[/sup] = -0.0001213L[sup]4[/sup]

These results seem to be in agreement with those posted by Celt83 and IDS.
Equations (1) and (2) provide a standard formula for deflection at midspan of beam and end of cantilever.

BA

 

[rb1957]

FWIW (which ain't much) I get the same results (I did make a math and an english stumble ...
math ... I calc'd -0.21^4 as (-0.21)^4 and not -(0.21^4) ... sigh
english ... I know +ve deflection is down (by convention), I know I had a -ve calc, yet I wrote "down" ... sigh

this "problem" is now flogged to death ! We have solutions from FEA and hand calcs.

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