Complex beam bending problem
Complex beam bending problem
(OP)
Hi
Wondering if anyone could point me in the right direction for solving this. I am looking work out the approximate reaction forces (and eventually bending moments) for a trailer design I'm putting together and have approximated it to a simply supported beam with multiple reactions (2 x wheels, 1 x tow hitch) and multiple loads. I've so far tried removing the R2 reaction and solving through superposition, but I'm struggling using beam theory to calculate the deflection at its position (x=1.724).
For section EF I have:
M(x) = -798.758392x + 1211.009551
Which integrating gives:
EI (dv/dx) = -399.379196x^2 + 1211.009551x + C1
EIv = -133.1263987x^3 + 605.504775 x^2 + C1x + C2
But I'm unsure what boundary conditions to apply to this section to get C1 and C2.
Does anyone know of a good way to solve this easily without having to go into FEA methods?

EI = 8890
Thanks
Wondering if anyone could point me in the right direction for solving this. I am looking work out the approximate reaction forces (and eventually bending moments) for a trailer design I'm putting together and have approximated it to a simply supported beam with multiple reactions (2 x wheels, 1 x tow hitch) and multiple loads. I've so far tried removing the R2 reaction and solving through superposition, but I'm struggling using beam theory to calculate the deflection at its position (x=1.724).
For section EF I have:
M(x) = -798.758392x + 1211.009551
Which integrating gives:
EI (dv/dx) = -399.379196x^2 + 1211.009551x + C1
EIv = -133.1263987x^3 + 605.504775 x^2 + C1x + C2
But I'm unsure what boundary conditions to apply to this section to get C1 and C2.
Does anyone know of a good way to solve this easily without having to go into FEA methods?

EI = 8890
Thanks






RE: Complex beam bending problem
Calculate the deflection at x=1.724m for the determinate structure, then calculate deflection at the same point for a unit load. From this, calculate the magnitude of R2 and its effect on R1 and R3.
BA
RE: Complex beam bending problem
I've removed R2 and solved for the reactions (R1' = 1975.958392N , R3' = 427.4916081N) but I cant find the deflection at x=1.724.
I have M(x)= -798.758392x + 1211.009551 for the section containing x=1.724, but integrating this leaves me with unknown constants. Is there a way to determine where the maximum deflection for the entire system is (i.e. dv/dx = 0) to easily find these constants? Or is there a better way to find the deflection?
Thanks
RE: Complex beam bending problem
RE: Complex beam bending problem
Mike McCann
MMC Engineering
RE: Complex beam bending problem
The problem is indeterminate to the first degree which is why you remove one reaction. You could have removed R1 or R3, but R2 seemed easier. After removing R2, you are left with a simple span beam from D to I (R1 to R3) with cantilever to the left of D.
The loads on the cantilever produce a negative moment at point D which I'll call MD. MD = P1*(a+b) where P1=490.5kN and a, b are the distance from each load to D. Considering only the loads on the cantilever, the moment varies linearly from D to I and the deflection at Point F is a function of MD. By inspection, it is upward. Its magnitude may be found using moment-area principles or from any text which includes a beam diagram with moment applied to one end of a simple beam.
For each of the loads inside the span, deflection at F is given in the Steel Handbook as well as many other texts.
You add the deflection at F from MD and each of the loads inside the span. That is the deflection of the determinate beam.
Now place a unit load at point F and calculate the deflection at F. Reaction R2 which was removed is numerically equal to the deflection due to all loads divided by deflection due to a unit load. In other words, R2 is the reaction required to bring the net deflection to zero.
The values of R1 and R3 will be changed by R2*a/L and R2*b/L where a, b are distance from the opposite support.
Another method is to draw the Shear Force and Bending Moment diagrams for the determinate beam, then calculate deflection at F using Moment-Area principles and finally solve for R2 as suggested above.
BA
RE: Complex beam bending problem
RE: Complex beam bending problem
RE: Complex beam bending problem
RE: Complex beam bending problem
Michael.
"Science adjusts its views based on what's observed. Faith is the denial of observation so that belief can be preserved." ~ Tim Minchin
RE: Complex beam bending problem
The jobs not finished until the paperwork is done...
By the way, civeng80 vindicated me...
Mike McCann
MMC Engineering
RE: Complex beam bending problem
RE: Complex beam bending problem
I get:
R1 = 1535.49 kN
R2 = 572.67 kN
R3 = 295.28 kN
Spreadsheet file attached.
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Complex beam bending problem
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Complex beam bending problem
Dik
RE: Complex beam bending problem
Quando Omni Flunkus Moritati
RE: Complex beam bending problem
I did want to add the following thought though. In my world trailer suspension systems are load distributing. If this trailer is using a tandem suspension package than the problem is statically determinate because the load is distributed equally to the two, three, etc axles through the spring hangers (more or less). Just solve for the trailer suspension reaction to the center of the axle group, balance the load to each axle, then carry the forces up to the frame through the spring hangers. If the trailer is not using suspension support then it's still statically determinate because the trailer will lift one axle off the ground when you go up and down hills so it has to span the load to one axle or the other.