Continue to Site

Eng-Tips is the largest engineering community on the Internet

Intelligent Work Forums for Engineering Professionals

Register Log in
  • Congratulations waross on being selected by the Eng-Tips community for having the most helpful posts in the forums last week. Way to Go!

Derivation of deflection at any point formula for a clamped - clamped beam 2

Status
Not open for further replies.
Replies continue below

Recommended for you

Sort by date Sort by votes
you could derive it by hand … the problem is doubly redundant but not impossible (nor very difficult) to solve.

the other tack would be the model the fixed ends with the moments that you can find in Roark, solving the redundancy for you, and work through the math … SF(x) > BM(x) > v(x) > d(x) … integrating the BM equation.


another day in paradise, or is paradise one day closer ?
 
Upvote 0 Downvote
You could start off with a simple beam, calculate the moment, Pab/L.
Calculate area under the M/EI curve which is (Pab/L)*L/2EI = Pab/2EI
Find c.g. of the moment curve (or the M/EI curve) = (0+a+L)/3 = (a+L)/3 or (2a+b)/3
Find slope at ends of beam, namely reaction of M/EI curve treated as load.
Slope A = (Pab/2EI)(a+2b)/3L = Pab(a+2b)/6EIL
Slope B = Pab(2a+b)/6EIL

Consider moment applied at A
Ma/EI varies from Ma down to 0. Area under Ma/EI curve = MaL/2EI
Similarly for moment applied at B, Area under Mb/EI curve = MbL/2EI

Solve for Ma and Mb such that slope A and slope B = 0

Calculate deflection in the usual way, knowing Ma, Mb and shape of moment curve.



BA
 
Upvote 0 Downvote
You may find it easier to see, using the method of Column Analogy which is explained in the text and diagrams below:

Scan_20200506_srtvmk.png


BA
 
Upvote 0 Downvote
You need to define what is a "clamp" in your thought. To me, a clamped support has no degree of freedom, so a beam with clamped ends behave the same as the fixed end beam. You can prove me wrong, then you need to show a physical example of clamped support.
 
Upvote 0 Downvote
I think "clamped" is a synonym for "fixed". The OP's sketch indicates fixed ends.

BA
 
Upvote 0 Downvote
Got you.

OP:

BA's method will help to strengthen your understanding of the classic method. My approach will save time.

Fix_End_Beam_w-Conc_Load_rhfqxc.png


Fix_End_Beam_w-Uniform_Load_aqzs4t.png
 
Upvote 0 Downvote
Thanks for all of your replies. I was able to find the support reactions easily by applying Castigliano's theorem. and then as rb1957 said integrating starting from v(x) solves the problem. But the main problem I was trying to solve was the deflection of fix-fix CURVED beam, which I'm not sure that integrating from v(x)is correct. The moment area method can be applied as shown in the attached figure. but here ∅/2 means the centroid has to be at the mid span right? or Am I correct that it is actually the distance to the centroid? If so, to find the deflection for a system with a load at random location I just have to replace the ∅/2 with correct radial distance to centroid right?
 
 https://files.engineering.com/getfile.aspx?folder=0bce0771-b17b-4623-8f30-71b87f01d2ec&file=deflectionCurvedBeam.jpg
Upvote 0 Downvote
Below I attached the arrangement of the beam.

curvedBeam_zm4npz.jpg
 
Upvote 0 Downvote
I believe you can also use superposition if you know a few perhaps more common relationships already. If you know the deflection and end slope of a simply supported beam, and the deflection of a beam with one end fixed under an imposed rotation at the free end (being the end rotation of the simply supported span). Add together all the individual formulas for these cases to give you the exact formula:-

Simply supported span deflection under your imposed loading.
Same end rotation as simply supported span imposed at ends of a one end fixed span (to give no net rotation). Repeat for both ends.
 
Upvote 0 Downvote
Forgot to tell its a horizontally curved beam. And gonna try superposition method also after trying moment area method to verify. Also for superposition can I use pin-pin with load and pin-pin with imposed end rotations instead of pin-fix as u suggested?
 
Upvote 0 Downvote
For a curved beam I really think a frame analysis with short straight members is the easiest, and probably most accurate, way to go.

Doug Jenkins
Interactive Design Services
 
Upvote 0 Downvote
horizontally curved beam : and the load in plane of curvature or perpendicular to plane of curvature ?
 
Upvote 0 Downvote
Perpendicular to the plane of curvature so there will be torsion.
 
Upvote 0 Downvote
theory : Roark's formulas for stress and strain - 7th edition - sec. 9.5 Curved beams loaded normal to plane of curvature and sec. 9.6 tables
 
Upvote 0 Downvote
"Formulas for stress, strain, and structural matrices" - Walter Pilkey

curved_beam_deflection_m4pj8w.jpg
 
Upvote 0 Downvote
thx for leaving out critical information !

I think trying to hand calc the deflected shape will fail. I think the interaction between the two section moments will make this difficult for all but the simplest sections.

I'd use FEA.

another day in paradise, or is paradise one day closer ?
 
Upvote 0 Downvote
@rb1957 No I initially wanted to study the derivation of fixed straight beam and then moved on to curved beams. In the book that roby suggested has everything I need. But seems like there's no derivation in it so I can study it. I will use FEA (abaqus) later but for now I just need to study it in a theoretical level.

@agent666 I solved it using superposition. Don't know if it is correct but I can verify it using expressions that roby suggested.

Thank you all for your time!
 
Upvote 0 Downvote
Vimukthi said:
I will use FEA (abaqus) later but for now I just need to study it in a theoretical level.
Click to expand...

Seems like overkill to me. One of the nice things about analysing relatively simple systems with a frame analysis is that you get clear graphs of member actions along the length of the beam, that you can relate back to fundamental beam bending theory.


Doug Jenkins
Interactive Design Services
 
Upvote 0 Downvote
Status
Not open for further replies.

Similar threads

metalchair
  • Question
Adding a New Support Point on a Loaded Beam
Replies
8
Views
189
desertfox
desertfox
Arun4567777
  • Locked
  • Question
Cantilever Beam Brace Point
Replies
13
Views
206
lexpatrie
lexpatrie
Rigger12
  • Locked
  • Question
Residential beam sizing for deflection 3
2 3 4
Replies
60
Views
518
Eng16080
Eng16080
anchorengineer
  • Question
Connecting Wide flange beam to a Beam reinforced with WT section at bottom 1
Replies
7
Views
202
cliff234
cliff234
dik
  • Locked
  • Question
Clamped Column 8
2
Replies
22
Views
104
dik
dik

Part and Inventory Search

Sponsor

Back
Top Bottom