Derive bending equation, Mechanical Engineering

Q - Derive bending equation that is,; M/I =  σ /y = E/R.                                                                          

Sol.: With reference to the figure given to us, consider any two normal sections AB and CD of a beam at small distance   δ L apart (that is, AC = BD = δ L). Let AB and CD intersect neutral layer at the points M and N respectively.

Let;

M = bending moment acting on beam

θ = Angle subtended at centre by the arc.

R = Radius of curvature of neutral layer M' N' .

At any distance 'y' from neutral layer MN, consider layer EF.

As shown in the figure the beam because of sagging bending moment. After bending, A' B', C' D' , M' N'  and

E'F' represent final positions of AB, CD, MN and EF in that order.

When produced, A' B' and C' D' intersect each other at the O subtending an angle θ radian at point O, which is centre of curvature.

As   L is quite small, arcs A' C' , M' N' , E' F'  and B' D'  can be taken as circular.

Now, strain in layer EF because of bending can be given by e = (E F  - EF)/EF = (E F  - MN)/MN

As MN is the neutral layer, MN = M' N'

 

2366_bending equation.png 
Let; σ  = stress set up in layer EF  because of bending

E = Young's modulus of material of beam.
1131_bending equation1.png
Equate the equation (i) and (ii);
1553_bending equation2.png  


Let;       σ = stress set up in layer EF because of bending

E = Young's modulus of material of beam.

704_bending equation3.png

1134_bending equation4.png

At distance 'y', let us consider an elementary strip of quite small thickness dy. We have already assumed that 'σ ' is bending stress in this strip.

Let dA = area of the elementary strip. Then, force developed in this strip =   σ.dA.

Then the, elementary moment of resistance because of this elementary force can be
given by dM = f.dA.y

Total moment of resistance because of all such elementary forces can be given by
1355_bending equation5.png
From the Equation (iii),
185_bending equation6.png
By putting this value of  f in Equation (iv), we get
1918_bending equation7.png
But
2036_bending equation8.png
where  I = Moment of inertia of whole area about neutral axis N-A.
2439_bending equation9.png

Where;

M = Bending moment

I  = Moment of Inertia about axis of bending that is; Ixx

y  = Distance of the layer at which the bending stress is consider

(We take always the maximum value of y, that is, distance of extreme fiber from N.A.)

E = Modulus of elasticity of beam material.

R = Radius of curvature

Posted Date: 10/20/2012 8:04:39 AM | Location : United States







Related Discussions:- Derive bending equation, Assignment Help, Ask Question on Derive bending equation, Get Answer, Expert's Help, Derive bending equation Discussions

Write discussion on Derive bending equation
Your posts are moderated
Related Questions
Calculate the displacement of a particle: The displacement of a particle is given by the equation, x = 10 t 3  - 7 t 2   + 6 t + 2       I.            Calculate its d


A tensile test on a specimen having an initial diameter of 13.11 mm and an initial gauge length of 200.0 mm, gave the following data:

b. Assuming a 20 MW simple cycle gas turbine with a firing temperature of 1600K develop an analysis of a combined cycle. Assume that Tmax in the Rankine cycle is 560 C, Pmax = 60

Determine ratio between mass flow rate to condensing steam: In steam power plant, the steam 0.1 bar and 0.95 dry enters condenser and leaves as saturated liquid at 0.1 bar an

(a) A geometrically similar model of spillway is to be laid to a scale of 1 in 50. Evaluate the velocity ratio, discharge ratio and acceleration ratio. (b) If model prototype ra

Single Cylinder Vertical Engine has a bore of 30cm and a stroke of 40cm. The connecting rod is 100cm long. The mass of the reciprocating parts is 140Kg on the expansion stroke with

two forces are acting at a point''o'',determine magnitude and direction by using parallelogram law of forces and triangular law of forces

hey ...!!!! i am looking for hydraulic car lift circuit diagram. can any 1 plzzz help me did this?

what is the relation ship between tolrancr grade and manufacturing process ?