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
Illustrate any three of the following : (i) Critical Damping (ii) Logarithmic decrement (iii) Vibration Isolation (iv) Equivalent stiffness of spring in series and in p

Describe with suitable diagram the governing of impulse turbines. What are design aspects of Pelton wheel turbine and also explain all the terms. Derive mathematical formula

Compute the ratio of the torque: Compute the ratio of the torque transmitted by a hollow & a solid shaft of the similar material, length & weight. Solution d i = inte


Thermal Conductivity The amount of heat passing throughout unit cross-section of a solid material in unit time for unit thermal gradient is termed as thermal conductivity. The

example with its properties

Just-In-Time: This unit begin with the discussion of Just-In-Time strategy. Several terms used in the process are also explained. JIT is an encompassing philosophy considering

Impact Factor on Vessel rigging analysis? Unless otherwise specified by the user, a minimum impact factor of 1.5 shall be applied to the lift weight for designing the lifting d

Hazardous Areas are formally ranked as being a designated hazardous area using the relevant regional or country standards that are directly applicable to the plant design under con

Deflection at the centre: A simply supported beam of span 6 m is subjected to Udl of 24 kN/m for a length of 2 m from left support. Discover the deflection at the centre, maxi