# Moments of deflection/ curvature of radius of a supported beam

1. Nov 17, 2012

I was wondering if there were any mechanical engineers that can answer a few questions I have regarding an assignment that I have been set. We have to choose a suitable beam to support a monorail. we are looking for a moment of deflection of around 10mm. Using the universal beams table bs 4 1993 I have found a beam that satisfies the requirements when using the formulas:

R=E.I/M

We're R is radius

E is Elastic modulus (given figure for steel is 210 x 10^9 pa)

I is second moment of area about the neutral axis (taken from Bs4 and converted to m^4)

M is bending moment (worked out to be 262.5 x 10^3 Nm)

And

H = R-√R^2-m^2

Were H is deflection in m (then converted to mm for purpose of the exercise)

m is the half the length of the original beam (20/2)

Using the beam with dimensions 1016 x 305 x 314, we get the following:

R=E.I/m

R= 210 x 10^9Pa x 6442 x 10^-6 m^4/262.5 x 10^3Nm

R=5153.6

So to get the moment of deflection we can use the formula

H=R√R^2-m^2

H=5153.6-√5153.6^2-10^2

Multiplying answer by 1000 to convert from m to mm we get the figure

9.701 mm

This answer seems to satisfy the requirements.

But since

R=y.E/s

Were y is distance from the neutral axis to edge or surface of the beam (m) (0.5m)

s(should be sigma but I am using ipad and don't have access to Greek symbols)=stress due to bending at distance y from the neutral axis (given as 80 MPa)

I am now getting the following

R=0.5 x 210x10^9/80x10^6

R=1312.5 this figure is drastically different to original 5153.6

Entering this figure into deflection formula we have

H=1312.5-√1312.5^2-10^2

H=38mm

No longer satisfactory

Do these figures and calculations look correct and if so, how come they are so drastically different. Is the second set of equations taking into account degradation of the beam/wear and tear? And as an engineer which figures/equation would you use... Obviously you would go for the worst case scenario, but what would the reasons be

I know it's a long winded question, thank you for taking the time out to read it

Regards Joe

2. Nov 17, 2012

### pongo38

What is the meaning of 'moment of deflection'?

3. Nov 17, 2012

### SteamKing

Staff Emeritus
I think you are assuming that the deflected shape of the beam is a circular arc whose radius is constant and equal to the radius of curvature determined for some value M of the bending moment. M is usually a function of position along the beam, and the radius of curvature is changing with position also.

4. Nov 18, 2012

Hi, thanks for e replies, sorry I haven't responded sooner... It should of read mid point deflection, not moment of deflection. I am an electrical engineer student and not totally au fait with mechanical terminology. I wasn't really assuming the radius as being a constant, my confusion is:

If you take the formula for elastic bending theory as being:

(S)stress/y=M/I=E/R

This gives two possible formulas for determining the radius of curvature:

R=E.I/M

And

R=y.E/s

The figures I have calculated previously or taken from BS4:1993 are as follows:

M=262.5x10^3Nm (previously calculated)
E=210x10^9Pa (generally used value for mild steel)
I=6442x10^-6m^4 (taken from Bs4)
Y=0.5m (taken from Bs4)
S=80x10^6Pa (given as part of the assignment question)

When calculated, the following results are calculated:

R=E.I/M

R=5153.6m

R=y.E/s

R=1312.5m

As you can see the results are vastly different, when I calculate the midpoint deflection, I get the results

Midpoint deflection=9.701mm and 38mm respectively.

Now as an engineer, considering there is a choice of two formulas that can beused, which formula for radius of curvature would you use, why and why are the results so different. I am thinking the given value for s is taking into account a safety margin to allow for wear/degradation of the beam

5. Nov 26, 2012

### pongo38

There are two criteria to determine the right beam. One is deflection, for which "R=E.I/M" seems appropriate. The other criterion is strength, for which "R=y.E/s" seems right. The question does not assume that the two criteria will be met at the same time. One or other will be critical. Which one is it? And, I think you should post your working. These calculations are notorious for being 10^6 wrong (or similar). State your units on every line.