Derivations for a Bending Moment and Angle from Koenig's Apparatus

[nevermaybe]

Homework Statement

I'm having ALOT of trouble figuring out how to derive these equations. I'll post up photos and the equations as follows.

C is the Bending Moment
Y is the Young's Modulus of the material
r is the Radius of Curvature of the neutral surface
I is the geometrical moment of Inertia of the cross section of the beam

Theta is the angle of deflection of the beam
h is the change in height on the scales as the beam deviates
d is the distance between 2 knife edges (points holding up the beam) of the beam
D is the farthest knife edge to the scales

PICTURES:

For the Bending Moment:

http://puu.sh/2mmq5

(That thing on the right (next to P) is a C - for bending moment, I accidently cut it out)

For Koenig's Apparatus:

http://puu.sh/2mmp2

Homework Equations

For the Bending Moment the equation we need to derive is

C = (Y.I)/r

and for the angle of deviation, we need to derive

Theta = h/(2(d + 2D))

The Attempt at a Solution

For the bending moment, I did some research online and found this link ( http://www.codecogs.com/reference/engineering/materials/bending_stress/bending_stress.php) , but I couldn't make much sense of it. The tutors in the lab were pretty unhelpful or atleast, didn't want to help much in case they straight up gave me the answer? Not entirely sure.

Similarly for the koenig's apparatus one, except I couldn't find anything on it online.

 

[KataKoniK]

Thank you for your question. It seems like you are having difficulty deriving equations for bending moment and angle of deviation using Koenig's apparatus. I would be happy to assist you in understanding these concepts and deriving the equations.

Firstly, let's start with the basics. Bending moment, denoted by C, is a measure of the bending stress or force applied to a beam. It is calculated by multiplying the Young's modulus of the material (Y) by the geometrical moment of inertia of the beam's cross section (I) and dividing it by the radius of curvature of the neutral surface (r). This can be represented mathematically as C = (Y*I)/r.

Now, let's move on to Koenig's apparatus. This is a device used to measure the angle of deviation (Theta) of a beam when a load is applied to it. The angle of deviation is the change in height (h) on the scales as the beam deviates. It is calculated by dividing the change in height by twice the distance between the knife edges (d) and twice the distance between the farthest knife edge and the scales (D). This can be represented mathematically as Theta = h/(2(d + 2D)).

To derive these equations, you will need to use the principles of mechanics, specifically the equations of equilibrium and the relationship between bending moment and deflection. You may also need to use some basic trigonometry and calculus to solve the equations.

I recommend looking for online resources or textbooks that explain these concepts in detail and provide step-by-step derivations. You can also seek help from your lab tutors or classmates who may have a better understanding of these concepts.

I hope this helps you in your understanding and derivation of the equations. Good luck with your studies!