Question

solve with explansion please

BONUS PROBLEM: Calculate the required diameter, d, of the bar if the allowable bending stress is 150 MPa. A and B can be assumed to be rollers (15 pts). 12 kN/m A B 3 m 1.5 m

Answer 1

First we will find out the reaction at supports,

Take summition of forces in Y direction = 0

Ra + Rb = 12×3+ 0.5× 12 ×1.5

Now take moment about B = 0,

Ra×3 - 12×3×1.5 + 0.5×12×1.5×0.33× 1.5= 0

Therefore Ra = 17.5 KN

So, Rb= 27.5 KN

Now we will find out maximum bending moment in the beam

For that take a section at a distance of x from A

Therefore, Ra ×x- 12×x×0.5×x= Moment at X

To find out the value of at for maximum bending moment , differentiate this equation

So, Ra - 2×12×0.5×x = 0

From here X = 1.458

So bending moment at X= 1.458 is , 12.76 KN-m

Now we will use bending equation to find out required diameter

Bending equation is,

M/I= ¶/y

Where, M is bending moment which is equal to 12.76 KN-m

I is second moment of area which is equal to π×D^4/64

¶ is allowable bending stresses whose will is given as 150mpa

Y is distance of extreme fiber from neutral axis which is equal to D/2

Where D is diameter

So put the values in equation and get the value of diamete

12.76×10^6/(π×D^4/64)= 150/(D/2)

Therefore required diameter is 95.33 mm