All of the information is in the above image. Thanks! Three structures with rect

Question

All of the information is in the above image.  Thanks!

Three structures with rectangular cross-sections (thickness = 10 cms), with the same heights and widths, are subject to loads in compression of 100 kN applied top and bottom at the middle of their widths, as shown in the figure. Even though all of them have the same cross-section areas, the distribution of material in the center is different. Without doing any calculation, state briefly which structure would have a higher stress than the other 2 and explain why 7 points. Calculate the stress in all structures and calculate the ratio of the maximum to minimum stress. 8 points.

Explanation / Answer

a) Stress will be = Moment*y/I

y = h/2 =32/2 =16 for all the sections

and M will be constant ......

so the structure with minimum I i.e. Moment of Inertia will have higher stress....

Now see the middle section of every cross-section

I is proportional to d^3 .....

FOR 1 : d^3 = 18^3 =5832 (max)

FOR 2 : d^3 = 9^3 + 9^3 = 1458 (min)

FOR 3: d^3 = 6^3 + 12^3 = 1944

SO For Structure 2 .....I will be minimum

hence Structure 2 will have higher stress

b) I will provide you with moment of intertias of each structure :

For 1 :

I = 16*18^3/12 + 2*36*8*(16-4)^2 + 2*36*8^3/12 = 93792 cm^4

For 2 :

I = 16*(2*9^3)/12 + 2*36*8*(16-4)^2 + 2*36*8^3/12 =87960 cm^4

For 3:

I = 16*(6^3 + 12^3)/12 + 2*36*8*(16-4)^2 + 2*36*8^3/12 = 88608 cm^4

Stress is inversely proportional to I

Ratio of Maximum to minimum stress = I max / I min =93792/ 87960 =1.066

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