Micromechanics - Neural Networks and Micromechanics - page 147

From these equations we have:

S 2 q A E A

L A ¼

q B E B

r A ¼

L B ¼

S r B : (8.27)

S

Equation (8.27) shows that the rigidity of bar A is S times greater than the rigidity
of bar B .

8.3.2.2 Bending of the bar. Case 1

The bar of length L is loaded at one end by the force F z . The other end is fixed to the
wall (Fig. 8.4 ). The bar has a constant section with the moment of inertia I . In this
case, the deflection D z of the bar end is:

F Z L 3

D z ¼

I ;

(8.28)

3 E

and the rigidity r is:

3 E I

r ¼ F Z =D z ¼ : (8.29)

L 3

We have:

E A ¼ E B ;

L A ¼ S L B ; (8.30)

S 4 I A ¼ I B :

Substitution into equation (8.29) gives:

3 E A I A 3 E B S 4 I B

r A ¼ ¼ ¼ S r B (8.31)

L A L B S 3

We can conclude from equation (8.31) that a size reduction by S times gives a
rigidity reduction by S times.

F z

L

Δ z

Fig. 8.4 Bending of the
console bar