Question

A certain structural member of uniform cross section has its strain limited to 0.6mm/m. The member has a tensile strength of 29 MPa and is 8 m long.

a. Determine the smallest permissible diameter of the cross section if the cross section is circular.

b. Determine the smallest permissible measure of one side of the cross section if the cross section is a square

Answer 1

Solution:- the values given in the question are as follows:

limited strain=0.6 mm/m

limited strain(e)=0.0006 mm/mm

tensile strength(t)=29 MPa , or 29 N/mm^2

length of member(L)=8 m

(a)

Cross-section is circular:-

let diameter of circular cross-section member is D

total strain in member(e)=0.0006

Apply hooke's law-

stress=strain*young's modulus of elasticity

, [Eq-1]

values put in above equation-(1)

29=0.0006*E

E=29/0.0006

E=48333.3333 MPa

E=4.8333*10^4 MPa

initia volume of member before apply tensile force=(Pi/4)*D^2*L

final volume of member after applied of tensile force=(Pi/4)*d^2*(L+change in length)

where, d=D-D

D=change in diameter after applied tensile force

initia volume of member before apply tensile force=final volume of member after applied of tensile force

Pi/4*D^2*L=Pi/4*d^2*(L+)

where, =longitudinal strain(e)*length of member

=0.0006*L

d=D-

Pi/4*D^2*L=Pi/4*(D-)^2*(L+0.0006*L)

Pi/4*D^2*L=Pi/4*(D-)^2*L*(1+0.0006)

D^2=(D-)^2*(1+0.0006)

1/1.0006=[(D-)/D]^2

0.9994^0.5=[(D-)/D]

0.9997=1-D/D

D/D=1-0.9997

D/D=0.0003

lateral strain(D/D)=0.0003 mm/mm

but, limited strain=0.6 mm/m

maximum strain in 1 m=0.6 mm

similarly calculate distance when maximum strain is 0.0003

maximum strain in 0.5 m=0.0003 mm

diameter of member(D)=0.5 m , or 500 mm

mimimum diameter of member is 0.5 m

(b)

Cross-section of member is square:-

let side of square is b

initia volume of member before apply tensile force=b^2*L

final volume of member after applied of tensile force=(b-b)^2*(L+change in length)

where, b=change in side of member after applied tensile force

initia volume of member before apply tensile force=final volume of member after applied of tensile force

b^2*L=(b-b)^2*(L+L*0.0006)

1/1.0006=[(b-b)/b]^2

0.9994^0.5=[(b-b)/b]

0.9997=1-b/b

b/b=1-0.9997

b/b=0.0003

lateral strain(b/b)=0.0003 mm/mm

but, limited strain=0.6 mm/m

maximum strain in 1 m=0.6 mm

similarly calculate distance when maximum strain is 0.0003

maximum strain in 0.5 m=0.0003 mm

side of square member(D)=0.5 m , or 500 mm

mimimum side of square member is 0.5 m or 500 mm

[Ans]