Question

Technical report (80 Points) A bar with one end being fixed is subject to force F 10KN. The entire length of the bar is 1 m The distance from the place where F is applied to the fixed end is 0.6 m. Initially, there is a gap A between the bar and the right wall. If there is a temperature increase of 20'C,calculate the stress at point a and the total strain of the bar when (1). A-0.3 mm (2). A =0.2 mm , and (3) Δ-0, 1 mm , respectively. Lo 0.6 m E=100GPa 200 mm v 0.25 F=10 KN a-10-5/ C 200 mm

This is a Mechanics of Materials (Deformable Bodies) question.

Answer 1

stress due to load 10 KN is for c/s area of 0.2m*0.2m

stress = force / area

= 10*1000/(0.2*0.2)

= 250000 N/m²

strain due to above stress is = stress/E

=250000/100*10⁹

=2.5*10^-6

increase in length due to the stress is 2.5*10^-6*1000 mm

=2.5*10^-3 mm

strain due to increase in temperature is =$\alpha \Delta$T

= 10^-5*20

=2*10^-4

therefore increase in length = strain * length

= 2*10^-4*1 m

= 0.2 mm

1

when delta is 0.3 mm

the total increase in length due to temperature raise is 0.2 mm

which is less than 0.3 mm

so there will be no thermal stress and only stress is due to 10KN load

therefore stress at A is 250000N/m²

total strain at A is 0.2+0.0025 mm/m= 0.2025 mm/m

2

when delta is 0.2 mm

the total increase in length due to temperature raise is 0.2mm

which is equal than 0.2 mm

as the expanded length is compensated there will be no thermal stress

total stress at A is 250000N/m² ie due to 10 KN load only

strain is change in length to original length as the maximum increase in length is 0.2 mm only

strain at A is 0.2 mm/m

3

when delta is 0.1 mm

increase in length due to temperature raise is 0.2 mm which is greater than 0.1 mm

the expansion is restricted as a result thermal stress is developed  

now the thermal strain is 0.2-0.1 mm/m ( as only restriction induces thermal stress)

thermal stress is = thermal strain* E

= -0.1*10^-3*100*10⁹ (compressive)

= -10*10⁶N/m² (compressive)

total stress at point A is +250000-10000000

=-9750000N/m² (compressive)

total strain at point A is change in length to original length as the maximum increase in length is 0.1 mm only

strain at A is 0.1 mm/m