Calculate force required for equilibrium, Mechanical Engineering

Calculate force required for equilibrium:

A member ABCD is subjected to the point loads P1, P2, P3  and P4  as shown in the figure given below

700_Calculate force required for equilibrium.png

Calculate force P3 required for equilibrium if P1 = 120 kN, P2 = 220 kN and P4 = 160 kN. Determine the net change in length of member. Take E = 200 GN/m2.
Sol.: Modulus of elasticity E = 200 GN/ m2 = 2 × 105 N/mm2.

By considering equilibrium of forces along axis of the member.

P1 + P3 = P2 + P4;

120 + P3 = 220 + 160

Force P3 = 220 + 160 - 120 = 260 kN

The forces which are acting on each segment of member are shown in free body diagrams shown below:

Let  δL1,  δL2 and  δL3,  be extensions in parts 1, 2 and 3 of steel bar respectively. Then,

 

845_Calculate force required for equilibrium1.png 
As, Tension in AB and CD but compression in BC,
So, Total extension of the bar,
   1318_Calculate force required for equilibrium2.png 
Extension of segment AB = [(120 × 103) × (0.75 × 103)]/[1600 × (2 × 105)] = 0.28125 mm

Compression of segment BC = [(100 ×103) × (1 × 103)]/[625 × (2 × 105)] = 0.8 mm

Extension of segment CD = [(160 ×103) × (1.2 × 103)]/[900 × (2 × 105)] = 1.0667 mm

Net change in length of the member = δl =0.28125 - 0.8 + 1.0667 = 0.54795 mm (increase)        .......ANS

Posted Date: 10/23/2012 1:45:59 AM | Location : United States







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