Introduction to Optimization - Numerical Methods in Engineering with MATLAB - page 406

13.

a b

A C

B

v

B'

u

P

The elasticcord ABC has an extensionalstiffness k .When the verticalforce P is
appliedat B , the cord deforms to the shape AB C . The potentialenergy of the
systeminthe deformedpositionis

+ +

k ( a b ) k ( a b )

2
AB

2
BC

V =− Pv + δ + δ

2 a 2 b

where

( a

δ AB = + u ) 2 + v 2 − a

( b

δ BC = − u ) 2 + v 2 − b

are the elongationsof AB and BC .Determine the displacements u and v by min-
imizing V (this is an application of the principle of minimum potentialenergy:
a systemis in stable equilibrium if its potentialenergy is at a minimum). Use
a = 150 mm, b = 50 mm, k = 0 . 6 N/mm and P = 5 N.

14.

b = 4 m

P = 50 kN

θ

Each member of the truss has a cross-sectional area A . Find A and the angle

that minimize the volume

bA
cos

V =

θ

of the material in the truss without violating the constraints

σ ≤ 150MPa δ ≤ 5mm