542 Geometric Programming
Figure 8.5 Floor consisting of a plate with supporting beams [8.36].8.21 A rectangular area of dimensionsAandBis to be covered by steel plates with supporting
beams as shown in Fig. 8.5. The problem of minimum cost design of the floor subject to a
constraint on the maximum deflection of the floor under a specified uniformly distributed
live load can be stated as [8.36]Minimizef (X)=cost of plates+cost of beams=kfγ ABt+kbγ Ak 1 nZ^2 /^3 (1)subject to56. 25 WB^4
EA
t−^3 n−^4 +(
4. 69 WBA^3
Ek 2)
n−^1 Z−^4 /^3 ≤ 1 (2)whereWis the live load on the floor per unit area,kfandkbare the unit costs of plates
and beams, respectively,γthe weight density of steel,tthe thickness of plates,nthe
number of beams,k 1 Z^2 /^3 the cross-sectional area of each beam,k 2 Z^4 /^3 the area moment
of inertia of each beam,k 1 andk 2 are constants,Zthe section modulus of each beam,
andEthe elastic modulus of steel. The two terms on the left side of Eq. (2) denote the
contributions of steel plates and beams to the deflection of the floor. By assuming the data
asA=10 m,B=50 m,W=1000 kgf/m^2 ,kb=$0.05/ kgf,kf=$0.06/ kgf, γ= 7850
kgf/m^3 , E= 2. 1 × 105 MN/m^2 , k 1 = 0 .78, andk 2 = 1 .95, determine the solution of the
problem (i.e., the values oft∗,n∗, andZ∗).
8.22 Solve the zero-degree-of-difficulty bearing problem given by Eqs. (E 8 ) and (E 9 )of
Example 8.12.
8.23 Solve the one-degree-of-difficulty bearing problem given by Eqs. (E 12 )and (E 13 )of
Example 8.12.
8.24 The problem of minimum volume design of a statically determinate truss consisting ofn
members (bars) withmunsupported nodes and subject toqload conditions can be stated
as follows [8.14]:Minimizef=∑ni= 1lixi (1)