Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

6.4Matrix Analysis of Pin-jointed Frameworks 179

Hence,

uj−ui=λ(uj−ui)+μ(vj−vi)

SubstitutinginEq.(6.31)andrewritinginmatrixform,wehave

Sij=

AE

L

[

λμ

ij

]{

uj−ui

vj−vi

}

(6.32)

Example 6.1
Determine the horizontal and vertical components of the deflection of node 2 and the forces in the
members of the pin-jointed framework that is shown in Fig. 6.4. The productAEis constant for all
members.

We see in this problem that nodes 1 and 3 are pinned to a fixed foundation and are therefore not
displaced.Hence,withtheglobalcoordinatesystemshown,

u 1 =v 1 =u 3 =v 3 = 0

Theexternalforcesareappliedatnode2suchthatFx,2=0,Fy,2=−W;thenodalforcesat1and3are
thenunknownreactions.
Thefirststepinthesolutionistoassemblethestiffnessmatrixforthecompleteframeworkbywriting
down the member stiffness matrices referred to the global coordinate system using Eq. (6.30). The

Fig.6.4

Pin-jointed framework of Example 6.1.