5.6 Flexibility Method 141Similarly,
(^) y=wl^4
(
11
24 EI
+
1
2 GJ
)
(^) z=wl^4
(
1
6 EI
+
1
2 GJ
)
5.6 FlexibilityMethod...................................................................................
An alternative approach to the solution of statically indeterminate beams and frames is to release
thestructure—thatis,removeredundantmembersorsupports—untilthestructurebecomesstatically
determinate.Thedisplacementofsomepointinthereleasedstructureisthendeterminedby,say,the
unit load method. The actual loads on the structure are removed and unknown forces applied to the
pointswherethestructurehasbeenreleased;thedisplacementatthepointproducedbytheseunknown
forcesmust,fromcompatibility,bethesameasthatinthereleasedstructure.Theunknownforcesare
thenobtained;thisapproachisknownastheflexibilitymethod.
Example 5.8
DeterminetheforcesinthemembersofthetrussshowninFig.5.20(a);thecross-sectionalareaAand
Young’smodulusEarethesameforallmembers.
The truss in Fig. 5.20(a) is clearly externally statically determinate but has a degree of internal
staticalindeterminacyequalto1.Wethereforereleasethetrusssothatitbecomesstaticallydeterminate
by“cutting”oneofthemembers,sayBD,asshowninFig.5.20(b).Becauseoftheactualloads(Pin
thiscase),thecutendsofthememberBDwillseparateorcometogether,dependingonwhetherthe
forceinthemember(beforeitwascut)wastensileorcompressive;weshallassumethatitwastensile.
Fig.5.20
Analysis of a statically indeterminate truss.