Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

5.8 The Principle of the Stationary Value of the Total Potential Energy 149

Fig.5.24

States of equilibrium of a particle.

Itmayalsobeshownthatifthestationaryvalueisaminimum,theequilibriumisstable.Aqualitative
demonstrationofthisfactissufficientforourpurposes,althoughmathematicalproofsexist[Ref.1].
In Fig. 5.24, the positions A, B, and C of a particle correspond to different equilibrium states. The
TPE of the particle in each of its three positions is proportional to its heighthabove some arbitrary
datum,sinceweareconsideringasingleparticleforwhichthestrainenergyiszero.Clearlyateach
position, the first-order variation,∂(U+V)/∂u, is zero (indicating equilibrium), but only at B where
theTPEisaminimumistheequilibriumstable.AtAandC,wehaveunstableandneutralequilibrium,
respectively.
Tosummarize,theprincipleofthestationaryvalueoftheTPEmaybestatedasfollows:

Thetotalpotentialenergyofanelasticsystemhasastationaryvalueforallsmalldisplacementswhen

thesystemisinequilibrium;further,theequilibriumisstableifthestationaryvalueisaminimum.

Thisprinciplemayoftenbeusedintheapproximateanalysisofstructureswhereanexactanalysis
doesnotexist.WeshallillustratetheapplicationoftheprincipleinExample5.11following,wherewe
shallsupposethatthedisplacedformofthebeamisunknownandmustbeassumed;thisapproachis
calledtheRayleigh–Ritzmethod.

Example 5.11
Determinethedeflectionofthemidspanpointofthelinearlyelastic,simplysupportedbeamshownin
Fig.5.25;theflexuralrigidityofthebeamisEI.

The assumed displaced shape of the beam must satisfy the boundary conditions for the beam.
Generally,trigonometricorpolynomialfunctionshavebeenfoundtobethemostconvenient,butthe
simplerthefunction,thelessaccuratethesolution.Letussupposethatthedisplacedshapeofthebeam
isgivenby

v=vBsin

πz

L

(i)