130 CHAPTER 5 Energy Methods
betweenFandBitis
M=
P
4
(L−z)−√
3
2
Rz hence∂M
∂R
=−
√
3
2
zandbetweenBandCthebendingmomentis
M=
P
4
(L−z)−√
3
2
R(L−z) hence∂M
∂R
=−
√
3
2
(L−z)Thus,
∫L0M
EI
∂M
∂R
dz=1
EI
⎧
⎪⎨
⎪⎩
∫L/^4
0−
(
3
4
Pz−√
3
2
Rz)√
3
2
zdz+∫L/^2
L/ 4[
P
4
(L−z)−√
3
2
Rz](
−
√
3
2
z)
dz+
∫L
L/ 2−
[
P
4
(L−z)−√
3
2
R(L−z)]√
3
2
(L−z)dz⎫
⎪⎬
⎪⎭
giving
∫L0M
EI
∂M
∂R
dz=− 11
√
3 PL^3
768 EI
+
RL^3
16 EI
(v)SubstitutingfromEqs.(iv)and(v)intoEq.(iii)
−
11
√
3 PL^3
768 EI
+
RL^3
16 EI
+
RL
4 E
(
A+ 10 AB
ABA
)
= 0
fromwhich
R=
11
√
3 PL^2 ABA
48[L^2 ABA+ 4 I(A+ 10 AB)]
hencetheforcesineachmemberoftheframework.Thedeflection oftheloadPoranypointonthe
frameworkmaybeobtainedbythemethodofSection5.3.Forexample,thestationaryvalueofthetotal
complementaryenergyofEq.(i)gives ,thatis,
∂C
∂P=
∫
ABCdθ∂M
∂R
+
∑ki= 1λi∂Fi
∂P− = 0
Althoughbracedbeamsarestillfoundinmodernlightaircraftintheformofbracedwingstructures,
amuchmorecommonstructuralcomponentistheringframe.Theroleofthisparticularcomponentis
discussedindetailinChapter11;itisthereforesufficientforthemomenttosaythatringframesform
thebasicshapeofsemimonocoquefuselagesreactingshearloadsfromthefuselageskins,pointloads
fromwingsparattachments,anddistributedloadsfromfloorbeams.Usuallyaringistwo-dimensional,
supportingloadsappliedinitsownplane.Ouranalysisislimitedtothetwo-dimensionalcase.