152 CHAPTER 5 Energy Methods
Fig.5.26
Linearly elastic body subjected to loadsP 1 ,P 2 ,P 3 ,...,Pn.
thecompletesystemofloadsarethen
1 =a 11 P 1 +a 12 P 2 +a 13 P 3 +···+a 1 nPn
2 =a 21 P 1 +a 22 P 2 +a 23 P 3 +···+a 2 nPn
3 =a 31 P 1 +a 32 P 2 +a 33 P 3 +···+a 3 nPn
..
.(^) n=an 1 P 1 +an 2 P 2 +an 3 P 3 +···+annPn
⎫
⎪⎪
⎪⎪
⎪⎬
⎪⎪
⎪⎪
⎪⎭
(5.25)
or,inmatrixform
⎧
⎪⎪
⎪⎪
⎪⎨⎪⎪
⎪⎪⎪
⎩(^1)
(^2)
(^3)
..
.
(^) n
⎫
⎪⎪
⎪⎪
⎪⎬
⎪⎪
⎪⎪⎪
⎭
=
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
a 11 a 12 a 13 ... a 1 n
a 21 a 22 a 23 ... a 2 n
a 31 a 32 a 33 ... a 3 n
..
...
.
..
.
..
.
an 1 an 2 an 3 ... ann⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
⎧
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪⎪
⎩
P 1
P 2
P 3
..
.
Pn⎫
⎪⎪
⎪⎪
⎪⎬
⎪⎪
⎪⎪⎪
⎭
whichmaybewritteninshorthandmatrixnotationas
{ }=[A]{P}SupposenowthatanelasticbodyissubjectedtoagraduallyappliedforceP 1 atapoint1,andthen,
whileP 1 remainsinposition,aforceP 2 isgraduallyappliedatanotherpoint2.Thetotalstrainenergy
Uofthebodyisgivenby
U 1 =P 1
2
(a 11 P 1 )+P 2
2
(a 22 P 2 )+P 1 (a 12 P 2 ) (5.26)Thethirdtermontheright-handsideofEq.(5.26)resultsfromtheadditionalworkdonebyP 1 asitis
displacedthroughafurtherdistancea 12 P 2 bytheactionofP 2 .Ifwenowremovetheloadsandapply