M e c h a n i c s o f M a t e r i a l s
CHAPTER 11 ENERGY METHOD
§ 11–1 GENERAL EXPRESSIONS OF THE STRAIN ENERGY
§ 11–2 MOHR’S THEOREM(METHOD OF UNIT FORCE)
§ 11–3 CATIGLIANO’S THEOREM

§ 11–1 变形能的普遍表达式
§ 11–2 莫尔定理 (单位力法 )
§ 11–3 卡氏定理
§ 11–1 GENERAL EXPRESSIONS OF THE STRAIN ENERGY
1,Principle of energy,
2,Calculation of the strain energy of rods,
1),Calculation of the strain energy of rods in tension or compression,
?? L xEA xNU d2 )(
2
?
?
?
n
i ii
ii
AE
LNU
1
2
2
??21?u
Strain energy stored in the elastic body is equal to the work done by external
forces,that is,
WU ?
Method to analyze and calculate displacements, deformations and internal
forces of deformable bodies by this kind of relation is called energy method,
or Density of the strain energy,
§ 11–1 变形能的普遍表达式

1.轴向拉压杆的变形能计算,
?? L xEA xNU d2 )(
2 ?
?
?
n
i ii
ii
AE
LNU
1
2
2

2
1,?u比能

WU ?

2,Calculation of the strain energy of rods in torsion,
?? L
P
n x
GI
xM
U d
2
)(
2
?
?
?
n
i Pii
ini
IG
LMU
1
2
2
??21?u
3,Calculation of strain energy of rods in bending,
?? L xEI
xM
U d
2
)(
2
?
?
?
n
i ii
ii
IE
LMU
1
2
2
??21?u
or
Density of the strain energy,
or
Density of the strain energy,
2.扭转杆的变形能计算,
?? L
P
n x
GI
xM
U d
2
)(
2
?
?
?
n
i Pii
ini
IG
LMU
1
2
2

??
2
1,?u比能
3.弯曲杆的变形能计算,
?? L xEI
xM
U d
2
)(
2
?
?
?
n
i ii
ii
IE
LMU
1
2
2

??
2
1,?u比能
3,General expressions of the strain energy,
mutually independent load may be summed up each other,
For slender columns,the strain energy due to shearing forces may be neglected,
x
EI
xM
x
GI
xM
x
EA
xN
U
LL
P
n
L
d
2
)(
d
2
)(
d
2
)( 222
??? ???
?? L xEA xQ d2 )(
2
S? ?S?
x
EI
xM
x
GI
xM
x
EA
xN
U
LL
P
n
L
d
2
)(
d
2
)(
d
2
)( 222
??? ???
Deflection factor of shear

?? L xEA xQ d2 )(
2
S? 剪切挠度因子?S?
x
EI
xM
x
GI
xM
x
EA
xN
U
LL
P
n
L
d
2
)(
d
2
)(
d
2
)( 222
??? ???
x
EI
xM
x
GI
xM
x
EA
xN
U
LL
P
n
L
d
2
)(
d
2
)(
d
2
)( 222
??? ???
Solution,In energy method（ work done by
external forces is equal to the strain energy）
① Determine internal forces
?? s i n)( PRM T ?
)c o s1()( ?? ?? PRM N
A
Bending moment,
Torque,
Example 1 A semicircle rod as shown in the figure is lie in horizontal plane,
A vertical force P act at its point A,Determine the displacement of point A in
vertical direction,
P R
O
Q
MN
MT
A
A
P N
B
?
T
O
MN
[例 1 ] 图示半圆形等截面曲杆位于水平面内,在 A点受铅垂力 P

① 求内力
?? s i n)(,PRM T ?弯矩
)c o s1()(,?? ?? PRM N扭矩
A
P R
O
Q
MT
A
A
P N
B
?
T
O
③ Work done by external forces is equal to the strain energy
② Strain energy,
??? ??? LL
P
L
x
EI
xM
x
GI
xMx
EA
xNU d
2
)(
d
2
)( d
2
)( 22n2
?? ?
?
?
??
?
?
?
?
0
222
0
222
d
2
)( s i n
d
2
)c o s1(
R
EI
RP
R
GI
RP
P
EI
RP
GI
RP
P 44
3 3232 ??
??
,
2
UfPW A ??
Let
EI
PR
GI
PR
f
P
A 22
3 33 ??
??
then
③ 外力功等于应变能
② 变形能,
??? ??? LL
P
L
x
EI
xM
x
GI
xMx
EA
xNU d
2
)(
d
2
)( d
2
)( 22n2
?? ?
?
?
??
?
?
?
?
0
222
0
222
d
2
)( s i n
d
2
)c o s1(
R
EI
RP
R
GI
RP
P
EI
RP
GI
RP
P 44
3 3232 ??
??
UfPW A ??
2
?
EI
PR
GI
PR
f
P
A 22
3 33 ??
???
Example 2 Determine the deflection of point C by the energy
method,where the beam is of equal section and straight,
CPfW 2
1?
Solution,Work done by external
forces is equal to the strain energy
?? L xEI xMU d2 )(
2
)0(; 2)( axxPxM ???
By using symmetry we get,
EI
aPxxP
EI
U
a
12
d)
2
(
2
12 32
0
2 ?? ?
EI
Pat h e n fUW
C 6,
3
??Thinking：
we determine the displacement of point C
by this method？
q
C a a
A
P
B
f
Let
[例 2 ] 用能量法求 C点的挠度。梁为等截面直梁。
CPfW 2
1?

?? L xEI xMU d2 )(
2
)0(; 2)( axxPxM ???

EI
aPxxP
EI
U
a
12
d)
2
(
2
12 32
0
2 ?? ?
EI
PafUW
C 6
3
????

C a a
A
P
B
f
§ 11–2 MOHR’S THEOREM(METHOD OF UNIT FORCE)
AC fUUU ???? 10
?? L xEI xMU d2 )(
2
?? L xEI xMU d2 )(
2
0
0
? ?? LC xEI xMxMU d2 )]()([
2
0
?? LA xEI xMxMf d)()( 0
Determine the displacement f A of an
arbitrary point A,
1,Provement of the theorem,
a
A
Fig
fA
q(x)
Figc
A
0 P =1 q(x)
f
A
Figb
A
=1 P0
§ 11–2 莫尔定理 (单位力法 )
AC fUUU ???? 10
?? L xEI xMU d2 )(
2
?? L xEI xMU d2 )(
2
0
0
? ?? LC xEI xMxMU d2 )]()([
2
0
?? LA xEI xMxMf d)()( 0

a
A

fA
q(x)

A
0 P =1 q(x)
f
A

A
=1 P0
Mohr’s theorem(method of unit force)
2,General form of Mohr’s theorem
x
EI
xMxMf
LA
d)()( 0??
?? ??? L
P
nn
LA
x
GI
xMxM
x
EA
xNxN d)()(d)()( 00? x
EI
xMxM
L
d)()( 0?

x
EI
xMxMf
LA
d)()( 0??
?? ??? L
P
nn
LA
x
GI
xMxM
x
EA
xNxN d)()(d)()( 00? x
EI
xMxM
L
d)()( 0?
3,What we must pay attention to as we apply Mohr’s theorem,
④ Coordinate of M0(x) must be coincide with that of M(x),For each segment
the coordinate may be set up freely,
⑤ Mohr’s integrationmust be through the whole structure,
② M0,The internal force of the structure as we act a generalized unit force
along the direction,of the generalized displacement that is to be determined,
where the applied force is taken out,
① M(x),The internal force of the structure acted by original loads,
③ The product of the applied generalized unit force and the generalized
displacement to be determined determined must be of the dimension of work,

④ M0(x)与 M(x)的坐标系必须一致，每段杆的坐标系可

⑤ 莫尔积分必须遍及整个结构 。
② M0——去掉主动力，在所求 广义位移 点,沿所求

① M(x)：结构在原载荷下的内力。
③ 所加广义单位力与所求广义位移之积，必须为功的量纲。
Example 3 Determine the displacement and the angle of rotation of point C by
the energy method,
2
)(
2qx
a q xxM ??
?
?
?
??
?
?
???
??
?
)2(; )2(
2
)0(;
2
)(
0
axaxa
x
ax
x
xM
Solution,① Plot the diagram of the structure acted by the unit load
② Determine the internal force
B A
a a C
q
B A
a a C
0 P =1
x
[例 3] 用能量法求 C点的挠度和转角。梁为等截面直梁。
2
)(
2qx
a q xxM ??
?
?
?
??
?
?
???
??
?
)2(; )2(
2
)0(;
2
)(
0
axaxa
x
ax
x
xM

② 求内力
B A
a a C
q
B A
a a C
0 P =1
x
d)()(d)()(
2
0
0
0 ?? ??
a
a
a
C xEI
xMxMx
EI
xMxMf
?
a
x
EI
xMxM
0
0 d)()(2
Symmetry
EI
qaxxqxqax
EI
a
24
5d
2
)
2
(2
4
0
2
??? ?
③ Deformation
B A
a a C
0 P =1 B A
a a C
q
x
( )
d)()(d)()(
2
0
0
0 ?? ??
a
a
a
C xEI
xMxMx
EI
xMxMf
?
a
x
EI
xMxM
0
0 d)()(2

EI
qaxxqxqax
EI
a
24
5d
2
)
2
(2
4
0
2
??? ?
③ 变形
B A
a a C
0 P =1 B A
a a C
q
x
( )
④ Determine the angle of rotation,Set up the coordinate again (as shown in the figure）
?? ?????
aa
x
a
xqxqax
EI
x
a
xqxqax
EI 0 2
2
2
2
2
0
1
1
2
1
1 d2)2(
1d
2
)
2
(1
2
)(,
2
1
1
qxq a xxMAC ??
a
xxM
2)(
1
0 ??
2
)(,
2
2
2
qxq a xxMBC ??
a
xxM
2)(
2
0 ?
q
B A
a a C
x2 x1
B A
a a
C
MC0=1
d ) ( ) (
) ( ) (
) ( 0
0
) ( 0
0
?
?
?
?
a
BC
a
AB
xEI x M x M
dx EI x M x M
c?
=0
④ 求转角，重建坐标系（如图）
?? ?????
aa
x
a
xqxqax
EI
x
a
xqxqax
EI 0 2
2
2
2
2
0
1
1
2
1
1 d2)2(
1d
2
)
2
(1
2
)(,
2
1
1
qxq a xxMAC ??
a
xxM
2)(
1
0 ??
2
)(,
2
2
2
qxq a xxMBC ??
a
xxM
2)(
2
0 ?
q
B A
a a C
x2 x1
B A
a a
C
MC0=1
d ) ( ) (
) ( ) (
) ( 0
0
) ( 0
0
?
?
?
?
a
BC
a
AB
xEI x M x M
dx EI x M x M
c?
=0
PxxM AB ?)(
xxM AB ?)(0
PxM n C A 3.0)( 1 ?
3.0)( 10 ?xM CAn
Solution,① Plot the diagram of
the structure acted by a unit load
② Determine
the internal force
5
10
20
A
P=60N
B
x 500
C
x1
5
10
20
A
B
x 500
C
=1 P0
Example 4 A folding rod is shown in the figure,A bearing is at position A and
the rod may rotate freely in the bearing but can not move up and down,Knowing：
E=210Gpa,G=0.4E,Determine the vertical displacement of point B,
[例 4 ] 拐杆如图,A处为一轴承，允许杆在轴承内自由转动，但不

PxxM AB ?)(
xxM AB ?)(0
PxM n C A 3.0)( 1 ?
3.0)( 10 ?xM CAn

② 求内力
5
10
20
A
P=60N
B
x 500
C
x1
5
10
20
A
B
x 500
C
=1 P0
PxxM AB ?)( xxM AB ?)(0 PxM n C A 3.0)( 1 ? 3.0)( 10 ?xM CAn
?? ?? LL
P
nn
B xEI
xMxMx
GI
xMxM d)()( d)()( 0
1
101?
?? ?
?
?
3.0
0
25.0
0
1 dd
3.03.0
x
EI
Px
x
GI
P
P P
ACAB
AB
AB
GI
LPLL
EI
PL ??
3
3
??
???
??? 3
3
3
10
1052103
123.060 3
4 10202 1 04.0
325.03.0603.0 ?
??
????
?
mm22.8?
③ Determine the deformation
( )
PxxM AB ?)( xxM AB ?)(0 PxM n C A 3.0)( 1 ? 3.0)( 10 ?xM CAn
?? ?? LL
P
nn
B xEI
xMxMx
GI
xMxM d)()( d)()( 0
1
101?
?? ?
?
?
3.0
0
25.0
0
1 dd
3.03.0
x
EI
Px
x
GI
P
P P
ACAB
AB
AB
GI
LPLL
EI
PL ??
3
3
??
???
??? 3
3
3
10
1052103
123.060 3
4 10202 1 04.0
325.03.0603.0 ?
??
????
?
mm22.8?
③ 变形
( )
§ 11–3 CATIGLIANO’S THEOREM
Give Pn an increment dPn, then,
),...,,( 21 nPPPUU ?
n
n
P
P
UUU d
1 ?
???
1)First apply forces P1,P2,???,Pn on the
body, then,
2),First apply the force dPn on
the body,then,
)d()d(
2
1
2 nnPU ???
1,Provement of the theorem
1P
2P
?n
nP
§ 11–3 卡氏定理

),...,,( 21 nPPPUU ?
n
n
P
P
UUU d
1 ?
???
1,先给物体加 P1,P2,???,Pn 个力，则,
2.先给物体加力 dPn,则,
)d()d(
2
1
2 nnPU ???

1P
2P
?n
nP
Again apply forces P1,P2,???,Pn, then,
)d(21 nn PUUU ???? ?
n
n P
U
?
???
1P
2P
?n
nP
n ?
n P
U
?
? ? Second Castigliano’s
theorem
Italian engineer— Alberto
Castigliano,1847～ 1884

)d(21 nn PUUU ???? ?
n
n P
U
?
???
1P
2P
?n
nP
n ?
n P
U
?
? ? 第二卡氏定理

Castigliano,1847～ 1884)
2,what we must pay attention to as we apply
Catigliano’s theorem,
① U—Linear elastic strain energy of the whole
② Pn is considered as a variable,The reactions
and the strain energy of the structure and so on
must be all expressed as the function of Pn,
③ ?n is the deformation of the point acted by Pn
and it isalong the direction of Pn,
④ If there is no Pn corresponding to ?n we may
first act a Pn along ?n and determine the
partial derivative and then let Pn be zero,
1P
2P
?n
nP

① U——整体结构在外载作用下的线

② Pn 视为变量，结构反力和变形能

③ ?n为 Pn 作用点的沿 Pn 方向的 变形。
④ 当无与 ?n对应的 Pn 时，先加一沿 ?n

1P
2P
?n
nP
3,Castigliano’s theorem for special structures(rods),
??? ??? LL
P
L
x
EI
xM
x
GI
xM
x
EA
xN
U d
2
)(
d
2
)(
d
2
)(
22
n
2
??
?
?
?
?
?
?
?
?
?
?
?
?
?
L
n
L
n
n
P
L
nn
n
x
P
xM
EI
xM
x
P
xM
GI
xM
x
P
xN
EA
xN
P
U
d
)()(
d
)()(
d
)()(
n
?

??? ??? LL
P
L
x
EI
xM
x
GI
xM
x
EA
xN
U d
2
)(
d
2
)(
d
2
)(
22
n
2
??
?
?
?
?
?
?
?
?
?
?
?
?
?
L
n
L
n
n
P
L
nn
n
x
P
xM
EI
xM
x
P
xM
GI
xM
x
P
xN
EA
xN
P
U
d
)()(
d
)()(
d
)()(
n
?
Example 5 The structure is shown in the figure,Determine the deflection and
the angle of rotation of the section A by Catigliano’s theorem,
③ Determine
the deformation
① Determine the internal force
Solution,Determine the deflection,
Set up the coordinate
xPxPxM A ??)(
EI
PL
3
3
?
② Determine the partial derivative
of the internal force with respect to PA xP xM
A
?
?
? )(
? ?????? L
AA
A xP
xM
EI
xM
P
Uf d)()(
??
L
x
EI
Px
0
2
d
A
L
P EI
x O
( )
[例 5 ] 结构 如图,用卡氏定理 求 A 面的挠度和转角。
③ 变形
① 求内力

xPxPxM A ??)(
EI
PL
3
3
?
② 将内力对 PA求偏导
x
P
xM
A
?
?
? )(
? ?????? L
AA
A xP
xM
EI
xM
P
Uf d)()(
??
L
x
EI
Px
0
2
d
A
L
P EI
x O
( )
Determine the angle ?A of rotation
① Determine the internal force
AMxPxM ??)(
There is no the generalized force
corresponding to ?A,we may act one,
EI
PL
2
2
??
“Negative sign”expresses that ?A is contrary to the
direction of the acted generalized force MA( )
② Determine the partial derivative of the internal
force M(x) with respect to MA and let M A =0,
1
)(
0
??
?
?
?
A
MAM
xM
? ??? L
A
A xM
xM
EI
xM d)()( ? ? ??
L
x
EI
Px
0
d
③ Determine the deformation
（ Note,M A=0）
L x O
A
P M A
EI
PL
A 2
2
???

① 求内力
AMxPxM ??)(

EI
PL
2
2
??
“负号, 说明 ?A与所加广义力 MA反向 。 ( )
EI
PL
A 2
2
??
② 将内力对 MA求偏导后，令 M A=0
1
)(
0
??
?
?
?
A
MAM
xM
? ??? L
A
A xM
xM
EI
xM d)()( ? ? ??
L
x
EI
Px
0
d
③ 求变形（ 注意,M A=0）
L x O
A
P M A
Example 6 Determine the deflection curve of the beam shown in
the figure by Castigliano’s theorem,
Solution,Determine the deflection curve—the
deflection of an arbitrary point on the beam f（ x）,
① Determine the internal forces
② Determine the partial derivative of the internal
force M(x) with respect to Px and let Px =0,
There is no the generalized force
corresponding to f（ x）,we may act one,
)()()( 111 xxPxLPxM xAB ?????
)()( 11 xLPxM BC ???
xx
P
xM
P
x
AB ??
?
?
? 10
)(
x
0)( 0 ?
?
?
?x
x
BC
PP
xM
P
A
L
x
B
Px
C
f
x O
x1
[例 6 ] 结构 如图,用卡氏定理 求 梁的挠曲线。

① 求内力
② 将内力对 Px 求偏导后,令 Px=0

)()()( 111 xxPxLPxM xAB ?????
)()( 11 xLPxM BC ???
xx
P
xM
P
x
AB ??
?
?
? 10
)(
x
0)( 0 ?
?
?
?x
x
BC
PP
xM
P
A
L
x
B
Px
C
f
x O
x1
③ Determine the deformation（ Note,Px=0）
? ?
??
?
??
L
xx
x
P
xM
EI
xM
P
Uxf d)()( )(
? ????
x
xxxxLP
EI 0 111
d))((
1
)
2
)(
3
( 2
23
LxxxLx
EI
P ????
③ 变形（ 注意,Px=0）
? ?
??
?
??
L
xx
x
P
xM
EI
xM
P
Uxf d)()( )(
? ????
x
xxxxLP
EI 0 111
d))((
1
)
2
)(
3
( 2
23
LxxxLx
EI
P ????
Example 7 A beam with equal section is shown in the figure,Determine the
deflection f(x) of point B by Catigliano’s theorem,
② determine internal forces
Solution,1.Determine redundant reactions
according to 0?
Cf
③ Determine the partial derivative of
the internal force with respect to RC,
)5.0()()( xLPxLRxM CAB ????
)()( xLRxM CBC ??
xL
R
xM
C
AB ??
?
? )(
① Take a primary beam as shown in the
xL
R
xM
C
BC ??
?
? )(
P
C
A L
0.5 L B
f
x O
P
C
A L
0.5 L B
RC
figure,
[例 7 ] 等截面梁 如图,用卡氏定理 求 B 点的挠度 。
② 求内力

Cf
③ 将内力对 RC求偏导
)5.0()()( xLPxLRxM CAB ????
)()( xLRxM CBC ??
xL
R
xM
C
AB ??
?
? )(
① 取静定基如图
xL
R
xM
C
BC ??
?
? )(
P
C
A L
0.5 L B
f
x O
P
C
A L
0.5 L B
RC
④ Deformation
? ?????? L
CC
C xR
xM
EI
xM
R
Uf d)()(
? ? ?
?
?
?
?
?
?????? ??
L
C
L
xxLRxxLxLP
EI 0
2
5.0
0
d)(d)()5.0(
1
0)
348
5(1 33 ???? LRPL
EI
C
16
5 PR
C ?
So
④ 变形
? ?????? L
CC
C xR
xM
EI
xM
R
Uf d)()(
? ? ?
?
?
?
?
?
?????? ??
L
C
L
xxLRxxLxLP
EI 0
2
5.0
0
d)(d)()5.0(
1
0)
348
5(1 33 ???? LRPL
EI
C
16
5 PR
C ??
2.Determine
Bf
② Determine the partial derivative of the internal force with respect
)5.0()(165)( xLPxLPxM AB ????
)(165)( xLPxM BC ??
16
311)( Lx
P
xM AB ??
?
?
16
)(5)( xL
P
xM BC ??
?
?
① Determine the internal forces
to P,
2.求
Bf
② 将 内力对 P求偏导
)5.0()(165)( xLPxLPxM AB ????
)(165)( xLPxM BC ??
16
311)( Lx
P
xM AB ??
?
?
16
)(5)( xL
P
xM BC ??
?
?
① 求内力
③ Deformation
? ?????? LB xP xMEI xMPUf d)()(
?
?
?
?
?
?
??
?
? ??
L
L
L
xxLPx
Lx
P
EI 5.0
22
5.0
0
2 d)()
16
5
(d)
16
311
(
1
EI
PL
768
7 3? ( )
③ 变形
? ?????? LB xP xMEI xMPUf d)()(
?
?
?
?
?
?
??
?
? ??
L
L
L
xxLPx
Lx
P
EI 5.0
22
5.0
0
2 d)()
16
5
(d)
16
311
(
1
EI
PL
768
7 3? ( )
③ Determine the deformation
Solution,① Plot the diagram of the structure acted by unit load
② Determine the internal force
Example 8 A frame is shown in the figure,Determine the
distance between section A and section B after the deformation,
P
P A
B
1
1
③ 变形

② 求内力
[例 8 ] 结构 如图，求 A,B两面的拉开距离。
P
P A
B
1
1
59
Chapter 11 Exercises
1,A straight rod with the tension (compression) rigidity EI is
subjected forces shown in the figure,May the strain energy be
expressed as
2,Try to explain how to determine the deflection of the free
end of the beam shown in the figure by Castigliano’s theorem,
3,As shown in the figure,a rigid frame is subjected to forces,
Knowing EI is a constant,Try to determine the relative
displacement between point A and point B by Mohr’s theorem
(neglecting the tensile deformation of Section CD),
22 2
2
21
2
1
EA
LP
EA
LPU ??
60

22 2
2
21
2
1
EA
LP
EA
LPU ??
61
Solution,
? ? ? ? ? ? ? ?? ?? ? ?? ?? a a
EI
xMxM
EI
xMxM
AB dxdx0
2/
0 21
2
0
21
0
12?
? ? ? ?????? ? ? ??
EI
Pa
dxdx
a a
EI
aPa
EI
xPx
3
0
2/
0 21
3
5
2 11
62

? ? ?? ?? a a EI xMxMEI xMxMAB dxdx0 2/0 21 2021012?
? ? ? ?????? ? ? ??
EI
Pa
dxdxa a EI aPaEI xPx
3
0
2/
0 21 3
5
2 11
63
4,A beam with the bending rigidity EI is shown in the
figure,The rigidity of the spring at the end B is k,Try to
determine the deflection of the point where the force P is
applied by Castigliano’s theorem,
Solution,① The strain energy of the system is
② The deflection of Section C is
? ? ? ?? ? kPEILPkPPL L PPEI dxxdxU 182 4 3233213/0 3/20 232322 1 232 ??????? ? ?
? ????? ?? kPEIPLPUcf 92434 3
64

② C截面的挠度
? ? ? ?? ? kPEILPkPPL L PPEI dxxdxU 182 4 3233213/0 3/20 232322 1 232 ??????? ? ?
? ????? ?? kPEIPLPUcf 92434 3
65
66