结构力学 能量法

2
2E
M (x) y
Iz
U
udV
V
A
l
1 2E
M 2(x)
I
2 z
y 2dAdx
U M 2 (x)dx
l 2EI z
4) 组合变形杆的应变能
Strain-energy of combined deformationed bars
M(x)
N(x) T(x)
U FN 2 (x)dx T 2 (x)dx M 2 (x)dx
机械能守恒 在等温绝热平衡的（缓慢的）加载过程中，外 力对弹性体所做的功全部转化为应变能
外力功 = 应变能
W=U
A
a
–√–2
P P
D Pa
B0 C P
例 图示桁架各杆件的 抗拉刚度均为 EA，求结 点 C 的竖向位移。
各杆应变能
U AD

P2a 2EA
P2a U DC 2EA
UBD (
2P)2 ( 2EA
U T 2 (x)dx M 2 (x)dx
l 2GI p
l 2EIz
U
( T 2 (x)dx
M 2 (x)dx )
n l 2GI p
l 2EIz
5) 杆件结构系统的应变能
Strain-energy of a bars system
A bars system , having some simple tension or compression bars and some beams . Its strain-energy can be attained as follows :
应变比能 (strain-energy density)


u d u d
0
0
σ
ε dε
Simple state of stress
x
y
z
u x d x y d y z d z
0
0
0
xy
yz
zx
xy d xy yz d yz zx d zx
For example :
Strain-energy caused by FN and Fs can be neglected .
FN Fs M
So the strain-energy of combined deformationed bars can be written as follows :
P
B
A
P
A
Vertical displacement at point A ? Rotation angle of section B ?
Vertical and level displacement at point A ?
Attention : energy method is only a
method to determine internal forces , reactions , displacements and rotation angles of a structure in Mechanics of Materials .
P



m

m





δ w P δ W P d
P
0
线弹性情况
w m
W 1 P
2

W m d 0
线弹性情况
W 1 m
2
dΔ
ΔΔ
P
dΔ
Δ
杆件中外力的功
a. 杆件的拉压
P
l
W 1Pl
2
b. 圆轴的扭转
m φ
W 1 m
c.应变能关于载荷是非线性的
Example : 杆件拉伸
P1

U1

P12l 2EA
P2

U2

P22l 2EA
P1 P2

U (1 2)

(P1 P2 )2l 2EA
P12l P22l P1P2l 2EA 2EA EA

U1
U2

P1P2l EA
Energy methods (1)
u

1
2

2
2E


FN ( x) A( x)
内力功
=
应变能
U
u dV
V
l
1 2E
FN 2 ( x) A2 (x)
A( x)dx
U FN 2 ( x) dx
l 2EA
c. 桁架的应变能
U
FN2i li
i 2Ei Ai
2) 圆轴扭转的应变能
Strain-energy of shaft
2
c. 梁的弯曲
P
W 1 P
2
Δ
m θ
q y(x)
W 1 m
2
W 1 l qy(x)dx 20
4) 应变能计算
外力功 = 应变能
外力功 = 内力功
W=U
W Wint
W U u d
u ------- called strain-energy density 应变比能
2a)
结构总应变能 U Ui (
UBC 0
2P2a EA 2 1) P2a
EA
由结构平衡可 得各杆内力
N AD P NCD P NBD 2P NBC 0
P 力的功
W

1 2
PvC
由功能关系 W U
vC 2(
2 1) Pa EA
Energy methods (1) 的评述
U M 2 (x)dx l 2EIz
For a system of beams , we can attain :
U
M 2 (x)dx
l n
2EIz
b. Method 2 to determine the strain- energy of beams
u 1 2
l 2GI p
3) 梁弯曲的应变能
Strain-energy of beams
a. Method 1 to determine the strain- energy of beams
dU 1 M (x)d
2
d dx
d 1 dx M (x)dx

EI z
d
M(x)
dx
dU M 2 (x)dx 2EI z
应变能
动能
外力功
应变能
外力功 动能
外力功
应变能
动能
Don’t consider the loss of energy , we can attain :
= 外力功
应变能
W=U
W ------- work done by external forces U -------- strain-energy of the structure
l 2EA
l 2GI p
l 2EIz
For a system of bars , we have :
U ( FN 2 (x)dx T 2 (x)dx M 2 (x)dx )
n l 2EA
l 2GI p
l 2EIz
The strain-energy caused by shearing forces was negligible when transeverse loadins acted on a beam .
U
FN2ili
[ T 2 (x)dx
M 2 (x)dx ]
m 2Ei Ai n l 2GI p
l 2EIz
Bars : m Beams : n
P
bars m=2
beam n=1
应变能的特点
PP1 1 P2
a.应变能是状态函数
Δl1 Δl2
b.结构总应变能等于各部件应变能之和
Only the elastic strain-energy can be released out .
(2) 应变能计算 ( Calculation of strain-energy )
外力的功 ( Work done by external forces ) The external loadings must be static loadings
For the elas-plastic materials , the strainenergy will be divided into two parts :
应变能 = 弹性应变能 +塑性应变能
U Ue U p
P
弹性区
塑性区 U p ----- energy consumed to form the plastic region .

FN 2 ( x) d x 2EA
U FN 2 ( x)dx l 2EA
FN(x) dx
l
For a system of bars , we have :
U
FN 2 (x) d x
l n
2EA
b. Method 2 to determine the strain- energy of bars
energy for the barsБайду номын сангаас)
1) 杆件拉压的应变能 Strain-energy of bars
a. Method 1 to determine the strain- energy of bars
1 dU 2 FN (x)(dx)
= 外力功
应变能
dU

1 2
FN
(
x)
FN
( x)dx EA
优点 ：概念及方法简单
P B xB
F C yc
缺点 ：只能求单个载荷作用下
A
的杆件结构系统在载荷作用点
处沿载荷作用方向的位移
F
B
C yc
B
W
U

1 2
Fyc

1 2
PxB
Cm
A
1 W U 2 Fyc
A
C
W
U

1 2
m
C
2U yc F
C

2U m
7.2 互等定理 ( reciprocal theorem )
U
T 2 (x)dx
n l 2GI p
b. Method 2 to determine the strain- energy of shafts
u 1 2
2
2G
T (x)
Ip
U
udV
V
A
l
1 2G
T
2 ( x)
I
2 p

2dAdx
U T 2 (x)dx
第11章 能量法
Energy Methods
Preface
Energy method is a very convenient method to determine the internal forces or reactions of a bars system and displacement of a structure at one point or rotation angle of a cross section in the bars system .
(1) 功的互等定理 ( reciprocal-work theorem )
一种变形状态
Displacement at the point C
C A
B
Rotation of section B
D Reaction at the point D
7.1 应变能 ( strain-energy )
(1) 应变能概念 (Concept of strain-energy )
Elastic-bodies have an ability to do work when deformationed by external forces . That means in deformationed elastic-bodies that reserve some kind of energy . This kind of energy is called strain-energy .
0
0
0
Complex of stress
state
线弹性体的应变比能
u 1 u 1
2
2
σ
u

1 2
(
x x

y
y

z z

xy xy yz yz zx zx )
ε dε
(3) 杆件应变能计算 ( Calculation of strain-
a. Method 1 to determine the strain- energy of shafts
dU 1 T (x)d
2
T(x)
1 T (x) T (x)dx T 2 ( x)dx
dx
2
GI p
2GI p
U T 2 (x)dx
l 2GI p
For a system of shafts , we have :
For example :
Strain-energy caused by Fs(x) can be neglected .
P M(x) Fs(x)
Futhermore , for the combined deformationed bars , the strain-energy caused by axial forces ordinarily can be neglected .