结构力学 能量法

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
动能 应变能 外力功 外力功 应变能 动能
外力功
应变能
Baidu Nhomakorabea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
Energy methods (1)
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 P – –√2 P
D P a
各杆应变能
U AD
B 0 C
P 2a 2 EA
U DC
P 2a 2 EA
U BC 0
P
例
图示桁架各杆件的
Rotation angle of section B ?
Attention : energy method is only a
method to determine internal forces , reactions , displacements and rotation angles of a structure in Mechanics of Materials .
σ
u
u d d
0


0
ε
dε
Simple state of stress
u x d x y d y z d z
0 0 0
x
y
z
xy d xy yz d yz zx d zx
0 0 0
l
1 W P l 2
l
m θ
b. 圆轴的扭转
m φ
q
1 W m 2
y(x)
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
应变比能 (strain-energy density)
xy
yz
zx
Complex state of stress
线弹性体的应变比能
1 1 u u 2 2
1 u ( x x y y z z 2
σ
xy xy yz yz zx zx )
ε
dε
(3) 杆件应变能计算 ( Calculation of strainenergy for the bars )
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 . P B A P A Vertical displacement at point A ? Vertical and level displacement at point A ?
T(x)
2
FN ( x)dx T 2 ( x)dx M 2 ( x)dx U l l l 2 EA 2GI p 2 EI z
For a system of bars , we have :
FN ( x)dx T ( x)dx M ( x)dx U ( ) l l 2GI l 2 EA 2 EI z n p
1) 杆件拉压的应变能
1 d U FN ( x)(dx) 2
Strain-energy of bars
a. Method 1 to determine the strain- energy of bars
外力功
=
dx l
应变能
FN ( x)dx FN 2 ( x ) d x 1 d U FN ( x) 2 EA 2 EA
P
M(x) Fs(x)
Futhermore , for the combined deformationed bars , the strain-energy caused by axial forces ordinarily can be neglected .
For example : Strain-energy caused by FN and Fs can be neglected .
P




m m





δ w Pδ
W
P d
0

P
线弹性情况
1 W P 2
W
dΔ P
Δ Δ
w m
线弹性情况
m d
0

1 W m 2
dΔ
Δ
杆件中外力的功
a. 杆件的拉压
P
c. 梁的弯曲
P
Δ
1 W P 2
1 W m 2
1 W q y ( x)dx 20
M ( x)dx U l 2 EI z
For a system of beams , we can attain :
M ( x)dx U l 2 EI z n
2
b. Method 2 to determine the strain- energy of beams
1 2 u 2 2E
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 .
F l T ( x)dx M ( x)dx U [ ] l l 2GI p 2EI z m 2 Ei A n i
P
2 Ni i
2
2
Bars : Beams :
m n bars
m=2 beam n=1
应变能的特点
PP 1 1 P2
a.应变能是状态函数 b.结构总应变能等于各部件应变能之和 c.应变能关于载荷是非线性的
U
FN(x)

l
FN ( x)dx 2 EA
2
For a system of bars , we have :
FN ( x) d x U l 2 EA n
b. Method 2 to determine the strain- energy of bars
2
1 2 FN ( x) u A( x) 2 2E
FN Fs M
So the strain-energy of combined deformationed bars can be written as follows :
T ( x)dx M ( x)dx U l l 2GI p 2EI z
2
2
T 2 ( x)dx M 2 ( x)dx U ( ) l l 2 GI 2EI z n p
2
3) 梁弯曲的应变能
Strain-energy of beams
a. Method 1 to determine the strain- energy of beams
1 d U M ( x)d 2
d
M(x)
d dx
2
M ( x)dx d dx EI z
2
1
dx
M ( x)dx dU 2 EI z
W U
Pa vC 2( 2 1) EA
Energy methods (1) 的评述
P
F
xB
优点 ：概念及方法简单 缺点 ：只能求单个载荷作用下 的杆件结构系统在载荷作用点 处沿载荷作用方向的位移
Example : 杆件拉伸
2 P 1 l P U1 1 2 EA
Δl1 Δl2
P2
P22l U2 2 EA
P P12l P22l P1 P2l ( P1 P2 ) 2 l 1P 2l U1 U 2 P1 P2 U (1 2) EA 2 EA 2 EA 2 EA EA
U BD
( 2 P ) 2 ( 2a ) 2P 2a 2 EA EA
抗拉刚度均为 EA，求结
点 C 的竖向位移。 由结构平衡可 得各杆内力
P2a 结构总应变能 U U i ( 2 1) EA
P 力的功 由功能关系
1 W PvC 2
N AD P N CD P
N BD 2P N BC 0
M ( x) y Iz
U udV
V
A l

1 M ( x) 2 y dAdx 2 2E I z
2
M ( x)dx U l 2 EI z
2
4) 组合变形杆的应变能
Strain-energy of combined deformationed bars
M(x) N(x)
Displacement at the point C C A D
B
Rotation of section B 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 .
The strain-energy caused by shearing forces was negligible when transeverse loadins acted on a beam .
2
2
2
For example :
Strain-energy caused by Fs(x) can be neglected .
b. Method 2 to determine the strain- energy of shafts
T ( x) 1 u Ip 2 2G
2
U udV
V
A l

1 T ( x) 2 dAdx 2 2G I p
2
T ( x)dx U l 2GI p
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 :
U
内力功
=
应变能
u dV
V



l
1 FN ( x) A( x)dx 2 2 E A ( x)
FN ( x ) dx 2 EA
2
2
U
l
c. 桁架的应变能
F l U 2 Ei Ai i
2) 圆轴扭转的应变能
Strain-energy of shaft
2 Ni i
a. Method 1 to determine the strain- energy of shafts
1 d U T ( x)d 2
T(x)
dx
1 T ( x)dx T 2 ( x )dx T ( x) 2 GI p 2GI p
T 2 ( x)dx U l 2GI p
For a system of shafts , we have :
T 2 ( x)dx U l 2GI p n