弹性力学 第一章 1
弹性力学及有限元
Elasticity and Finite Element Method
The lectures will be given both in
English and Chinese
采用中英文双语讲授
弹性力学 第一章 2
Give me a fish and I will eat today,
Teach me to fish and I will eat for a
life time.
授人以鱼，不如授人以渔。
弹性力学 第一章 3
Textbook:
Applied Elasticity
徐芝纶
中文教材：
弹性力学简明教程
徐芝纶
弹性力学 第一章 4
Chapter 1,Introduction
第一章 绪论
弹性力学 第一章 5
1.1 Contents of Theory of Elasticity
1.1 弹性力学的内容
? NAME
Theory of elasticity is often called elasticity
for short,It is the branch of solid
mechanics,
弹性力学的理论简称为弹性理论或弹性力学,
它是固体力学的一个分枝
弹性力学 第一章 6
Three branches of solid mechanics
固体力学的三个分 枝
? Mechanics of materials 材料力学,
Structural Mechanics 结构力学
Elasticity 弹性力学
弹性力学 第一章 7
? What does the Elasticity deal with?
It deals with the stresses,deformations and
displacements in elastic solids produced by
external forces or changes in temperature.
研究弹性体 由于外力和温度改变而引起 的应力，
形变和位移。
? It analyzes the stresses,deformations and
displacements of structural elements within the
elastic range and thereby to check the sufficiency
of their strength,stiffness and stability,
分析结构的应力，形变和位移，检查是否满足强
度，刚度和稳定性条件 。
弹性力学 第一章 8
Comparison among the three courses
in solid mechanics
固体力学三门学科的比较
? Three branches have the same purpose and do
differ from one another both in objects studied
and the methods of analysis used.
? 1,Objects studied 研究对象
2,Methods of analysis 研究方法
弹性力学 第一章 9
to deal with the elastic solids
都是研究弹性体
1,objects studied:--研究对象,
(1) Similarity---------相同点
弹性力学 第一章 10
(2)objects studied--difference研究对象 --不同点
Mechanics of materials, bar element
材料力学 单根杆件
Structural bar systems:--
Mechanics, truss,rigid frame
结构力学 杆件系统,桁架, 刚架 。
Elasticity,1,plates and shells 板, 壳
弹性力学 2.blocks,块体
e.g,dams,foundations 坝, 基础
3.analyze bar element precisely
对杆件作精确分析
弹性力学 第一章 11
? Mechanics of materials deals essentially with
the stresses and displacements of structural
element in the shape of a bar,straight or
curved,which is subjected to tension,
compression,shear,bending,or torsion.
材料力学研究受到拉、压、剪、弯或扭的直杆
或曲杆的应力和位移。
弹性力学 第一章 12
? Structural Mechanics deals with the stresses and
displacements of a structural in the form of a bar
system,such as a truss or a rigid frame.
? 结构力学研究杆件系统（例：桁架或刚架）的应力
和位移
弹性力学 第一章 13
? Elasticity deals with the stresses and displacements
of the structural elements such as blocks,plates and
shells,which are not in the form of a bar.
? 弹性力学研究块体、板和壳体的应力和位移。
? Elasticity also analyze a bar element thoroughly and
precisely.
? 弹性力学对杆件作更精确分析
弹性力学 第一章 14
2,methods of analysis:研究方法
(1) Similarity--- 相同点,
equilibrium aspects 静力学方面
geometrical aspects 几何学方面
physical aspects 物理学方面
弹性力学 第一章 15
?equilibrium aspects --equilibrium of forces of an
isolated body
静力学方面 --脱离体力的平衡
? geometrical aspects --the relations between
displacements and strains.
几何学方面 --位移和应变的关系
? physical aspects-- --
the relations between stresses and strains
物理学方面 --应力和应变的关系
弹性力学 第一章 16
(2) methods of analysis:------ difference
研究方法 -------- 不同点,
Mechanics of materials,some assumptions on the
strain condition or the stress condition are made
材料力学,对应变或应力情况作某些假定
Elasticity,no assumptions on the strain condition
or the stress condition are made,
弹性力学, 对应变或应力情况不作假定
弹性力学 第一章 17
Mechanics of materials,
some assumptions on the strain condition or the
stress condition are made
The assumptions simplify the mathematical derivation
to a certain extent.
The assumptions inevitably reduce the degree of
accuracy of the results obtained..
弹性力学 第一章 18
Elasticity：
no assumptions on the strain condition or the stress
condition are made.
The results obtained in elasticity are more accurate
and may be used to check the approximate results
obtained in Mechanics of materials.
弹性力学 第一章 19
?The problem of bending of a straight beam under
transverse loads.
?It is assumed in mechanics of materials that a plane
section of the beam remains plane after bending,which
leads to the linear distribution of bending stresses,
?No assumption,that a plane section of the beam
remains plane after bending,is made in Elasticity.
弹性力学 第一章 20
?A prismatical tension member with a small hole
?It is assumed in mechanics of materials that the tensile
stresses are uniformly distributed across the net section
of the member.
?The analysis in elasticity shows that the stresses are by
no means uniform,but are concentrated near the hole,
弹性力学 第一章 21
1.2 some important concepts in theory of
elasticity
1.2 弹性力学中的几个重要概念
A,External Forces 外力
B,Stress 应力
C,Deformation(Strain) 形变 (应变 )
D,Displacement 位移
弹性力学 第一章 22
A,external forces 外力
1,Body forces 体积力，体力
2,Surface forces 表面力，面力
弹性力学 第一章 23
1,Body forces 体力 。
External forces or the loads,distributed over the
volume of the body,are called body forces.
分布在物体体内的外力叫体力
E.g,gravitational forces,or inertia forces in the case
of a body in motion.
例如, 重力,惯性力
弹性力学 第一章 24
Body force Fig,体力定义图 。
弹性力学 第一章 25
F=lim ?Q/ ?V
? v 0
F--body force vector at p,The vector quantity F is
the intensity of body force at P
F-- P点的体力矢量
?V--an elementary volume of the body around
point p
?V--包含 P点的小体积
?Q--body force acting on ? V
?Q--作用在 ? V上的体力的合力
弹性力学 第一章 26
Body force components 体力分量
? F=X i+Y j+Z k=(X,Y,Z)
? The projections of F on the x,y,and z
axes are called the body force
components at P.
体力在坐标轴上的投影叫体力分量。
? The body force components will be
denoted by X,Y and Z
体力分量用 X,Y,Z表示
弹性力学 第一章 27
Sign Conventions,dimension
正负约定,因次
? It is considered positive (negative) when it
acts in the positive (negative) direction of the
corresponding coordinate,
与坐标轴正向一致为正。
? Its dimension is [force]/[length]3
因次：力 /长度 3
? for example:
弹性力学 第一章 28
2,Surface forces 面力
? Definition,external forces,or the loads,
distributed over the surface of a body,
are called surface forces.
分布在物体表面的外力叫面力
? e.g,hydrostatic pressure,the pressure
of one body on another
例：水压力，接触力
弹性力学 第一章 29
Surface force Fig,面力定义图 。
弹性力学 第一章 30
? F=lim ? Q/ ? S
? F---surface force vector at P,the vector quantity
is the intensity of surface force at P,
F--- P点的面力矢量 。
? ? S--an elementary area of the surface around
point P,? S --包含 P点的小面积
? ? Q--the surface force acting on ? S
作用在 ?S上的面力矢量
弹性力学 第一章 31
Surface force components 面 力分量
? F=X i+Y j+Z k=(X,Y,Z)
? Definition,the projections of F on the x,y and z
axes are called the surface force components at
P,
面 力在坐标轴上的投影叫面力分量
? The surface force components will be denoted
by X,Y and Z
面 力分量用 X,Y,Z表示
弹性力学 第一章 32
Sign Conventions,dimension
正负约定,因次
? It is considered positive (negative)
when it acts in the positive (negative)
direction of the corresponding
coordinate,
与坐标轴正向一致为正。
? Its dimension is [force]/[length]2
因次：力 /长度 2
? for example:
弹性力学 第一章 33
写面力分量
? 3-9 3-10
弹性力学 第一章 34
1,Internal forces,under the action of
external forces,internal forces will be
produced between the parts of a body.
内力：在外力作用下，物体各部分间产生相
互作用的力叫内力
2,Stresses are the internal forces acting on
the per unit area
应力：作用在单位面积上的内力
B,Stress 应力
弹性力学 第一章 35
3,s= lim ?Q/?A （ ?A 0）
(1) s--the stress at point P on the section mPn
s-- 截面 mPn上 P点的应力 。
(2) ?A--an elementary area on the section mPn
around P,
?A-- mPn面上包含 P点的微小面积
(3) ?Q--the internal force acted by part B on part
A across ?A
?Q--B 部分作用在 A部分上的 ?A上的内力。
弹性力学 第一章 36
Stress Fig,应力定义图 。
弹性力学 第一章 37
Stress components 应力分量
? s=XN i+YN j+ZN k=(XN,YN,ZN)
? The projections of s on the x,y,and z axes
are called the stress components at P,
应 力在坐标轴上的投影叫应力分量
? the stress components will be denoted by
XN,YN and ZN
应 力在坐标轴上分量用 XN,YN and ZN 表示
弹性力学 第一章 38
Sign Conventions,dimension
正负约定,因次
? It is considered positive (negative) when
it acts in the positive (negative) direction
of the corresponding coordinate,
XN,YN and ZN 与坐标轴正向一致为正
? Its dimension is [force]/[length]2
因次：力 /长度 2
弹性力学 第一章 39
Another Stress components另一应力分量
? s=?N N+?N T=(?N,? N)
? N is the outward normal to the plane mPn,
T is on the plane mPn,axes N,T and s are
on the same plane,
N 是 mPn 的外法线,T 在 mPn 面上，
N,T 和 s在同一面上
? The projections of s on the N and T axes
are called the stress components at P,
s 在 N 和 T 上投影是 另一应力分量
弹性力学 第一章 40
Normal component and tangential component
法向和切向分量,
? The stress is resolved into a normal component
and a tangential component,
应力分解为法向和切向分量
? The normal component is called the normal
stress,The tangential component is called the
shearing stress,
法向分量叫法向 应力，切向分量 叫剪应力
? the stress components will be denoted by ?N and
?N, Its dimension is [force]/[length]2
应 力分量用 ?N ?N 表示,因次：力 /长度 2
弹性力学 第一章 41
Coordinate plane 坐标面
? Coordinate plane--a plane with the outward
normal parallel to the coordinate axes,
坐标面 --截面的外法线平行于坐标轴的面
? Coordinate plane--x plane,y plane,z plane
坐标面 --x面,y面,z面
? positive(negative) coordinate plane-- a plane with
the outward normal in the positive(negative)
direction of the coordinate axis,
正（负）面 --外法线为坐标轴 正（负）向 的面
弹性力学 第一章 42
Stress Notation 1,坐标面上应力记号 1.
? We associate the stress with two coordinate
subscripts,first coordinate subscript indicates
the coordinate plane the stress is acting,the
second subscript indicates the direction in which
the stress is acting.
? 应力用 ?加二个下标表示，第一个下标表示应力
的作用面,第二个下标表示应力的作用方向 。
弹性力学 第一章 43
Stress Notation 1 坐标面上应力记号 1
x-direction y-direction z-direction
? X plane,?xx ?xy ?xz
? y plane,?yx ?yy ?yz
? z plane,?zx ?zy ?zz
弹性力学 第一章 44
Stress Notation 2,坐标面上应力记号 2.
? Normal stress--? with a coordinate subscript
which indicates the coordinate plane the stress is
acting and the direction in which the stress is
acting.
? 正 应力用 ?加一个下标表示，该下标既表示应力
的作用面，又表示应力的作用方向。
? Shearing stress--? with two coordinate subscripts
? 剪应力 ?用加二个下标表示，第一个下标表示应
力的作用面,第二个下标表示应力的作用方向。
弹性力学 第一章 45
Stress Notation 2 坐标面上应力记号 2
x-direction y-direction z-direction
? X plane,?x ?xy ?xz
? y plane,?yx ?y ?yz
? z plane,?zx ?zy ?z
弹性力学 第一章 46
Sign Conventions 正负约定
? A stress component on positive (negative)
coordinate plane will be considered positive as it
acts in the positive (negative) direction of the
corresponding axis.
正（负）面上沿坐标轴正（负）向的应力为正。 ---
正面正向为正,负面负向为正
? A normal stress is positive (negative) for tension
(compression)
正应力拉为正，压为负
弹性力学 第一章 47
The fig,of stress notation 坐标面上应力记号图
弹性力学 第一章 48
The Equality of Shearing Stresses
剪应力互等
? ?xy=?yx ?xz= ?zx ?yz= ?zy
? The six shearing stresses are mutually
equal in pairs,hence,the subscript letters
of the notation of shearing stress may be
interchanged at will.
? 剪应力互等,当只考虑大小时，下标顺序可随
心所欲。
弹性力学 第一章 49
To precisely define the stress condition
完全确定一点的应力状态
? The stresses on any section through point P can
be evaluated if the stress components ?x ?y ?z ?xy=
?yx ?xz=?zx ?yz=?zy at that point are known.
? Consequently,the six stress components ?x ?y ?z
?xy= ?yx ?xz=?zx ?yz=?zy at any point P precisely
define the stress condition at that point.
? 一点的六个坐标面上的应力分量 ?x ?y ?z ?xy= ?yx
?xz=?zx ?yz=?zy 已知后,可求得经过该点的任意截
面上的应力。 ------称为 一点的六个坐标面上的应
力分量可完全确定该点的应力状态 。
弹性力学 第一章 50
1,体力分量、面力分量、应力分量的正负
号规定？
2,坐标面上应力记号?
弹性力学 第一章 51
C,Deformation 形变
? By deformation we mean the change of
the shape of a body,which may be
expressed by the changes in lengths and
angles of its parts.
? 形变 --物体形状（各部分长度和角度）的
改变
弹性力学 第一章 52
? To study deformation condition at a
certain point P,we consider line segments
PA,PB,PC
研究一点的变形，考虑通过 P点的三个
正向微段 PA,PB,PC
? PA//x PA=dx P A --positive x direction
PB//y PB=dy P B --positive y direction
PC//z PC=dz P C -positive z direction
弹性力学 第一章 53
Fig,PA PB PC 图 PA PB PC
z
C
P B
A
y
x
弹性力学 第一章 54
Normal strain--a change in length per unit length
正应变 --单位长度的长度改变
? ?x--change in length per unit length of PA
? ?y--change in length per unit length of PB
? ?z--change in length per unit length of PC
? positive (negative) for elongation (contraction)
? ?x--x向微段 PA的相对伸缩
? ?y--y向微段 PB的相对伸缩 伸长为正
? ?z-- z向微段 PC的相对伸缩 缩短为负
弹性力学 第一章 55
Shearing strain--the change of a right angle（ radian)
剪应变 --直角的改变量 (弧度 )
? rxy--the change of a right angle APB
? ryz--the change of a right angle BPC
? rzx--the change of a right angle APC
? positive (negative) for a decrease(increase) of
the right angle
? rxy-- 直角 APB的改变量
? ryz-- 直角 BPC的改变量 直角 变小为正
? rzx-- 直角 APC的改变量 直角 变大为负
弹性力学 第一章 56
D,Displacements 位移
? By displacement,we mean the change of position,
位置的移动叫位移
? Displacement components u,v,w---the
projections of the displacement on the x,y and
z axes.位移在坐标轴上投影叫 位移分量 u,v,w
? It is considered positive as it is in the positive
direction of the corresponding coordinate axis,
沿坐标正向的位移分量为正 。
? The dimension is [length],因次为长度
弹性力学 第一章 57
1,研究一点的变形，考虑通过 P点的三个
正向微段 PA,PB,PC，为何要 正向微段?
2,正应变、剪应变的定义和正负号规定？
弹性力学 第一章 58
1.3 Basic assumptions 基本假定
? The body is continuous 物体是连续的
? The body is perfectly elastic,
物体是完全弹性的
? The body is homogeneous,物体是均质的
? The body is isotropic 物体是各向同性的
? the displacements and strains are small,
位移和应变是微小的。
弹性力学 第一章 59
The body is continuous 物体是连续的
? The whole volume of the body is filled with
continuous matter without any void.假定整 个
物体的体积都被组成这个物体的介质所充满，
不留下任何孔隙。
? Under this assumption,the physical quantities
in the body,such as stresses,strains and
displacements,can be expressed by continuous
functions of coordinates in the space.物理量
（例：应力，应变，位移）能用坐表的连续函
数表示 。
弹性力学 第一章 60
The body is perfectly elastic,物体是完全弹性的
? The body wholly obeys Hook’s law of elasticity,
----The relations between the stress components
and the strain components are linear,
物体遵守虎克定律 ---应力分量和应变分量是线
性关系 ---线性本构关系 ------物理线性。
? The elastic constants will be independent of
the stress or strain components under this
assumption.弹性常数与应力和应变的大小无关。
弹性力学 第一章 61
The body is homogeneous,物体是均质的
? The elastic constants will be independent
of the location in the body,
弹性常数与位置无关。
? 物体由同一种材料组成。
? 物体由多种材料组成，但每一种材料的
颗粒远小于物体且在物体内均匀分布 。
弹性力学 第一章 62
The body is isotropic 物体是各向同性的
? The elastic constants will be independent
of the orientation of the coordinate axes,
弹性常数与坐标轴的方向无关。
? Steel structure-------isotropic
钢 --各向同性
? wooden structure----not isotropic
木 --各向异性
弹性力学 第一章 63
The displacements and strains are small,
位移和应变是微小的 ---几何线性。
? The displacement components are very small in
comparison with its original dimensions---We may
use the lengths and angles of the body before
deformation when we formulate the equilibrium
equations.位移远小于 物体尺寸 ---可用变形前的尺寸
代替变形后的尺寸 。
? The strain components and the rotations of all line
elements are much smaller than unity.---we may
neglect the squares and products of the small
quantities,应变分量和转角远小于 1，其乘积及二次
幂可忽略 。
? Linear equations and problems 线性方程,线性问题
弹性力学 第一章 64
作业,1.写图 3-9、图 3-10的面力分量 3-9 3-10
2,体力分量、面力分量、应力分量的正负号规定？
3,何为坐标面？何为正（负）坐标面？试述坐标
面上的应力记号。
4,正应变、剪应变的定义和正负号规定？
5,研究一点的变形，考虑通过 P点的三个正向微
段 PA,PB,PC，为何要 正向微段?
6,设 x=-10为负 x面,该面上沿 x正向作用 100KN/m2
的荷载,沿 y负向作用 200KN/m2的荷载,设该面
为物体表面，写出 该面上面力分量； 设该面原
位于物体内部，写出 该面上应力分量。试述应
力和面力的异同点？
弹性力学及有限元
Elasticity and Finite Element Method
The lectures will be given both in
English and Chinese
采用中英文双语讲授
弹性力学 第一章 2
Give me a fish and I will eat today,
Teach me to fish and I will eat for a
life time.
授人以鱼，不如授人以渔。
弹性力学 第一章 3
Textbook:
Applied Elasticity
徐芝纶
中文教材：
弹性力学简明教程
徐芝纶
弹性力学 第一章 4
Chapter 1,Introduction
第一章 绪论
弹性力学 第一章 5
1.1 Contents of Theory of Elasticity
1.1 弹性力学的内容
? NAME
Theory of elasticity is often called elasticity
for short,It is the branch of solid
mechanics,
弹性力学的理论简称为弹性理论或弹性力学,
它是固体力学的一个分枝
弹性力学 第一章 6
Three branches of solid mechanics
固体力学的三个分 枝
? Mechanics of materials 材料力学,
Structural Mechanics 结构力学
Elasticity 弹性力学
弹性力学 第一章 7
? What does the Elasticity deal with?
It deals with the stresses,deformations and
displacements in elastic solids produced by
external forces or changes in temperature.
研究弹性体 由于外力和温度改变而引起 的应力，
形变和位移。
? It analyzes the stresses,deformations and
displacements of structural elements within the
elastic range and thereby to check the sufficiency
of their strength,stiffness and stability,
分析结构的应力，形变和位移，检查是否满足强
度，刚度和稳定性条件 。
弹性力学 第一章 8
Comparison among the three courses
in solid mechanics
固体力学三门学科的比较
? Three branches have the same purpose and do
differ from one another both in objects studied
and the methods of analysis used.
? 1,Objects studied 研究对象
2,Methods of analysis 研究方法
弹性力学 第一章 9
to deal with the elastic solids
都是研究弹性体
1,objects studied:--研究对象,
(1) Similarity---------相同点
弹性力学 第一章 10
(2)objects studied--difference研究对象 --不同点
Mechanics of materials, bar element
材料力学 单根杆件
Structural bar systems:--
Mechanics, truss,rigid frame
结构力学 杆件系统,桁架, 刚架 。
Elasticity,1,plates and shells 板, 壳
弹性力学 2.blocks,块体
e.g,dams,foundations 坝, 基础
3.analyze bar element precisely
对杆件作精确分析
弹性力学 第一章 11
? Mechanics of materials deals essentially with
the stresses and displacements of structural
element in the shape of a bar,straight or
curved,which is subjected to tension,
compression,shear,bending,or torsion.
材料力学研究受到拉、压、剪、弯或扭的直杆
或曲杆的应力和位移。
弹性力学 第一章 12
? Structural Mechanics deals with the stresses and
displacements of a structural in the form of a bar
system,such as a truss or a rigid frame.
? 结构力学研究杆件系统（例：桁架或刚架）的应力
和位移
弹性力学 第一章 13
? Elasticity deals with the stresses and displacements
of the structural elements such as blocks,plates and
shells,which are not in the form of a bar.
? 弹性力学研究块体、板和壳体的应力和位移。
? Elasticity also analyze a bar element thoroughly and
precisely.
? 弹性力学对杆件作更精确分析
弹性力学 第一章 14
2,methods of analysis:研究方法
(1) Similarity--- 相同点,
equilibrium aspects 静力学方面
geometrical aspects 几何学方面
physical aspects 物理学方面
弹性力学 第一章 15
?equilibrium aspects --equilibrium of forces of an
isolated body
静力学方面 --脱离体力的平衡
? geometrical aspects --the relations between
displacements and strains.
几何学方面 --位移和应变的关系
? physical aspects-- --
the relations between stresses and strains
物理学方面 --应力和应变的关系
弹性力学 第一章 16
(2) methods of analysis:------ difference
研究方法 -------- 不同点,
Mechanics of materials,some assumptions on the
strain condition or the stress condition are made
材料力学,对应变或应力情况作某些假定
Elasticity,no assumptions on the strain condition
or the stress condition are made,
弹性力学, 对应变或应力情况不作假定
弹性力学 第一章 17
Mechanics of materials,
some assumptions on the strain condition or the
stress condition are made
The assumptions simplify the mathematical derivation
to a certain extent.
The assumptions inevitably reduce the degree of
accuracy of the results obtained..
弹性力学 第一章 18
Elasticity：
no assumptions on the strain condition or the stress
condition are made.
The results obtained in elasticity are more accurate
and may be used to check the approximate results
obtained in Mechanics of materials.
弹性力学 第一章 19
?The problem of bending of a straight beam under
transverse loads.
?It is assumed in mechanics of materials that a plane
section of the beam remains plane after bending,which
leads to the linear distribution of bending stresses,
?No assumption,that a plane section of the beam
remains plane after bending,is made in Elasticity.
弹性力学 第一章 20
?A prismatical tension member with a small hole
?It is assumed in mechanics of materials that the tensile
stresses are uniformly distributed across the net section
of the member.
?The analysis in elasticity shows that the stresses are by
no means uniform,but are concentrated near the hole,
弹性力学 第一章 21
1.2 some important concepts in theory of
elasticity
1.2 弹性力学中的几个重要概念
A,External Forces 外力
B,Stress 应力
C,Deformation(Strain) 形变 (应变 )
D,Displacement 位移
弹性力学 第一章 22
A,external forces 外力
1,Body forces 体积力，体力
2,Surface forces 表面力，面力
弹性力学 第一章 23
1,Body forces 体力 。
External forces or the loads,distributed over the
volume of the body,are called body forces.
分布在物体体内的外力叫体力
E.g,gravitational forces,or inertia forces in the case
of a body in motion.
例如, 重力,惯性力
弹性力学 第一章 24
Body force Fig,体力定义图 。
弹性力学 第一章 25
F=lim ?Q/ ?V
? v 0
F--body force vector at p,The vector quantity F is
the intensity of body force at P
F-- P点的体力矢量
?V--an elementary volume of the body around
point p
?V--包含 P点的小体积
?Q--body force acting on ? V
?Q--作用在 ? V上的体力的合力
弹性力学 第一章 26
Body force components 体力分量
? F=X i+Y j+Z k=(X,Y,Z)
? The projections of F on the x,y,and z
axes are called the body force
components at P.
体力在坐标轴上的投影叫体力分量。
? The body force components will be
denoted by X,Y and Z
体力分量用 X,Y,Z表示
弹性力学 第一章 27
Sign Conventions,dimension
正负约定,因次
? It is considered positive (negative) when it
acts in the positive (negative) direction of the
corresponding coordinate,
与坐标轴正向一致为正。
? Its dimension is [force]/[length]3
因次：力 /长度 3
? for example:
弹性力学 第一章 28
2,Surface forces 面力
? Definition,external forces,or the loads,
distributed over the surface of a body,
are called surface forces.
分布在物体表面的外力叫面力
? e.g,hydrostatic pressure,the pressure
of one body on another
例：水压力，接触力
弹性力学 第一章 29
Surface force Fig,面力定义图 。
弹性力学 第一章 30
? F=lim ? Q/ ? S
? F---surface force vector at P,the vector quantity
is the intensity of surface force at P,
F--- P点的面力矢量 。
? ? S--an elementary area of the surface around
point P,? S --包含 P点的小面积
? ? Q--the surface force acting on ? S
作用在 ?S上的面力矢量
弹性力学 第一章 31
Surface force components 面 力分量
? F=X i+Y j+Z k=(X,Y,Z)
? Definition,the projections of F on the x,y and z
axes are called the surface force components at
P,
面 力在坐标轴上的投影叫面力分量
? The surface force components will be denoted
by X,Y and Z
面 力分量用 X,Y,Z表示
弹性力学 第一章 32
Sign Conventions,dimension
正负约定,因次
? It is considered positive (negative)
when it acts in the positive (negative)
direction of the corresponding
coordinate,
与坐标轴正向一致为正。
? Its dimension is [force]/[length]2
因次：力 /长度 2
? for example:
弹性力学 第一章 33
写面力分量
? 3-9 3-10
弹性力学 第一章 34
1,Internal forces,under the action of
external forces,internal forces will be
produced between the parts of a body.
内力：在外力作用下，物体各部分间产生相
互作用的力叫内力
2,Stresses are the internal forces acting on
the per unit area
应力：作用在单位面积上的内力
B,Stress 应力
弹性力学 第一章 35
3,s= lim ?Q/?A （ ?A 0）
(1) s--the stress at point P on the section mPn
s-- 截面 mPn上 P点的应力 。
(2) ?A--an elementary area on the section mPn
around P,
?A-- mPn面上包含 P点的微小面积
(3) ?Q--the internal force acted by part B on part
A across ?A
?Q--B 部分作用在 A部分上的 ?A上的内力。
弹性力学 第一章 36
Stress Fig,应力定义图 。
弹性力学 第一章 37
Stress components 应力分量
? s=XN i+YN j+ZN k=(XN,YN,ZN)
? The projections of s on the x,y,and z axes
are called the stress components at P,
应 力在坐标轴上的投影叫应力分量
? the stress components will be denoted by
XN,YN and ZN
应 力在坐标轴上分量用 XN,YN and ZN 表示
弹性力学 第一章 38
Sign Conventions,dimension
正负约定,因次
? It is considered positive (negative) when
it acts in the positive (negative) direction
of the corresponding coordinate,
XN,YN and ZN 与坐标轴正向一致为正
? Its dimension is [force]/[length]2
因次：力 /长度 2
弹性力学 第一章 39
Another Stress components另一应力分量
? s=?N N+?N T=(?N,? N)
? N is the outward normal to the plane mPn,
T is on the plane mPn,axes N,T and s are
on the same plane,
N 是 mPn 的外法线,T 在 mPn 面上，
N,T 和 s在同一面上
? The projections of s on the N and T axes
are called the stress components at P,
s 在 N 和 T 上投影是 另一应力分量
弹性力学 第一章 40
Normal component and tangential component
法向和切向分量,
? The stress is resolved into a normal component
and a tangential component,
应力分解为法向和切向分量
? The normal component is called the normal
stress,The tangential component is called the
shearing stress,
法向分量叫法向 应力，切向分量 叫剪应力
? the stress components will be denoted by ?N and
?N, Its dimension is [force]/[length]2
应 力分量用 ?N ?N 表示,因次：力 /长度 2
弹性力学 第一章 41
Coordinate plane 坐标面
? Coordinate plane--a plane with the outward
normal parallel to the coordinate axes,
坐标面 --截面的外法线平行于坐标轴的面
? Coordinate plane--x plane,y plane,z plane
坐标面 --x面,y面,z面
? positive(negative) coordinate plane-- a plane with
the outward normal in the positive(negative)
direction of the coordinate axis,
正（负）面 --外法线为坐标轴 正（负）向 的面
弹性力学 第一章 42
Stress Notation 1,坐标面上应力记号 1.
? We associate the stress with two coordinate
subscripts,first coordinate subscript indicates
the coordinate plane the stress is acting,the
second subscript indicates the direction in which
the stress is acting.
? 应力用 ?加二个下标表示，第一个下标表示应力
的作用面,第二个下标表示应力的作用方向 。
弹性力学 第一章 43
Stress Notation 1 坐标面上应力记号 1
x-direction y-direction z-direction
? X plane,?xx ?xy ?xz
? y plane,?yx ?yy ?yz
? z plane,?zx ?zy ?zz
弹性力学 第一章 44
Stress Notation 2,坐标面上应力记号 2.
? Normal stress--? with a coordinate subscript
which indicates the coordinate plane the stress is
acting and the direction in which the stress is
acting.
? 正 应力用 ?加一个下标表示，该下标既表示应力
的作用面，又表示应力的作用方向。
? Shearing stress--? with two coordinate subscripts
? 剪应力 ?用加二个下标表示，第一个下标表示应
力的作用面,第二个下标表示应力的作用方向。
弹性力学 第一章 45
Stress Notation 2 坐标面上应力记号 2
x-direction y-direction z-direction
? X plane,?x ?xy ?xz
? y plane,?yx ?y ?yz
? z plane,?zx ?zy ?z
弹性力学 第一章 46
Sign Conventions 正负约定
? A stress component on positive (negative)
coordinate plane will be considered positive as it
acts in the positive (negative) direction of the
corresponding axis.
正（负）面上沿坐标轴正（负）向的应力为正。 ---
正面正向为正,负面负向为正
? A normal stress is positive (negative) for tension
(compression)
正应力拉为正，压为负
弹性力学 第一章 47
The fig,of stress notation 坐标面上应力记号图
弹性力学 第一章 48
The Equality of Shearing Stresses
剪应力互等
? ?xy=?yx ?xz= ?zx ?yz= ?zy
? The six shearing stresses are mutually
equal in pairs,hence,the subscript letters
of the notation of shearing stress may be
interchanged at will.
? 剪应力互等,当只考虑大小时，下标顺序可随
心所欲。
弹性力学 第一章 49
To precisely define the stress condition
完全确定一点的应力状态
? The stresses on any section through point P can
be evaluated if the stress components ?x ?y ?z ?xy=
?yx ?xz=?zx ?yz=?zy at that point are known.
? Consequently,the six stress components ?x ?y ?z
?xy= ?yx ?xz=?zx ?yz=?zy at any point P precisely
define the stress condition at that point.
? 一点的六个坐标面上的应力分量 ?x ?y ?z ?xy= ?yx
?xz=?zx ?yz=?zy 已知后,可求得经过该点的任意截
面上的应力。 ------称为 一点的六个坐标面上的应
力分量可完全确定该点的应力状态 。
弹性力学 第一章 50
1,体力分量、面力分量、应力分量的正负
号规定？
2,坐标面上应力记号?
弹性力学 第一章 51
C,Deformation 形变
? By deformation we mean the change of
the shape of a body,which may be
expressed by the changes in lengths and
angles of its parts.
? 形变 --物体形状（各部分长度和角度）的
改变
弹性力学 第一章 52
? To study deformation condition at a
certain point P,we consider line segments
PA,PB,PC
研究一点的变形，考虑通过 P点的三个
正向微段 PA,PB,PC
? PA//x PA=dx P A --positive x direction
PB//y PB=dy P B --positive y direction
PC//z PC=dz P C -positive z direction
弹性力学 第一章 53
Fig,PA PB PC 图 PA PB PC
z
C
P B
A
y
x
弹性力学 第一章 54
Normal strain--a change in length per unit length
正应变 --单位长度的长度改变
? ?x--change in length per unit length of PA
? ?y--change in length per unit length of PB
? ?z--change in length per unit length of PC
? positive (negative) for elongation (contraction)
? ?x--x向微段 PA的相对伸缩
? ?y--y向微段 PB的相对伸缩 伸长为正
? ?z-- z向微段 PC的相对伸缩 缩短为负
弹性力学 第一章 55
Shearing strain--the change of a right angle（ radian)
剪应变 --直角的改变量 (弧度 )
? rxy--the change of a right angle APB
? ryz--the change of a right angle BPC
? rzx--the change of a right angle APC
? positive (negative) for a decrease(increase) of
the right angle
? rxy-- 直角 APB的改变量
? ryz-- 直角 BPC的改变量 直角 变小为正
? rzx-- 直角 APC的改变量 直角 变大为负
弹性力学 第一章 56
D,Displacements 位移
? By displacement,we mean the change of position,
位置的移动叫位移
? Displacement components u,v,w---the
projections of the displacement on the x,y and
z axes.位移在坐标轴上投影叫 位移分量 u,v,w
? It is considered positive as it is in the positive
direction of the corresponding coordinate axis,
沿坐标正向的位移分量为正 。
? The dimension is [length],因次为长度
弹性力学 第一章 57
1,研究一点的变形，考虑通过 P点的三个
正向微段 PA,PB,PC，为何要 正向微段?
2,正应变、剪应变的定义和正负号规定？
弹性力学 第一章 58
1.3 Basic assumptions 基本假定
? The body is continuous 物体是连续的
? The body is perfectly elastic,
物体是完全弹性的
? The body is homogeneous,物体是均质的
? The body is isotropic 物体是各向同性的
? the displacements and strains are small,
位移和应变是微小的。
弹性力学 第一章 59
The body is continuous 物体是连续的
? The whole volume of the body is filled with
continuous matter without any void.假定整 个
物体的体积都被组成这个物体的介质所充满，
不留下任何孔隙。
? Under this assumption,the physical quantities
in the body,such as stresses,strains and
displacements,can be expressed by continuous
functions of coordinates in the space.物理量
（例：应力，应变，位移）能用坐表的连续函
数表示 。
弹性力学 第一章 60
The body is perfectly elastic,物体是完全弹性的
? The body wholly obeys Hook’s law of elasticity,
----The relations between the stress components
and the strain components are linear,
物体遵守虎克定律 ---应力分量和应变分量是线
性关系 ---线性本构关系 ------物理线性。
? The elastic constants will be independent of
the stress or strain components under this
assumption.弹性常数与应力和应变的大小无关。
弹性力学 第一章 61
The body is homogeneous,物体是均质的
? The elastic constants will be independent
of the location in the body,
弹性常数与位置无关。
? 物体由同一种材料组成。
? 物体由多种材料组成，但每一种材料的
颗粒远小于物体且在物体内均匀分布 。
弹性力学 第一章 62
The body is isotropic 物体是各向同性的
? The elastic constants will be independent
of the orientation of the coordinate axes,
弹性常数与坐标轴的方向无关。
? Steel structure-------isotropic
钢 --各向同性
? wooden structure----not isotropic
木 --各向异性
弹性力学 第一章 63
The displacements and strains are small,
位移和应变是微小的 ---几何线性。
? The displacement components are very small in
comparison with its original dimensions---We may
use the lengths and angles of the body before
deformation when we formulate the equilibrium
equations.位移远小于 物体尺寸 ---可用变形前的尺寸
代替变形后的尺寸 。
? The strain components and the rotations of all line
elements are much smaller than unity.---we may
neglect the squares and products of the small
quantities,应变分量和转角远小于 1，其乘积及二次
幂可忽略 。
? Linear equations and problems 线性方程,线性问题
弹性力学 第一章 64
作业,1.写图 3-9、图 3-10的面力分量 3-9 3-10
2,体力分量、面力分量、应力分量的正负号规定？
3,何为坐标面？何为正（负）坐标面？试述坐标
面上的应力记号。
4,正应变、剪应变的定义和正负号规定？
5,研究一点的变形，考虑通过 P点的三个正向微
段 PA,PB,PC，为何要 正向微段?
6,设 x=-10为负 x面,该面上沿 x正向作用 100KN/m2
的荷载,沿 y负向作用 200KN/m2的荷载,设该面
为物体表面，写出 该面上面力分量； 设该面原
位于物体内部，写出 该面上应力分量。试述应
力和面力的异同点？
