材料力学(双语)Bending弯曲
材料力学(双语)弯曲变形
1 3 EIw(0) = Pa + C2 = 0 6 1 2 EIθ (0) = − Pa + C1 = 0 2 θ (a− ) = θ (a+ ) ∴C1 = D1
a L f
P x
w(a ) = w(a )
−
+
∴C1a + C2 = D1a + D2
1 2 1 3 ∴ C1 = D1 = Pa ; C 2 = D2 = − Pa 2 6
EIw ′′( x ) = − M ( x )
7
2. Determine the equation of the deflection curve (elastic curve) 1). Integration of the differential equation
EIw′′( x) = − M ( x)
4
1. Two basic displacements（两个基本位移量）
1). Deflection(挠度)： The displacement of the centroid of a section in a direction perpendicular to the axis of the beam. It is designated by w . It is positive if its direction is the same as y, otherwise it is negative. 2). Angle of rotation (转角)：
EIw′( x) = ∫ (− M ( x))dx + C1
EIw( x) = ∫ ( ∫ (− M ( x))dx)dx + C1 x + C2
材料力学-第四章弯曲应力教学
FS
x
dx
0
FS
x
dM x
dx
qx
dM 2x
dx 2
注:q(x)向上为正,反之为负。
●简易法作剪力图和弯矩图
①梁上无分布荷载作用：q（x）=0
qx dFS x 0
dx
FS x cont
剪力图斜率为零，FS（x）图为平行于x轴的直线。
dM x
B 1kN
A FAx
FB
FAy
FAx=-3kN FAy=3kN
FB=5kN
2）剪力图: 简易法 BC杆：取一点（水平线） DC杆：取两点（水平线） DA杆：取两点（斜直线）
D 3kN
C
1kN E
5kN
1kN B
3kN A
q=1kN/m 4m 3m
8kN
1m D
2m C
E
B 1kN
A FAx
A
A
ydA Sz 0 中性轴z必通过截面形心
A
横截面对z轴的静矩
My
z dA 0
A
zE
A
y dA
E
A
zydA
0
zydA I yz 0
A
截面对yz轴的惯性积
*由于y为对称轴, 上式自然满足。
M z
y dA
A
M
例5.作外伸梁的内力图
q
FA
ql 8
A
FB
5ql 8
FA
FS
B
lC
l
FB 2
ql / 2
材料力学(英汉对照)06_Bending_Deformation
REVIEW REVIEWBending shear stressQS τ = bI* z zQ zτy A*τ max = kτ m = 1.5Q AStrength condition of shear stressτ maxQ = k ≤ [τ ] APart one: Basic DeformationsChapter 6. Bending DeformationContaining Sections: Containing Sections:6.1 Introduction 6.1 Introduction 6.2 Equation of Elastic Curve 6.2 Equation of Elastic Curve 6.3 Direct Integration Method 6.3 Direct Integration Method 6.4 Method of Superposition 6.4 Method of Superposition 6.5* Moment-Area Method 6.5* Moment-Area Method 6.6 Stiffness Condition of Beams and Ways to Improve the Stiffness of 6.6 Stiffness Condition of Beams and Ways to Improve the Stiffness of Beams Beams 6.7 Statically Indeterminate Beams 6.7 Statically Indeterminate BeamsChapter 6. Bending Deformation6.1 IntroductionTensional deformation Torsional deformationNL ∆l = EAM nL ϕ= GI pϕxBending deformationdx vDeflection Deflection 截面形心的竖向位移 挠度 vv 挠度θElastic CurveSlope Slope 转角 θ 转角 θ截面绕中性轴的转角Chapter 6. Bending Deformation6.2 Equation of Elastic CurveWhen deduce the bending normal stress formula, we get : Consider the small deformation : From mathematical relation :1 M (x) = ρ (x) EI z1 ρ (x)Curvature 曲率 Flexural rigidity 抗弯刚度EI zdv θ ≈ tan θ = = v' ( x) dx1ρ=v" ≈ v" 2 3 2 (1 + v ′ )Deduce further, we can get the approximate differential equation of elastic curve (挠曲线近似微分方程):M (x) v ′′ ( x ) = θ ′ = E IZ公式的适用条件： 1) 材料服从虎克定律； 2) 小变形，忽略剪力对挠度的影响。
材料力学第五章弯曲内力
CA和DB段：q=0，Q图为水平线， M图为斜直线。
AD段：q<0， Q图为向下斜直线， M图为上凸抛物线。
3、先确定各分段点的Q 、M 值，用相应形状的线条连接。
32
§5-6 纯弯曲时的正应力
• 纯弯曲(Pure Bending):某段梁的 内力只有弯矩没有剪力时，该段 梁的变形称为纯弯曲。
如图（b）示。
qL A
x1Q1
图（a） M1
图（b）
Y qL Q1 0 Q1 qL
mA(Fi) qLx1 M1 0 M1 qLx1
17
2--2截面处截取的分离体如图（c） qL
Y qL Q2 q(x2 a) 0 Q2 qx2 a qL
剪力等于梁保留一侧横向外
②写出内力方程
Q(x)
P
Q( x ) YO P
M(x) PL
x
M( x ) YOx MO
P( x L ) x
③根据方程画内力图
20
F
a
b
A
C
x1 x2
FAY
l
FS Fb / l
Fa / l
Fab/ l
M
[例]图示简支梁C点受集中力作用。
试写出剪力和弯矩方程，并画 B 出剪力图和弯矩图。
4. 标值、单位、正负号、纵标线
31
例 外伸梁AB承受荷载如图所示，作该梁的Q---M图。
3kN
6kN m 2kN/m
A C
B D
1m
4m
FA
Q 4.2
(kN) +
E
_
3
x=3.1m
1m
FB
_
3.8
(完整版)材料力学中英对照词汇
材料力学中英对照词汇A安全因数safety factorB半桥接法half bridge闭口薄壁杆thin-walled tubes比例极限proportional limit边界条件boundary conditions变截面梁beam of variable cross section 变形deformation变形协调方程compatibility equation 标距gage length泊松比Poisson’s ratio补偿块compensating blockC材料力学mechanics of materials冲击荷载impact load初应力，预应力initial stress纯剪切pure shear纯弯曲pure bending脆性材料brittle materialsD大柔度杆long columns单位荷载unit load单位力偶unit couple单位荷载法unit-load method单向应力，单向受力uniaxial stress等强度梁beam of constant strength低周疲劳low-cycle fatigue电桥平衡bridge balancing电阻应变计resistance strain gage电阻应变仪resistance strain indicator 叠加法superposition method叠加原理superposition principle 动荷载dynamic load断面收缩率percentage reduction in area多余约束redundant restraintE二向应力状态state of biaxial stressF分布力distributed force复杂应力状态state of triaxial stress复合材料composite materialG杆，杆件bar刚度stiffness刚架，构架frame刚结点rigid joint高周疲劳high-cycle fatigue各向同性材料isotropical material功的互等定理reciprocal-work theorem工作应变计active strain gage工作应力working stress构件structural member惯性半径radius of gyration of an area惯性积product of inertia惯性矩，截面二次轴距moment of inertia广义胡克定律generalized Hook’s lawH横向变形lateral deformation胡克定律Hook’s law滑移线slip-linesJ基本系统primary system畸变能理论distortion energy theory畸变能密度distortional strain energydensity极惯性矩，截面二次极矩polar moment of inertia极限应力ultimate stress极限荷载limit load挤压应力bearing stress剪力shear force剪力方程equation of shear force剪力图shear force diagram剪流shear flow剪切胡克定律Hook’s law for shear剪切shear交变应力，循环应力cyclic stress截面法method of sections截面几何性质geometrical properties ofan area截面核心core of section静不定次，超静定次数degree of astatically indeterminate problem静不定问题，超静定问题staticallyindeterminate problem静定问题statically determinate problem静荷载static load静矩，一次矩static moment颈缩neckingK开口薄壁杆bar of thin-walled open crosssection抗拉强度ultimate stress in tension抗扭截面系数section modulus in torsion抗扭强度ultimate stress in torsion抗弯截面系数section modulus inbendingL拉压刚度axial rigidity拉压杆，轴向承载杆axially loaded bar理想弹塑性假设elastic-perfectly plasticassumption力法force method力学性能mechanical properties连续梁continuous beam连续条件continuity condition梁beams临界应力critical stress临界荷载critical loadM迈因纳定律Miner’s law名义屈服强度offset yielding stress莫尔强度理论Mohr theory of failure敏感栅sensitive gridN挠度deflection挠曲轴deflection curve挠曲轴方程equation of deflection curve挠曲轴近似微分方程approximatelydifferential equation of the deflectioncurve内力internal forces扭力矩twisting moment扭矩torsional moment扭矩图torque diagram扭转torsion扭转极限应力ultimate stress in torsion扭转角angel of twist扭转屈服强度yielding stress in torsion扭转刚度torsional rigidityO欧拉公式Euler’s formulaP疲劳极限，条件疲劳极限endurancelimit疲劳破坏fatigue rupture疲劳寿命fatigue life偏心拉伸eccentric tension偏心压缩eccentric compression平均应力average stress平面弯曲plane bending平面应力状态state of plane stress平行移轴定理parallel axis theorem平面假设plane cross-section assumptionQ强度strength强度理论theory of strength强度条件strength condition切变模量shear modulus切应变shear strain切应力shear stress切应力互等定理theorem of conjugate shearing stress屈服yield屈服强度yield strength全桥接线法full bridgeR热应力thermal stressS三向应力状态state of triaxial stress三轴直角应变花three-element rectangular rosette三轴等角应变花three-element delta rosette失稳buckling伸长率elongation圣维南原理Saint-Venant’s principle实验应力分析experimental stress analysis塑性变形，残余变形plastic deformationductile materials塑性材料，延性材料塑性铰plastic hingeT弹簧常量spring constant弹性变形elastic deformation弹性模量modulus of elasticity体积力body force体积改变能密度density of energy ofvolume change体应变volume strainW弯矩bending moment弯矩方程equation of bending moment弯矩图bending moment diagram弯曲bending弯曲刚度flexural rigidity弯曲正应力normal stress in bending弯曲切应力shear stress in bending弯曲中心shear center位移法displacement method位移互等定理reciprocal-displacementtheorem稳定条件stability condition稳定性stability稳定安全因数safety factor for stabilityX细长比，柔度slenderness ratio线性弹性体linear elastic body约束扭转constraint torsion相当长度，有效长度equivalent length相当应力equivalent stress小柔度杆short columns形心轴centroidal axis形状系数shape factor许用应力allowable stress许用应力法allowable stress method许用荷载allowable load许用荷载法allowable load methodY应变花strain rosette应变计strain gage应变能strain energy应变能密度strain energy density应力stress应力速率stress ratio应力比stress ratio应力幅stress amplitude应力状态state of stress应力集中stress concentration应力集中因数stress concentration factor应力－寿命曲线，S-N曲线stress-cyclecurve应力－应变图stress-strain diagram应力圆，莫尔圆Mohr’s circle for stressesZ正应变normal strain正应力normal stress中面middle plane中柔度杆intermediate columns中性层neutral surface中性轴neutral axis轴shaft轴力axial force轴力图axial force diagram轴向变形axial deformation轴向拉伸axial tension轴向压缩axial compression主平面principal planes主应力principal stress主应力迹线principal stress trajectory主轴principal axis主惯性矩principal moment of inertia主形心惯性矩principal centroidalmoments of inertia主形心轴principal centroidal axis转角angel of rotation转轴公式transformation equation自由扭转free torsion组合变形combined deformation组合截面composite area最大切应力理论maximum shear stresstheory最大拉应变理论maximum tensile straintheory最大拉应力理论maximum tensile stresstheory最大应力maximum stress最小应力minimum stress。
材料力学第八章-弯曲变形
L
A
L
解：建立静定基 确定超静定次数 用反力代替多余约束 得新结构 —— 静定基
或
q0
A
B L RB
32
q0 A L B RB
几何方程——变形协调方程
f B f Bq f BRB 0
物理方程
=
A B RB q0 A B
qL RB L f Bq ; f BRB 8EI 3EI
A A 铰连接
P
C D
C
D
B
A点：f A 0, A 0
B点： f B左 f B右
C点： f C左 f C右 C左 C右
D点：f D 0
21
边界条件、连续条件应用举例
P
弯矩图分二段，
共积分常数 需4个边界条件 和连续条件
A B
C
(+)
A点： A 0 B点： f B左 f B右 , C点：f C 0
解：载荷分解如图
=
P A B
查梁的简单载荷变形表，
得到变形
Pa PA 4 EI
q B
2
Pa f PC 6 EI
3
+
A
qa qA 3EI
3
5qL f qC 24 EI24
4
P
A
C a a
q B
Pa PA 4 EI
qa 3 qA 3EI
2
Pa 3 f PC 6 EI
Differential Equation of beam deformation 1 M ( x) 已知曲率为 EI z x
M>0
材料力学第6章 弯曲内力
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6.1 梁的内力—剪力和弯矩
例题 6-2
（2）计算(jìsuàn)指定截面上的剪力和 弯矩
C截截面面C左（以侧梁的左力半：边为研究对象）：
FAy 2 kN () (+)
FSC Fy FAy 2kN
C截面左侧的力矩：
FAy * 2m (+)
M e 8kN m (-)
M C
M F 2m - M -4kN m O
19
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6.2 剪力图和弯矩图
例题 6-3
(2) 作剪力图(lìtú)和弯矩图
由剪力、弯矩方程画剪力、弯矩图。
注意： 画图时应将剪力图、弯矩图与计算简图 对齐，并注明图名（FS图、M图）、 峰值点的值及正负号。
秦飞 编著《材料力学》 第6章 弯曲(wānqū)内
20
力
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6.2 剪力图和弯矩图
(plane bending)。当所有外力均作用在纵向对称面内时，梁只发生平面弯曲。
秦飞 编著《材料力学》 第6章 弯曲(wānqū)内力
6
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6.1 梁的内力(nèilì)—剪力和弯 矩
梁在外力作用下，其任一横截面上的内力可用截面法确定。
（1）截：在横截面m-m处假想地将梁分为两段
原来处于平衡状态的梁，被截出的任意段也处于平衡状态。
秦飞A编y 著《材料力学(cái lieào lìxué)》 第6章 弯
16
曲内力
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6.1 梁的内力(nèilì)—剪力和弯矩 例题 6-2
截面B（以梁右半边为研究对象）：
B左截面
F 2kN (+)
FBy 4kN (-)
FSB左 F FBy -2kN
材料力学第12章弯曲补充
A A
例 试画出下列各薄壁截面弯曲中心的大致位置; 若剪力FS的方 试画出下列各薄壁截面弯曲中心的大致位置; 若剪力F 向垂直向下,试画出切应力流的方向. 向垂直向下,试画出切应力流的方向. A A A A A
A
A
A
A
A
弯曲剪应力组成的合力向截面形心简化
FT FT MC 2 FQ FT MC 1 FQ FT FQ MC
z
y
剪应力流的概念
对于薄壁截面，剪应力方向必须平行于截面 周边的切线方向，形成剪应力流。
通过考察微段 的局部平衡确定切 应力流的方向
τ=
Fsy S I zδ
* z
第8~第9页
剪应力流的概念
将与剪应力相对应的分布力系向横截面所在 平面内不同点简化， 平面内不同点简化，将得到不同的结果。
可以只是一个力——在这种情形下 可以只是一个力——在这种情形下，将只产 在这种情形下， 生弯曲，而不发生扭转； 生弯曲，而不发生扭转； 也可以是一个力和一个力偶——这时不仅产 也可以是一个力和一个力偶——这时不仅产 生弯曲，而且会发生扭转。 生弯曲，而且会发生扭转。 剪应力流及其合力
对称轴 y 梁的轴线 梁在水平纵向对称面 xz 平面内 弯曲，y 轴为中性轴。 弯曲， 轴为中性轴。
图示为一根没有纵向对称面的梁。 图示为一根没有纵向对称面的梁。为了表示 横截面上点的位置， 横截面上点的位置，先确定横截面的形心 C， ， 通过形心的杆轴为 x 轴，外力作用线所在的 在载荷作用下， 平面为 x y 面。在载荷作用下，纯弯曲变形 时的横截面的中性轴为 z 轴.
f =
f y2 + f z2 = 3.882 + 17.12 = 17.5mm
f z 17.1 = = 4.4 tgβ = f y 3.88
材料力学中英对照词汇
拉压刚度 axial rigidity 拉压杆，轴向承载杆 axially loaded bar
理想弹塑性假设 elastic-perfectly plastic assumption 力法 force method 力学性能 mechanical properties 连续梁 continuous beam 连续条件 continuity condition 梁 beams 临界应力 critical stress 临界荷载 critical load
泊松比 Poisson’s ratio
ing 补偿块 compensating block e C b 材料力学 mechanics of materials ir 冲击荷载 impact load e 初应力，预应力 initial stress th 纯剪切 pure shear in 纯弯曲 pure bending s 脆性材料 brittle materials g D in 大柔度杆 long columns th 单位荷载 unit load ll 单位力偶 unit couple A 单位荷载法 unit-load method d 单向应力，单向受力 uniaxial stress n 等强度梁 beam of constant strength a 低周疲劳 low-cycle fatigue e 电桥平衡 bridge balancing tim 电阻应变计 resistance strain gage
平面弯曲 plane bending
d 平面应力状态 state of plane stress
oo 平行移轴定理 parallel axis theorem
g 平面假设
re plane cross-section assumption
材料力学专业词汇中英对照
材料力学Mechanics of Materials构件Structure member强度strength刚度rigidity稳定性stability变形固体solid deformation body基本假设fundamental assumption各向同性Isotropy内力internal force外力external force二力杆two force bar刚体rigid body轴向拉伸Axial tension轴向压缩Axial compression拉力tensile force内力Internal force轴力axial force轴力图diagram of axial force集中载荷concentrate load应力stress平均应力Average stress全应力whole stress平面假设Hypothesis of plane section拉应力Tensile stress应力集中stress concentration剪切shearing正应力normal stress剪应力shearing Stress线应变linear strain剪应变angular strain剪力shearing force弹性应变能Elastic strain energy静定问题statically determinate problem 静不定问题statically indeterminate problem 平衡方程equilibrium equations比例极限Proportional limit弹性极限Elastic limit屈服极限Yielding limit塑性材料ductile material 强度极限Strength limit铸铁cast iron轴Shaft扭转Torsion扭转角The angle of twist扭矩Internal torque扭矩图Internal torque diagram剪应力互等定Theorem of complementary shearingstresses弯曲bending梁Beam固定铰支座Fixed hinged support可动铰支座Movable hinged support固定端Rigidly fixed end简支梁simply supported beam悬臂梁Cantilever beam外伸梁Overhanging beam静定梁Statically determinate beam超静定梁Statically indeterminate beam弯曲内力Internal force in bending弯矩Bending moment内力方程Internal force equation剪力图shearing force diagram弯矩图bending moment diagram分布荷载the distributed load荷载集度the density of the distributed load平面刚架Planar rigid frame曲杆Curved rod纯弯曲Pure Bending中性轴Neutral axis曲率半径the radius of curvature挠度Deflection转角Angle of rotation挠曲线deflection curve位移边界条Boundary conditions of thedisplacement支点位移条件Displacement conditions at thesupports连续条件Continuity conditions 光滑条件Sliding conditions积分常数integral constant 一点的应力状态Stressed state at a point原始单元体Original element 主单元体Principal element主面principal plane 主应力principal stress三向应力状态Three Dimensional State of Stress二向应力状态Plane State of Stress 应力圆Stress Circle单向应力状态Unidirectional State of Stress 主应力迹线Principal stress trajectories应变能密度strain-energy density 强度理论Theories of strength最大拉应力（第一强度）理论：Theory of maximum tensile stress（first strength）最大伸长线应变（第二强度）理论：Theory of maximum tensile strain（second strength）最大剪应力（第三强度）理论Theory of maximum shearing stress(third strength)形状改变比能（第四强度）理论Theory of maximum torsional (shearing)strain energy (fourth strength）莫尔强度理论Mohr’s strength theory相当应力Equivalent stress组合变形Composite deformation 斜弯曲Skew bending中性轴方程Equation of the neutral axis 不稳定平衡instable equilibrium拉(压)弯组合变形Bending and tension or compression稳定平衡Stable equilibrium 压杆失稳Loss of stability of the column临界压力the critical pressure of the column 理想压杆Ideal compressive columns长度系数Length coefficient 约束系数constraint coefficient临界应力Critical stress 柔度Flexibility应变能Strain energy 卡氏定理Catigliano’S Theorem莫尔定理(单位力法) Mohr’S theorem(Method of unit force)。
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Chapter 4 Internal Forces in Bending
MECHANICS OF MATERIALS 材料力学（双语）
Content
§4.1 Concept of symmetrical bending and calculation sketch of the beam §4.2 The shearing force and bending moment of the beam §4.3 The shearing-force and bending-moment equations · the shearing-force and bending-moment diagrams §4.4 Relations among the shearing force、the bending moment and the density of the distributed load and their applications §4.5 Plot the bending-moment diagram by the theorem of superposition §4.6 The internal-force diagrams of the planar rigid frames and curved rods
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目录
§4.1 对称弯曲的概念及梁的计算简图 §4.2 梁的剪力和弯矩 §4.3 剪力方程和弯矩方程 ·剪力图和弯矩图 §4.4 剪力、弯矩与分布荷载集度间的关系及应用 §4.5 按叠加原理作弯矩图 §4.6 平面刚架和曲杆的内力图
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4.1 Concept of symmetrical bending and calculation sketch of the beam
1. Concepts of Bending 1). BENDING(弯曲): The action of the external force or the external couple vector perpendicular to the axis of the rod makes the axis of the rod change into curve from original straight lines, this deformation is called bending. 2). BEAM（梁): The member of whose deformation is mainly bending is generally called beam.
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3). Practical example in engineering about bending
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4). Symmetric bending(平面弯曲) After deformation the curved axis of the beam is still in the same plane with the external forces. P1 q P2
M
The plane of symmetry
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Unsymmetrical bending： If a beam does not possess any plane of symmetry, or the external forces do not act in a plane of symmetry of the beam with symmetric planes, this kind of bending is called unsymmetrical bending.
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2. Calculation sketch of the beam
In general supports and external forces of the beam are very complex. We should do some necessary simplification for them for our convenient calculation and obtain the calculation sketch.
1). Simplification(简化) of the beams
In general case we take the place of the beam by its axis.
2). Simplification of the loads (荷载)
The loads (including the reaction) acting on the beam may be reduced into three types：concentrated force、concentrated force couple and distributed force.
3). Simplification of the supports (支座条件)
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①Fixed hinged support
固定铰支座
2 constraints，1 degree of freedom. Such as the fixed hinged support under bridges.
②Movable hinged support
可动铰支座
1 constraint，2 degree of freedom. Such as the movable hinged support under the bridge.
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θ
A B
D
B R R
A A A
B B B
0.50.50.5P
0.5P
P
0.50.50.5P
0.5P
P 0。