工程力学英文版课件10 Shear Stresses and Strains,Torsion
工程力学双语课件ch13-Shear Force and Bending Moments in Beams
perpendicular to the section when the beam is bending.
2). Shearing force：Q
Internal force which the acting line in the cross-section parallel to the section, when the beam is bending.
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. In later chapters we will mainly discuss the bending stresses and
Uniformly distributed force
P—
Concentrated force
5). Statically determinate and statically indeterminate beams
Statically determinate beams：Reactions of the beam can be determined only by static equilibrium equations，such as the above three kinds of basic beams. Statically indeterminate beams： Reactions of the beam cannot be determined or only part of reactions can be determined by static equilibrium equations.
工程力学全英文Engineering Mechanics (30)
12
The Procedure of Analysis
to determine the relative displacement between two points A and B on axially loaded member 1. Internal Force
The method of sections; How to determine the location of the section?
Part II： Mechanics of Materials
Axial load
AXIAL LOAD
Average normal stress in axial loaded member
• The localization of the stress distribution
• The deformation of the axial loaded member • Analysis of indeterminate axial load member • Analysis of thermal stress and stress concentration • The stress on the inclined surface of an axial loaded member
If the force varies continuously along the member’s length, arbitrary location and P(x); If several constant external forces, each segment between any two external forces
AB 0
Since the end supports are fixed. Thus relative displacement of one end of the bar with respect to the other end to be equal to zero
机械类专业英语ppt课件
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❖ cumulative ['kjuːmjʊlətɪv]
❖ adj. 累积的
❖ cumulative error 累积误差 ; 积累误差 ; 累积差误 ❖ cumulative frequencies 累积频数 ; 累计频数 ❖ cumulative learning 累积学习
❖ dimensionless
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❖ uniaxial ❖ adj. 单轴（向）的，同轴的
念 ; 压力集中 ❖ axial stress 轴向应力 ; 轴应力 ; 多轴向应力 ❖ ; 热应力试验 ❖ stress corrosion 应力腐蚀 ; 超应力腐蚀 ; 应力腐化 ; 应 ❖ Stress ball 压力球 ; 减压球 ; 美国原装 ; 握力球 ❖ stress amplitude 应力幅 ; 应力幅值 ; 应力幅度 ; 应力振幅
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❖ quantity
❖ n. 参数，值；数量；总量
❖ physical quantity 物理量 ; 物理量 ; 实物量 ; 要的实物量 ❖ dimensionless quantity 无量纲量 ; 无因次量 ; 无因次值 ❖ average quantity 平均数量 ; 平均量 ; 均匀数量 ; 均匀数目 ❖ additional quantity 追加数量 ; 额外数量 ; 附加量 ❖ considerable quantity 大数量 ; 可观的
❖ magnitude
❖ n. 数量、数值；大小；量级
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❖ equilibrium ❖ n. 平衡；稳定；保持平衡的能力
❖ [,iːkwɪ'lɪbrɪəm]
Transformations of Stress and Strain
CHAPTER
MECHANICS OF MATERIALS
Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University
Transformations of Stress and Strain
Introduction Transformation of Plane Stress Principal Stresses Maximum Shearing Stress Example 7.01 Sample Problem 7.1 Mohr’s Circle for Plane Stress Example 7.02 Sample Problem 7.2 General State of Stress Application of Mohr’s Circle to the Three- Dimensional Analysis of Stress Yield Criteria for Ductile Materials Under Plane Stress Fracture Criteria for Brittle Materials Under Plane Stress Stresses in Thin-Walled Pressure Vessels
τ xy = +40 MPa
θ p = 26.6°, 116.6°
• Determine the principal stresses from
σ max,min = σx +σ y
2 σ x −σ y 2 + τ xy ± 2
工程力学英文版课件07 Stress for Axial Loads
F1
F2
F3
Fn
10
Axial force FN, Shear force FQ,
Bending moment MB,
Torque or twisting moment MX
FQ
FR
Mx
FN
MB
M
11
Procedure for the method of section
Calculation of the internal forces is foundation to analyze the problems of strength, rigidity, stability etc. General method to determine internal forces is the method of section. The following steps should be performed:
8
A section was used to cut the member into two parts.
F1Biblioteka F2F3Fn
9
The internal forces of the member cut became the external forces, and can be found by the equilibrium equations.
1
Internal Reactions: Stress for Axial Loads
§10–1 Introduction §10–2 Problems Involving Normal,
Shear and Bearing Stress §10–3 Allowable Stress §10–4 Stresses on Oblique Sections
机械工程专业英语第二课应力和应变Simple Stress and Strain
Simple Stress and StrainIn any engineering structure the individual components or members will be subjected to external forces arising from the service conditions or environment in which the component works. If the component or member is in equilibrium, the resultant of the external forces will be zero but, nevertheless, they together place a load on the member which tends to deform that member and which must be reacted by internal forces set up within the material.There are a number of different ways in which load can be applied to a member. Loads may be classified with respect to time:(a) A static load is a gradually applied load for whichequilibrium is reached in a relatively short time.(b) A sustained load that is constant over a long period of time,such as the weight of a structure (called dead load). Thistype of load is treated in the same manner as a static load;however, for some materials and conditions of temperatureand stress, the resistance to failure may be different undershort time loading and under sustained loading.(c)An impact load is a rapidly applied load (an energy load).Vibration normally results from an impact load, andequilibrium is not established until the vibration iseliminated, usually by natural damping forces.(d)An repeated load is a dead that is applied and removedmany thousands of times.(e) A fatigue of alternating load is a load whose magnitude andsign are changed with time.It has been noted above that external force applied to a body in equilibrium is reacted by internal forces set up within the material. If, therefore, a bar is subjected to a uniform tension or compression, i.e.a force, which is uniformly applied across the cross-section, then the internal forces set up are also distributed uniformly and the bar is said to be subjected to a uniform normal tress, the stress being defined asStress=load/area=P/AStress may thus be compressive or tensile depending on the nature of the load and will be measured in units of newtons per square meter (N/m2) or multiple of this.If a bar is subjected to an axial load, and hence a stress, the bar will change in length. If the bar has an original length L and changed in length by an amount QL, the stain produced is defined as follow: Train=change in length/original length=QL/LStrain is thus a measure of the deformation of the material and is non-dimensional, i.e. it has no units; it is simply a ratio of two quantities with the same unit.Since, in practice, the extensions of materials under load are very small, it is often convenient to measure the strains in the form of strain *10-6, i.e. microstrain, when the symbol used becomes UE.Tensile stresses and strains are considered positive in sense. Compressive stresses and strains are considered negative in sense. Thus a negative train produces a decrease in length.A material is said to be elastic if it returns to its original, unloaded dimensions when load is removed. A particular form of elasticity which applies to a large range of engineering materials, at least over part of their load range, produces deformations which are proportional to the loads producing them. Since loads are proportional to the stresses they produce and deformations are proportional to the strains, this also implies that, whilst materials are elastic, stress is proportional to strain, Hooke’s law therefore states thatStress trainThis law is obeyed within certain limits by most ferrous alloys and it can even be assumed to apply to with reasonable accuracy.Whilst a material is elastic the deformation produced by any load will be completely recovered when the load is removed; there is no permanent deformation.Within the elastic limits of materials, i.e. within the limits inwhich Hooke’s law applies, it has been shown thatTress/train=constantThis constant is given the symbol E and modulus of elasticityOr Young’s modulus. ThusE=tress/strainYoung’s modulus E is generally assumed to the same in tension or compression and for most engineering materials had a high numerical value. Typically, E=200*109N/m2 for steel, so that it will be observed from Eq. that trains are normally very small.In most common engineering applications strains rarely exceed 0,1%. The actual of Young’s modulus for any material is normally determined by carrying out a standard test on a specimen of the material.。
应力应变英语
应力应变英语Stress and Strain in EngineeringStress and strain are fundamental concepts in the field of engineering, as they are crucial in understanding the behavior of materials and structures under various loading conditions. These two interrelated quantities are essential for the design, analysis, and optimization of engineering systems, from small components to large-scale structures.Stress, in the context of engineering, can be defined as the internal force per unit area acting within a material or structure. It is a measure of the intensity of the internal forces that arise due to the application of external loads or constraints. Stress can be classified into different types, such as normal stress, shear stress, and torsional stress, depending on the direction and nature of the forces acting on the material.Normal stress is the stress that acts perpendicular to the surface of a material, and it can be either compressive or tensile. Compressive stress occurs when the material is subjected to forces that tend to push it together, while tensile stress occurs when the material issubjected to forces that tend to pull it apart. Shear stress, on the other hand, is the stress that acts parallel to the surface of a material, causing the material to slide or deform in a particular direction.Strain, on the other hand, is a measure of the deformation of a material or structure due to the application of stress. It is the change in the size or shape of a material relative to its original dimensions. Strain can be classified into different types, such as normal strain and shear strain, just like stress.Normal strain is the change in the length of a material divided by its original length, and it can be either compressive or tensile. Shear strain, on the other hand, is the change in the angle between two originally perpendicular lines in the material, caused by the application of a shear stress.The relationship between stress and strain is governed by the material's mechanical properties, which can be determined through various testing methods. The most fundamental relationship between stress and strain is described by Hooke's law, which states that for small deformations, the stress in a material is proportional to the strain. This linear relationship is valid for many materials, such as steel and aluminum, within their elastic range.However, not all materials exhibit a linear stress-strain relationship.Some materials, such as rubber and certain polymers, exhibit a non-linear relationship, where the stress-strain curve is curved and the material exhibits more complex behavior, such as viscoelasticity or plasticity.Understanding the stress-strain relationship of materials is essential for the design and analysis of engineering structures and components. By knowing the stress and strain values, engineers can determine the safety and reliability of a structure under various loading conditions, as well as optimize the design to ensure that the material is being used efficiently and effectively.For example, in the design of a bridge, engineers need to consider the stresses and strains that will be experienced by the structure due to the weight of the bridge, the traffic load, and environmental factors, such as wind and earthquakes. By analyzing the stress and strain distribution within the bridge, engineers can ensure that the structure is designed to withstand the expected loads without exceeding the material's strength or deformation limits.Similarly, in the design of a mechanical component, such as a gear or a shaft, engineers need to consider the stresses and strains that will be experienced by the component during operation. By analyzing the stress and strain distribution within the component, engineers can optimize the design to minimize the risk of failure and ensurethe component's reliability and longevity.In conclusion, stress and strain are fundamental concepts in engineering that are essential for the design, analysis, and optimization of engineering systems. By understanding the relationship between stress and strain, engineers can ensure the safety, reliability, and efficiency of their designs, ultimately contributing to the advancement of technology and the betterment of society.。
麻省理工工程力学课件4
Vmax ,τ
9-11: engineering questions
(cont’d)
• Kausel’s Discrete Mass Formulation
m0 ,V0
h
m1 = m0 + m V1 = ?
m2 = m0 + 2m V2 = ?
mi = m0 + im Vi = ?
mN − M = m0
North Tower: South Tower: 8:46 am above 96th floor, failed at 10:28 am 9:03 am above 80th floor, failed at 9:59 am
Immediate question: How did the towers fail?
Application of Dynamic Resultant
Theorem
• Linear Momentum before collision
p i −1 = mi −1Vi −1 e z
mi −1 ,Vi −1 mi = mi −1 + m Vi
m i = m i −1 + m Vi
• Linear Momentum after collision
1.050 Engineering Mechanics
Lecture 4: Stresses and Strength
Stresses and Equilibrium
Discrete Model
1.050 – Content overview
I. Dimensional analysis
1. 2. On monsters, mice and mushrooms Similarity relations: Important engineering tools
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14
O1
dx
d
O2
E
C
G G H H
F
A
C B
D D
The angle , is indicated on the element. It can be related to the length dx of the element and the difference in the angle of rotation, d , between the shaded faces.
a
a’
d
d’
b
c
This angle is denoted by (gamma) and is measured in radians (rad).
7
Hooke’s Law for Shear The behaviour of a material subjected to pure shear can be studied in a laboratory by using specimens in the shape of thin circular tubes and subjecting them to a torsion loading. Experiments show that as for normal stress and strain, shear stress is proportional to shear strain as long as stress does not exceed the proportional limit. Hooke’s law for shear can be written as
E G 2(1 )
9
§12-2 Torsion of a Circular Shaft
Torque is a moment that tends to twist a member about its longitudinal axis. Its effect is of primary concern in the design of axles or drive shafts used in vehicles and machinery.
( x)
deformed plane y x x
T
undeformed plane
12
The angle ( x) , so defined, is called the angle of twist. It depends on the position x and will vary along the shaft as shown. In order to understand how this distortion strains the material, a small piece located at a distance x from the fixed end is now isolated from the shaft. The back face will rotate by ( x), and the front face by ( x) d . As a result, the difference in these rotations, d , causes the element to be subjected to a shear strain.
11
If the shaft is fixed at one end and a torque is applied to its other end, the shaded plane in figure will distort into a skewed form as shown. Here a radial line located on the cross section at a distance x from the fixed end of the shaft will rotate through an angle ( x). z
1
Shear Stresses and Strains; Torsion
§12–1 Shear Stresses and Strains;
Hooke’s Law for Shear
§12–2 Torsion of a Circular Shaft
§12–3 Problems Involving Deformation
zy xy z yz xz y 0
zy
zy
zy yz
zy zy yz yz
yz
x
∆y
yz
∆z
∆x
y
All four shear stresses must have equal magnitude and be directed either toward or away from each other at opposite edges of the element.
4
Complementary property of shear
z Consider a volume element of material taken at a zy point located on the surface yz ∆z yz of any sectioned area on ∆x zy which the average shear ∆y stress acts. Consider force x equilibrium in the y direction, then,
and Stress in a Circular Shaft
2
§12-1 Shear Stresses and Strains; Hooke’s Law for Shear
Shear Stress The intensity of force, or force per unit area, acting tangent to ∆A is called the shear stress. The average shear stress distributed over each sectioned area that develops this shear force is defined by V
By inspection, twisting causes the circles to remain circles, and each longitudinal grid line deforms into a helix that intersects the circles at equal angles. Also, the cross sections at the ends of the shaft remain flat ― that is, they do not warp or bulge in or out ― and radial lines on these ends remain straight during the deformation. From these observations we can assume that if the angle of rotation is small, the length of the shaft and its radius will remain unchanged.
y
zy xy zy xy 0
zy zy
5
And in a similar manner, force equilibrium in the z direction yields yz yz . Finally, taking moments about the x axis, z
When the torque is applied, the circles and longitudinal grid lines originally marked on the shaft tend to distort into the pattern shown in Fig. 10 (b).
15O1dx NhomakorabeadO2
E
C
G G H H
F
A
From the figure, we have
C B
D D
GG d GG d EG dx
16
d
dx
Since dx and d are the same for all elements located at points on the cross section at x, then d / dx is constant, and the equation states that the magnitude of the shear strain for any of these elements varies only with its radial distance from the axis of the shaft. In other words, the shear strain within the shaft varies linearly along any radial line, from zero at the axis of the shaft to a maximum max at its outer boundary.
17
If the material is linear-elastic, then Hooke’s law applies, G , and consequently a linear variation in shear strain, as noted in the previous section, leads to a corresponding linear variation in shear stress along any radial line on the cross section. Hence, like the shear strain variation, for a solid shaft, will vary from zero at the shaft’s longitudinal axis to a maximum value, max , at its outer surface.