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梁的剪切应力模型

Shear in Beams Model

This model was designed to help students understand
此模型是为帮助学生理解
how shear stresses arise in beams,
剪切应力是如何在梁中产生
why those stresses vary from place to place over a beam cross section,
为什么这些力在梁的横截面各处均不相同
and why they produce specific shear flow patterns.
以及这些力为什么会产生特殊的剪切流型
To provide context for the model,
为了更好地理解这个模型
you might imagine that it is part of a cantilever beam
你可以将它想象成悬臂梁的一部分
like this one
就像我手中的这个
If that were the case,
如果悬臂梁是如图的盒子
the bending moment acting on the model at section B
作用在B段的弯矩
would be larger than that acting at section A.
比作用在A段的弯矩要大
We represent the tensile bending stresses that these moments would produce
我们用画在有机玻璃块上的红色剪头
using red arrows overlaid on Plexiglas blocks
来表示这些力矩产生的拉伸曲力
and the compressive ones
与其反向的力
with blue arrows overlaid on the beam.
则用画在梁上的蓝色箭头表示
As you can see
正如你所看到的
the bending stresses at B are also larger than those at A.
B段的弯曲应力同样比A段的大
As we will discover shortly
我们很快就会发现
differences in bending stresses from one section
正是沿着梁的长度从一个截面
to another along the length of a beam
到另一个截面
are the reason that shear stresses arise.
的弯曲应力的差异产生剪切应力
Before we begin that discussion, however,
但是 在我们讨论之前
notice that the length of each arrow is proportional
要注意到每个箭头的长度
to the magnitude of the stress that it represents.
与它所表示的应力的大小是对应的
As a result, the volume of each Plexiglas block
因此 每个有机玻璃块的体积
is proportional to the force generated
与空间上产生的力
over its base area.
是对应的
As you can see,
可以看到
large forces are generated in the flanges
翼缘处产生的是较大的应力
and smaller ones in the web.
而较小的应力则产生在腹板
Not only that, but the flange forces are
不仅如此 翼缘处的应力
relatively far from the neutral axis of the beam,
相对远离梁的中轴
and so they contribute much to the bending moment.
它们正是弯曲力距的主要部分
The forces on the web are much smaller
而作用在腹板上的力更小
and they act at a shorter moment arm.
对应的力臂距更短
As a result, they contribute relatively little to the moment.
因此它们对力矩的贡献相对而言是很小的
To understand how shear stresses arise in a beam,
接下来解释梁的剪切应力是如何产生的
imagine that we used a saw
假设我们用锯子
将翼缘的一个纤维与其他剩余部分分离开
As you can see, the force from the bending stresses on end B of the fiber
可以看到 该纤维上末端B的弯曲应力
is larger than that produced at end A.
要比末端A所产生的力要大
In the worked example included at the end of this video,
在视频结尾提到的有效范例中
the difference in these forces is 15 units.
可知这两个力的大小相差15个单位
So, for this fiber to remain in equilibrium,
所以为了该纤维保持平衡
the rest of the beam must exert a force of 15 units on it.
梁其余部分须作用15个单位的力在它上
We treat the fiber as if it were a free body,
我们假设该纤维部分是灵活的
and therefore draw the arrow on the fiber
在该纤维上画上箭头
in the direction of the force that [the beam] exerts on it.
表示作用在纤维上的力的方向
At the same time that the beam is exerting a force of 15 units on the fiber,
同时梁上有15单位的力作用在该纤维上
the fiber is exerting an equal and opposite force on the rest of the beam,
该纤维对梁的其余部分作用等大反向的力
as shown by this arrow.
正如这个箭头所示
As you can see,
可以看到
the forces between the fiber and the rest of the beam
纤维和梁其余部分之间的力
act parallel to the cut,
正好与切口平行
and so we call them shear forces,
所以我们称之为剪切应力
and we represent them using arrows that have a single-sided head.
我们用单边箭头来表示这些剪切应力
In contrast, forces that act normal to a surface,
相反 作用在表面的力是垂直方向的
like those produced by these bending stresses,
像由这些弯曲应力所产生的力
are called normal loads,
称为正常负载
and we represent them using arrows that have symmetrical heads.
我们用双边剪头表示它们
If we now imagine a cut that removes two fibers of the flange,
我们假设现在切掉翼缘的两个纤维
that pair will be out of balance in the axial direction by twice
这一对在轴向上将失去平衡
as much as the single fiber we originally considered.
力的大小是我们刚刚讨论的一个纤维时的两倍
So, to keep this new free body in equilibrium,
同理 为了保证这个新的自由体保持平衡
a shear force of 30 units must act,
必须有30个单位的剪切力与其对应
as shown by these arrows.
分别用这两个箭头表示出来
If we cut off all 5 of the fibers
如果我们把构成上翼缘的
that make up the top flange from the rest of the beam
五根纤维从梁的其余部分拿掉
the total axial imbalance is 5 times 15
总的轴向上的不平衡力为5倍的15
or 75 units.
即75单位
And that is the magnitude of the shear force
这便是剪切应力的大小
that must act on the newly cut surface
该力作用在新的切面上
that is between them and the rest of the beam.
即上翼缘与梁的剩余部分之间
A fiber closer to the neutral axis of the beam
离梁的中心轴较近的纤维
experiences bending stresses that are smaller.
承受的弯曲应力较小
The difference between the forces on the two ends of this fiber
该纤维的两个末端的力相差
is only 9 force units.
仅9个单位
Adding 9 to 75 gives a total shear of 84 units.
9加75即得到84单位的总的剪切应力
The difference in the end forces for the fiber just above the
在中性轴上方的纤维的末端力
neutral axis is only 3 units.
差值仅仅为3
Adding that to 84 gives a total of 87 units.
加上84总和即为87
Fibers below the neutral axis
处于中性轴下方的纤维
carry compression, rather than tension.
代表压力而不是张力
As a result, those fibers are out of balance in the opposite axial direction
相比于中性轴上方 下方这些纤维在
compared to those above it.
相反的轴向上失去了平衡
If we imagine a cut below the neutral axis
假设我们从中性轴下方切开
some of these fibers are now included
包括其中一些纤维
and they reduce the total axial imbalance
与在梁中性轴切割相比
compared to a beam cut right at the neutral axis.
它降低了总的轴向不平衡
That is why the maximum shear force
这就是为什么最大的剪切应力
is always found at the neutral axis.
是在中性轴处
Notice that these shear forces arise
注意这些剪切应力是
because the bending stresses
由于弯曲应力所产生的
and the moments that cause them
而且产生剪切应力的力
vary with position along the length of the beam.
沿着梁长的各个位置是不同的
It is customary to report the shear stresses on a beam cross-section
通常记述的是在梁的横截面上的剪切应力
rather than the shear forces.
而不是剪切力
For the sake of simplicity,
为了理解简单
the web and flanges in this model are assumed to be of unit thickness.
这个模型当中的腹板和翼缘假定为单位厚度
Also, the two cross-sections are considered to be separated by a unit amount,
此外这两个截面被认为是由一个单位长度分开
even though the physical dimensions of the model may suggest otherwise.
尽管模型的实际尺寸可能并非如此
As a result of these dimensional choices,
在这些尺寸假定下
the shear stress along any of the cuts we have made
沿着我们切过的任何一切面上的剪切应力
will be numerically equal to the shear force it carries.
将与它所承受的剪切力相等
Notice too that our cuts always
同样要注意我们切割时
go across the thickness of the web or flanges.
是沿着腹板和翼缘的厚度切割
That way all of the points in the cut are very near to each other
这样的话 切口上的所有点都非常接近
and would generally be expected to carry similar stresses.
通常默认它们承受相同的压力
We never cut the flange parallel to its top surface, for example.
比如我们并没有沿着平行于翼缘顶部表面的方向切割
Points along such a cut would be relatively distant from each other
这样的话任何两个切点之间距离太远
and the stresses at various points along the cut could differ substantially.
而且沿着切口每个切点承受的力是不同的
One can plot the magnitude of the shear stress
我们可以画出剪切应力的大小
as a function of the distance between the cut and the outer extremity of the flange.
它是从切口到翼缘外缘的距离的函数
As you can see, the graph is linear,
可以看到 图像是线性的
a result confirmed by the calculations
在屏幕右侧的计算结果
shown on the right.
也证明了这一点
Similar graphs can be made
同样翼缘的其他地方也可以
for other regions of the flanges
作出类似的图像
and, by convention,
另外 按照惯例
the axes are typically oriented as shown here.
坐标轴的方向如图所示
As these calculations show,
从这些计算数据中可以看出
the shear stress in the web takes a parabolic form.
腹板处的剪切应力大小呈抛物线型
Plotting all of these graphs on a single figure reveals what is called
将这些图形绘制在一张图上 则揭示了
the “shear stress distribution” for that cross section.
在那个截面的剪切应力的分布
In order to understand shear flow,
为了理解剪切流
we must transfer the shear stresses we just calculated
我们必须将我们所计算的剪切应力
from their respective longitudinal cutting planes, to the beam cross section.
从各自的纵切面传递到梁的横截面
We do that using the simple fact that
我们用一个简单的例子
shear in a plane always involves four matched stresses
平面上的剪切应力始终包括四个匹配的应力
with arrows that go head-to-head and tail-to-tail.
用循环的剪头表示
If we concentrate on this inside corner of a flange fiber,
如果我们注意观察这个翼缘纤维的靠里的角
the shear on the cross-section must go tail-to-tail
横截面上的剪切应力必
with the longitudinal stress arrow
与纵向应力箭头方向一致
and so it must point outwards, and its magnitude must be 15.
所以它必须是指向外面且大小为15
If we focused on the outside corner of the adjacent flange fiber,
再看邻近的翼缘纤维的内角处
we would get exactly the same result.
我们能够得到相同的结果
We can show the shear stress on the cross section using a single-sided arrow.
我们用单头箭头表示横截面的剪切应力
At two fibers in, the shear stress is 30,
在两个纤维之间 剪切应力为30
and so that is the stress shown on the cross section at that location.
这就是该位置的横截面上的应力
To analyze the web, we note that its shear operates in a vertical plane,
接着分析腹板 注意在垂直平面的剪切应力
unlike that in the flanges, which operates in a horizontal plane.
不同于翼缘是作用于水平面的
Collectively, these shear arrows
这些所有的切力箭头
show how the shear flows over the cross section,
展示了横截面上的切力是如何流动的
and when taken together,
当连在一起时
they reveal what is known as the “shear flow”.
它们就展现了我们所知道的剪切流
These arrows represent forces or stresses that must act on the beam
这些箭头代表作用在梁上的力或者应力
for it to remain in equilibrium.
使它保持平衡
So, clearly, an upwards external force must act
所以 很显然向上的朝外的力
on the end of the beam that carries the larger bending stresses
必作用至梁的末端B即承受更大的弯曲力
And a downwards force must act on the other end.
而向下的力则作用在另一个末端
If you examine the beam from which this model was taken,
如果你仔细看这个模型的梁
you can see that those shear directions are indeed correct.
你可以发现这些切力的方向的确是正确的
Not only that,
不仅如此
but if we estimate the total vertical load on the beam cross section
如果我们通过将垂直方向的力进行求和
by summing the indicated vertical forces
评估得到在梁的横截面
we get 405 units.
总的垂直荷载为405
This value is quite close to the shear of 412 units
这个值与412的切力十分接近
noted in the worked example that follows the credits.
412是实际工作中所记录的权威数字
It may seem surprising that the shear forces acting on a beam
可能令人惊讶的是作用在梁上的剪切应力
could be determined solely from the bending stresses
是由作用在邻近的两个横截面的
acting on two nearby cross sections.
的弯曲应力所唯一决定的
However, the differences between the stresses on those two sections
但是 这两部分之间应力的差异
are a direct result of changes in the bending moment M
是由轴向位置x的弯矩M变化
with axial position x.
的直接结果
And that rate of change can be used to calculate the beam shear
而这一变化率可以用著名的公式
by using the well-known formula, V=dM/dx.
V=dM/dx来计算剪切应力
This relationship explains, mathematically,
从数学角度 该关系式解释了
why shear and changes in bending moment are so closely related.
切力和弯矩变化之间的密切联系
We hope this video helped you to understand
我们希望这个视频能帮助大家理解
how shear stresses arise in beams,
梁的剪切应力是如何产生的
why they vary over a beam cross section,
为什么在同一个横截面上是不同的
and how they produce specific shear flow patterns.
已经它们是如何形成特殊的剪切流模型
Thanks for watching.
谢谢观看

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视频概述

利用模型解释了关于梁的剪切应力的三个问题,如何产生,怎样产生以及相关的影响因素

听录译者

收集自网络

翻译译者

巷陌繁花

审核员
视频来源

https://www.youtube.com/watch?v=aivDhiLwu8E

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