In our previous topics, we have seen some important concepts such asdeflection and slope of a simply supported beam with point load那简单地支撑光束的偏转和斜率携带均匀分布式负载and悬臂梁的偏转和悬臂梁在自由端点载荷in our previous post.
Now we will start here, in this post, another important topic i.e. deflection and slope of a cantilever beam loaded with uniformly distributed load throughout the length of the beam with the help of this post.
我们已经见过terminologies and various termsused in deflection of beam with the help of recent posts and now we will be interested here to calculate the deflection and slope of a cantilever beam loaded with uniformly distributed load throughout the length of the beam with the help of this post.
悬臂梁基本上被定义为光束的一端将固定，并且梁的其他端将是自由的
。
Uniformly distributed load is the load which will be distributed over the length of the beam in such a way that rate of loading will be uniform throughout the distribution length of the beam
B.asic concepts
基本上有三种重要方法，我们可以在其中可以轻松地确定装载光束的任何截面的偏转和斜率。
Double integration method
Moment area method
Macaulay’s method
双积分法和时刻区域方法基本上用于确定当梁将以单个负载加载时装载梁的任何截面的偏转和斜率。
虽然MAMaulay的方法基本上用于确定加载梁的任何部分的偏转和斜率，当梁将加载多个负载时。
We will use double integration method here to determine the deflection and slope of a cantilever beam which is loaded with uniformly distributed load throughout the length of the beam.
梁弹性曲线的微分方程将用于双积分法，以确定加载光束的偏转和斜率，因此我们必须必须召回这里梁弹性曲线的微分方程。
梁弹性曲线的微分方程
在差分方程首次集成之后，我们将具有斜率的值I.. DY / DX。类似地，在微分方程的第二集成后，我们将具有偏转的值I. Y.Y。
L.et us come to the main subject i.e. determination of deflection and slope of a cantilever beam which is loaded with uniformly distributed load throughout the length of the beam.
让我们考虑长度L的悬臂梁AB，其固定在支撑件A处并自由B处于B点并加载均匀分布的负载，如下图所示。
We have following information from above figure,
w = n / m中的加载率
AB =装载前悬臂梁的位置
ab'=装载后悬臂梁的位置
θ一种= Slope at support A
θB.= Slope at support B
yB.= Deflection at free end B
Total load due to udl = W = w.L
边界条件
We must be aware with the boundary conditions applicable in such a problem where beam will be a
悬臂梁udl
。我们在此提到的边界条件如下。
一种t point A, Deflection will be zero
一种t point A, Slope will be zero
一种t point B, Deflection will be maximum
在B点，斜率也将是最大的
L.et us consider one section XX at a distance x from end support A, let us calculate the bending moment about this section.
我们采取了符号公约的概念，为上述XX的上述弯矩提供了合适的标志。有关用于弯曲时刻的符号约定的更多详细信息，我们请您找到帖子“Sign conventions for bending moment and shear force”.
让我们考虑较早确定的弯曲瞬间关于XX部分和弯曲矩的弯矩表达式。我们将在下面的图中显示以下等式。
我们现在将集成这个方程式，我们也将应用边界条件，以确保斜率的表达式以及梁的一部分的偏转，我们可以为此显示加载光束的斜率和偏转的方程式。
在哪里，C.1and C2是集成的常数，我们可以确保这些常数c的值1and C2通过考虑和应用边界条件。
L.et us use the boundary condition as we have seen above.
一种t point A i.e. x = 0, Slope will be zero i.e. dy/dx =0
在点a i.e. x = 0时，偏转将为零I.e = 0
在将边界条件应用于斜率和偏转的偏转方程中，我们将具有以下常数C的值1and C2as mentioned here.
C1= - WL.3./6
C2= wL4./ 24.
L.et us insert the values of C1and C2in slope equation and in deflection equation too and we will have the final equation of slope and also equation of deflection at any section of the loaded beam. We can see the slope equation and deflection equation in following figure.
Slope at the free end
一种t x = L,
θB.=斜率斜率b
让我们使用斜率方程并插入x = l的值，我们将在支持b上具有斜率的值。θB.
θB.= -
W.L.
3.
/ 6EI.
θB.= -W.L.2/ 6EI.
负符号表示端部B处的切线与梁轴AB处于逆时针方向的角度。
最大偏转
在点B i.e. x = l，偏转将最大
L.et us use the deflection equation and insert the value of x = L in deflection equation, we will have value of deflection at point B.
yB.= - WL.4./ 8EI.
yB.= - W
L.
3.
/ 8EI.
负标志在此表示加载光束中的偏转将是向下方向的。
We will see another topic in our next post.
Please comment your feedback and suggestions in comment box provided at the end of this post.
Reference:
物质的强度，r.k.淘士
图片礼貌：谷歌
一种lso read
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