They are used in bridges, antenna towers, cranes, even in parts of the International Space Station.

它們被用於橋樑、天線塔、起重機，甚至是國際空間站的一部分。

And for good reason.

這是有道理的。

They allow us to create strong structures while using materials in a very efficient and cost-effective way.

它們使我們能夠創造出堅固的結構，同時以非常高效和具有成本效益的方式使用材料。

So, what exactly is a truss?

那麼，桁架究竟是什麼呢？

It is essentially a rigid structure made up of a collection of straight members.

從本質上講，它是由一系列直線構件組成的剛性結構。

But that's not a complete definition.

但這並不是一個完整的定義。

There are two important assumptions we need to be able to make for a structure to be considered as a truss.

要將結構視為桁架，我們需要做出兩個重要假設。

First, we need to be able to assume that all of the joints in the structure can be represented by a pinned connection, meaning that members are free to rotate at the joints.

首先，我們需要假設結構中的所有連接點都可以用銷軸連接來表示，這意味著構件可以在連接點上自由旋轉。

The members of a truss are often rigidly connected using what is known as a gusset plate.

桁架的構件通常使用所謂的桁架板進行剛性連接。

But if the center lines of all the members at a joint intersect at the same point, like they do here, it's reasonable to assume that the joint behaves like a pinned connection.

但是，如果連接處所有構件的中心線都相交於一點，就像這裡一樣，那麼就可以合理地認為連接處的行為就像銷釘連接一樣。

The second assumption we need to be able to make is that loads are only ever applied at joints of the truss.

我們需要做出的第二個假設是，荷載只施加在桁架的連接處。

We never have loads acting in the middle of a member, for example.

例如，我們從不在成員中間加載行動。

Because all joints are pinned, the members cannot carry bending moments.

由於所有連接處都用銷釘固定，是以構件無法承受彎矩。

They can only carry axial loads.

它們只能承受軸向載荷。

This simplifies the analysis of a truss significantly.

這大大簡化了對桁架的分析。

Each member has to be in equilibrium, so the forces acting at each end of a member must be equal and opposite.

每個構件都必須處於平衡狀態，是以作用在構件兩端的力必須相等且相反。

Each member is either in tension or in compression.

每個構件要麼處於拉伸狀態，要麼處於壓縮狀態。

These assumptions are what differentiate a truss from a frame.

這些假設是桁架與框架的區別所在。

Unlike trusses, frames don't necessarily have pinned joints, and so members can carry bending moments.

與桁架不同，框架不一定有銷釘連接，是以構件可以承受彎矩。

A frame can also have loads applied directly to its members.

框架也可以直接在其構件上施加載荷。

The base shape of a truss is three members connected to form a triangle.

桁架的基本形狀是三個構件連接成一個三角形。

If a load is applied, the angles of the triangle won't be able to change if the length of each of the members stays the same.

如果施加荷載，如果每個構件的長度保持不變，三角形的角度就不會發生變化。

This means that the triangle is a very stable shape which won't deform when loads are applied to it, and so it is a great base from which to build a larger structure.

這意味著三角形是一種非常穩定的形狀，在受到載荷時不會變形，是以它是構建更大結構的良好基礎。

Joining four members together does not form a stable structure.

將四個成員連接在一起並不能形成一個穩定的結構。

The angles between members can change without any change in the length of the members, and so using a four-sided shape as the base for building a truss would be a terrible choice.

構件之間的角度可以在不改變構件長度的情況下發生變化，是以使用四邊形作為桁架的基礎是一個糟糕的選擇。

An easy way to stabilize this configuration is to add a diagonal bracing member to split it into triangles.

穩定這種結構的簡單方法是增加一個斜撐構件，將其分割成三角形。

We can start with our triangle and build it out to form a structure.

我們可以從我們的三角形開始，將其擴展成一個結構。

There are a lot of different ways to build a truss, but there are some particularly popular truss designs that you will see again and again, and so they are referred to by specific names.

搭建桁架的方法有很多種，但有一些特別流行的桁架設計你會反覆看到，是以它們有特定的名稱。

The one shown here is a Fink roof truss, but there are many more as you can see here.

這裡展示的是芬克屋頂桁架，但還有更多的屋頂桁架，您可以在這裡看到。

Later on in this video, I'll cover how these different designs carry loads in different ways.

在本視頻的稍後部分，我將介紹這些不同的設計如何以不同的方式承載負荷。

The members of these trusses are all located in the same plane.

這些桁架的構件都位於同一平面內。

These are called planar trusses, and we can analyze them as two-dimensional structures.

這些被稱為平面桁架，我們可以將其作為二維結構進行分析。

Even seemingly three-dimensional structures can often be analyzed as planar trusses.

即使是看似三維的結構，通常也可以作為平面桁架進行分析。

Take a look at this bridge for example.

比如這座橋。

The loads are transmitted from the horizontal floor beams to the two vertical trusses on each side of the bridge.

荷載從水準地板梁傳遞到橋樑兩側的兩個垂直桁架上。

Each of these trusses only carries loads acting in its plane, and so we can analyze it as a two-dimensional structure.

每個桁架只承受其平面內的荷載，是以我們可以將其作為二維結構進行分析。

To be able to design or analyze a truss, we need to be able to determine the force in each of its members.

為了能夠設計或分析桁架，我們需要能夠確定其每個構件的受力情況。

This allows us to check that the member can carry the loads without failing, or gives us the information we need to select the best cross-section for each of the members.

這樣，我們就可以檢查構件是否能承受荷載而不發生故障，或者為我們提供所需的資訊，以便為每個構件選擇最佳截面。

There are two main methods we can use to do this, the method of joints, and the method of sections.

我們可以使用兩種主要方法來做到這一點，即連接法和分段法。

Let's look at the method of joints first, using the Fink roof truss we saw earlier.

首先，讓我們用前面看到的芬克屋頂桁架來看看連接方法。

The method is really simple.

方法其實很簡單。

First, you draw a free body diagram, showing all of the external loads acting on the truss, and you use the three equilibrium equations to calculate the reaction forces.

首先，繪製自由體圖，顯示作用在桁架上的所有外部載荷，然後使用三個平衡方程計算反作用力。

Then, you draw a free body diagram for every single joint, and work through them one by one to solve the unknown forces acting at each Since all of the joints are pinned connections, there are no moments, and so you only need to consider equilibrium of the horizontal and vertical forces.

然後，為每個關節繪製自由體示意圖，並逐一求解作用在每個關節上的未知力。由於所有關節都是銷釘連接，不存在力矩，是以只需考慮水準力和垂直力的平衡。

Remember that we are calculating the forces acting at each joint, not the forces in the member.

請記住，我們計算的是作用在每個關節上的力，而不是構件上的力。

If a member is in tension, the internal forces will be acting to make the member longer.

如果構件處於拉伸狀態，內力將使構件變長。

For every action, there is an equal and opposite reaction, which means that a member in tension will be exerting a force on the joint which is acting away from the joint.

每一個作用力都有一個相等和相反的反作用力，這意味著處於拉伸狀態的部件會對關節施加一個作用力，而這個作用力是遠離關節的。

For members in compression, the force will be acting towards the joint.

對於受壓構件，力將作用在連接處。

Let's work through an example for a slightly simpler truss.

讓我們以稍簡單的桁架為例進行說明。

First, let's draw the free body diagram and determine the reaction forces using our three equilibrium equations.

首先，讓我們繪製自由體圖，並利用三個平衡方程確定反作用力。

Taking equilibrium of the horizontal forces, the horizontal force at joint A must be equal to zero, because it is the only force in the horizontal direction.

在水準力平衡的情況下，關節 A 處的水準力必須等於零，因為它是水準方向上唯一的力。

Taking equilibrium of the vertical forces, the reaction forces at joints A and E must sum up to 20 kilonewtons.

在平衡垂直力的情況下，A 和 E 接頭處的反作用力總和必須達到 20 千牛頓。

Both joints are located at the same distance from joint C, so taking equilibrium of the moments acting about joint C, we can calculate that they both equal 10 kilonewtons.

兩個關節與關節 C 的距離相同，是以，在平衡作用於關節 C 的力矩時，我們可以計算出兩個關節的力矩都等於 10 千牛頓。

Now, let's determine the forces acting on each joint.

現在，讓我們來確定作用在每個關節上的力。

Since we don't know yet which members are in tension and which are in compression, it's easiest to just assume that all of the members are in tension, and so we'll draw the internal forces as pointing away from each joint.

由於我們還不知道哪些構件處於拉伸狀態，哪些處於壓縮狀態，是以最簡單的方法就是假設所有構件都處於拉伸狀態，這樣我們就可以把內力畫成從每個連接處指向遠處。

If we end up with negative values for these forces, it just means that we guessed wrong, and the member is actually in compression.

如果這些力最終為負值，則說明我們猜錯了，實際上構件處於壓縮狀態。

Now we can work through each of the joints, starting with joint A.

現在，我們可以從關節 A 開始，逐一檢查每個關節。

Analyzing trusses involves a lot of resolving forces to different angles, so if you want to be good at it, you're going to need to remember your trigonometry.

分析桁架時，需要將大量的力轉換成不同的角度，是以，如果你想在這方面有所建樹，就必須牢記三角函數。

Here's a quick reminder.

在此提醒大家

Back to our joint.

回到我們的接頭處

All we have to do is apply the equilibrium equations to determine the forces acting on each joint.

我們要做的就是應用平衡方程來確定作用在每個關節上的力。

Taking equilibrium of the vertical forces, the 10 kilonewton reaction force must balance the vertical component of the force FAB, and so we can calculate FAB as negative 10 divided by sine of the angle of 60 degrees.

考慮到垂直力的平衡，10 千牛頓的反作用力必須與力 FAB 的垂直分量平衡，是以我們可以計算出 FAB 為負 10 除以 60 度角的正弦值。

Taking equilibrium of the horizontal forces, we get that the force FAC must balance the horizontal component of the force FAB, and so is equal to 5.8 kilonewtons.

在平衡水準力的情況下，我們得到 FAC 力必須平衡 FAB 力的水準分量，是以等於 5.8 千牛頓。

That's all of the forces acting on joint A calculated.

這就是作用在關節 A 上的所有力的計算結果。

The force in each member is constant, and so we now also know the forces acting on the joints at the other ends of these two members.

每個構件上的力都是恆定的，是以我們現在也知道了作用在這兩個構件另一端關節上的力。

We can repeat the process for joint B.

我們可以對接頭 B 重複上述過程。

We can start by considering equilibrium of the vertical forces, which allows us to calculate the force FBC.

我們可以先考慮垂直力的平衡，這樣就可以計算出力 FBC。

And then we can consider equilibrium of the horizontal forces, to calculate FBD.

然後，我們可以考慮水準力的平衡，計算 FBD。

We then need to work through all of the remaining joints, but we can save a bit of time by noticing that the truss and the applied loads have an axis of symmetry, and so the forces on the other side of the truss must be identical.

然後，我們需要對所有剩餘的連接點進行計算，但我們可以注意到，桁架和施加的荷載有一個對稱軸，是以桁架另一側的受力必須相同，這樣可以節省一些時間。

That gives us all of the forces at the joints.

這就得出了關節處的所有力。

We can show which members are in tension, and which are in compression, like this.

我們可以像這樣顯示哪些構件處於拉伸狀態，哪些處於壓縮狀態。

One thing you'll notice as you analyze trusses is that some members don't carry any loads at all.

在分析桁架時，你會發現有些構件根本不承載任何荷載。

We call these zero force members.

我們稱其為 "零力量成員"。

There are two main configurations where we have zero force members.

我們有兩種主要的零受力構件配置。

The first is where we have three members connected at a single joint, and two of the members are aligned.

第一種情況是，有三個構件連接在一個接頭處，其中兩個構件是對齊的。

Here, only one member has a component in the vertical direction, and so to maintain equilibrium of forces at the joint in the vertical direction, the force in this member must be zero.

在這裡，只有一個構件在垂直方向上有一個分量，是以要保持連接處在垂直方向上的力平衡，該構件上的力必須為零。

The second configuration is when we have only two members connected at a joint, and the members are not aligned.

第二種情況是，只有兩個構件連接在一個接頭處，而且構件沒有對齊。

Only one member has a component in the vertical direction, and so both must be zero force members.

只有一個構件在垂直方向上有分量，是以這兩個構件都必須是零力構件。

By the way, this is true regardless of how the members are oriented, because we can rotate the orientation of the coordinate system we are using to apply the equilibrium equations.

順便說一下，無論構件的方向如何，這一點都是正確的，因為我們可以旋轉用於應用平衡方程的座標系的方向。

These two configurations only contain zero force members if there are no external loads acting at the joints.

這兩種結構只有在連接處沒有外荷載作用的情況下才會包含零力構件。

If we have external loads, there will be components in the vertical direction, and so these will not be zero force members.

如果有外部荷載，垂直方向上會有分量，是以這些構件不會是零力構件。

Let's look at an example.

我們來看一個例子。

At this joint here, we have two connected members.

在這個接頭處，我們有兩個相連的成員。

The members are not collinear, and there are no external loads, so they must be zero force members.

這些構件並不共線，也沒有外荷載，是以它們必須是零力構件。

And at this joint, we have three members, of which two are collinear.

在這個關節處，我們有三個成員，其中兩個是共線的。

The vertical member must be a zero force member.

垂直構件必須是零力構件。

We can remove these members, and so have a much easier starting point for solving the truss.

我們可以移除這些構件，這樣就有了一個更容易解決桁架問題的起點。

You'll notice that we haven't removed the two members at this joint.

你會注意到，我們還沒有拆除這個連接處的兩個部件。

That is because there is an external load acting here, and so these can't be zero force members.

這是因為這裡有外力作用，所以這些構件不可能是零力構件。

You might be wondering why anyone would bother including zero force members in a truss if they carry no loads.

你可能會想，既然桁架中的零力構件不承載任何荷載，為什麼還要在其中加入零力構件呢？

They are definitely not useless.

它們絕對不是一無是處。

They are usually included to provide stability, for example to prevent buckling of long members which are under compression.

它們通常用於提供穩定性，例如防止受壓長構件發生彎曲。

Or they may be used to make sure that unexpected loads won't cause the structure to fail.

或者，它們可以用來確保意外荷載不會導致結構失效。

We've covered the method of joints.

我們已經介紹了接頭的方法。

Let's look at the other method we can use to solve trusses, which is the method of sections.

我們再來看看另一種解決桁架問題的方法，即截面法。

The first step is the same as the method of joints.

第一步與連接方法相同。

We draw the free body diagram, and use the equilibrium equations to solve the reaction forces.

我們繪製自由體圖，並利用平衡方程求解反作用力。

Next, we make an imaginary cut through the members of interest in our truss, and we draw the internal forces in the cut members.

接下來，我們對桁架中的相關構件進行假想切割，並繪製切割構件的內力圖。

The internal and external forces must be in equilibrium, and so we can apply the equilibrium equations to solve the internal forces.

內力和外力必須處於平衡狀態，是以我們可以應用平衡方程來求解內力。

When choosing how to cut your truss, remember that we only have three equilibrium equations.

在選擇如何切割桁架時，請記住我們只有三個平衡方程。

If you cut through too many members, you will have too many unknowns and not enough equations.

如果切入的成員過多，就會出現未知數過多而方程不足的情況。

You can choose which side of the cut you want to assess.

您可以選擇要評估的切割面。

The left side looks easier to solve because there are less forces.

左側看起來更容易解決，因為受力較小。

But we could have chosen to solve the right side instead.

但我們可以選擇解決右邊的問題。

The method of sections is best used when you have a truss which has a lot of members, but you are only interested in the loading in a few specific members.

當桁架有很多構件，但你只對幾個特定構件的荷載感興趣時，最好使用截面法。

Let's look at an example.

我們來看一個例子。

We need to determine the internal forces in these three members.

我們需要確定這三個部件的內力。

First, let's draw the free body diagram and apply the equilibrium equations to calculate the reaction forces.

首先，讓我們畫出自由體圖，然後應用平衡方程計算反作用力。

The horizontal reaction force at joint A is the only force acting in the horizontal direction, so it must be equal to zero.

關節 A 處的水準反力是水準方向上的唯一作用力，是以它必須等於零。

By considering equilibrium of forces in the vertical direction and equilibrium of the moments acting about joint A, we can figure out that F A is equal to 19, and F H is equal to 21.

考慮到垂直方向的力的平衡和作用在關節 A 上的力矩的平衡，我們可以算出 F A 等於 19，F H 等於 21。

Next, let's make our imaginary cut through members F D, F E, and G E, and draw the internal forces.

接下來，讓我們對 F D、F E 和 G E 構件進行假想切割，並繪製內力圖。

Like we did earlier for the method of joints, we will assume that all unknown forces are tensile.

就像我們之前做的連接方法一樣，我們將假設所有未知力都是拉力。

Next, we just need to apply the equilibrium equations.

接下來，我們只需應用平衡方程即可。

The force in member F E is the only unknown force with a component in the vertical direction, so that's a good place to start.

構件 F E 中的力是唯一一個在垂直方向上有分量的未知力，是以這是一個很好的起點。

The diagonal members are all at 45 degree angles, so by considering equilibrium in the vertical direction, we get that the force in member F E is equal to 12.7 kN.

對角線構件都成 45 度角，是以考慮到垂直方向上的平衡，我們可以得出構件 F E 所受的力等於 12.7 千牛。

Now, let's consider equilibrium of moments acting about joint F.

現在，我們來考慮作用於關節 F 的力矩平衡。

This is a good joint to choose because three of the five forces in this free body diagram have a line of action passing through F, and so only the force in member G E and the 21 kN reaction force generate a moment about this joint.

這是一個很好的連接點，因為在這個自由體圖中，五個力中有三個力的作用線都經過 F，是以只有構件 G E 中的力和 21 kN 的反作用力會對這個連接點產生力矩。

Both forces are located at a force in member G E is equal to 21 kN.

這兩個力都位於構件 G E 等於 21 kN 的位置。

Finally, we can take equilibrium in the horizontal direction to calculate that the force in member F D is equal to negative 30 kN.

最後，我們可以利用水準方向上的平衡來計算構件 F D 所受的力等於負 30 千牛。

And that's it, we've calculated the internal forces in the three members we were interested in.

就這樣，我們計算出了我們感興趣的三個成員的內力。

One member is in compression, and two are in tension.

一個構件處於壓縮狀態，兩個構件處於拉伸狀態。

If it is possible to determine the reaction forces and the members of a truss by applying the equilibrium equations, the truss is said to be statically determinate.

如果可以通過應用平衡方程來確定桁架的反力和構件，則稱該桁架為靜力確定桁架。

Real life structures sometimes contain more members than are needed for the structure to be stable, as this makes them safer.

現實生活中的結構有時會包含比結構穩定所需的更多的構件，因為這樣會更安全。

This means we may not be able to apply the method of joints or the method of sections, because we have too many unknowns and not enough equilibrium equations, either to determine the reaction forces, or to determine the internal forces within the truss.

這意味著我們可能無法使用連接法或截面法，因為我們有太多的未知數，沒有足夠的平衡方程來確定反作用力或桁架內的內力。

These trusses are said to be statically indeterminate, and would need to be solved using other methods like the force method or the displacement method, which I won't get into in this video.

這些桁架被稱為靜力不確定桁架，需要使用力法或位移法等其他方法進行求解，本視頻中我將不對此進行討論。

Now that we know how to calculate the loads in a truss, let's explore some of the differences between truss designs.

既然我們已經知道如何計算桁架的荷載，那就讓我們來探討一下桁架設計之間的一些差異。

Here we have three different bridge trusses, the Howe, Pratt, and Warren trusses.

這裡有三種不同的橋桁架：豪氏桁架、普拉特桁架和沃倫桁架。

These trusses were all patented in the 1840s, at a time when new bridge designs were being developed to accommodate the expansion of the railroad industry.

這些桁架都是在 19 世紀 40 年代獲得專利的，當時人們正在開發新的橋樑設計，以適應鐵路行業的擴張。

They were typically constructed from a combination of wood and iron.

它們通常由木頭和鐵組合而成。

We can learn a lot about truss design by figuring out which members are in tension and which are in compression.

通過了解哪些構件處於拉伸狀態，哪些處於壓縮狀態，我們可以學到很多關於桁架設計的知識。

Let's start with the Howe truss.

讓我們從豪氏桁架開始。

We can see that its vertical members are in tension, and its diagonal members are in compression.

我們可以看到，其垂直構件處於拉伸狀態，而對角線構件處於壓縮狀態。

Members in compression usually need to be thicker than members in tension, to reduce the risk of buckling.

受壓構件通常需要比受拉構件更厚，以降低彎曲風險。

This means that the Howe truss isn't very cost effective, since the diagonal members, which need to be thicker, are quite long.

這意味著豪氏桁架的成本效益不高，因為需要加厚的對角線構件很長。

The Pratt truss addresses this issue.

普拉特桁架解決了這一問題。

Its vertical members are mostly in compression, and its inner diagonal members are in tension.

其垂直構件大部分處於壓縮狀態，內部對角線構件處於拉伸狀態。

This is more cost effective than the Howe truss, since the longer diagonal members can be thinner.

這種桁架比豪氏桁架更具成本效益，因為對角線較長的構件可以做得更薄。

Longer members are also more susceptible to buckling under compressive loading than shorter ones.

與較短的構件相比，較長的構件在壓縮荷載作用下也更容易發生屈曲。

So it's a good idea for long members to be in tension.

是以，長期的成員關係緊張是個好主意。

The design of the Warren truss was based on equilateral triangles.

沃倫桁架的設計基於等邊三角形。

The fact that all of the members are the same length is an advantage for construction, and it uses less members overall than the Howe and Pratt trusses, so it is more efficient.

所有構件的長度都相同，這對施工來說是一個優勢，而且與 Howe 和 Pratt 桁架相比，它使用的構件更少，是以效率更高。

The diagonal members alternate between tension and compression, so it does have some quite long members in compression.

對角線構件在拉伸和壓縮之間交替使用，是以確實有一些相當長的構件處於壓縮狀態。

It can also be interesting to observe how the loading in members changes as a load moves across a bridge.

觀察荷載在橋上移動時構件中荷載的變化也很有趣。

In this simplified model of a load moving across the Pratt bridge, we can see that some members alternate between tension and compression, and so will need to be designed accordingly.

在這個載荷在普拉特橋上移動的簡化模型中，我們可以看到一些構件在拉伸和壓縮之間交替，是以需要進行相應的設計。

The three-dimensional bridge we looked at earlier could be assessed as a collection of structures, and sometimes a truss will need to be assessed in three dimensions.

我們之前看到的三維橋樑可以作為一個結構集合進行評估，有時也需要對桁架進行三維評估。

This type of truss is called a space truss.

這種桁架被稱為空間桁架。

These can be analyzed in essentially the same way as planar trusses, using the method of joints and the method of sections.