Structures — trusses and frames
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| truss/trʌs/ | khung giàn |
| tension/ˈtenʃn/ | lực kéo |
| compression/kəmˈpreʃn/ | lực nén |
| method of joints/ˈmeθəd ɒv dʒɔɪnts/ | phương pháp nút |
| zero-force member/ˈzɪərəʊ fɔːs ˈmembə/ | thanh không lực |
| method of sections/ˈmeθəd ɒv ˈsekʃnz/ | phương pháp cắt |
| frames and machines/freɪmz ænd məˈʃiːnz/ | khung và máy |
Hundreds of slim bars, and none of them bends
- A railway bridge carries a train on a web of thin steel bars. Each bar is only pulled or pushed along its length, never bent.
- The secret is the triangle. A pinned triangle cannot change shape, so every bar is a two-force member, and every bar force can be found.
What makes a truss
- A truss 桁架 is straight members joined by frictionless pins and loaded only at the joints. Every member is a two-force member.
- So each member carries pure tension 拉力 (it is stretched, and pulls on its joints) or compression 压力 (it is squeezed, and pushes on them). Take tension as positive.
A straight member that is pinned at both ends and loaded only at its joints carries only a force along its own axis.
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
It is a two-force member, so the two end forces are equal, opposite and along the line through both pins, which is its axis.
Method of joints: two equations at each pin
- Method of joints 节点法: draw one joint as a particle. Two equations, $\Sigma F_x = 0$ and $\Sigma F_y = 0$, find two member forces.
- Draw every unknown member force as tension (pulling away from the joint). A negative answer then means compression.
- A joint at the end of a wall bracket carries $3.6\ \text{kN}$ hanging down. A horizontal member and a member rising on 3-4-5 geometry hold it.
- Vertically: $F \times \tfrac{3}{5} = 3.6$, so the inclined member carries $6\ \text{kN}$ tension: the joint hangs from it.
- Horizontally: the horizontal member takes $6 \times \tfrac{4}{5} = 4.8\ \text{kN}$ compression: it pushes the joint away from the wall.
A $6\ \text{kN}$ load hangs at a truss joint held by a horizontal member and a member rising on 3-4-5 geometry (3 up, 4 across). What is the force in the inclined member, in kN?
Độ khó 2/5 · 140 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
Only the inclined member has a vertical part: $F \times \tfrac{3}{5} = 6$, so $F = 10\ \text{kN}$, in tension.
In the same joint, what does the horizontal member carry?
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
The inclined member pulls the joint towards the wall with $10 \times \tfrac{4}{5} = 8\ \text{kN}$. The horizontal member must push back with $8\ \text{kN}$, so it is in compression.
Where to start
- Find the support reactions first, using the whole truss as one rigid body.
- Then start at a joint with no more than two unknown members, and walk joint to joint. Each solved member gives one known force to the next joint.
Zero-force members, found by eye
- At an unloaded joint with only two members that are not in line, both carry zero. At an unloaded joint with three members, two of them in line, the third carries zero.
- These zero-force members 零杆 are not waste. They stop long slim members from buckling and carry load when the loading changes.
In which cases must a member carry zero force? Select all that apply.
Độ khó 2/5 · 140 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
With no load, the two in-line forces cancel each other, so nothing is left to balance the third. A load at the joint, or a support reaction, changes that.
Worked example: a three-panel truss
- A Pratt truss spans $6\ \text{m}$ in three $2\ \text{m}$ panels and is $2\ \text{m}$ tall: bottom joints A, C, D, B; top joints E above C and F above D; diagonals A–E, E–D and F–B. It carries $6\ \text{kN}$ at C and at D.
- By symmetry $R_A = R_B = 6\ \text{kN}$. Joint A: $6 + F_{AE}\sin 45^{\circ} = 0$ gives $F_{AE} = -8.49\ \text{kN}$, so compression.
- Working on: bottom chord $6\ \text{kN}$ tension, top chord E–F $6\ \text{kN}$ compression, verticals $6\ \text{kN}$ tension, inner diagonal E–D zero.
- The truss sags, so its bottom chord stretches and its top chord shortens. A bottom chord in compression under downward loads means a sign slip.
In the three-panel Pratt truss with $6\ \text{kN}$ at C and D, what is the size of the force in end diagonal A–E, in kN?
Độ khó 2/5 · 140 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
At joint A only A–E has a vertical part: $6 + F_{AE}\sin 45^{\circ} = 0$, so $F_{AE} = -8.49\ \text{kN}$, which is $8.49\ \text{kN}$ compression.
Method of sections: cut straight to the member you want
- Method of sections 截面法: cut through at most three members and use one piece as a free body. Three equations find three forces.
- Take moments about the point where two cut members meet. Only the third force is left. For the truss above, cut E–F, E–D and C–D; moments about E on the left piece: $-6 \times 2 + 2F_{CD} = 0$, so $F_{CD} = 6\ \text{kN}$ tension.
Put the method of sections in order.
Độ khó 3/5 · 200 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
The reactions come first because the piece you keep usually includes a support.
Frames come apart into pieces
- Frames and machines 刚架与机构 contain members loaded along their length. Their pin forces do not act along the member, so the joint method fails.
- Take the frame apart: one free-body diagram per member, with equal and opposite forces at each shared pin. Solve any two-force member first, because its direction is known.
Match each structure or task to the best method.
Độ khó 2/5 · 140 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
Choosing the method is half the work. The section method saves walking through every joint to reach one member.
Marks that slip away
- Mixing up the arrow directions. Draw every unknown as tension, pulling away from the joint, and let the sign decide.
- Cutting four members in a section. Three equations can only find three forces.
- Loading a member between its joints. Then it bends, it is not a truss member, and the truss method does not apply.
You've got it
- Truss members carry only tension or compression; take tension as positive.
- Joints: two equations per pin. Sections: cut at most three members and take moments where two of them meet.
- Find zero-force members first; take frames apart member by member.