Training Civil Engineering Truss Analysis — Method of Joints
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Truss Analysis — Method of Joints

24 min Civil Engineering

Truss Analysis — Method of Joints

A truss is a structure made of straight members connected at joints (nodes). Each member carries only axial force (tension or compression).

Method of Joints

At each joint, apply:

$$\sum F_x = 0 \qquad \sum F_y = 0$$

Start at a joint with at most two unknowns. Solve one joint at a time, working through the truss.

Convention

Assume all members are in tension (pulling away from the joint). If the result is negative, the member is in compression.

Example 1

A simple triangular truss has joints A (pin, left), B (roller, right), and C (top). Span AB = 6 m, height = 4 m. A vertical load of 20 kN acts at C. Find all member forces.

By symmetry: $R_A = R_B = 10$ kN.

At joint A: $\tan\alpha = 4/3$, so $\alpha = 53.13°$.

$$\sum F_y: F_{AC}\sin 53.13° + 10 = 0 \implies F_{AC} = -12.5 \text{ kN (compression)}$$

$$\sum F_x: F_{AB} + F_{AC}\cos 53.13° = 0 \implies F_{AB} = 7.5 \text{ kN (tension)}$$

By symmetry: $F_{BC} = -12.5$ kN (compression).

Example 2

For the truss above, verify equilibrium at joint C.

At C: $F_{CA}$ and $F_{CB}$ both push up and inward (compression), and the 20 kN load acts down.

$$\sum F_y: 12.5\sin 53.13° + 12.5\sin 53.13° - 20 = 10 + 10 - 20 = 0 \;\checkmark$$

Example 3

A Pratt truss has bottom chord members AB and BC each 3 m, height 3 m. Load of 15 kN at the midpoint of the top chord. Is the bottom chord in tension or compression?

In a Pratt truss under downward loading, the bottom chord is in tension and the top chord is in compression. Diagonal members carry tension.

Practice Problems

1. A triangular truss: span = 8 m, height = 3 m, load 24 kN at the apex. Find all member forces.
2. Can you solve a joint with three unknown member forces? Why or why not?
3. A truss has 7 joints and 11 members. Is it statically determinate?
4. A Warren truss (equilateral triangles) has span 12 m with 4 panels. A 10 kN load at each of the 3 interior bottom joints. Find the reaction forces.
5. In the Warren truss of #4, find the force in the bottom chord member at midspan.
6. A zero-force member occurs when two non-collinear members meet at a joint with no external load. Identify zero-force members in a given truss.
7. Triangular truss: AB = 5 m (horizontal), height = 5 m. Load 30 kN at C. Find reactions and member forces.
8. A simple truss has members meeting at 45° angles. Side length = 4 m. Vertical load 20 kN. Find the diagonal member force.
9. If a truss has $m$ members, $j$ joints, and $r = 3$ reactions, what condition ensures static determinacy?
10. A roof truss spans 10 m with 5 m height. Snow load equivalent to 8 kN at each of the 3 top joints. Find the maximum member force.
11. Use the method of sections to find the force in a specific member of a Pratt truss.
12. A Howe truss has vertical web members and inclined diagonals. Under gravity loading, are the diagonals in tension or compression?
Show Answer Key

1. $R_A = R_B = 12$ kN; $\alpha = \arctan(3/4) = 36.87°$; $F_{AC} = F_{BC} = -20$ kN (C); $F_{AB} = 16$ kN (T)

2. No — only 2 equilibrium equations per joint in 2-D; need at most 2 unknowns

3. Check: $m + r = 11 + 3 = 14 = 2j = 14$. Yes, statically determinate.

4. By symmetry: $R_A = R_B = 15$ kN

5. Using method of sections at midspan: $F_{\text{bottom}} \approx 17.3$ kN (T)

6. Look for joints where only two members meet with no external load — those members are zero-force.

7. $R_A = R_B = 15$ kN; $F_{AC} = F_{BC} = -21.2$ kN (C); $F_{AB} = 15$ kN (T)

8. $F_{\text{diag}} = 20/\sin 45° = 28.28$ kN

9. $m + r = 2j$

10. Total load 24 kN; reactions 12 kN each; maximum compression in top chord ≈ 13.4 kN

11. Cut through 3 members including the target; take moments about a point where 2 unknown forces intersect.

12. In a Howe truss, diagonal members are in compression under gravity loading (opposite of Pratt).

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