CivilStructural AnalysishardFE ExamCourseworkPE / Advanced

Top Chord Force by the Method of Sections

A simply supported truss spans $12\ \text{m}$ in four equal $3\ \text{m}$ panels, with a constant height of $4\ \text{m}$ .

Bottom-chord joints are at $L_0(0,0)$ , $L_1(3,0)$ , $L_2(6,0)$ , $L_3(9,0)$ , $L_4(12,0)$ . Top-chord joints are at $U_1(3,4)$ , $U_2(6,4)$ , $U_3(9,4)$ .

A downward load of $20\ \text{kN}$ acts at each of $L_1$ , $L_2$ , and $L_3$ . Supports are at $L_0$ (pin) and $L_4$ (roller).

Find the force in top chord member $U_1U_2$ , and state tension or compression.

Given

P	20 kN	Load at each interior bottom joint
h	4 m	Truss height
L	12 m	Span

Hint 1

The method of sections lets you reach an interior member directly, without working joint by joint.

Hint 2

Cut through the panel containing U₁U₂, then pick a moment centre that eliminates the other two cut members.

Hint 3

Moments about L₂ kill both the diagonal and the bottom chord, leaving only the top chord.

Worked solution — try the problem first

Reactions. Total load $= 3 \times 20 = 60$ kN, symmetric, so

$R_{L_0} = R_{L_4} = 30\ \text{kN}$

Cut vertically between $x = 3$ and $x = 6$ , severing $U_1U_2$ , the diagonal, and $L_1L_2$ . Keep the left segment.

Take moments about $L_2(6,0)$ . This is the key choice: the diagonal and the bottom chord both pass through $L_2$ , so they contribute no moment and $U_1U_2$ is the only unknown left.

Forces on the left segment, with counterclockwise positive:

$R_{L_0} = 30$ kN up at $x = 0$ , lever arm 6 m: $-30(6) = -180$
Load 20 kN down at $L_1$ , lever arm 3 m: $+20(3) = +60$
$F_{U_1U_2}$ acting horizontally at height 4 m: $-4F$ (taking tension positive)

$\sum M_{L_2} = 0: \quad -4F - 180 + 60 = 0$ $-4F = 120 \quad \Longrightarrow \quad F = -30\ \text{kN}$

The negative sign means the assumed tension is wrong:

$\boxed{F_{U_1U_2} = 30\ \text{kN compression}}$

Sanity check. The top chord of a simply supported truss under gravity load is always in compression, and the bottom chord in tension — the truss behaves like a beam, with the top in compression above the neutral axis.

Concepts:Method of sectionsTruss analysisStatic equilibrium

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