Trusses: Method of Sections

Introduction

The method of sections is used to determine forces in specific truss members without solving the entire truss joint-by-joint. By passing an imaginary cut through the truss and analyzing one section using equilibrium equations, internal member forces can be solved directly. This method is especially efficient when only a few member forces are required.

Method of Sections Procedure

1. Solve support reactions using global equilibrium.

2. Cut a section through desired members.

3. Isolate one side of the truss.

4. Draw section free-body diagram.

5. Assume unknown member forces are in tension.

6. Apply equilibrium equations:

   1. $\Sigma F_{x} = 0$

   2. $\Sigma F_{y} = 0$

   3. $\Sigma M = 0$

7. Positive result $\rightarrow$ tension

8. Negative result $\rightarrow$ compression

Choosing the Best Cut

• Pass through the target members.

• Cut through no more than 3 unknown members (3 unknown forces).

• Use moments around intersection points of unknown members or joints where multiple unknowns pass through.

Example:
Find the force in member LK and if the member is in tension or compression.

1) Start with Global Equilibrium to solve for Reaction Forces at points A and G:

$\Sigma F_{x} = 0 = A_{x}$

$\Sigma F_{y} = 0 = A_{y} + G_{y} - 10\ kN - 20\ kN - 10\ kN$

$A_{y} + G_{y} = 40\ kN$

$⟳^{+}\ \Sigma M_{G} = 0 = (A_{y})(12\ m) - (10\ kN)(8\ m) - (20\ kN)(6\ m) - (10\ kN)(4\ m)$

Solving: $A_{y} = 20\ kN$

Substituting into $A_{y} + G_{y} = 40\ kN$: $G_{y} = 20\ kN$

2) Cut the truss vertically through member LK, and take moments about point C to cancel out LC and BC

Moment arm from $F_{LK}$ to point C is length LC, or $\sqrt{2^2+2^2}$

Moment arm from $A_{y}$ to point C is length AC, or $4\ m$

$⟳^{+}\ \Sigma M_{C} = 0 = (20\ kN)(4\ m) + (F_{LK})(\sqrt{2^2+2^2})$

Solving: $F_{LK} = - 28.28\ N$

$F_{LK} = 28.28\ N\ (C)$