Method of Joints

  • Analyze internal forces of trusses using the method of joints.

In this method, the free body diagram of each joint is separately analyzed to obtain internal forces in the truss members. The unknown forces are determined by equilibrium equations.

  • Determine the support reactions.

  • Draw the free body diagram for each joint.

  • Write the equations of equilibrium for each joint,

  • Begin solving the equilibrium equations at a joint where only two unknown reactions exist. Work your way from joint to joint, selecting the new joint using the criterion of two unknown reactions.

  • Solve the joint equations of equilibrium simultaneously, typically using a computer or an advanced calculator.

Method of Joints

First, determine the support reactions for the truss.

\[\begin{aligned} \Sigma M_A &= 0\\ -500 \times 10 + B_y \times 10 &= 0\\ B_y &= 500~KN \end{aligned}\] \[\begin{aligned} \Sigma F_y &= 0\\ A_y + B_y &= 0\\ A_y &= -500~KN \end{aligned}\] \[\begin{aligned} \Sigma F_x &= 0\\ A_x + 500 &= 0\\ A_x &= -500~KN \end{aligned}\]

The equations of equilibrium for Joint A:

FBD of Joint A

\[\begin{aligned} \Sigma F_x &= 0 \\ F_{AC} - 500 &= 0 \\ F_{AC} &= 500~KN \end{aligned}\] \[\begin{aligned} \Sigma F_y &= 0 \\ F_{AB} - 500 &= 0 \\ F_{AB} &= 500~KN \end{aligned}\]

The equations of equilibrium for Joint B:

FBD of Joint B

\[\begin{aligned} \Sigma F_x &= 0\\ F_{BC} \cos 45^\circ + 500&= 0\\ F_{BC} &= 707.2~KN \end{aligned}\]

The forces in the truss can be summarized as: \[\begin{aligned} F_{AB} &= 500~KN (T)\\ F_{BC} &= 707.2~KN (C)\\ F_{AC} &= 500~KN (T) \end{aligned}\]

Solved Examples

Solved Example:

25-1-01

A structure is __________ when the static equilibrium equations are not sufficient for determining the internal forces and reactions on that structure.

Solution:
Statically indeterminate structures indicate that there's at least one more unknown reaction force than there are equations of equilibrium, meaning that the sum of forces and moments in each direction is equal to zero.

Correct Answer: B