Certain elements of a truss system are not loaded with force. This is referred to as a zero-force member, or simply a zero member.
Identifying these members simplifies the truss analysis procedure. Truss analysis using the method of joints is considerably simplified if one first determines which members do not sustain any load. Still, a zero-force member is not necessarily an unnecessary member. It may be required to ensure the truss stability during construction, to provide support when the applied loading is altered, or to restrain a long compression member.
What Is a Zero-Force Member?
For an ideal pin-jointed truss, each member is treated as a two-force member. It carries axial tension or compression along its own centreline. If equilibrium of a joint requires the axial force in a connected member to be zero, that member is a zero-force member for that load case.
The two usual rules come directly from the joint equilibrium equations:
Rule 1: Two Non-Collinear Members at an Unloaded Joint
If only two non-collinear members meet at a joint and no external load or support reaction acts at that joint, both members are zero-force members.
Consider joint J with members JA and JB. Since the two members are not on one straight line, a force in one member has a component that cannot be balanced by the other member. The only solution that satisfies equilibrium in both directions is:
The angle between the members does not need to be 90 degrees. It only has to be different from 0 or 180 degrees.
Rule 2: Three Members With Two Collinear Members
If three members meet at an unloaded joint and two of them are collinear, the third non-collinear member is a zero-force member.
At joint K, assume KL and KM lie on the same straight line while KN is inclined or perpendicular to them. The forces in KL and KM act along the same line. Therefore, the force component perpendicular to that line can come only from member KN. Equilibrium requires:
This rule does not say that the two collinear members are zero. They may carry equal or different axial forces depending on the rest of the truss and its loading.
Quick Decision Table
| Joint condition | Conclusion |
|---|---|
| Two non-collinear members; no load or reaction | Both members are zero-force members |
| Three members; two collinear; no load or reaction | The third, non-collinear member is zero |
| External load or support reaction acts at the joint | Do not apply the two basic rules directly; draw the joint free-body diagram |
| Four or more active members meet at the joint | No immediate conclusion unless a known zero member reduces the joint |
| Members only look collinear in a rough sketch | Check the geometry or coordinates before using Rule 2 |
Worked Example: Finding Zero Members Step by Step
Look at the lower part of the figure. At joint C, members BC and CD are collinear. Member CE is the third member, and no external load or reaction acts at C. From Rule 2:
Now move to joint E. Member CE is already known to be zero. This leaves only members EA and EF at the unloaded joint. They are non-collinear, so Rule 1 can now be used:
This is why the check must be repeated. One zero-force member can create a new two-member or three-member condition at the next joint.
How to Identify Zero-Force Members in a Full Truss
- Draw all external loads and support reactions. A support symbol can be easy to overlook, especially at an end joint.
- Start at unloaded joints. Look for joints with only two or three connected members.
- Check collinearity. For a three-member joint, confirm that two members share the same straight line.
- Mark each zero member clearly. Do not mentally remove it and lose track of the joint connection.
- Scan the joints again. A known zero member may reveal another zero member at the adjacent joint.
- Use joint equilibrium when a rule is not obvious. Resolve forces parallel and perpendicular to the collinear members.
- Continue with the method of joints or sections. Start from a joint with no more than two unknown active member forces.
Cases Where the Simple Rules Do Not Apply Directly
A Load Acts at the Joint
An external load changes the joint equilibrium. A member can still have zero force, but you must include the load direction in the free-body diagram. For example, when an external load is exactly collinear with one connected member, a transverse member may be zero. That conclusion comes from equilibrium, not from blindly applying Rule 1 or Rule 2.
A Support Reaction Acts at the Joint
A pin can provide two reaction components and a roller normally provides one reaction in its restraint direction. Treat these reactions as external joint forces. The support joint is not unloaded.
More Than Three Members Meet
The two inspection rules normally do not identify a zero member immediately when four or more active members meet at the same joint. First solve another joint or use equilibrium to reduce the number of unknowns.
The Truss Is Not Ideal
Real truss joints have gusset plates, member self-weight, connection eccentricity, fabrication tolerance, and sometimes partial fixity. The term zero-force member belongs to the ideal analysis model. A real member may still experience small secondary forces.
Why Are Zero-Force Members Used?
- They can stabilize the truss during erection before the complete loading system is present.
- They can carry force under another load case, such as wind from the opposite direction, moving load, or an unsymmetrical roof load.
- They may reduce the unbraced length of a compression member and help control buckling.
- They can maintain the intended geometry and provide convenient panel points for later loading or bracing.
- They may be needed for redundancy, vibration control, or service conditions not shown in one simple analysis case.
Common Mistakes
- Calling a member zero without checking for a support reaction at the joint
- Applying Rule 2 when the two supposed collinear members have a small change in direction
- Assuming all three members at an unloaded joint are zero
- Stopping after the first scan instead of checking the truss again
- Removing a zero member from the actual design because it is zero in one analysis case
- Using the inspection rules for a frame whose members carry bending, not an ideal pin-jointed truss
Summary
There are two main arrangements to remember. Two non-collinear members at an unloaded joint are both zero. When three members meet at an unloaded joint and two are collinear, the third member is zero.
Always check loads and reactions first, mark the zero members, and scan the truss again. If the joint does not match one of the rules exactly, use ΣFx = 0 and ΣFy = 0 instead of guessing.
Reference
Hibbeler, R. C., Structural Analysis, 10th Edition in SI Units, Section 3.4: Zero-Force Members.
Read Also:
Equations of Static Equilibrium