Statically Determinate Trusses

Truss analysis determines the support reactions and axial forces in the members of a pin-jointed structural system. Under the classical idealization, straight truss members carry axial tension or compression rather than bending and shear.

Ideal truss model

Classical static analysis of a plane truss is based on several assumptions:

• joints are ideal frictionless pins;
• external loads and support reactions act at the joints;
• member centerlines meet at the joint centers;
• member self-weight is neglected or converted into equivalent joint loads;
• each member behaves as a two-force member and carries only an axial force $N$.

With these assumptions, ideal truss members have no bending moment or shear force. Real joint rigidity, loads applied between joints, and eccentric connections can introduce secondary bending effects.

Spatial structure of a truss

The interactive 3D model below helps visualize joints, longitudinal and diagonal members, spatial bracing, supports, and joint loads. Rotate the model to see how triangulation works beyond a single plane.

Static determinacy of a plane truss

For a simple plane truss with $m$ members, $j$ joints, and $r$ external reaction components, the necessary counting condition for static determinacy is:

$$m+r=2j.$$

For the common case of three independent support reactions, this becomes $m=2j-3$. The count alone does not guarantee geometric stability: an unfavorable member arrangement can still form a mechanism.

Truss analysis procedure

1. Check the structural model. Identify joints, members, supports, geometry, and applied loads.
2. Calculate support reactions. Treat the entire truss as a rigid body and apply $\sum F_x=0$, $\sum F_y=0$, and $\sum M=0$.
3. Identify zero-force members. Recognizing them early can simplify the calculation considerably.
4. Select an analysis method. Use the method of joints when forces in many members are required; use the method of sections when only a few selected member forces are needed.
5. Use one sign convention. A convenient approach is to assume each unknown member force is tensile; a negative result then indicates compression.
6. Check equilibrium. The final member forces must satisfy joint equilibrium and global equilibrium.

Method of joints

In the method of joints, individual pin joints are isolated successively. All forces acting at a joint are concurrent, so a plane joint provides two independent equilibrium equations:

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

After the support reactions are known, begin with a joint containing no more than two unknown member forces. Solve those forces, then move to an adjacent joint where the number of remaining unknowns has been reduced.

It is convenient to draw an unknown member force pointing away from the isolated joint, initially assuming tension. If the calculated value is positive, the member is in tension under that convention; if it is negative, the actual member force is compression.

Method of sections

The method of sections determines selected member forces without solving the complete truss joint by joint. Pass an imaginary cut through the members of interest, isolate one side of the cut, and apply:

$$\sum F_x=0,\qquad \sum F_y=0,\qquad \sum M=0.$$

For a statically determinate plane truss, a useful section normally cuts no more than three members with unknown forces. If the lines of action of two cut-member forces intersect, taking moments about their intersection eliminates both and can give the third member force directly.

Zero-force members

Several common zero-force cases can be recognized without numerical calculation:

• if an unloaded joint connects only two non-collinear members, both member forces are zero;
• if an unloaded joint connects three members and two are collinear, the non-collinear member has zero force.

A zero-force member is not necessarily unnecessary. It may stabilize the geometry, become active under another load case, or provide construction and bracing functions.

Method of joints or method of sections?

Use the method of joints when the goal is to determine forces throughout most of the truss. The method of sections is usually faster when only one or several particular member forces are required, especially for members far from the supports. In practical calculations, the two methods are often combined.

Common mistakes in truss analysis

• applying distributed load directly to an ideal truss member instead of converting it to joint loads;
• starting the method of joints at a joint with more than two unknown member forces;
• confusing a negative calculated force with an error rather than interpreting it as compression under the assumed tension-positive convention;
• using incorrect direction cosines for inclined members;
• cutting too many unknown members in the method of sections;
• treating $m+r=2j$ as sufficient proof that the truss is geometrically stable.

Checking the result

After solving the truss, verify equilibrium at one or more joints that were not used to obtain the corresponding member forces. The complete structure must satisfy global equilibrium, and every isolated joint must satisfy horizontal and vertical force balance. These checks are effective for finding sign, angle, and support-reaction errors.