Statics · Trusses

The roof truss

On the right stands the truss carrying the hall roof: span 8.0 m, ridge height 3.0 m, nine members between six joints. The load acts at the ridge — the accumulated snow load of the adjoining roof area.

For the model to hold, two idealising assumptions are made: the members are connected at the joints concentrically and by pin joints — the joints are therefore frictionless hinges — and the external forces act only at the joints.

That is the whole point of a truss. A member loaded only at its two ends is a two-force member: its force lies on the member axis. There is no bending moment for the cross-section to carry — which is why a truss gets by with remarkably little material.

Static determinacy

A truss is statically determinate if the member and support forces can be determined from the equilibrium conditions alone. Counting is particularly simple here, because at every joint a concurrent force system acts — no moments arise there. Per joint there are therefore only two equations available:

Here is the number of joints, that of the members and that of the support reactions. For the truss: six joints, nine members, three support reactions — , the count works out.

As with the support reactions, this is a necessary condition. A truss is additionally kinematically determinate if it is immovable; if it can perform finite or even merely infinitesimal motions, it is kinematically indeterminate — and thus useless, however neatly the count works out.

Assembly of a truss

Instead of counting, a truss can also be built that it is determinate from the outset. For this the textbook gives three formation rules:

Following the first formation rule one starts from a single member and adds two further members such that a triangle is formed. Repeating this grows a truss that always remains statically and kinematically determinate — triangles are the only polygons that cannot be distorted without deforming a side.

The second formation rule connects two trusses built by the first rule with three members whose lines of action must not intersect at a single point. The third permits removing a member and reinserting it elsewhere — this is how the so-called non-simple trusses arise.

The roof truss on the right is built by the first formation rule and is therefore its own example: follow in the view how triangle joins triangle.

Finding zero-force members

Before calculating, it pays to look for members whose force is zero from the outset. They are called zero-force members; every zero-force member identified is one unknown less. Three rules help:

1. Rule 1: At an unloaded joint with exactly two members whose axes do not lie on the same straight line (non-collinear), both are zero-force members (an “unloaded two-member joint”).
2. Rule 2: At a loaded joint with exactly two non-collinear members, if the external force acts along the axis of one member, the other is a zero-force member.
3. Rule 3: At an unloaded joint with exactly three members, if two member axes lie on the same straight line and the third member axis does not, the third is a zero-force member.

Collinear means that the member axes lie on the same straight line, even when the members extend from the joint in opposite directions. At an unloaded joint with two collinear members, the forces are not necessarily both zero; non-zero forces can also balance each other there. Their values must be determined from equilibrium of the rest of the truss.

All three follow directly from the equilibrium conditions at the joint — they need not be memorised but can be re-derived at any time: if no other force acts transverse to the common direction, the only remaining one must be zero.

Hands-on task: On the roof truss the load acts only at the ridge. In the view on the right, click every member that becomes a zero-force member as a result.

The method of joints

The method of joints is the systematic procedure: all joints are isolated and the equilibrium conditions are written for each of them. For statically and kinematically determinate trusses it always leads to the goal.

The convention applies that tensile forces are positive. Positive member forces therefore act away from the joints — they pull on them; negative ones indicate compression and act towards the joints. All member forces are therefore assumed as tension, and the type of loading is read off from the sign.

The only trick is the order: one starts at a joint with at most two unknown member forces — otherwise two equations face three unknowns. On this truss that is the support joint .

Ritter's method of sections

If only individual member forces are sought, it is usually expedient to apply Ritter's method of sections . Instead of working from joint to joint, the truss is separated into two parts by one section, and the equilibrium conditions are written for one of them.

The section must satisfy two conditions: it must sever the truss Total , and it may cut at most three members whose lines of action do not meet at a single point — otherwise three equations face more than three unknowns.

The actual trick lies in the reference point: placing it at the intersection of two of the three cut members makes their moments vanish, and the third member force stands alone in the equation. For the three members of one section only three individual moment balances are needed — and not a single joint beforehand.

The methods compared

Both methods give the same result — they differ in effort, and the choice hinges on one question:

All member forces sought? Then the method of joints is right. It works through systematically, tabulates well and is hard to get wrong — although an error at an early joint propagates through all the following ones.

Individual member forces sought, for instance to design the most heavily loaded lower chord? Then Ritter section saves almost all the work: one balance instead of a chain of joints. It also serves as a Specimen — confirming a member force found by the method of joints with a Ritter section checks it independently.

The textbook also names a third, graphical method, the Cremona diagram; it is not covered in this course.