Torce Lines

GSA gives the option of plotting ‘Torce lines’, which are similar to thrust lines but include the effects of torsion. They are equal for plane frames.

Good structural design requires a clear thinking head, well-presented information, and some creative flair. At the detailed level designs progress by iterative evolution and the quality of the design depends on the effectiveness of the interaction between the designer and the information which describes the behaviour of the structure in the current cycle. In today’s world this information is almost always presented by a computer. Graphical plots of the parameters used in the numerical analyses are often ergonomically inefficient, meaning that although they contain the information they fail to transmit insight or understanding into the head of the designer. Better representations are needed.

For many applications thrust lines meet this need. They are useful where members are subjected to combined axial load and bending, particularly for compression members made of a material that takes no tension. In these cases the adequacy of the member can often be reasonably described by a limit on the eccentricity of the thrust line. Thrust lines are frequently used by the designers of masonry arches.

A thrust line is the locus along a member of points drawn at an eccentricity of

$$\frac{M}{F_{x}}$$

from the centroidal axis of the member, where $M$ is the moment and $F_{x}$ the axial force. It follows the line of action of the force carried by the member and two related points follow from this:

• No scale is needed when drawing a thrust line. It is not a diagram superimposed on a view of a structure. It occupies the position where it is shown.
• It is not subject to any sign conventions.

However the key feature of the thrust line which makes it so useful is that it is a complete description of the forces carried by the member. Because of this completeness property a designer can, with no loss of accuracy, substitute a thrust line, which he or she can visualise, for the combination of the three numbers representing the axial force, moment, and shear, which remains stubbornly abstract.

In a computer model of a structure the thrust line is derived from the forces in a member but

because it occupies the actual position of the force transmitted through the structure being modelled it follows that its position remains fixed in space even if the member which ‘supports’ it is moved, providing the movement of the member does not change the force being transmitted. This can be referred to as the invariance property of thrust lines.

The above statements refer to thrust lines in two dimensions, as in a plane frame model. Thrust lines can also be drawn in three dimensions but unfortunately in shifting up a dimension they lose both their completeness and invariance properties and so they lose their usefulness to the designer.

The general state of force in a section of a member in three dimensions cannot be reduced to a single force, but it can be reduced to a combination of a force and a torque or, more completely, to any one of an infinite number of force/torque pairs. There is just one force/torque pair, hereafter referred to as the torce, for which the force axis and the torque axis are parallel. In three dimensions it is torce lines that have the completeness and invariance properties and so it is torce line which can be used by designers to visualise structural actions.

The position of a torce line is derived from the forces in a section of a member as follows

The resultant shear force, $Q$, and its angle to the $z$ axis, $\alpha$, are given by

$$\begin{aligned} Q &= \sqrt{{F_{y}}^{2} + {F_{z}}^{2}} \\ \alpha &= \tan^{- 1}\left( \frac{F_{z}}{F_{y}} \right) \end{aligned}$$

The angle $\beta$ between the member axis and the thrust line is given by

$$\beta = \tan^{- 1}\left( \frac{Q}{F_{x}} \right)$$

The torque, $T$, of the torce is resolved into moments about the three member axes. The component about the $x$ axis is the torque $M_{x}$ in the member. The other two components modify the bending components $M_{x}$ and $M_{x}$.

Because the force components of the torce (as opposed to the moment components) are the same as those on the member, the thrust line and the torce are parallel.

Hence

$$T\cos\beta = M_{x}$$

Therefore

$$T = M_{x}\sec\beta$$

Eccentricities of the torce, $t_{y}$ and $t_{z}$, are given by

$$\begin{aligned} t_{y} &= - \frac{M_{z} - T\sin\beta\sin\alpha}{F_{x}} \\ t_{z} &= \frac{M_{y} - T\sin\beta\sin\alpha}{F_{x}} \end{aligned}$$

hence

$$\begin{aligned} t_{y} &= - \frac{M_{z} - M_{x} \cdot \frac{F_{z}}{F_{x}}}{F_{x}} \\ t_{z} &= \frac{M_{y} - M_{x} \cdot \frac{F_{y}}{F_{x}}}{F_{x}} \end{aligned}$$

How do torces behave? A torce is a unique representation of the force state in a section of structure. Torces can be added or subtracted. There is always a unique result. Note that if two torces lie in the same plane their addition or subtraction is not necessarily in that plane. In three dimensions four forces on a body in equilibrium are necessarily coincident. The same is not true for torces. Consider four coincident torces in equilibrium. If one of the torces is translated without changing its direction then the body is subjected to a moment. Equilibrium can be restored by adjusting the torques in the other three torces. Hence equilibrium is achieved with four non-coincident torces.

Often it will be known that one or more of the torces have zero torque. For example gravity imposes zero torque on a body. How many torqueless torces are needed to restore the coincidence rule? The answer is four. This is the same as the four forces on a body rule. If just one of the four carries a torque it can be resisted by a combination of lateral shifts of the other three forces and coincidence is lost. The usefulness of the concept of coincidence is limited to statements such as: The sum of two coincident torces, one of which is torqueless, is coincident with its components.

For the designer the lesson from this is that, unless a structure is conceived as being truly three dimensional, it is often better to analyse it as a two dimensional plane frame during the design evolution phase so the coincidence rule can be used to understand what is driving the magnitudes of the forces.

This item was written by Angus Low for Feedback Notes [an Arup internal publication] (1999 NST/7) and is reproduced here with permission