Geometry, developed length & sign conventions

This is a static copy of the chapter for search engines. The interactive version has animated figures, check questions, and buttons that load the example into the calculator.

Geometry, developed length & sign conventions

Before anything can be computed, the shape has to be described unambiguously. FrameLab uses the simplest description that survives contact with real problems: a chain of segments, each with a length L_i and an absolute direction angle α_i measured from the +x axis, counter-clockwise positive.

Geometry is entered segment by segment: a length and an absolute direction angle. The developed length s runs continuously along the bar and is the x axis of every diagram.

Absolute angle, not kink angle

Some textbooks describe the geometry by the kink at each corner. That is elegant on paper but painful in practice: change one angle early in the chain and every later member rotates with it. Absolute angles keep each segment independent, and the kink angle is then just the difference

Δα_i = α_i+1 − α_i

which FrameLab displays for you next to each segment.

The developed length s

Diagrams need a horizontal axis. For a bent bar the natural choice is the developed length — the distance measured along the bar from the start:

s ∈ ⟨0, S⟩, S = ΣL_i

Unroll the bar and it becomes a straight beam again; the diagrams then read exactly like beam diagrams, except that they may jump at each corner. FrameLab marks the corners with dashed amber lines on every chart, so you can always see where you are.

Local axes and the sign convention

Each member has its own local frame that travels with it:

t — along the member, in the direction of increasing s;
n — t rotated by +90° (to the left of travel), still in the frame plane;
z — out of the plane, towards the viewer.

Quantity	Positive means
N	tension (the bar is being pulled apart)
V	acts along −n on the cut face whose outward normal is +t (the usual beam shear rule)
M	tension on the −n side (the right-hand side of travel)
T	torque vector pointing along +t (right-hand rule)
M_⊥	tension on the −z side

The sign of M and V therefore depends on which way you decided to walk along the bar — reverse the segment order and they flip. This is not a defect of the method; it is inherent to bent bars, and it is exactly why FrameLab also draws the moment diagram on the structure, on the side that is in tension. That picture is independent of the direction of travel and is what you should trust when detailing a joint.

💡 Loads are placed by choosing a segment and a position along it, not by a global s value. That way your loads stay where you put them when you later change a length earlier in the chain.

Example

A shallow two-segment bar (+30° then −30°) on a pin and a roller, with a force at the crown. Notice that the diagrams are continuous in M but jump in N and V at the kink.