Statics · Support reactions
Statics · Support reactions
What holds the structure
So far the systems have somehow hung on the wall. Now the connection itself is the subject: supports hold a structure in place and transmit forces in doing so. How many and which — that is decided by the valency of the support, and on that depends whether the reactions can be calculated at all. The guiding system of this module is the crane runway girder of the hall; it grows through the chapters — from the simple beam on two supports to the Gerber beam.
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Plane structures and their classification
Before turning to supports, the terms for what is supported. The textbook classifies components by shape and loading:
A bar has cross-sectional dimensions much smaller than its length and is loaded only along its axis — in tension or compression. If the same component is loaded transverse to its axis, it is called a beam. A curved beam is an arch; several rigidly connected, angled beams form a frame. Plane components are called disc when loaded in their plane, and plate when loaded transversely. A curved surface structure is a shell.
The crane rail girder on the right is therefore a beam: slender, and loaded transversely to its axis — by its own weight as a uniformly distributed load and by the trolley as a concentrated load.
What distinguishes bar and beam?
Types of support and their valency
Supports connect a structure to its surroundings. They establish a desired position in space and transmit forces in doing so. The number of reactions a support transmits is called its valency:
A single-valued support transmits one reaction — the roller bearing transmits only the normal force perpendicular to the bearing plane, the pendulum support only a force along its axis. A two-valued support transmits two: the pinned support holds in both directions but allows rotation; the sliding sleeve is transversely movable but rotationally rigid and transmits a normal force and a moment. A three-valued support transmits three — the fixed support holds in both directions and additionally prevents rotation.
Try it out: below the view on the right both support points of the crane rail girder can be switched. The symbol changes, and with it the number of released reactions.
No real support rotates without friction or is perfectly rigid. The support types are models: one decides which movements are practically prevented and which are not, and chooses the symbol accordingly. A steel girder on an elastomeric bearing is neither a pinned nor a roller support — for the calculation it is idealised as whichever is closer to reality.
Separating the structure from its supports
The procedure given in the textbook is always the same: isolating the structure from its supports and separating the individual sub-bodies · Sketch the free-body diagrams with all applied forces and moments as well as all internal forces and moments · Set up the equilibrium conditions · Solve for the unknowns.
The only novelty compared with the previous modules is that the released forces now follow from the valency : at a roller support one force appears, at a pinned support two, at a fixed support two forces and a moment.
Hands-on task: Isolate the crane rail girder. Mind the valency — if you draw too few reactions at a support, the feedback tells you what valency it has.
Static and kinematic determinacy
A structure is called statically determinate if the support reactions can be calculated from the equilibrium conditions alone. In the plane three equations are available for this — so there may be exactly three support reactions.
But that is not enough. A structure is kinematically determinate if it is immovable ; if it can perform finite or even merely infinitesimal movements, it is kinematically indeterminate. The textbook requires both together:
These three may be: three forces that are neither all parallel nor concurrent — or two forces and one moment, where the forces must not be parallel.
A beam on three parallel support bars has three reactions — and is nevertheless useless: horizontally the force equilibrium cannot be satisfied at all. And a disc whose three support lines of action meet at one point can rotate infinitesimally about that point — there the moment condition about fails. In both cases the number is right, the arrangement is not.
And a note the textbook gives explicitly: static determinacy depends only on the supports , not on the loading. A statically determinate girder stays determinate no matter what stands on it.
Hands-on task: Set the two supports so that the count is correct — exactly three reactions — but the structure is still not usable.
Calculating the support reactions
Now the calculation on the crane rail girder. Two things are new compared with the module “General force systems”: the distributed load must first be replaced by its resultant — magnitude equal to the area under , point of application at its centroid, exactly as developed in the previous module. And the reference point is again chosen wisely: on one of the two supports, so that its reactions drop out.
Hands-on task: Move the trolley so that both supports carry the same load.
Principle of superposition
The crane rail girder carries two loads: its own weight as a distributed load and the trolley as a concentrated load. Because all equilibrium conditions are linear in the forces, both load cases may be calculated separately and the results added:
This is more than a computational convenience: when the trolley travels, only its share changes — the share of the own weight stays constant. That is precisely why, for a crane rail girder, one can determine the influence line of the travelling load once and simply add the constant share.
The view on the right shows both shares separately — switch the load case and watch how the bars in the balance sheet divide.
How large is the share of the own weight in the support reaction when the trolley travels?
Spatial structures
A body free to move in space has six degrees of freedom: one translation each in the -, - and directions and one rotation each about these three axes. Accordingly six equilibrium conditions are available, and a spatial structure is statically determinate when exactly six reactions occur and it is immovable.
The support types are the same ideas in three dimensions: the pendulum support remains single-valued, a spatial pin holds three directions, a spatial fixed support six. The view on the right shows the crane rail girder with an additional wind load transverse to the girder axis — fixed viewing direction as in the textbook.
In space the moment equilibrium is formed not about one point, but about axes . Here too a clever choice pays: placing a reference axis through as many support reactions as possible makes their moments vanish — the same principle as in the plane, one dimension higher.
Multi-part structures
Structures often consist not of a single rigid body but of several connected to one another. The connecting elements transmit forces and moments that are made visible by cuts — just as with the supports, only between two sub-bodies instead of against the surroundings:
A pendulum bar transmits only a force along its axis, hence . A hinge transmits a force in any direction, hence two components — — but no moment. A parallel guide transmits a normal force and a moment, likewise . For all of them actio = reactio holds: the connecting reactions act in opposite directions on the two sub-bodies.
For every isolated sub-body three equilibrium conditions hold. With sub-bodies these are equations; they face support reactions and connecting reactions. Necessary for static determinacy is therefore
The status line above the balance sheet computes this along — switch the supports and watch when it jumps from “determinate” to “indeterminate”.
As a check, a multi-part structure may be regarded as solidified — that is, as if the joints were welded. Then only three equilibrium conditions apply to the whole system, and the connecting reactions do not appear in them at all: they cancel in pairs. If the calculated support reactions satisfy those too, the calculation was correct.
three-hinged arch
The simplest multi-part case: two sub-bodies, supported by two pinned supports and connected by a joint. The count works out — equations face support reactions and hinge reactions.
The name is misleading: they need not be arches. Frames and even trusses can be built to the same scheme — what matters are the three hinges: two at the supports, one in between.
The calculation runs over the sub-systems: three conditions per sub-body, together six equations for six unknowns. A useful device is the moment balance about the hinge — there the joint forces drop out, because their lever arm is zero.
Which reference point is particularly useful for the three-hinged arch?
hinged beam (Gerber beam)
Finally the form into which the crane rail girder grows: a continuous girder over several spans would be statically indeterminate. By building in hinges it becomes determinate — such a girder is called a hinged beam or Gerber beam.
For joints the girder falls into sub-bodies, and each joint provides connecting reactions. The condition holds unchanged.
The solution path has a clear beginning: start with the sub-system that is determinate on its own — usually a cantilever hanging only on one support and one joint. From it the joint forces follow, and with them the next sub-system becomes accessible.
Here too the count is not enough. If the joints sit unfavourably — for instance two joints and both supports on one straight line — the structure remains movable even though is satisfied. The check is always the same question: can any part move without anything deforming? If so, the structure is kinematically indeterminate and therefore useless.
Control accounts
All results for the example values of the guiding system — for checking by hand.
support force B
Moment balance about A — there both reactions at A drop out.
support force A
The force balance gives A once B is known.
check about B
A different reference point, the same statement — the calculation is correct.
Sources: Gross/Hauger/Schröder/Wall, Technische Mechanik 1 (Springer), chapter 5 · Engineering Mechanics I course (HSO, summer term 2026).
On to the three exercises — from assessing determinacy through the girder under distributed and inclined loads to the Gerber beam.
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