Statics · Support reactions
Condition: The theoretical background develops types of support and valency, static and kinematic determinacy, the calculation of support reactions and the Gerber beam on the crane rail girder.
Open theoretical foundations →Work through every exercise first on paper: isolate the structure, draw the free-body diagram, set up the balances. The sub-steps here are your check.
Exercise 1 · Assessing determinacy
Assign to each support or connecting element its valency.
single-valued · transmits only the normal force
two-valued · holds in both directions, allows rotation
triple-valued · two forces and one moment
· one force along the axis
The valency is the number of transmitted reaction forces and reaction moments. The roller support transmits one normal force, the pinned support two force components, the fixed support additionally a moment. The two-force member is a connecting element between two part bodies — hence the .
face — one reaction too many. The beam is statically indeterminate to the first degree; the equilibrium conditions alone are no longer sufficient.
A beam rests on three parallel support bars. The counting formula works out. Is the structure usable?
All three reactions are parallel — the force balance transverse to them cannot be satisfied at all. The structure is kinematically indeterminate and therefore useless, although the number is right. Answer c) misleads: determinacy depends only on the supports , not on the loading.
Conclusion: counting is not enough
is a necessary condition. Whether the structure is also immovable has to be checked separately — the two classics are parallel reactions and lines of action meeting at one point.
Exercise 2 · Girder under two loads
A girder of length rests on the left on a pinned support A and on the right on a roller support B. Over its entire length a uniformly distributed load . At a force under to the horizontal additionally acts, directed down and to the right.
, point of application at midspan at — area under the load curve and its centroid.
Only the pinned support can hold horizontally — the roller support cannot. Hence .
Only those vertical Component has a lever arm about A; the horizontal one runs through the girder axis.
Check about B: ✓
First replace the distributed load by its resultant — after that it is an ordinary problem with concentrated forces.
Resolve into components. For the moment balance about A only the vertical one counts.
Conclusion: two loads, one method
Distributed load to its resultant, inclined force into components, reference point on a support — then only one unknown remains in each equation. And the check with the other reference point costs two lines.
Exercise 3 · Gerber beam
A Gerber beam is pin-supported at and rests at and on roller bearings. At sits a hinge. On the span acts , at the concentrated load .
Part step a: Fix the order — which sub-system do you calculate first, and which quantity does it give?
The right part G–C carries only the joint force, and — three unknowns fewer than the left one. And the moment balance about the hinge G makes both hinge forces drop out: stands alone.
From on the right part yields upwards. By actio = reactio the same force acts on the left part downwards.
Check on the solidified whole system: ✓ — the joint forces cancel in pairs in doing so.
Cut the structure at the joint into two parts and draw a separate free-body diagram for each.
Start with the part carrying the fewest unknowns — not necessarily the left one.
The joint force changes sign when passing to the other part — actio = reactio.
Conclusion: the order is half the solution
Starting with the left part means facing four unknowns. Starting with the right one gives the joint force after two lines — and thereby opens up the left part. The check on the rigidified whole system costs one line and reliably finds sign errors.
Report
The completion status of all three tasks is exported as a PDF report with a verification token — checkable via the platform's report verification.
| Exercise | Status |
|---|---|
| Exercise 1: Assessing determinacy | Open |
| Exercise 2: Girder under two loads | Open |
| Exercise 3: Gerber beam | Open |