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Types of Supports on a Plane
and in 3D Space
From this text you will learn:
- what is a support in structural mechanics
- what degrees of freedom of movement we have on a plane
- what types of supports we have on a plane
- what types of supports we have in 3D space in systems such as:
- how to write boundary conditions in differential notation (finite difference method)
You will also find computational examples, because, in reality, every task with beams, frames, trusses begins with marking the reactions in supports and their calculation.
On a plane, there are three main degrees of freedom of movement:
It is worth noting that degrees of freedom may vary depending on the context and specifics of the construction. For example, the degrees of freedom of a complex mechanism may include additional movements, such as tilting or rotating around other axes.
In the analysis of the movement of objects on a plane, these three basic degrees of freedom are a key reference point.
- popular name/names,
- graphic symbol/symbols,
- what reaction occurs in it,
- what degree of freedom of movement it takes away.
A horizontal line under the triangular symbol of the hinge support shows in which direction there is possibility of movement. The direction perpendicular to the possibility of movement is blocked, and in this direction, a reaction arises. Of course, a hinge support can be used at various angles, as shown below:
If the support and reaction are at an angle, often for simplicity of calculations, it will be more convenient for us to decompose this reaction into components:
After decomposition into components, in calculations, we use either one, resultant reaction (red) or both components (blue) – never both at the same time. That means when writing the equation of static equilibrium or determining internal forces, we use either the components or the resultant reaction.
Note! If the outgoing rod is additionally ended with a hinge also from the other side, then we are dealing with a truss bar/tendon/cable which you will find more information about below.
Often the dilemma regarding marking reactions arises when we see a non-sliding support rotated at a different angle. It is important to remember that in such a support we always have two perpendicular reactions to each other, and it is up to us in which directions we mark these reactions. Every variant shown below is OK.
or with tethering the beam on the cables themselves. (See free example 2 )
For us to talk about a cable/tendon/truss bar the following assumptions must be met:
Below on the left an example of two truss bars, on the right due to the lack of a hinge in the middle the bar is not linear, so we do not deal with a truss bar but with a frame system supported by two non-sliding hinge supports.
This is a fundamental difference between these two constructions, and they are calculated completely differently. In hinge supports, we show reactions as shown earlier, but as for reactions in a truss bar, we have two possibilities of marking reactions
On our site, you will rather find the first variant, as it seems to be better
In the case above:
Rotation lock is a constraint that blocks, as the name suggests, only the possibility of rotation of the point (node). We can encounter such markings of rotation lock:
In this support, there is only a reaction in the form of a bending moment, the node has freedom of movement in every direction, but it cannot rotate – that means the shape of the node after displacement must be the same as before displacement. The first from the left graphic marking resembles a wedge, and we can also encounter such a name for this constraint. The third marking also resembles a roller both vertically and horizontally.
All these degrees of freedom together define how an object can move and rotate in 3D space. Thanks to this classification, engineers and designers are able to analyze and model the movements of objects and structures in three dimensions.
It will be difficult to classify supports in 3D space, therefore, below we will rather provide general principles regarding supports and the most popular supports in constructions such as:
A grillage is called a flat rod structure, which is loaded perpendicular to the plane of the grillage beams. Depending on the way beams of the grillage are connected, we can distinguish:
Since the load on the grillage acts in the direction perpendicular to the plane of the beams and given that beams transfer only transverse force to each other, only transverse forces and bending moments occur in a hinged grillage. This affects the type of reactions that can occur in supports, and we can distinguish such supports as:
Solutions for tasks with hinged grillages can be found in the online Course structural mechanics – specifically here, they are covered by subscription (one subscription provides access to materials on the entire site – i.e., to all online courses).
In rigid grillages, we have the same supports as in hinged grillages, however, here the difference is that we also deal with torsion, so in fixation besides the reaction in the form of a bending moment there will also be a reaction in the form of a torsional moment. We will also have more options for rollers (either it blocks bending, or torsion, or both).
Types of supports in rigid grillages:
And here we have a different way of marking a roller than before, generally, I could not find anywhere in literature a roller for a rigid grillage, but the marking shown below seems to be sensible, it also matches the marking which appears in programs like Robot Structural Analysis.
Well, the first marking is often used as a full fixation, so it's best to specify if there's such a possibility what type of support the person who drew the task topic had in mind.
In one place, one, two, or three directions can be blocked, so now a variant with two blocks in one place
With other types of support than those listed above in spatial systems, you will hardly encounter (I have not), generally, as a rule, any direction of movement and the possibility of rotation in any direction can also be blocked in any combination, for example
and the like.
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- what is a support in structural mechanics
- what degrees of freedom of movement we have on a plane
- what types of supports we have on a plane
a) sliding hinge support
b) fixed hinge support
c) truss bar/tendon/cable
d) fixed support/cantilever
e) support with slide/roller
f) rotation lock/wedge
- what types of supports we have in 3D space in systems such as:
a) hinged frames,
b) rigid frames,
c) 3D frames,
- how to write boundary conditions in differential notation (finite difference method)
You will also find computational examples, because, in reality, every task with beams, frames, trusses begins with marking the reactions in supports and their calculation.
What is a support in structural mechanics?
Support in structural mechanics is an element that provides reaction to forces and moments acting on a structure or construction. In the context of engineering and structural design, supports are essential components that allow for the transfer of loads, ensure stability and balance of the structure, and enable it to function safely and efficiently.The number of independent movements an object can make on a plane
Different types of supports limit different possibilities of movement on a plane or in space.On a plane, there are three main degrees of freedom of movement:
• Translation along the X-axis: This movement along the horizontal X-axis allows an object to move on the plane left or right.
• Translation along the Y-axis: This movement along the vertical Y-axis allows an object to move on the plane up or down.
• Rotation around the vertical axis (Z): This is the movement of the object rotating around an axis perpendicular to the plane, usually called the Z-axis. This movement allows the object to rotate around the center of the plane.
It is worth noting that degrees of freedom may vary depending on the context and specifics of the construction. For example, the degrees of freedom of a complex mechanism may include additional movements, such as tilting or rotating around other axes.
In the analysis of the movement of objects on a plane, these three basic degrees of freedom are a key reference point.
Types of supports on a plane
Given that on a plane we have three degrees of freedom of movement, supports can take away these degrees of freedom in various combinations. Removing the possibility of movement, e.g., horizontal, by a support results in the creation of a reaction in that direction in the support. First, we will briefly characterize the types of supports on a plane, and then for clarity, we will compile the following information regarding supports on a plane in a table:- popular name/names,
- graphic symbol/symbols,
- what reaction occurs in it,
- what degree of freedom of movement it takes away.
a) sliding hinge support
Blocks linear movement in one direction, allows movement in the direction perpendicular to the blocked one, and rotation. Graphic symbols we encounter in literature:
Fig1. Sliding hinge support
A horizontal line under the triangular symbol of the hinge support shows in which direction there is possibility of movement. The direction perpendicular to the possibility of movement is blocked, and in this direction, a reaction arises. Of course, a hinge support can be used at various angles, as shown below:
Fig2. Sliding hinge support at different angles with marked reaction
If the support and reaction are at an angle, often for simplicity of calculations, it will be more convenient for us to decompose this reaction into components:
Fig3. Decomposition of the reaction into components
After decomposition into components, in calculations, we use either one, resultant reaction (red) or both components (blue) – never both at the same time. That means when writing the equation of static equilibrium or determining internal forces, we use either the components or the resultant reaction.
b) fixed hinge support
In a fixed hinge support, there is also essentially one reaction, but we do not know its direction. Therefore, it is best to immediately mark two components of this reaction. They do not necessarily have to be vertical and horizontal, but they must be perpendicular to each other. It is also worth mentioning that the directions of reactions never matter and are just a matter of our assumption. The drawing below shows, in addition to the support symbol itself, a rod coming out of it (of course, it also does not matter at what angle the rod comes out of the support)
Fig4. Fixed hinge support
Note! If the outgoing rod is additionally ended with a hinge also from the other side, then we are dealing with a truss bar/tendon/cable which you will find more information about below.
Often the dilemma regarding marking reactions arises when we see a non-sliding support rotated at a different angle. It is important to remember that in such a support we always have two perpendicular reactions to each other, and it is up to us in which directions we mark these reactions. Every variant shown below is OK.
Fig5. Fixed hinge support with marked reactions
c) truss bar/tendon/cable
We might encounter such support for a beam/frame – fixed hinge support + tethering with a cable (See free example 1 )
Fig6. Frame system tethered with a cable
or with tethering the beam on the cables themselves. (See free example 2 )
Fig7. Beam tethered on three tendons
For us to talk about a cable/tendon/truss bar the following assumptions must be met:
- the bar must be ended with hinges on both sides (these are the nodes of the bar),
- the bar must be linear (there can be no bends, see example below),
- there can be no load between the nodes,
- the load can be applied at most to the node (while remembering that if it is a concentrated moment, it cannot be applied strictly to the hinge, it must be applied from one side or the other of the hinge)
- only axial (normal) force occurs in it,
- there are no shear forces and bending moments in it.
Below on the left an example of two truss bars, on the right due to the lack of a hinge in the middle the bar is not linear, so we do not deal with a truss bar but with a frame system supported by two non-sliding hinge supports.
Fig8. Example classification of a truss bar and frame system
This is a fundamental difference between these two constructions, and they are calculated completely differently. In hinge supports, we show reactions as shown earlier, but as for reactions in a truss bar, we have two possibilities of marking reactions
1. Cutting the bar and showing internal forces in the bar (axial) and discarding the part outside the construction
2. Showing reactions at the end of the bar, outside the construction, i.e., at the support; the reaction occurs only in the axis of the bar
Fig9. Second variant of reaction in a truss bar and reactions in a frame system hingedly supported.
Fig10. First variant of marking reactions/forces in a truss bar
On our site, you will rather find the first variant, as it seems to be better
In the case above:
a) we cut through the bar,
b) we show forces in the bar (always either towards each other or away from each other – it's best to assume as above, i.e., towards each other – then we assume that the bar is being stretched),
c) we discard those parts outside the construction.
d) fixed support/cantilever
Fixed support, or the popular cantilever takes away all possibilities of movement on the plane, thus possibility of linear movement and possibility of rotation. Therefore, there are three reactions in it, two perpendicular to each other reactions in the form of concentrated forces and a concentrated bending moment. Just like in the case of a non-sliding hinge support, directions and directions of reactions are a matter of our assumption (we just remember that concentrated reactions should be perpendicular to each other).
Fig11. Ways to mark a cantilever and reactions in it
e) support with slide/roller
Support with slide allows for linear movement in one direction, blocking linear movement in the perpendicular direction and rotation at the same time. We can distinguish vertical roller and horizontal roller, but support with slide can be just as well under any angle, below a few possibilities of marking a roller along with reactions in it.
Fig12. Vertical and horizontal roller along with reactions in them
Fig13. Other possibilities of support with slide along with reactions
f) rotation lock/wedge
This type of support is very rarely encountered and is not strictly marked in the task topic, but we can encounter it most often solving a beam/frame by the principle of virtual work and of course, we deal with it almost in every task of the structural mechanics section – displacement methods.Rotation lock is a constraint that blocks, as the name suggests, only the possibility of rotation of the point (node). We can encounter such markings of rotation lock:
Fig14. Graphic marking of support/constraint – rotation lock
In this support, there is only a reaction in the form of a bending moment, the node has freedom of movement in every direction, but it cannot rotate – that means the shape of the node after displacement must be the same as before displacement. The first from the left graphic marking resembles a wedge, and we can also encounter such a name for this constraint. The third marking also resembles a roller both vertically and horizontally.
Degrees of freedom of movement in 3D space
In three-dimensional space (3D), there are six main degrees of freedom of movement, which determine the number of independent movements an object can perform. Here are these degrees of freedom:• Translation along the X-axis,
• Translation along the Y-axis,
• Translation along the Z-axis,
• Rotation around the X-axis,
• Rotation around the Y-axis,
• Rotation around the Z-axis.
All these degrees of freedom together define how an object can move and rotate in 3D space. Thanks to this classification, engineers and designers are able to analyze and model the movements of objects and structures in three dimensions.
Types of supports in 3D space
In a planar system having 3 degrees of freedom of movement (translation X, translation Y, rotation Z) we were able to distinguish 6 types of supports depending on the combinations in which supports took away permissible degrees of freedom.It will be difficult to classify supports in 3D space, therefore, below we will rather provide general principles regarding supports and the most popular supports in constructions such as:
a) hinged frames,
b) rigid frames,
c) 3D frames.
A grillage is called a flat rod structure, which is loaded perpendicular to the plane of the grillage beams. Depending on the way beams of the grillage are connected, we can distinguish:
a) hinged frames
Beams of the grillage are connected to each other in a hinged manner, allowing them to rotate freely relative to each other. Such a connection allows only the link connecting the beams to transfer vertical force (perpendicular to the plane of the grillage beams).Since the load on the grillage acts in the direction perpendicular to the plane of the beams and given that beams transfer only transverse force to each other, only transverse forces and bending moments occur in a hinged grillage. This affects the type of reactions that can occur in supports, and we can distinguish such supports as:
- hinged support
Where it does not matter whether it is drawn as sliding or not – we do not mark the reaction in the plane of the grillage, only the reaction from the plane.
Fig15. Ways to mark a hinged support in a hinged grillage
- vertical roller
Below a vertical roller marked in two directions along with the reaction in the form of a moment – which is marked in two ways – in the form of a moment vector and in the form of a moment on a "sheet of paper" – both notations are equivalent. In the roller, there is no vertical reaction, in grillages in general, we omit reactions in the axis of beams, so here we have only a reaction in the form of a moment – and it is a bending moment for the bar, because hinged grillages do not transfer torsion (due to hinged connections).
Fig16. Vertical roller along with reactions for a hinged grillage
- full fixation
In the fixation at a hinged grillage, we have a reaction in the form of a bending moment for the beam and a vertical force. Below, the reaction in the form of a moment is marked on one drawing in two ways
Fig17. Full fixation along with reactions for a hinged grillage
Solutions for tasks with hinged grillages can be found in the online Course structural mechanics – specifically here, they are covered by subscription (one subscription provides access to materials on the entire site – i.e., to all online courses).
b) rigid frames
Beams of the grillage are connected to each other rigidly (you can imagine it as if the rods were concreted into each other). Bending of rods arranged in one direction causes bending and torsion of rods in the other direction.In rigid grillages, we have the same supports as in hinged grillages, however, here the difference is that we also deal with torsion, so in fixation besides the reaction in the form of a bending moment there will also be a reaction in the form of a torsional moment. We will also have more options for rollers (either it blocks bending, or torsion, or both).
Types of supports in rigid grillages:
- hinged support
Exactly the same as in a hinged grillage. It does not matter whether it is drawn as sliding or not – we do not mark the reaction in the plane of the grillage, only the reaction from the plane.
Fig18. Ways to mark a hinged support in a rigid grillage
- vertical roller
As mentioned earlier, we have three types of rollers in a rigid grillage:- with a reaction in the form of a torsional moment,
- with a reaction in the form of a bending moment,
- with both mentioned above.
And here we have a different way of marking a roller than before, generally, I could not find anywhere in literature a roller for a rigid grillage, but the marking shown below seems to be sensible, it also matches the marking which appears in programs like Robot Structural Analysis.
Fig19. Vertical rollers along with reactions below for a rigid grillage
Well, the first marking is often used as a full fixation, so it's best to specify if there's such a possibility what type of support the person who drew the task topic had in mind.
- full fixation
If we wanted to consistently apply a marking similar to that at the roller, it would be necessary to mark the full fixation in the way shown on the left side, however, I think the most popular will still be the simplest marking we encountered earlier – presented on the right side
Fig20. Full fixation in a rigid grillage along with reactions
c) 3D frames
In a spatial frame, we have 6 constraints, so reactions can take these constraints in various configurations. Most often, we will deal with full fixation and with ordinary supporting rods taking away the possibility of movement in the direction in which the rod is positioned.- supporting rod/sliding hinge support
Fig21. Movement block in one direction along with reaction
In one place, one, two, or three directions can be blocked, so now a variant with two blocks in one place
- supporting rods/sliding hinge support
Fig22. Movement block in two directions along with reactions
- non-sliding hinge support
Fig23. Non-sliding hinge support along with reactions
- full fixation
Fig24. Full fixation along with reactions
With other types of support than those listed above in spatial systems, you will hardly encounter (I have not), generally, as a rule, any direction of movement and the possibility of rotation in any direction can also be blocked in any combination, for example
Fig25. Various combinations of linear movement and rotation blocks along with reactions
and the like.
What are kinematic boundary conditions and where do we use them
More information coming soon.Boundary conditions in differential notation (finite difference method)
More information coming soon.