What is Propped Cantilever Beam?
A propped cantilever beam is a beam with one end fixed and the other end provided with simple support. Such type of beam is cast by adding one pinned support at the free end of a cantilever beam.
Propped cantilever beam provides more rigidity than simply supported or cantilever beams. It has one redundant reaction, which provides more strength and lesser deflection as compared to a simply supported beam or cantilever beam.
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Propped Cantilever Beam Formula
When we take an assignment to get a propped cantilever beam formula, we need to know the beam's modulus of the material, length of beam, area moment of inertia, load intensity, and distance from the load.
All these factors considered the sign conventions as downwards loads are positive and upward loads are taken as negative. Similarly, anticlockwise reaction and anticlockwise moment caused positive signs, and clockwise direction reaction and clockwise direction moment caused negative signs.
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Propped Cantilever Beam With UDL
We know that a propped cantilever beam is an indeterminate beam, and it also has a redundant reaction. It can consist of any load or any combination of loads. The bending moment resulting from any combination of load for the propped cantilever beam is hogging in nature at some section of the beam, which increases the beam's strength. Here such a beam having UDL is shown along with their bending moment and shear force diagram.
Propped Cantilever Beam With Point Load
As we know that a propped cantilever beam is an indeterminate beam, So it has some redundant reaction. The loading effect on such a beam is found to be lesser than that for a simply supported beam or cantilever beam. The bending moment or shear force resulting from the combination of loads for the propped cantilever beam is less than that for other conventional beams, which results in increasing the strength of the beam. Here such a beam having a point load is shown, which makes a better understanding of the concepts.
Propped Cantilever Beam Reactions
To find out the reactions in a propped cantilever beam, first we need to know the type of reactions and their point of application. As we know, that propped cantilever beam has fixed support at one end, which results in three reactions named horizontal reaction, vertical reaction, and moment offered by fixed support, and at another end of the beam, roller support exerts only vertical reaction. These four reactions make it an indeterminate beam.
These four reactions of the beam can be calculated with the help of the superposition method of analysis. In this method of analysis, a beam is classified as a cantilever beam having a given load distribution, and deflection is found at the point of roller support. And this deflection is then equated with the deflection of the cantilever beam at the free end having point load at the free end. It is because there is roller support, so deflection should be zero.
Propped Cantilever Beam Examples
A propped cantilever beam is the extended form of the cantilever beam, having roller support at the free end. Such a type of beam provides more stability than the cantilever beam and simply the supported beam. Due to this advantage of the beam, it has many applications in the real world of structure.
Propped cantilever beam has many examples listed below.
- End beam of continuous beam
- Two-way slab
- Roof slab
What is the Degree Of Static Indeterminacy of Propped Cantilever Beam?
Static indeterminacy is the number of restraints in the beam or structure over the number of equilibrium equations. Static determinacy tells about the stability of the structure. Static determinacy can be calculated as D= R - 3 For a planar structure. Because in a plane, there are only 3 numbers of equilibrium equations which are known as a horizontal reaction, vertical reaction, and moment are zero about an axis perpendicular to the plane.
In a propped cantilever beam, there is fixed support at one end and pinned or roller support at the other end. In the case of inclined loading, at fixed support, there will be three reactions; at roller support, there will be only one. So, in a propped cantilever beam, there will be only four reactions hence the degree of static indeterminacy will be 4 - 3 = 1. Static determinacy 1 represents only one redundant reaction for the structure or beam.
Kinematic Indeterminacy For Propped Cantilever Beam
Kinematic Indeterminacy is the value related to a beam or structure, which tells about how the structure is free to move and in which direction. The degree of kinematic indeterminacy can be calculated by calculating the minimum possible directions in which the structure can move freely. The number of movements can be horizontal, vertical, or rotation at any point. These movements in the structure must be independent of each other.
The kinematic indeterminacy of the beam depends on either members or axially deformable or rigid. So, Considering the beam is not axially deformable, only rotation will be allowed at the pinned end. Hence Kinematic indeterminacy for the propped cantilever beam will be only one.
Order of Flexibility Matrix For a Propped Cantilever Beam
The flexibility matrix is the matrix that relates forces or moments to the displacements or deflections in the structure. The flexibility matrix also provides an understanding of the stiffness of the member in the structure. The flexibility matrix will be a square symmetric matrix based on Maxwell's reciprocal theorem. And as per Castigliano's theorem, diagonal elements of the flexibility matrix are always positive.
The flexibility of a beam is defined as the displacement caused by a unit force at the redundant force. As we know, the degree of static determinacy for the propped cantilever beam is one for the inclined loading, so there will be only one redundant reaction. The order of the flexibility matrix for a structure is the number of redundant reactions present in the structure or beam. Hence, the order of flexibility matrix for the propped cantilever beam is one.
For Propped Cantilever Beam Kinematic Indeterminacy With Axial Deformation
Kinematic Indeterminacy is related to a beam or structure, which tells about how the structure is free to move and in which direction. The degree of kinematic indeterminacy can be calculated by calculating the minimum possible directions in which the structure can move freely. The number of movements can be horizontal, vertical, or rotation at any point. These movements in the structure must be independent of each other.
Kinematic indeterminacy with considering the member of a propped cantilever beam is axially deformable is calculated by finding the number of directions in which the member can move freely. In the propped cantilever beam having an axially deformable member, there are two directions in which the beam can move and these directions include rotation about pinned support and horizontal movement of the member.
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