Cantilever Beams and their Functionalities
What is a Cantilever Beam: Cantilever Beams are members that are supported from a single point only; typically, with a Fixed Support. In order to ensure the structure is static, the support must be fixed; meaning it is able to support forces and moments in all directions. A cantilever beam is usually modelled like so:
A good example of a cantilever beam is a balcony. A balcony is supported on one end only, the rest of the beam extends over open space; there is nothing supporting it on the other side.
Cantilever Beam Deflection
Cantilevers deflect more than most other types of beams since they are only supported from one end. This means there is less support for the load to be transferred to. Cantilever Beam deflection can be calculated in a few different ways, including using simplified cantilever beam equations or cantilever beam calculators and software (more information on both is below).
Primary Cantilevered Structure Approach
The primary cantilevered structure ?sitting? on the plinth is made of a series of steel trusses. Maximum stiffness is achieved by keeping the ratio cantilever/backspan as small as possible and having the maximum truss depth. In this case, the site constraints and the shape of the building which had to be achieved dictated these values.
First of all, a cantilever/backspan ratio of 1 was provided, placing the four ?cores? on the outer edges of the plinth. This allows for a car park free of very large vertical supports within the plinth and a column-free platform area around the railway tracks. Both the cantilever and back span have a length of 42 cm.
Then, because the cantilevered upper structure had to have a ?spiral? shape, it was not possible to use trusses which were the full depth of the upper structure, i.e. 3-storey high. Pairs of 1-storey high were therefore used to create the desired shape. As it was expected that deflections would be a major issue; it was important to find a way to limit them.
This has been done by simply connecting the ends of the upper and lower trusses together with a steel tie. This principle was initially developed when both trusses are in the same plane. In order to achieve the spiral shape, the support points were connected to different sides of the cores, rotating both trusses about the steel ties connecting them.
Finally, the width of the building being approximately 35m, a way of limiting the floor spans had to be found in order to reduce their depth and weight - as much as possible. This has been achieved through the layout of the pairs of combined trusses. This way, a truss may start at the edge of the building at one end and be located at its centre at the other end. The upper and/or lower floors can then sit or hang from this truss. The following figures show the exact layout of the main trusses supported by four cores as well as the columns and hangers for the floors in the upper cantilevered structure.
Bending moment and shear force diagram of a cantilever beam
A shear force diagram is the graphical representation of the variation of shear force along the length of the beam and is abbreviated as S.F.D.
A bending moment diagram is the graphical representation of the variation of the bending moment along the length of the beam and is abbreviated as B.M.D.
We will take different cases of beams and loading for plotting S.F. D and B.M.D.
1) Cantilever : Point Load at the End: t section x from the end A, Fx = ? W1 and is constant for any position of the section. The S.F.D. will, therefore, be rectangle of height W. Bending moment at a section x from end A is given by
Mx = + W, x
At x=0, MA=D ; x=1, MB=WL.
The B.M.D. will thus be a triangle having zero ordinate at A and WL at B.
Cantilever: Several Point Loads
S.F.D. : Between A and C,
Fx= ? 3 kN (constant)
Between C and D,
Fx = -3-3=-6 kN (constant)
Between D and B,
Fx = -3-3-2=-8 kN (constant)
The S.F.D. will, therefore, consist of several rectangles having different ordinates, as shown in B.M.D.: Between A and C,
Mx = 3x
When x=0, MA=0. When x=2m, MC=3?2=6 kN.m
Between C and D.
Mx=3x+3(x-2)
When x=2, MC=6 kN.m (as before)
When x=4,
MD=(3?4)+(3?2)=18 kN.m
Between D and B,
Mx=3x+3(x-2) +2(x-4)
When x=4, MD=12+6=18 kN.m (as before)
When x=5, MB=15+9+2=26 kN.m.
The B.M.D. is shown in
Cantilever: U.D.L. Over Whole Span
S.F.D. At any section x from A
Fx=-wx
When x=0, FA=0;
when x = L, FB = ? wL.
S.F.D. At any section x from A
Fx=-wx
When x=0, FA=0;
when x = L, FB = ? wL.
The S.F.D. will be triangle
B.M.D.
When x=0, MA=0; When x=L,
The B.M.D. will be a parabola
Design of Cantilever Steel Beams: Refined Approach
A new set of effective-length factors is presented for designing doubly symmetric I-shaped built-in cantilever beams against lateral-tensional buckling when subject to different loading and restraint conditions at the cantilever tip.
This refined approach rectifies inherent problems of current solutions caused by overlooking the restraint conditions as well as the limitations that existed in their original derivation. Because of these problems, the effective-length factors currently available may result in either overly conservative designs, depending on the type of problem involved. Also, an interaction buckling design model is suggested for overhanging beams, in which the load is applied only at the cantilever tip.
This design model takes into account the ratio of the length of the cantilever span to that of the back span, a significant parameter that has not generally been considered. Finally, a design procedure is given for determining the elastic critical moments of crane-trolley beams.
Uses of Cantilever Beam
Without any supporting columns or bracing, cantilevers provide a clear space underneath the beam and with the introduction of steel and reinforced concrete; cantilevers became a popular structural form.
The cantilever beam is used in;
? In Buildings.
? Cantilever bridges.
? Overhanging projections and elements.
? Balconies such as at Frank Lloyd Wright?s Falling Water.
? Machinery and plant such as cranes.
? Overhanging roofs like shelters, and stadium roofs.
? Shelving and Furniture.
In building constructions, there are various applications of the cantilevered beam such as cantilevers carrying a gallery, roof, runway for an overhead travelling crane, or part of a building above and also used in various structures such as sun shed, shelves, large halls, exhibition buildings, and armoires.
