T-beam cross section

Design of Flange Beam to BS 8110: an Overview

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A flange section is that which has a rectangular extension from the main rectangular body. The extension is called a flange while the rectangular body is called web. This type of section mostly occurs when a beam is cast monolithically with a slab. 

The design of flange beam is similar to that of rectangular section only that the flange gives additional area of concrete which adds to the overall compressive strength of the section. So instead of using the breadth of the rectangular section only in estimating the compressive strength of the section, the breath of the rectangular section plus that of the flange is used thereby augmenting the compressive strength of the beam.

It is however very pertinent to note that this increase in compressive strength due to the flange area will be valid only when the flange is in compression. Whenever the flange is in tension, the section should be designed as a rectangular beam as more concrete area in tension will actually serve little or no purpose.

Types of Flange Section

Flange sections are however categorized based on their geometry. The two basic types are:

  1. L section
  2. T section

The L sections are mostly found in perimeter beams of building as the slab is only spanning from the beam on one side while the T section on the other hand is mostly found in buildings intermediate beams as slabs span from both sides of the beam.

EFFECTIVE WIDTH OF FLANGE

When the flange of a beam is exploited in designing a beam, the total span of the adjoining slab when large does not often act with the supporting beam to resist the loads on the beam, only some portion of the slab does. To calculate the portion of the slab that act together with the beam to resist loads results to the concept of effective width of flange.

The effective width of flange according to BS 8110 is stipulated below

For L section:     web width + (whichever is smaller between) Lz/10 or actual flange width

For T section:     web width + (whichever is smaller between) Lz/5 or actual flange width

Where Lz is defined as the distance between point where moment is zero. This can be taken as the entire span for a simply supported beam. As for a continuous beam Lz should be taken as 0.7x times the effective span of the beam.

Flexural design of Flange Beams

For the design of a flange in flexure, two cases are paramount to be considered:

  1. When the portion of concrete to resist compression lies within the flange (ie: Neutal axis lies within the flange)
  2. When the portion of concrete to resist compression extends beyond the flange.(ie: Neutral axis lies below the flange)

When Neutral axis lies within the Flange

In designing a flange section, when the area of concrete within the flange section is enough to develop the strength required to resist the compressive force, then the beam is designed as a rectangular section with breath bf (effective flange width). Therefore, the moment of resistance of the section can be calculated using:

        Mu  =             Kbfd2fcu             (k = 0.156)

When Mu < M, then the stress block is truly within the flange and the beam is designed as singly reinforced.

The area of tensile reinforcement in this case can be determined using:

$A_{s\,\,=\,\,\frac{M}{0.87f_yZ}}
$

When Neutral axis lies below the Flange

However, when the stress block to resist compressive force extends below the flange, then these can be designed by either

  1. Calculating the exact depth of the web below the flange that is harnessed to resist the compressive force.
  2. By conservatively assuming that the depth of the neutral axis of the section equals half of the effective depth of the beam section.

To calculate the actual depth of the web that partakes in resisting compression can be very tedious, as a work around, and in conformance with the formula given in BS 8110, the author prefers conservatively taking the depth of the neutral axis to be half of the effective depth.

Consequently, the moment of resistance of the section will be the moment of resistance of the flange plus the moment of resistance of the web.

ie: Mu = Mf + Mw

0.45fcu(bf – bw)(d – hf/2)hf + 0.156fcubd2

If the design moment (M) is greater than the moment obtained after evaluating the above formula, then the section shall be designed as doubly reinforced.

BS 8110:1:1997 gives the formula for calculating the area of tensile reinforcement when the neutral axis falls below the flange as:

$
A_s\,\,=\,\,\frac{M\,\,+\,\,0.1f_{cu}b_wd\left( 0.45d\,\,-\,\,h_f \right)}{0.95f_y\left( d\,\,-\,\,0.5h_f \right)}
$

Longitudinal Shear Design

There is always a shear at the connection between the web and the flange. The code, in table 3.25, stipulates that minimum area of steel of 0.15% of reinforcement should be provided over full effective flange width to resist horizontal shear.

Vertical Shear Design

The design of shear force for flange beam is exactly the same for a rectangular beam except that the breadth of the flange beam, (bv) is always taken as the breadth of the rib below the flange.

As in the case of rectangular beams, the shear stress (v) is checked against the shear strength of the concrete (vc). The shear stress (v) is calculated by dividing the shear force (V) in the section by the effective cross-section area of the section.

ie: v = V/bd

The shear strength (vc) of the concrete section is affected by a few numbers of variables. The strength of the concrete is a chief factor, likewise do the interlocking action and dowel action of the coarse aggregate and longitudinal reinforcement bars respectively contribute also considerably. This can be observed in table 3.8 of the code where the percentage longitudinal reinforcement is to be checked against the effective depth of the concrete section to obtain the shear strength of the section.

So when the shear stress (v) is lesser than the shear strength (vc) plus 0.4N/mm2 minimum links is to be used.

i.e.:  if v < Vc + 0.4  – Asvmin

where:

$
Asv_{\min}=\,\,\frac{\text{0.4}x\,\,b_v\,\,x\,\,s_v}{0.9fy_v}\,\,
$

However, if the shear stress (v) is greater than Vc + 0.4 then the shear reinforcement should be designed for using:

$
Asv_{}=\,\,\frac{b_v\,\,x\,\,s_v\left( v-v_c \right)}{0.95fy_v}\,\,
$

t is nonetheless pertinent to note that on no account should the shear stress be greater than the lesser of 5N and 0.8xfcu0.5 should this happen then the section must be resized and redesigned.

When the shear stress is extremely low such that it is lesser than half of the concrete shear strength as it happens in lightly loaded element such as lintel, the shear links should not be provided theoretically. However, for all practical purpose, it is pragmatic to provide minimum links so that a reinforcement cage can be formed and large cracks can be prevented.

Under no circumstances should the spacing of the links exceed 0.75d. And at right-angles to the span, the spacings of the main tension bar should be within 150 mm from a vertical leg. This spacing between the longitudinal bars should never be more than the effective depth of the beam for a shallow beam.

Deflection

Deflection is an important serviceability limit state that must be checked whether its limit is exceeded. Excessive deflection can compromise the aesthetic of the structure, give impression of unsafe structure, cause damage to brittle finishes, cause problem to fixtures etc. So ensuring deflection limit is not exceeded is one of the important criteria of beam design.

The factors affecting deflection of beams are numerous likewise is calculating deflection for heterogenic material like reinforced concrete very tedious. In order to simplify the problem, the code uses the concept of span-depth ratio to estimate deflection. Hence, table 3.9 gives limiting span to depth ratio for flange beams with bw/b which is ≤ 3. When bw/b > 3, the limiting or basic ratio can be found by interpolating linearly between the values for rectangular beams and flange beams.

Allowance for factors such as amount of tension and compression reinforcement are also catered for in tables 3.10 and 3.11 respectively. There are modification factors which modifies the limiting span-depth ratio base on amount of reinforcement bars.

Table 3.9 is based on limiting the deflection of the beam to L/250. This is expected to assume a lesser value of L/500 after the construction of cladding and other finishes which will consequently stiffen the member. The span of the member is expected not to be more than 10m, if more than it is expected then the value in table 3.9 should be multiplied by 10/actual span.

In determining whether the beam satisfy deflection criteria, the actual span to effective depth is to be calculated and checked against the limiting/basic value in table 3.9 after it has been duly modified with appropriate factors from table 3.10 and 3.11 respectively. If the actual value is lesser, then the beam has passed the deflection criteria but would have failed if otherwise. 

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