Fire Resistance Design of Steel Columns

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June 10, 2022

Table of Contents

At normal temperatures, structural design requires the structure to support the design ultimate loads (the ultimate limit state) while also limiting deformation and vibrations under serviceability circumstances (the serviceability limit state). The fire resistance design of steel columns and beams is aimed at maintaining the structural integrity of the structure at elevated fire temperature within a stipulated time. At room temperature, the structure’s design effort is primarily focused on preventing excessive deformations.

The basic goal of fire-resistant design is to prevent collapse before the stated fire resistance duration expires. During a fire, large deformations are normal and do not need to be calculated in the fire design. For a particular fire loading, the load-bearing function of a steel member is assumed to be lost at time t, when:

E_fi,d= R_fi,d,t ——– (1)

where;
E_fi,d is the design value of the relevant effects of actions in the fire situation;
R_fi,d,t is the design value of the resistance of the member in the fire situation at time t.

burning steel structure — **Figure 1**: Burning steel structure

R_fi,d,t represents the design resistance of a member in a fire situation at time t, which can be M_fi,t,Rd (design bending moment resistance in a fire situation), N_fi,t,Rd (design axial resistance in a fire situation), or any other force (separately or in combination), and the corresponding values of M_fi,Ed(design bending moment in a fire situation), N_fi,Ed (design axial force in the fire situation), etc. represent E_fi,d.

The design resistance R_fi,d,tat time t shall be determined by reducing the design resistance for normal temperature design according to EN 1993-1-1 to account for the mechanical properties of steel at elevated temperatures (assuming a consistent temperature across the cross section). The differences in the equations for cold and fire design are mostly related to the shape of the stress-strain diagram at ambient temperature and the shape of the diagram at elevated temperature. Figure 2 shows this schematically.

When employed at greater temperatures, some adaptation to the design equations established for room temperature circumstances is required. If a non-uniform temperature distribution is employed, the normal temperature design resistance to EN 1993-1-1 should be changed based on this temperature distribution.

Fire Resistance Design of Steel Columns

The most important design requirements of steel structures is the verification of the buckling and shear resistance of the column. For uniaxial or biaxial bending of steel columns, the interaction factors should be duly considered.

Buckling Resistance

The design value of the compression force in the fire situation, N_b,fi,Ed, at each cross section should satisfy the following condition:

N_b,fi,Ed/N_{b,fi,t ,Rd}≤ 1.0 ——– (2)

where the design buckling resistance N_b,fi,t,Rdat time t of a compression member with a Class 1, Class 2 or Class 3 cross section with a uniform temperature θ_a should be determined from:

N_{b,fi,t ,Rd}= χ_fiAk_y,θf_y / γ_M,fi ——– (3a)

and for Class 4 cross sections

N_{b,fi,t ,Rd}= χ_fiA_effk_0.2p,θf_y /γ_M,fi——– (3b)

where;
k_y,θ is the reduction factor for the yield strength of steel at uniform temperature θ_a ,reached at time t.
k_0.2p,θ is the reduction factor for the 0.2% proof strength of steel at uniform temperature θ_a, reached at time t,.
A_effis the effective area of the cross section when subjected only to uniform compression;
χ_fiis the reduction factor for flexural buckling in the fire design situation, given by Eq. (5.45).

The value of χ_fi should be taken as the lower of the values of χ_y,fi and χ_z,fidetermined according to:

χ_fi= 1/[φ_θ+ φ_θ² −λ_θ²] ——– (4)

where;
φ_θ = 0.5[1 + αλ_θ + λ_θ²] ——– (5)

and the imperfection factor, α , proposed by Franssen et al (2005) is given by;

α = 0.65 √235/f_y ——– (6)

The non-dimensional slenderness λ_θ for the temperature θ_a, is given, for Class 1, 2 and 3 by;

λ_θ = λ √(k_y,θ/k_E,θ) ——– (7a)

and for Class 4 cross sections

λ_θ= λ √k_{0.2 p,θ}/ k_E,θ) ——– (7b)

where;
λ is the non-dimensional slenderness at room temperature given by Eq. (8a) or Eq. (8b) using the buckling length in fire situation l_fi. The non-dimensional slenderness at room temperature, λ, is given by;

λ = √(Af_y/N_cr) ——– (8a)

for Class 1, 2 and 3 cross sections or;

λ = √(A_efff_y/N_cr) ——– (8b)

for Class 4 cross sections.

where N_cr is the elastic critical force for flexural buckling based on the gross cross sectional properties and in the buckling length in fire situation, l_fi given by;

N_cr = π²EI/l_fi² ——– (9)

where;
E is the Young’s modulus at room temperature;
I is the second moment of area about y-y or x-x axis based on the gross cross sectional properties;
l_fi is the buckling length in fire situation.

The buckling length l_fiof a column for the fire design situation should generally be determined as for normal temperature design. In the case of a braced frame, the buckling length l_fi of a continuous column may be determined by considering it as fixed to the fire compartments above and below, provided that the fire resistance of the building components that separate these fire compartments is not less than the fire resistance of the column.

**Figure 3:** High-rise building under fire

Shear Resistance

The design value of the shear force in a fire situation, V_fi,Ed at each cross section should satisfy;

V_fi,Ed/V_{fi,t ,Rd} ≤ 1.0 ——– (10)

where the design shear resistance V_{fi,t ,Rd}at time t for a Class 1, Class 2 or Class 3 cross section should be determined from:

V_{fi,t ,Rd}= k_y,θ,webV_Rd[γ_M0/ γ_M,fi] ——– (11)

where;
V_Rd is the shear resistance of the gross cross section for normal temperature design, according to EN 1993-1-1, and given in Eq. (12);
θ_web is the average temperature of the web;
k_y,θ,web is the reduction factor for the yield strength of steel at the web temperature θ_web.

It should be noted that when a uniform temperature is considered in the design, the average temperature in the web is equal to the uniform temperature in the section. Alternatively, the temperature in the web can be determined using the section factor of the web. For an I-section, the section factor can be approximated as k_shA_m / V = k_sh2/t_w , where the correction factor for the shadow effect is taken for the full section or, in a more accurate way, as the view factor evaluated as shown in Section 4.9.

The shear resistance of the gross cross section for normal temperature design is given (according to EN 1993-1-1) by;

V_pl,Rd= A_v(f_y/3)/γ_M0 ——– (12)

where A_vis the shear area

Substituting Eq. (11) into Eq. (12), and considering a uniform temperature distribution, gives the following expression for the design shear resistance;

V_{fi,t ,Rd} = A_vk_y,θf_y/√3γ_M,fi ——– (13)

Design Example (Franssen and Real, 2015)

Consider a 3.5 m long HE 180 B column in S275 grade steel, located in an intermediate storey of a braced frame and subject to a compression load of N_fi,Ed = 495 kN in the fire situation. Assuming that the column doesn’t have any fire protection and that the required fire resistance is R30, verify the fire resistance in each of the following domains:

a) Temperature;
b) Time;
c) Resistance.

Solution:
Classification of the cross section:
The relevant geometrical characteristics of the profile for the cross section classification are;

h = 180 mm
b = 180 mm
t_w = 8.5 mm
t_f = 14 mm
r = 15 mm
c = b/2 − t_w/2 − r = 70.75 mm (flange)
c = h − 2t_f− 2r = 122 mm (web)

As the steel grade is S275
ε = 0.85 √(235/f_y) = 0.786
The class of the flange in compression is
c/t_f = 70.75/14 = 5.1 < 9ε = 7.07 ⇒ Class 1

The class of the web in compression is
d/t_w = 122/8.5 = 14.4 < 33ε = 25.9 ⇒ Class 1

The cross section of the HE 180 B in fire situation is Class 1. This classification could be directly obtained using the table for cross sectional classification of Annex F, Vila Real et al (2009b).

Evaluation of the critical temperature:
For the HE 180 B:
Area, A = 6525 mm²
Second moment of area, I_z = 13630000 mm⁴
The design value of the compression load in fire situation: N_fi,Ed= 495 kN
The buckling length for intermediate storey is: l_fi = 0.5L = 0.5 × 3.5 = 1.75 m

The Euler critical load takes the value:

N_cr= (π²EI)/l_fi² = 9224414 N

The non-dimensional slenderness at elevated temperature is given by Eq. (5.48)
λ_θ = λ ⋅√(k_y,θ/k_E,θ)

This is temperature dependent and an iterative procedure is needed to calculate the critical temperature. Starting with a temperature of 20ºC at which k_y,θ = k_E,θ= 1.0, equations (5.48), and (5.49) give:

λ_θ = λ ⋅√(k_y,θ/k_E,θ) = λ =√(Af_y/N_cr) = √((6525 × 275)/9224414) = 0.441

The reduction factor for flexural buckling χ is evaluated using Eq. (5.45):

α = 0.65 √(235/f_y) = 0.65 × √(235/275) = 0.601
and
φ = 0.5 × (1 + 0.601 × 0.4361 + 0.436²) = 0.730

Therefore the reduction factor for flexural buckling is:
χ_fi= 1/[0.730 + √(0.730² − 0.441²)] = 0.763

The design value of the buckling resistance N_b,fi,t,Rd at time t = 0, is obtained from Eq. (5.44):

N_b,fi,0,Rd = χ_fiAf_y / γ_M,fi= 1368 kN

and the degree of utilisation takes the value:

μ₀ =N_fi,Ed/N_fi,0,Rd = 495/1368 = 0.362

For this degree of utilisation Eq. (5.104) gives a critical temperature, θ_a,cr= 635 ºC. Using this temperature, the non-dimensional slenderness λ_θ can be corrected, which leads to another critical temperature. The iterative procedure should continue until convergence is reached, as illustrated in the next table:

fire design of steel columns — Adapted from Franssen and Real (2015)

After three iterations a critical temperature of θ_a,cr= 623 ºC is obtained.

The verification of the fire resistance of the column may be now made.
a) The section factor of the HE 180 B is A_m/V = 159 m⁻¹ .

The box value for the section factor;

[A_m/V]_b = (2 × (b + h))/A = [2 × (0.18 + 0.18)]/(65.25 × 10⁻⁴) = 110.3 m⁻¹

and the shadow factor k_sh
k_sh = 0.9 [A_m/V]_box / [A_m/V] = (0.9 × 110.3 ]/159 = 0.624

The modified section factor is: k_sh[A_m/V]_b = 0.624 × 159 = 99.2 m⁻¹

This value could be directly obtaining from the table of Annex E, Vila Real et al (2009a). Interpolating, from table of the Annex A.4 yields the following temperature after 30 minutes:

θ_d = 766 ºC
and
θ_d > θ_a,cr⇒ not satisfactory.

b) By double interpolation of table of the Annex A.4 the time needed to reach a temperature of 623 ºC is;

t_fi,d = 17.4 min
and
t_fi,d< t_fi,requ⇒ not satisfactory.

c) The reduction factors for the yield strength and the Young’s modulus after 30 minutes of fire exposure are, interpolating in Table 5.2 for a temperature of 766 ºC:
k_y,θ = 0.1508 and k_E,θ= 0.1036

The design value of the buckling resistance is obtained from;

N_{b,fi,t ,Rd}= χ_fiAk_y,θf_y/γ_M,fi

The non-dimensional slenderness at 766 ºC, is
λ_θ = λ ⋅√(k_y,θ/k_E,θ) = 0.441√(0.1508/0.1036) = 0.532

and using

φ_θ = 0.5[ 1 + αλ_θ +λ_θ²]
with
α = 0.65 √(235/f_y)
gives
φ_θ = 0.8014
and the reduction factor for the flexural buckling is:
χ_fi= 1 / [φ_θ + φ_θ²− λ_θ²] = 0.714

The design value of the buckling resistance after 30 minutes of fire exposure, takes the value:
N_{b,fi,t ,Rd}= χ_fiAk_y,θf_y /γ_M,fi = (0.714 × 6525 × 0.1508 × 275 × 10⁻³)/1.0 = 193 kN

and

N_{b,fi,t ,Rd} < N_fi,d⇒ not satisfactory.
The column does not fulfil the required fire resistance R30.

References
Franssen J. and Real P. V. (2015): Fire Design of Steel Structures (2nd Edition). ECCS – European Convention for Constructional Steelwork

Bearing Capacity of Pile Groups

By

Ubani Obinna

-

June 9, 2022

In many cases, the load bearing capacity of a group of vertically loaded piles is smaller than the sum of the capacities of the individual piles that make up the group. The elastic and consolidation settlements of the group are always bigger than a single pile carrying the same working load as each pile within the group. This is due to the fact that the zone of soil or rock that is stressed by the entire group is significantly wider and deeper than the zone beneath a single pile.

Even while loading tests on a single pile have shown good performance, group action in piled foundations has resulted in several recorded examples of failure or excessive settlement. It is therefore very important to check the settlement and bearing capacity of pile groups.

**Figure 1:** Comparison of stressed zones beneath single pile and pile group (a) Single pile (b) Pile group

A single pile driven to a satisfactory depth in a compact or stiff soil layer underlain by soft compressible clay is a classic case of foundation failure. When a single pile is loaded (Figure 1(a)), the latter formation is not significantly stressed, but when the weight from the superstructure is applied to the entire group, the stressed zone spreads down into the soft clay. The group may then experience excessive settlement or full shear failure (Figure 1(b)).

The allowable load on pile groups is frequently established by ‘efficiency formulas,’ in which the group’s efficiency is defined as the ratio of the average load per pile when the entire group fails to the load at failure of a single comparable pile. While there are so many of pile group efficiency equations to choose from, these equations should be used with caution, as they may be little more than a fair guess in many circumstances. The Converse-Labarre Formula is one of the most extensively used group-efficiency formulae;

E_g = 1 – {[θ(n – 1)m + (m – 1)n]/90mn} ——- (1)

Where;
m = number of columns in the pile group
n = number of rows in the pile group
θ = tan^-1(d/s) in degrees
d = diameter of the piles
s = spacing of the piles

Most of the varying efficiency ratios are solely developed based on personal experience, with no connection to soil mechanical theory. As a result, Tomlinson (2004) argues that this is not a desirable or logical approach to the problem, and instead prefers to design methods based on the assumption that the pile group behaves as a block foundation with a degree of flexibility that is determined by the rigidity of the capping system and the superimposed structure.

When applying soil mechanics methods to the design of pile groups, it’s important to keep in mind that, whereas the installation method influences the selection of design parameters for skin friction and end bearing in the case of a single pile, the installation procedure has less of an impact when considering group behaviour.

This is due to the fact that the zone of soil disturbance occurs only within a few pile diameters surrounding and beneath each individual pile, whereas the soil is considerably stressed to a depth equal to or higher than the group’s breadth (Figure 1(b)). The majority of this zone is located below ground level, which has been disrupted by the pile construction.

If two or more piles depart from alignment and come into close contact at the toe, there is a risk of severe base settlement when piles are erected in small numbers. As shown in Figure 2, the toe loads are concentrated over a limited region, and while failure would not occur if the end bearing safety factor was appropriate, the settlement would be greater than when the piles were spaced at their design spacing. As a result, the piles in the group would undergo differential settlement.

pile group deviation from alignment — **Figure 2:** Effect of deviation of piles from correct alignment in group

Spacing of Piles in a Group

In clays, the selection of a centre-to-centre spacing of at least three pile diameters, with a minimum of 1m, is a precaution against severe base settlement due to close alignment of piles. For friction piles, BS 8004 requires a centre-to-centre spacing of not less than the perimeter of the pile or three times the diameter of circular piles.

For piles carrying their load mostly in end bearing, closer spacing can be used, but the distance between neighbouring piles must not be less than their minimum width. The spacing of piles with larger bases requires special attention, including a study of the interaction of stresses and the impact of construction tolerances.

The Swedish piling code gives the following minimum centre-to-centre spacing for end-bearing and friction piles;

Pile Length (m)	Circular Pile	Square Pile
Less than 10 m	3 x Diameter	3.4 x width
10m – 25m	4 x Diameter	4.5 x width
> 25 m	5 x Diameter	5.6 x width

In all cases the centre-to-centre spacing should not be less than 0.8m.

Ultimate Bearing Capacity of Pile Group in Clay

There is no risk of general shear failure of the group if piles in groups are driven through soft clays, loose sand, or fill to end in a stiff clay, as long as there is an enough safety factor against single pile failure. However, the group’s settlement must be determined.

If a group of piles must be terminated totally within a soft clay (which is not recommended), the group’s safety factor against ‘block failure’ must be determined. Equation (2) is used to compute the ultimate bearing capacity of the block of soil enclosed by the group.

Q = 2D(B + L)c_a+ 1.3c_bSN_cBL ——- (2)

where;
D is the depth of the piles below ground level,
B is the overall width of the group,
L is the overall length of the group,
c_a is the average cohesion of the clay over the pile embedment depth,
c_b is the cohesion of the clay at the pile base level or within the zone of soil below the base affected by the loading,
s is a shape factor, and
N_c is the bearing capacity factor.

The remoulded shearing strength should be considered if the pile group is required to handle the full working load within a few days or weeks of the piles being put. Because the majority of the zone in which general shear failure would occur remains undisturbed, undisturbed cohesion can be employed for c_b in most circumstances. N_c values are determined by the group’s depth to width ratio (Figure 3). The length to width ratio determines the shape factor s, and suitable values are indicated in Figure 4.

Bearing Capacity of Pile Groups — **Figure 3:** Bearing capacity factor N_c(after Meyerhof)

Shape factor of pile group — **Figure 4:** Shape factor for rectangular pile groups (Meyerhof-Skempton)

Ultimate Bearing Capacity of Pile Group in Sand

There is no risk of block failure of a pile group terminated in and applying stress to a cohesionless soil if each individual pile has an acceptable safety factor against failure under compressive pressure. The piles must be designed with high-end bearing loads for economy.

When piles deviate from their intended line, as with piles terminating in clay, there is a potential of differential settlement between adjacent piles if the toe loads of a small group become concentrated in a small location. The simplest way to avoid this is to keep the piles separated by a reasonable amount of space.

Waterproofing of Basements

By

Ubani Obinna

-

June 7, 2022

Table of Contents

During the waterproofing of basements, the architect or structural engineer, or another party such as the contractor or specialised subcontractor, could perform a number of duties related to achieving basement watertightness. Generally, waterproofing systems should be designed to resist the passage of water and moisture to internal surfaces.

As a result, it will be necessary to explicitly identify the roles of each member of the design team in respect to these challenges from the start, as well as to notify the client. The requirement for a resident engineer on large projects should be explored with the client.

The first step in planning a basement waterproofing programme is to ensure that the membrane or other waterproof barrier is raised to the appropriate height. Borehole data isn’t usually a good indicator of the actual level of ground water surrounding a finished basement’s walls. The basement, for example, could be built on a sloping slope to act as a barrier to ground water seepage over the property.

On the uphill side of the structure, this will result in a rise in groundwater level. Borings on a clay site may reveal only sporadic water seepages at depth. Water may gather in the backfilled space surrounding the walls once the basement is finished, especially if the backfilling was placed in a loose state. The compartment could operate as a sump for surface water that collects around the walls and rises to near ground level.

6A46D2F0 CF52 4F75 AA53 983A9621B98F — **Figure 1:** Basement ruined by ground water ingress

Generally, waterproofing of basements should reach 150mm above the external ground level and link with damp-proofing in the superstructure. This is usually accomplished by connecting a continuous cavity tray to the below-ground waterproofing system. The link between the below-ground and above-ground waterproofing systems should be connected and constructed with the right materials.

When waterproofing is connected to an above-ground structure via a cavity tray, the materials must be able to:

compress to form a watertight seal, and
bear the load.

Generally, construction works that are at risk of coming into contact with groundwater and generally require waterproofing include:

basements
semi-basements
below ground parking areas
underground water tanks and swimming pools
lift pits
cellars
storage or plant rooms
service ducts, or similar, that are connected to the below ground structure
stepped floor slabs where the retained ground is greater than 150mm.

Elements forming a waterproofing structure below ground including: foundations, walls and floors, shall adequately resist movement and be suitable for their intended purpose. Issues to be taken into account include:

a) site conditions
b) structural design
c) durability
d) movement
e) design co-ordination.

Waterproofing of basement wall — **Figure 2**: Waterproofing of a basement wall using coatings

Grades of Basements

During the design and waterproofing of basements, the client must describe the intended use of the basement space, as well as whether flexibility is required to allow for future changes of usage. The client’s final brief to the design team is usually developed through an interactive consultation process between the client and the design team. The term ‘waterproof’ basement should be avoided at all costs. Rather, acceptable levels of water and vapour penetration should be decided upon – see Table 1.

Grade of basement	Usage	Performance level	Relative humidity	Dampness	Wetness
1. (Basic utility)	Car parking Plant rooms (excluding electrical equipment) Workshops	Some leakage and damp areas tolerable. Local drainage may be required	65% normal UK external range	Visible damp patches may be acceptable	Minor seepage may be acceptable
2. (Better utility)	Workshops and plant rooms requiring drier environment than Grade 1 Retail storage	No water penetration but damp areas tolerable dependent on the intended use. Ventilation may be required to control condensation	35–50%	No visible damp patches, construction fabric to contain less than air dry moisture content	None acceptable
3. (Habitable)	Ventilated residential and commercial areas including offices restaurants etc. Leisure centres	Dry environment. No water penetration. Additional ventilation, dehumidification or air conditioning appropriate to intended use	40–60% 55–60% for restaurants in summer	None acceptable. Active measures to control internal humidity may be necessary	None acceptable
4. (Special)	Archives Landmark buildings and stores requiring controlled environment	Totally dry environment. Requires ventilation, dehumidification or air conditioning appropriate to intended use	50% for art storage 40% for microfilms and tapes 35% for books	Active measures to control internal humidity probably essential	None acceptable

Table 1: Guide to grades of basements: functional environmental requirements and levels of protection

The level of active and passive measures necessary to control the interior environment will be determined by this. BS 8102 includes a useful classification table as well as usage grades. The various grades are intended to qualitatively distinguish the various levels of performance. Table 1 reproduces this information along with recommendations from CIRIA Report R140 (Water-resisting basements), which details how to define the internal climate (temperature, humidity, and wetness) for various purposes within each basement grade.

Relative Humidity (RH) is determined by exterior and internal factors and managed internally by natural or mechanical ventilation within a basement. Waterproofing methods usually have no effect on it. The design team and client should discuss and agree on a plan for controlling RH. The recommended temperature levels are attained by the use of heating and insulation. They, like RH, are unaffected by waterproofing measures and hence are no longer a BS 8102 requirement. Special settings, such as archive or retail storage, require a heating/ventilation system as well as the right style of architecture. In the case of archival storage, BS 5454 provides useful recommendations.

Types of Water-resisting Construction/Protection

After determining the desired basement grade, the next step in the waterproofing of basements is to identify the right type of construction. Types A, B, and C of water-resistant construction/protection are identified in BS 8102. These are barrier, structurally integral, and drained protection, as discussed below.

The location of the water table is regarded crucial in terms of the eventual construction’s possible dangers. With any water table levels and basement grades, Type A, B, or C could potentially be suitable. It should be emphasised, however, that in areas with changeable or high water tables, additional procedures for Type A and piled wall construction are required. It should also be highlighted that reduced permeability of the external earth (where undisturbed) and primary structural wall lowers the risk.

Type A – Waterproofing Barrier Protection

This type of construction, as shown in Figure 3, is entirely reliant on a continuous barrier of a waterproofing membrane, which can be applied to the exterior faces of walls and floors, sandwiched within the structure, or applied to the inner faces of walls. In Type A waterproofing, the structure itself does not prevent water ingress. Protection is dependent on the total water barrier system or water and vapour barrier system applied internally or externally or sandwiched between structural elements in accordance with manufacturers’ instructions. Edge thickenings are to be discouraged with external waterproofing.

Type A Barrier Protection — **Figure 3:** Type A water-resisting construction (barrier protection)

Membranes are usually not applied to floor surfaces and left uncovered because they lack the necessary wear characteristics. If applied to the tops of slabs, a protective slab (or something similar) will be required to keep the membrane in place. A variety of waterproofing materials are available (see below).

Any chosen system should be able to withstand hydrostatic pressure and/or loading effects, as appropriate. Some waterproofing systems may also provide excellent vapour resistance. However, plain polyethylene sheet should not be used as a waterproofing system. The structure is not specifically designed to be watertight in this type of construction, but it may be designed to meet the requirements of BS EN 1992-1-1.

Type A waterproofing — **Figure 4:** Barrier protection of basement wall

External membranes (or ‘tanking’) will obviously only be suitable where the external face can be accessed for initial construction. Access will limit the scope of subsequent repairs, and locating the source of any defect in a system that is not continuously bonded will be difficult, especially since defects may not become apparent until after construction.

Internally applied membranes will be easier to maintain, but their performance may be harmed by hydrostatic pressures and post-construction attachments. External membranes prevent early-age cracks from autogenously healing and encourage drying shrinkage cracks in concrete. Membranes may be used to protect the concrete structure in extremely aggressive ground conditions.

Type B – Structurally Integral Protection

Type B structure is often a reinforced concrete box that does not rely on applied membranes for water tightness (see Figure 5). The box is designed in accordance with BS EN 1992-3 so that water infiltration is minimised. Crack width limits are determined by the water table and/or the planned grade of use. Design to BS EN 1992-1-1 should be acceptable where the water table and risk are classed as low.

Type B protection — **Figure 5:** Type B water-resisting construction (structurally integral protection).

Type B systems acceptable to NHBC include:

in-situ concrete with or without admixtures and crack widths limited by design
in-situ high-strength concrete with crack widths limited by design and post-construction crack injections
precast concrete systems assessed in accordance with Technical Requirement R3.

The structure is unlikely to be totally vapour resistant without membranes, and other measures may be required. As a result, a type B basement may require conversion to a type A or C structure. Alternatively, and more commonly, the consequences of vapour penetration can be easily mitigated by the use of heating and/or ventilation. With the inclusion of vapour barriers, Type B building can accomplish all levels of internal environment.

Design details for reinforced concrete structures should include:

Concrete specification.
The type of concrete.
Concrete strength.
Proportion of any admixture.
Proposals for limiting crack widths.
Consideration of temporary support to the formwork.
Type and position of reinforcement.
The method of making good holes in the concrete formed for shutter bolts and tie bars.
Positioning of structural elements.
Appropriate tolerances

To avoid faults that allow water to pass through, good workmanship is required. Permeable concrete is a common fault caused by poor craftsmanship, such as inadequate compaction, honeycombed concrete, improper water bar installation, and poor joint preparation and contamination. Under high water table conditions, any water penetration through minor faults can be rectified from the inside.

Type C – Drained Protection

Type C building contains a drained cavity within the basement, which collects any seepage water and drains it to sumps for pumping out (see Figure 6). If any flaws are repaired and the system is maintained, a dry internal environment can be produced with certainty using a drained cavity wall and floor construction.

cavity drained protection — **Figure 6**: Type C water-resisting construction (drained protection).

The cavity and pumps may not be able to cope with the flows if the external wall and base slab do not substantially limit water infiltration. Large flows may also cause particles to be lost in the soils around them. Even tiny amounts of drained water may become a problem that necessitates negotiations with authorities.

It should be noted that significant amounts of groundwater pumped into sewers or rivers will normally not be approved by water authorities or the Environment Agency (EA), and specific provisions may be required to avoid loss of fine materials. If a drainage solution is chosen, maintenance requirements must be taken into account in case the drain or filter becomes clogged or fails. If no room is made for maintenance, ineffective drains and filters are likely to cause difficulties.

The cavity should not be used to conceal major leaks, according to CIRIA Report R140. When using water bars, make sure they are continuous and cover all construction joints.

Flooding caused by the failure of drains or pumps, or drain blockage caused by silt or other sediments, are examples of defects that can occur with this type of construction. To collect water infiltration, proprietary channels are often incorporated at the base of the walls. In the case of a blockage, access should be accessible for cleaning the silt and rodding the drains. Some linings prevent access to the cavity behind them, thus it’s obvious that building any interior walls or linings as late as feasible will allow any problems to be seen and rectified.

Rule of Thumb for Structural Design of Water Resistant Basements

Minimum thickness
Preferred minimum thickness of walls and slabs: 300mm
Where thicker consider surface zones of 200mm each face for reinforcement to control shrinkage/thermal cracking

Reinforcement
Typically for water resistant walls:
T16 @ 200 c/c in both faces and in both directions, or
T12 @ 150 c/c in both faces and in both directions

Standard concrete cover
Assumed concrete grade 35 (this should be a minimum)
Put the horizontal reinforcement furthest from earth face.

Face	Concrete Cover (mm)
Earth face of walls where shuttered	50
Earth face of walls (cast against earth)	75
External exposed faces of walls	40
Bottom and sides to base	75
Internal faces	Greater of 25 or bar diameter

Waterstops / waterbars

Required by BS 8102 for grade 1 basements with concrete design to BS 8110
Give extra “comfort” at construction joints, otherwise total reliance on workmanship
Not essential but often desirable
Use external waterstop for basements (preferred)
Can use centrestop in vertical construction if necessary (e.g. swimming pool), must be carefully supported/kept in place.

Materials for Waterproofing of Basements

Materials for waterproofing of basements should be suitable for the desired site, weather conditions, and any expected movements. There are a lot of proprietary systems out there. Choosing systems with Agreement certifications is a good idea, but the designer should think about the implications of any constraints listed in the certificates. When a vapour-proof system is necessary, further caution should be exercised. In general, it is not a good idea to mix systems.

Structural waterproofing can be done with a variety of products. They have been divided into seven distinct categories based on product kind, form, and application for simplicity of understanding. They are considered barrier systems for Type A protection, with the exception of Category 2. (but may be combined with Type B protection). Category 2 is a Type C protective mechanism that generates a drainable cavity. Below is a brief description of each category.

Category 1 – Bonded sheet membranes
These are cold-applied or heat-bonded to the structure. They are flexible and can accommodate minor movements. There are also composite sheet membranes, which can be fixed to vertical formwork or laid on the ground prior to pouring the slabs.

25296286 6E0D 4721 AAC0 7B7E4E897653 — **Figure 7**: Typical bonded sheet membrane

Category 2 – Cavity drain membranes
These are high-density polyethylene sheets placed against the structure. The dimples form the permanent cavity. These are generally used internally. They are flexible and are able to adapt to minor settlement and shrinkage of substrate. These are not waterproofing membranes in themselves; but facilitate drainage of any water ingress (see Figure 8).

Cavity drain system — **Figure 8:** Cavity drain membrane

Category 3 – Bentonite clay active membranes
These are sheets of sodium bentonite clay sandwiched between two layers of geotextile or biodegradable cardboard. When the clay meets water, it can swell to many times its original volume sealing any gaps or voids in the membrane. This category of membrane is used externally. Bentonite systems can be either bonded or unbonded. Where bonded, the system is simple to apply with minimum preparation of the substrate. The efficacy of the system under alternating drying and wetting conditions should be verified with manufacturers. It should not be used in acidic or excessively alkaline soil.

Category 4 – Liquid applied membranes
These one- or two-part systems are applied cold as a bitumen solution, elastomeric urethane or modified epoxy. A loading coat (a layer of material designed to hold a Type A waterproofing compound in place when resisting water pressure) will be required if applied internally, and it must be strong enough to adhere to a suitable substrate and sustain hydrostatic pressure. In Type B protection, they can be utilised solely as a vapour barrier if the building can take the load. The membrane’s continuity is preserved due to its lack of joints. It is simple to apply, but proper surface preparation is required. They can protect the structure from aggressive soils and groundwater when applied externally. Minor substrate motions can be accommodated because the substrate is elastic and flexible.

3F8E4F0F 3C3F 45E4 8D8A 9BBB074447AF — **Figure 9**: Typical liquid applied membrane

Category 5 – Mastic asphalt membranes
As a hot mastic liquid, these are applied in three coats. They harden into a waterproof layer as they cool. The application can be external or interior, but if applied internally, a loading coat is required. The likelihood of a defect in one coat being carried across all of the membrane’s coats is low. For complex foundation profiles, externally applied membranes are often unsuitable.

asphalt tanking of basement — **Figure 10:** Mastic asphalt membrane

Category 6 – Cementitious crystallisation active systems
These slurry coatings react with free lime in concrete, renders or mortars and block hairline cracks and capillaries. The chemicals remain active and will self-seal leaks. These products will not waterproof defective concrete (e.g. honeycombed areas).

Category 7 – Proprietary cementitious multi-coat renders, toppings and coatings
These coatings usually incorporate a waterproofing component and are applied in layers generally internally but may also be external. They are effective against severe ground water infiltration. Mechanical fixings through the system should be avoided.

Finally, ancillary components should be assessed as part of the waterproofing system. Alternatively, an assessment of compatibility and satisfactory performance should be provided for materials and products that are interchangeable between different systems.

Ancillary components include:

preformed junctions and corners
reinforcement
water stops/water bars
hydrophilic strips.

Braced Frame Structures

By

Ubani Obinna

-

June 7, 2022

Table of Contents

A braced frame is a structural system that is prevented from undergoing excessive sidesway under the effect of lateral loads by the provision of diagonal steel members (for steel structures) or shear walls/cores (for reinforced concrete structures). Therefore, braced frames are effective structural solutions for resisting lateral loads due to wind or earthquake in civil engineering buildings and structures. In effect, they provide the lateral stability needed in structures.

The stabilising members in a braced frame are usually made of structural steel, which can be very effective in resisting tensile and compressive forces. Most of the multi-story braced frames in the UK are designed as ‘simple construction,’ with nominally pinned connections between beams and columns. The horizontal force resistance of buildings in simple construction is provided by the bracing systems or cores in the global analysis.

As a result, the beams are designed to be simply supported, and the columns are merely designed to withstand moments caused by a minimal eccentricity in the beam-to-column connection (in conjunction with the axial forces). As a result, there is no need to consider pattern loading when calculating design forces in the columns.

braced frame highrise building — **Figure 1**: A braced high-rise building

The Eurocodes take the ‘simple construction’ design approach into account. If the joint is categorised as ‘nominally pinned’ according to BS EN 1993-1-8, and this classification is based on previous satisfactory performance in similar instances, a ‘simple’ joint model, in which the joint is assumed not to transfer bending moments, may be utilised. The beam reactions are applied eccentrically to the columns in the regularly used joint configurations in the UK, which assume a pinned connection but also assume that the beam reactions are applied eccentrically to the columns.

The global analysis model may therefore assume pinned connections between the columns and the beams for braced frames constructed in accordance with BS EN 1993-1-1, provided that the columns are designed for bending moments due to eccentric reactions from the beams.

Bracing Systems

The beams and columns in a multi-story building are typically placed in an orthogonal pattern in both elevation and plan. Two orthogonal bracing systems provide horizontal force resistance in a braced frame building:

Vertical Bracing, and
Horizontal bracing

Vertical bracing

Vertical bracing (between lines of columns) provides load pathways that carry horizontal forces to the ground level while also providing stiff resistance to overall sway. The vertical bracing planes in a braced frame multi-story building are commonly provided by diagonal bracing between two lines of columns, as shown in Figure 2.

Single diagonals, as shown, must be designed for either tension or compression, however crossing diagonals can be utilised with narrow bracing components that do not resist compressive stresses (then only the tensile diagonals provide the resistance). The floor beams participate as part of the bracing system when crossing diagonals are used, and it is considered that only the tensile diagonals produce resistance (in effect a vertical Pratt truss is created, with diagonals in tension and posts in compression).

Vertical bracings must be designed to withstand the following forces:

Wind forces
Equivalent horizontal forces, which illustrate the effects of initial imperfections
Sway-induced second-order effects (if the frame is flexible).

For the right combinations of actions, forces in specific members of the bracing system must be determined. Design forces at ULS are anticipated to be the most onerous for bracing members due to the combination where wind load is the dominating action. Bracing members that are inclined at about 45 degrees are preferred whenever possible.

This results in an efficient system with low member forces compared to other configurations, as well as compact connection details where the bracing meets the beam/column joints. The sway sensitivity of the structure will be increased by narrow bracing systems with steeply inclined interior elements. Structures with extensive bracing will be more stable.

braced steel structure — **Figure 3:** Installation of braces in a steel building

In a building, at least two vertical bracing planes in each orthogonal direction must be provided to avoid disproportionate collapse. There should be no considerable component of the structure braced by only one plane of bracing in the direction being studied since there would be no other restraint system in that direction if the local failure occurred in one of its parts.

Types of Vertical Bracing

Different types of bracing systems can be adopted for the lateral stability of framed structures. Diagonally Braced Frames, V-Braced Frames, and Chevron Braced Frames are three popular forms of concentrically braced frames. Studies have shown that the seismic excitation due to earthquake can be efficiently resisted by concentrically braced frames. This means that the braces, columns, and beams resist lateral seismic acceleration predominantly through axial forces (tension and compression) and deformation.

In order to appreciate the effects of bracing, let us consider the unbraced frame loaded as shown in Figure 4;

Under the applied load, the deflection in the frame is shown in Figure 5;

displacement unbraced — **Figure 5:** Deflection of an unbraced frame

Now, let us consider the effects of different bracing systems on the deflection behaviour of the frame.

Single Diagonal Bracing

Single diagonal bracing is considered effective in resisting lateral loads. They are formed by introducing single diagonal members to the frame (trussing). When lateral load is applied to the braced frame, the diagonal braces are subjected to compression while the horizontal web acts as the axial tension member in order to maintain the frame structure in equilibrium. If we consider the frame shown in Figure 4, let us introduce diagonals with the same member property UB 152 x89x16 as shown in Figure 6;

Interestingly, the introduction of single diagonal bracing reduced the lateral displacement of the frame by 99.34% as shown in Figure 7. In this case, the columns on the left-hand side and the diagonals of the braced frame were in tension, while the horizontal beams and the columns on the right-hand side were in axial compression.

single diagonal deflection — **Figure 7:** Deflection of framed structure with single diagonal bracing

Cross Bracing (X-Braced Frames)

In cross-braced frames, two diagonal members cross each other to form an X-shape. These simply need to be tension-resistant, with one brace functioning at a time to resist lateral loading, depending on the loading direction. Steel cables can therefore be utilised for cross-bracing. The performance of the tension braces in the design of single diagonal braces and X-bracing depends on the stiffness, resistance and ductility.

When cross bracing is applied to the frame, the deflection is shown in Figure 9;

deflection of x braced frame — **Figure 9:** Deflection of framed structure with cross bracing (x-bracing)

The result shows that x-braced frame is more efficient than single diagonal cross bracing in reducing the lateral displacement of the structure. However, this usually comes at the expense of extra cost, and increased bending of the horizontal beams. From the structural behaviour, it was observed that one of the braces is in tension, while the other is in compression (depending on the direction of the load).

V-Bracing (Chevron Bracing)

A chevron bracing is formed by introducing v-shaped braces into the frame. Chevron braces are known for their high elastic stiffness and strength. Unlike cross-bracing, chevron bracing is also effective in increasing architectural functionality. This is necessary in order to arrange the window and entrances in the braces bay.

However, under the effect of lateral load, however, uneven forces are formed on the braces. This is because the compression bracing will deform while the tension braces will remain in place to maintain the tension force during lateral loading.

Chevron bracing in a steel building construction — **Figure 10:** Typical installation of a chevron bracing on site

Under earthquake load, the tension and compression braces have a substantial influence on the unbalanced distributed force, which can induce elastic deflection of the braced frame. As a result of the deformation of the braces, the entire braces frame performs poorly. This means that at each level of braces bay, one brace resists the tension while the other brace resists compression. Before the buckling point, they both distribute the lateral stress equally in the elastic range.

The tension braces, on the other hand, maintain their tension after buckling, but the compression braces lose all of their axial load capacity. This contributes to the unbalanced distributed lateral load and result in a significant bending moment at the beam-brace intersection. As a result, the mid-span beam develops a plastic hinge and collapses due to its inability to withstand downward stresses.

For the frame loaded as shown in Figure 4, Chevron braces are were introduced as shown in Figure 11;

V BRACING — **Figure 11:** Framed structure with v-bracing (Chevron bracing)

The deflection of the frame under lateral load is shown in Figure 12;

deflection of v braced frame — **Figure 12:** Deflection of framed structure with Chevron bracing

The deflection behaviour of the x-braced frame and the chevron bracing are quite comparable.

Summary of the lateral deflection is shown in the Table below;

Frame Type	Lateral Displacement (mm)
Unbraced frame	1111.635 mm
Single diagonal bracing	7.313 mm
Cross Bracing (X-bracing)	4.648 mm
C-bracing (Chevron Bracing)	4.732 mm

Horizontal bracing

At each floor level, horizontal bracing (usually provided by floor plate action) provides a load route for horizontal forces (mostly from perimeter columns owing to wind pressure on the cladding) to be transferred to the vertical bracing planes.

At each floor level, a horizontal bracing system is required to transfer horizontal forces (mostly those transferred from the ends of perimeter columns) to the vertical bracing planes that provide resistance to horizontal forces. In multi-story braced frames, there are two types of horizontal bracing systems:

Diaphragms
Discrete triangulated bracing.

In most cases, the floor system will operate as a diaphragm without the need for extra steel bracing. If there is no slab at roof level, bracing, also known as a wind girder, may be necessary to carry the horizontal forces at the top of the columns.

If the horizontal bracing system at each floor level is relatively stiff (as it is when the floor acts as a diaphragm), the forces carried by each plane of vertical bracing are determined by its relative stiffness and placement, as well as the location of the horizontal forces’ centre of pressure.

Horizontal Diaphragm

Permanent formwork, such as metal decking connected to the beams by through-deck stud welding, and in-situ concrete infill, provide an effective rigid diaphragm to transfer horizontal forces to the bracing system. If precast concrete planks are to operate as a diaphragm, due thought must be given to ensure effective force transfer.

Planks and steelwork can have a coefficient of friction as low as 0.1, and even lower if the steel is painted. This will allow the slabs to glide over the steelwork and move relative to one another. Grouting between the slabs will only solve part of the problem; for significant shears, a more positive fastening system between the slabs and from the slabs to the steelwork would be necessary.

Reinforcement in the topping might be used to connect the planks. This could be mesh or ties running along both ends of a row of planks to ensure the entire panel functions as one. In most cases, a 10 mm bar at half the depth of the topping will suffice.

One of two approaches can be used to connect to the steelwork:

Provide ties between the topping and an in-situ topping to the steelwork (known as a ‘edge strip’).
Enclose the slabs with a steel frame (on shelf angles, or particularly provided limitation) and fill the gap with concrete. Shear connections should be installed on the steel beam to transfer forces between the in-situ edge strip and the steelwork.

The capacity of the connection should be verified if plan diaphragm forces are transferred to the steelwork by direct bearing (usually the slab bears on the face of a column). The plank’s capacity is often restricted by local crushing. In every scenario, in-situ concrete should be used to fill the gap between the plank and the steel.

Without additional safeguards, timber floors and floors made of precast concreted inverted tee beams and infill blocks (often referred to as “beam and pot” floors) are not considered suitable diaphragms.

Discrete Triangulated Bracing

A horizontal system of triangulated steel bracing is indicated when diaphragm action cannot be relied upon. In each orthogonal direction, a horizontal bracing system may be required. Horizontal bracing systems often span between the ‘supports,’ which are the vertical bracing’s positions. This configuration frequently results in a truss that spans the entire width of the structure and has a depth equal to the bay centres. Warren trusses, Pratt trusses, and crossing members are all common floor bracing arrangements.

How to Identify Expansive Soils

By

Ubani Obinna

-

June 6, 2022

Table of Contents

The nature of the soil, its plasticity, clay content, soil structure, and other factors can all be used to determine whether or not a soil has the potential to be expansive. Expansive soils must be identified during the reconnaissance and preliminary stages of a site investigation in order to determine the best sample and testing methods to use.

Expansive soils can cause considerable damage to civil engineering structures and foundations. This is due to the high swelling pressure they exert on foundations as they absorb water. Furthermore, their high shrinkage on drying can also affect foundations negatively.

The methods for determining the swell potential of expansive soils can generally be divided into two types. The first category mainly involves measurement of physical properties of soils, such as Atterberg limits, free swell, and potential volume change. The second category involves measurement of mineralogical and chemical properties of soils, such as clay content, cation exchange capacity, and specific surface area (Nelson et al, 2015).

To detect expansive soils, practicing geotechnical engineers often rely solely on physical property measurements. However, agricultural and geological practitioners routinely measure mineralogical and chemical parameters, and the engineering community should not overlook them.

Many of the methods of identification simply detect the presence of minerals with the capacity to expand. Physical factors such as in situ water content and density are not taken into account. As a result, they do not always determine whether or not the natural soil deposit is genuinely expansive, nor do they assess the potential for expansion. Nonetheless, they serve as important indicators of the need to dig more into the expansivity potential.

**Figure 1:** Typical view of expansive soil

Identification of Expansive Soils Based on Physical Properties

These are the tests on the physical properties of soils that can be carried out in the laboratory to determine if the soil has the potential to be expansive. These tests are;

Methods Based on Plasticity

Expansive soils can be identified using the Atterberg limits. The plasticity index, PI, and the liquidity index, LI, are two indexes based on the Atterberg limits. One or both of these indices have been used in a variety of identification methods for expansive . More expansive minerals, on the whole, have a higher plasticity. According to Peck, Hanson, and Thornburn (1974), there is a general relationship between a soil’s plasticity index and its expansion potential, as illustrated in Table 1.

Table 1: Expansion Potential of Soils and Plasticity Index (Peck, Hanson, and Thornburn 1974)

Plasticity Index	Expansion Potential
0 – 15	Low
0 – 35	Medium
22 – 55	High
> 55	Very high

However, Zapata et al. (2006) observed that the plasticity index alone does not accurately predict the expansion potential of remoulded expansive soils. They concluded that associating expansion potential with the product of plasticity index and % passing the No. 200 (75 μm) sieve improves the correlation significantly. It’s vital to remember that, while a soil’s plasticity may indicate the presence of expansive minerals, it’s not a guarantee that the soil is expansive.

Atterberg limits and clay content can be combined into a parameter called activity, A_c. This term was defined by Skempton (1953) as;

Activity = Plasticity Index / (% by weight finer than 2μm)

According to Skempton, clays can be divided into three categories based on their activity. “Inactive” for activities less than 0.75, “normal” for activities between 0.75 and 1.25, and “active” for activities greater than 1.25. The biggest potential for expansion is found in active clays. Table 2 shows typical activity values for a variety of clay minerals. The largest growth potential is seen in sodium montmorillonite, as evidenced by the extremely high value of activity in Table 2.

Table 2: Typical Activity Values for Clay Minerals (Skempton 1953)

Mineral	Activity
Kaolinite	0.33 – 0.46
Illite	0.9
Montmorillonite (Ca)	1.5
Montmorillonite (Na)	7.2

Free Swell Test

The free swell test involves inserting a known volume of dry soil that has passed through the No. 40 (425 μm) sieve into a graduated cylinder filled with water and measuring the swollen volume after it has settled completely. The ratio of the change in volume from the dry to the wet condition over the initial volume, expressed as a percentage, is used to calculate the free swell of the soil.

The free swell value of a high-grade commercial bentonite (sodium montmorillonite) will range from 1200 to 2000 percent. According to Holtz and Gibbs (1956), soils with free swell values as low as 100 percent can expand significantly in the field when wetted under modest loading. In addition, according to Dawson (1953), certain Texas clays with free swell values in the 50 percent range have caused significant damage due to expansion. Extreme climate circumstances, along with the soil’s expansion tendencies, caused this.

The free swell index (FSI) is calculated as;

FSI = [(soil volume in water − soil volume in kerosene)/ soil volume in kerosene] × 100%

The expansion potential of the soil as classified according to the FSI is shown in Table 3.

Table 3: Expansion Potential Based on Free Swell Index

Free Swell Index (FSI)	Expansion Potential
< 20	Low
20 – 35	Medium
35 – 50	High
> 50	Very High

Potential Volume Change (PVC)

T. W. Lambe (1960) developed the potential volume change (PVC) method for the Federal Housing Administration. Figure 2(a) shows the PVC apparatus, which has been used by many states in the United States of America. The test consists of placing a remoulded soil sample into an oedometer ring. The sample is then wetted and allowed to swell against a proving ring in the device. The pressure on the ring is given as the swell index, which is connected to qualitative ranges of possible volume change using the chart in Figure 2(a) (Lambe 1960).

potential volume chane apparatus — **Figure 2:** (a) Potential volume change (PVC) apparatus; (b) swell index vs. PVC.

The test’s simplicity is a benefit. The downside is that the proving ring’s stiffness is not uniform, allowing for varying degrees of swelling depending on the stiffness of the proving ring. The quantity of swelling allowed by the proving ring will affect the swelling pressure that develops. The swell index and PVC values are more beneficial for identifying potential expansive behaviour and should not be utilised as design parameters for undisturbed in situ soils because the test uses remoulded samples.

Expansion Test (EI) Test

The expansion index test entails compacting a soil at a saturation level of 50% ± 2% under standard conditions. The sample is subjected to a vertical pressure of 144 psf (7 kPa) and then flooded with distilled water. Equation is used to calculate the expansion index, which is reported to the closest whole number.

EI = [(final thickness − initial thickness) / initial thickness] × 1000

The expansion potential of the soil is classified according to the expansion index, as shown in Table 4.

Table 4: Expansion Potential Based on Expansion Index

Expansion Index (EI)	Expansion Potential
0 – 20	Very low
21 – 50	Low
51 – 90	Medium
91 – 130	High
> 130	Very high

The International Building Code (2012) and the International Residential Code (2012) both adopted the expansion index test for identification of an expansive soil. Both of the codes state the following:

Soils meeting all four of the following provisions shall be considered expansive, except that tests to show compliance with Items 1, 2, and 3 shall not be required if the test prescribed in Item 4 is conducted:

Plasticity Index (PI) of 15 or greater, determined in accordance with ASTM D 4318.
More than 10 percent of the soil particles pass a No. 200 sieve, determined in accordance with ASTM D 422.
More than 10 percent of the soil particles are less than 5 micrometers in size, determined in accordance with ASTMD422.
Expansion Index greater than 20, determined in accordance with ASTM D 4829.

Coefficient of Linear Extensibility (COLE)

The coefficient of linear extensibility (COLE) test measures the linear strain of an undisturbed, unconfined sample as it is dried from 5 psi (34 kPa) suction to oven-dry suction (150,000 psi = 1,000 MPa). A flexible plastic resin is applied to undisturbed soil samples during the procedure. The resin is impermeable to liquid water, but permeable to water vapor. Natural clods of soil are brought to a soil suction of 5 psi (34 kPa) in a pressure vessel.

Using Archimedes’ principle, they are weighed in air and in water to determine their weight and volume. After that, the samples are oven-dried, and another weight and volume measurement is done in the same way. COLE is a measurement of how much a sample’s dimension changes from wet to dry.

The value of COLE is given by:

COLE = ΔL∕ΔL_D= (?_dD∕?_dM)^0.33 − 1

where:
ΔL∕ΔL_D= linear strain relative to dry dimensions,
?_dD= dry density of oven-dry sample, and
?_dM= dry density of sample at 5 psi (34 kPa) suction.

The value of COLE is sometimes expressed as a percentage. Whether it is a percentage or dimensionless is evident from its magnitude. COLE has been related to swell index from the PVC test and other indicative parameters. The linear extensibility, LE, can be used as an estimator of clay mineralogy.

The LE of a soil layer is the product of the thickness, in centimeters, multiplied by the COLE of the layer in question. The LE of a soil is defined as the sum of these products for all soil horizons. The ratio of LE to clay content is related to mineralogy as shown in Table 5.

Table 5: Ratio of Linear Extensibility (LE) to Percent Clay

LE/Percent Clay	Mineralogy
< 0.05	Kaolinite
0.05–0.15	Illite
0.15	Montmorillonite

Standard Absorption Moisture Content (SAMC)

Yao et al. (2004) proposed the SAMC test as a means of identifying expansive soils. It was suggested in China’s Specifications for Highway Subgrade Design (CMC 2004). The SAMC test has the advantage of being simple. The SAMC is the water content at which a soil will reach equilibrium under specified conditions.

An undisturbed soil sample is placed on a porous plate within a constant humidity container above a saturated sodium bromide solution. After measuring the weight of the soil sample at equilibrium, it is oven-dried. The SAMC is determined as follows:

SAMC (%) = (W_e −W_s)/W_s

where:
W_e = weight of sample at equilibrium (77 ^∘F and 60% relative humidity) and
W_s = weight of oven-dry sample.

China’s Specifications for Design of Highway Subgrades (CMC 2004) presents a method for classifying expansive soils based on the standard absorption moisture content, plasticity index, and free-swell values, as shown in Table 6.

Table 6: Classification Standard for Expansive Soils (CMC, 2004)

Standard Absorption Moisture Content (%)	Plasticity Index (%)	Free-Swell Value (%)	Swell Potential Class
< 2.5	< 15	< 40	Non-expansive
2.5 – 4.8	15 – 28	40 – 60	Low
4.8 – 6.8	28 – 40	60 – 90	Medium
6.8	> 40	> 90	High

Mineralogical Methods

The presence of montmorillonite in a soil can be used to determine whether or not the soil is potentially expansive. A clay’s mineralogy can be determined based on its crystal structure or by chemical analysis. X-ray diffraction (XRD), differential thermal analysis (DTA), and electron microscopy are all popular mineralogical identification procedures.

The amount by which X-rays are diffracted around crystals is used to calculate basal plane spacing in XRD. DTA involves heating a sample of clay and an inert material at the same time. The resulting thermograms are contrasted to those for pure minerals, which are plots of temperature difference vs applied heat. On the thermograms, each mineral exhibits distinct endothermic and exothermic reactions. The clay particles can be observed directly using electron microscopy.

The size and shape of the particles can be used to make a qualitative identification. X-ray absorption spectroscopy, petrographic microscopy, soil micromorphology, digital image analysis, atomic force microscopy, and diffuse reflectance spectroscopy are some more mineralogical techniques (Ulery and Drees, 2008). In engineering practice, mineralogical approaches are rarely applied. They’re especially beneficial for research.

Chemical Methods

The most common chemical methods that are used to identify clay minerals include measurement of cation exchange capacity (CEC), specific surface area (SSA), and total potassium (TP). These methods are described in the following sections;

Cation Exchange Capacity (CEC)

The total amount of exchangeable cations required to balance the negative charge on the surface of clay particles is known as the CEC. Milliequivalents per 100 grammes of dry clay are used to calculate CEC. Excess salts in the soil are first eliminated, then adsorbed cations are restored by saturating the soil exchange sites with a known cation during the test procedure. For a mineral with a higher imbalanced surface charge, the amount of known cation required to saturate the exchange sites is greater.

Chemical examination of the extract can reveal the nature of the cation complex that was removed. Clay mineralogy is linked to CEC. A high CEC value suggests the presence of a highly active clay mineral like montmorillonite, whereas a low CEC value indicates the presence of a non-expansive clay mineral like kaolinite. In general, as the CEC rises, so does the expansion potential. Table 7 shows typical CEC values for the three most common clay minerals (Mitchell and Soga, 2005).

Table 7: Typical Values of CEC, SSA, and TP for Clay Minerals (Mitchell and Soga, 2005)

Clay Mineral	Cation Exchange Capacity (CEC) (meq/100 g)	Specific Surface Area (SSA) (m²/g)	Total Potassium (TP) (%)
Kaolinite	1 – 6	5 – 55	0
Illite	15 – 50	80 – 120	6
Montmorillonite	80 – 150	600 – 800	0

To determine the CEC of a soil, a variety of methods can be used. CEC measurement necessitates complex and precise testing procedures that are rarely performed in soil mechanics laboratories. However, many agricultural soils laboratories undertake this test on a regular basis, and it is rather inexpensive.

expansive soil — **Figure 3**: Expansion potential as indicated by clay activity and CEAc (Nelson et al, 2015)

The graph in Figure 3 was created using data from the Natural Resources Conservation Service’s soil survey reports for soils in California, Arizona, Texas, Wyoming, Minnesota, Wisconsin, Kansas, and Utah. The CEAc (CEAc = CEC/clay content) versus A_c chart was used to create Figure 3, which was designed to be used as a generic classification method for potentially expanding soils.

Specific Surface Area (SSA)

The total surface area of soil particles in a unit mass of soil is described as the specific surface area (SSA) of a soil. The SSA of montmorillonite is substantially higher than that of kaolinite. Therefore, a clayey soil with a high SSA, will have a larger water holding capacity and better expansion potential (Chittoori and Puppala 2011).

A high SSA, on the other hand, does not always imply an expanding soil. If a soil has a high organic component, for example, that fraction may have a highly reactive surface with properties similar to those of a material with a large specific surface area (Jury, Gardner, and Gardner 1991). Several methods for determining a soil’s specific surface area have been devised. Adsorption of polar molecules, such as ethylene glycol, on the surfaces of clay minerals is the most prevalent approach.

The typical SSA values for the three basic clay minerals are shown in Table 7. The SSA of the montmorillonite minerals is around ten times that of the kaolinite group. Although within the same group of minerals, the range of typical SSA values indicated in Table 7 might vary by 100 percent or more, the variation in SSA between groups, notably for montmorillonite, is so great that mineral identification is usually attainable.

Total Potassium (TP)

The only clay mineral that includes potassium in its structure is illite. Therefore, the amount of potassium ions in a soil provides a direct indication of the presence of illite (Chittoori and Puppala 2011). Table 7 shows the differences in amount of total potassium between illite and the other two minerals. Thus, high potassium content is indicative of low expansion potential.

References

[1] China Ministry of Construction (CMC). 2004. “Specifications for the Design of Highway Subgrades.” JTG D 30, Beijing: Renmin Communication Press.
[2] Chittoori, B., and A. J. Puppala. 2011. “Quantitative Estimation of Clay Mineralogy in Fine-Grained Soils.” Journal of Geotechnical and Geoenvironmental Engineering, ASCE 137(11): 997–1008
[3] Dawson, R. F. 1953. “Movement of Small Houses Erected on an Expansive Clay Soil.” Proceedings of the 3rd International Conference on Soil Mechanics and Foundation Engineering, 1, 346–350.
[4] Holtz, W. G., and H. J. Gibbs. 1956. “Engineering Properties of Expansive Clays.” Transactions ASCE 121, 641–677.
[5] International Building Code (IBC). 2012. International Building Code. Falls Church, VA: International Code Council.
[6] International Residential Code (IRC). 2012. International Residential Code for One- and Two-Family Dwellings. Falls Church, VA: International Code Council.
[7] Jury, W. A.,W. R. Gardner, andW. H. Gardner. 1991. Soil Physics (5th ed.). New York: John Wiley and Sons.
[8] Lambe, T.W. 1960. “The Character and Identification of Expansive Soils, Soil PVC Meter.” Federal Housing Administration, Technical Studies Program, FHA 701.
[9] Mitchell, J. K., and K. Soga. 2005. Fundamentals of Soil Behavior (3rd ed.). Hoboken, NJ: John Wiley and Sons.
[10] Nelson J. D., Chao K. C., Overton D. D., Nelson E. J. 2015. Foundation Engineering for Expansive Soils. John Wiley & Sons, Inc
[11] Peck, R. B., W. E. Hanson, and T. H. Thornburn. 1974. Foundation Engineering (2nd ed.). New York: John Wiley and Sons.
[12] Skempton, A. W. 1953. “The Colloidal ‘Activity’ of Clays.” Proceedings of the 3rd International Conference on Soil Mechanics and Foundation Engineering, Switzerland, 1, 57–61.
[13] Ulery, A. L., and L. R. Drees. 2008. “Mineralogical Methods,” In Methods of Soil Analysis, Part 5. Madison, WI: Soil Science Society of America.
[14] Yao, H. L., Y. Yang, P. Cheng, and W. P. Wu. 2004. “Standard Moisture Absorption Water Content of Soil and Its Testing Standard.” Rock and Soil Mechanics 25(6): 856–859.
[15] Zapata, C. E., S. L. Houston, W. N. Houston, and H. Dye. 2006. “Expansion Index and Its Relationship with Other Index Properties.” Proceedings of the 4th International Conference on Unsaturated Soils, Carefree, AZ, 2133–2137.
[16] Zheng, J. L., R. Zhang, and H. P. Yang. 2008. “Validation of a Swelling Potential Index for Expansive Soils.” In Unsaturated Soils: Advances in Geo-Engineering, edited by D. G. Toll, C. E. Augarde, D. Gallipoli, and S. J. Wheeler. London: Taylor and Francis Group.

Rafters: Functions, Types, Design, and Installation

By

Ubani Obinna

-

May 31, 2022

Table of Contents

Rafters are loading bearing structural members that are used in roof construction. They typically run from the ridge board or hip of the roof at a sloping angle to the roof wall plate, columns, or roof beams, depending on the support system adopted for the entire roof structure.

By implication, rafters receive the load from the roof covering/sheeting, accessories, and other services that may be attached to it. They are also important members for resisting upward wind pressure on roofs. Rafters can be constructed using steel or timber. Furthermore, they are constructed in series and laid parallel to each other at usually a constant spacing. The spacing of rafters can be determined from the design requirements, type of roof sheeting to employed, type and spacing of purlins, availability of materials, etc.

parts of a roof — **Figure 1:** Typical components of a hipped roof

Under heavy gravity loads, rafters have a tendency to flatten outwards on the walls. This is generally a case of structural failure of rafters, and can lead to the collapse of the walls if the spans are longer and the walls are thinner. To overcome this problem, coupled rafters have been used, which are two opposing rafters joined together by a horizontal tie beam.

However, such roofs were structurally unstable, and since they lacked longitudinal support, they were prone to racking, or horizontal movement-induced collapse. Timber roof trusses were developed later, during the Middle Ages. A cross-braced timber roof truss creates a stable, rigid unit. It should, in theory, balance all lateral forces against one another and only thrust directly downwards on the supporting walls.

Trussed rafter section — **Figure 2:** Typical trussed rafter section

Functions of Rafters

The functions of rafters in a roof are as follows;

Serve as a load bearing member for the loads and services on the roof
Safely support the purlins and the roof sheeting/coverings
Provide rigidity and stability to the roof structure
Safely resist imposed loads from wind and snow
Be capable of resisting movements due to moisture or thermal variation
Be durable so as to give satisfactory performance and reduce maintenance to a minimum

simple rafter construction — **Figure 3:** Rafter roof framing for a simple structure

Types of Rafters

Rafters are usually constructed using timber or steel. Steel rafters are popular in the construction of portal frames, where they are directly supported by steel columns and stanchions. Timber rafters are more popular in the construction of residential homes, small offices, or other smaller structures.

Steel Rafters

Steel rafters in portal frames are usually subjected to significant bending moment and shear forces from the dead and imposed loads from the roof. In order to increase the rigidity of the rafters, haunches are introduced at the eaves and at the apex. Lateral stability of the rafters are enhanced by purlins or cross-bracings. In some cases steel curved rafters are used in the design of industrial steel structures.

erection of steel structures — **Figure 4:** Steel portal frame consisting of stanchions and rafters

Timber Rafters

Timber rafters are popular options in timber/wooden roof structures. Rafters are part of a basic wood framing system and are made of wood lumber. The common rafters form the sloped sides of the triangle on a traditional gable roof, which has a triangular shape. Many common, or general purpose, rafters make up each roof. The number of common rafter units needed for each project is mostly determined by the roof’s size and scope, as well as the distance each unit rafter must span. Timber rafters rely on nails or screws for connection.

Trussed rafters are generally employed for large scale timber roof construction, but direct timber rafters are more efficient for small scale constructions. Hence, timber rafters are usually constructed in form of A-shape, consisting basically of the rafters, rafter ties, and ridge board/hips.

The rafters and external walls are then connected with ceiling joists. As a result, the area in the roof is left as a vaulted ceiling that may be finished with insulation and drywall. It could also be left as open space in an attic.

rafter in roof construction — **Figure 6:** Well constructed roof rafter

While the most basic gable roof can be built with just one type of timber roof rafter, the most sophisticated roof designs can incorporate different types of timber rafter. The types of timber rafters often employed are;

types of rafters — **Figure 7**: Plan view of roof framing members

Principal rafter

Principal rafters are the largest form of rafter found at the ends of a roof structure in a timber-framed roof. They are commonly used to carry a purlin and sit directly on a tie beam. Principal rafters run from the roof’s ridge to the wall plate; they’re a little heavier than ordinary rafters, and they’re usually framed into a tie beam at a corner post, story post, or chimney post. The principal rafters, when combined with the principal purlins, constitute a very stable roof construction system.

Common rafter

The basic gable roof is constructed using a common rafter. This style of rafter starts at an outside wall and extends all the way to the roof’s ridge board or peak. The common roof rafter is used to calculate the roof’s height and where the ridge board should be installed. The roof is now ready for the next type of rafter once the ridge board has been installed. Smaller rafters located in between the principals at both ends.

hip valley roof — **Figure 8:** The components of a hip and valley roof

Hip rafter

These are rafters that runs diagonally between the roof ridge and the top of the wall plate, forming a hipped roof. A hip rafter connects to the ridge at a 45-degree angle, as opposed to ordinary roof rafters, which run perpendicular to the peak of the roof. Traditional stick-framing techniques can be used to make these rafters, or they can be integrated in a pre-engineered steel or timber truss system.

Valley rafter

The valley rafter is the rafter in the valley line that joins the ridge to the wall plate along the meeting line of two sloped sides of a roof that are perpendicular to each other in a roof framing system. To put it another way, it’s the main rafter at the bottom of a hip and valley roof.

Jack rafter

A jack rafter is any rafter that is shorter than the whole length of the sloping roof, such as one that begins or ends at a hip or valley. They extend up from the top of the wall plate at a right angle (90°) to abut into an existing hip rafter. A jack rafter is one that has been shortened by falling on a hip rafter or being interrupted by a dormer window.

95D10DDB DDDD 4719 84D8 F6C9DB66BF5C 1 — **Figure 9:** Valley jack rafter

A88A09CC C761 4A88 8EC0 A8ED6556E56B — **Figure 10:** Hip jack rafter

Barge rafter

This is the outermost rafter on a gable end and is occasionally utilised to form a roof overhang. It is one of the two rafters that support the portion of a gable roof that extends beyond the gable wall.

Rafter Design and Spacing

Rafters are designed to be structurally stable under gravity and horizontal loads, without undergoing excessive deflection or failure. The design of rafters often involves selecting the appropriate timber class, dimensions, and spacing that will safely support the roof load for a given roof span. In most cases, tables are available for the selection of rafters.

Definitions

Rafter span
This is the un-supported length of the rafter along its slope – the diagonal or hypotenuse of a right triangle.

Rafter run
This is the horizontal or level distance covered by the rafter – the bottom chord or base of a right triangle.

Roof span
A roof span is normally the same as the building width between the outer edges of the wall top plates.

Roof slope
Roof slope is the degree of change in height as a ratio of horizontal distance traveled, usually expressed as inches of rise per foot of horizontal run, or cm of rise per m of horizontal run.

Rafter Design Tables

Using regularised UK timber dimensions, it is possible to determine the maximum span of timber rafters subjected to different values of live loads according to BS 5268-7.5.

Table 1: Maximum Clear Span of Rafter when slope is more than 15 degrees but less than 22.5 degrees (Timber class: C16; Imposed load = 0.5 – 0.75 kN/m²)

Rafter Size Width x Depth (mm)	400 mm spacing	450 mm spacing	600 mm spacing
38 × 95	1.798	1.752	1.633
38 × 120	2.505	2.430	2.242
38 × 145	3.171	3.048	2.706
38 × 170	3.710	3.568	3.145
38 × 195	4.247	4.086	3.579

44 × 95	2.009	1.954	1.816
44 × 120	2.760	2.654	2.410
44 × 145	3.328	3.201	2.907
44 × 170	3.893	3.746	3.379
44 × 195	4.456	4.288	3.845

47 × 95	2.109	2.050	1.903
47 × 120	2.821	2.713	2.464
47 × 145	3.401	3.272	2.973
47 × 170	3.978	3.828	3.480
47 × 195	4.553	4.382	3.970

Table 2: Maximum Clear Span of Rafter (Timber class: C16; Imposed load = 0.75 – 1.00 kN/m²)

Rafter Size Width x Depth (mm)	400 mm spacing	450 mm spacing	600 mm spacing
38 × 95	1.682	1.633	1.509
38 × 120	2.319	2.242	2.053
38 × 145	2.969	2.862	2.503
38 × 170	3.522	3.354	2.909
38 × 195	4.033	3.816	3.312

44 × 95	1.873	1.816	1.674
44 × 120	2.565	2.477	2.251
44 × 145	3.160	3.038	2.692
44 × 170	3.698	3.556	3.129
44 × 195	4.234	4.072	3.561

47 × 95	1.964	1.903	1.753
47 × 120	2.678	2.575	2.325
47 × 145	3.230	3.106	2.782
47 × 170	3.779	3.635	3.233
47 × 195	4.326	4.162	3.679

Table 3: Maximum Clear Span of Rafter when slope is more than 22.5 degrees but less than 30 degrees (Timber class: C24; Imposed load = 0.5 – 0.75 kN/m²)

Rafter Size Width x Depth (mm)	400 mm spacing	450 mm spacing	600 mm spacing
38 × 95	2.285	2.197	1.993
38 × 120	2.880	2.769	2.514
38 × 145	3.472	3.339	3.034
38 × 170	4.062	3.908	3.551
38 × 195	4.649	4.473	4.068

44 × 95	2.399	2.307	2.095
44 × 120	3.022	2.907	2.642
44 × 145	3.643	3.505	3.186
44 × 170	4.260	4.100	3.729
44 × 195	4.874	4.692	4.270

47 × 95	2.452	2.358	2.142
47 × 120	3.089	2.971	2.701
47 × 145	3.722	3.582	3.257
47 × 170	4.352	4.189	3.812
47 × 195	4.978	4.794	4.364

Table 4: Maximum Clear Span of Rafter when slope is more than 22.5 degrees but less than 30 degrees (Timber class: C24; Imposed load = 0.75 – 1.0 kN/m²)

Rafter Size Width x Depth (mm)	400 mm spacing	450 mm spacing	600 mm spacing
38 × 95	2.166	2.082	1.888
38 × 120	2.731	2.625	2.382
38 × 145	3.293	3.167	2.874
38 × 170	3.854	3.706	3.366
38 × 195	4.412	4.244	3.856

44 × 95	2.275	2.187	1.985
44 × 120	2.867	2.757	2.503
44 × 145	3.457	3.325	3.020
44 × 170	4.044	3.891	3.536
44 × 195	4.629	4.454	4.050

47 × 95	2.325	2.236	2.029
47 × 120	2.930	2.818	2.560
47 × 145	3.533	3.398	3.088
47 × 170	4.132	3.976	3.615
47 × 195	4.729	4.551	4.140

Assumptions in the Preparation of the Table

The allowed clear spans were computed using the BS 5268-2:2002 standard and BS 5268-7.5:1990 Structural Use of Timber – Part 2: Code of Practice for Permissible Stress Design, Materials, and Workmanship Section 7.5 Domestic rafters (explains how to use wood for structural purposes).
The self weight of the rafters is not included in the dead loads given at the top of the span table above; however, the rafter self weights are included (in addition to the dead loads) in the calculations used to calculate permissible clear spans.
Roofs with trussed rafter roofs are not covered by these span tables.
Only roof systems with four or more rafters are covered by these span tables. Ceiling joists are also expected to be employed to transmit the horizontal component of eaves-level push to adjacent rafters.
The tile battens affixed to the tops of the rafters are assumed to provide enough lateral restraint and distribute lateral stresses in these span tables.
The calculations used to create these span tables presume that the rafters are not continuous over the purlins, but that they can be continuous over the supporting purlin if necessary.
Holes and notches in the rafters can only be drilled or cut if they are proven to be adequate by specialised calculations.
These span tables do not apply to wood that has been completely exposed to the outdoors.
Wane is allowed in all parts covered in these span tables, as approved by BS 4978:2007+A2:2017.
Rafters must have a 35mm minimum end bearing.
The imposed load should be calculated in accordance with BS 6399:Part 3:1988 Code of practise for imposed roof loads; as a rule of thumb, for altitudes not exceeding 100m, a uniformly distributed load of 0.75 kN/m² can be used, and for most other areas exceeding 100m but not exceeding 200m, a uniformly distributed load of 1 kN/m² can be used.
Because there are no brittle finishes on the underside of the rafters, such as plasterboard, the effects of deflection under concentrated (point) load are not need to be considered as per BS 5268-7.5 clause 4.3.

7C48223C 6FCB 4B13 8B17 E942B9184070 — **Figure 12:** Typical timber roof construction and framing

Installation of Rafters

The following steps may be followed in the installation of rafters;

Nail a 2-by-4 board up the centre of the gable-end wall to serve as a ridge board bracing. The board should be taller than the total height of the wall and roof rise.
Place your ridge beam or rafter ties perpendicular to your rafter pattern across the walls.
With the ridge ends facing up, lean your rafters along the outside walls. You’ll have easy access to them up on the roof as a result of this.
Bring your gable-end rafters up to the rafter ties and put one nail through them. Ensure that the heel cut is flush with the wall plate.
Lean the rafters that have been fastened in against each other. Nothing is holding them up, so your assistant will have to hold them in place.
As with the gable-end rafters, go to the opposite end of the ridge board and nail two opposing rafters to their respective rafter ties and lean them against each other.
Raise one end of the ridge board to the intersection of the two rafters.
Attach the rafters to the ridge board using nails.
Slip the ridge beam between the two rafters at the first rafter course and nail it in place.
These two common rafters have enough support at this time to stand up on their own. Nail the remaining rafters to the ridge board, making sure they’re evenly spaced.
Install collar ties, purlins, sway braces, and other supports as needed or required by code once the rafters are securely connected.

See how this Cantilever Design Problem was Solved

By

Ubani Obinna

-

May 30, 2022

Table of Contents

I came across an Instagram post on the design of reinforced concrete cantilever beams that forms part of a modern residential dwelling. According to the author of the post, the cantilever beam is about 5 m long.

Cantilevers are beams that are rigidly fixed at one end, and freely supported at the other end. By implication, they are very susceptible to excessive deflection and vibration.

The architectural rendering of the structure in question is shown in Figures 1 and 2.

8353324C 3AC7 494C B089 5E8CFE3FB397 — **Figure 1**: A 3D render of the proposed building

A7FCD682 45D3 4A86 B90E 0E4A8777A012 — **Figure 2**: Another 3D render of the proposed building

As can be seen from the images above, the architectural rendering depicts a contemporary building with a cantilever projection at the front. To solve the problem, the design engineer, decided to introduce a diagonal/slanted reinforced concrete column, which would act as tension members to support the cantilever beam. The structural modelling of the scheme adopted by the structural engineer is shown in Figures 3 to 5.

C078FC29 9017 451C 9A61 B7FF35FC93FE — **Figure 3:** Structural scheme of the building

779918AB 8667 47CC BF4D E411730D1885 — **Figure 4:** Typical analytical model of the building on a design software

1B402596 2B89 42FB AB5A 5A60F09ACE12 — **Figure 5:** *Another view of the structural scheme*

By all indications, the design has been completed and the contractor has gone to the site. The construction images of the models are shown in Figures 5 – 7.

4C788259 EFE3 473F BD71 68935A037E1F — **Figure 5:** Picture of the building under construction

08AB2F28 A33F 4CD7 87D4 C0721BAD55D8 — **Figure 6:** Another picture of the building under construction

A73B1FB0 1CA7 4901 9365 EAF1E132C553 — **Figure 7:** Another picture of the building under construction

As can be seen in Figure 7, the reinforced concrete diagonal columns were cast monolithically with the frame of the structure. Then bricks were used to cover the openings to create a plain wall that matches the architectural requirements.

Why did the Model Succeed?

The model adopted by the structural engineer succeeded because there are full walls without openings where the inclined columns can be hidden without any implications. Because there are no openings (doors, windows, or curtain walling) on the first bay of the first floor, the inclined columns can be introduced without affecting the architectural concept of the building.

If there had been need for openings in those walls, the adopted structural scheme would not have been an ideal solution. Therefore the ingenuity of the design engineer is appreciated.

How Efficient is the Model?

To check the structural efficiency of the model, let us carry out a comparative assessment of a cantilever beam with and without diagonal columns on Staad Pro software. The 3D rendering of the model without diagonal columns is shown in Figure 8.

3d model of the comparison — **Figure 8:** 3D rendering of the test model without diagonal columns

For simplicity, all the first floor beams were loaded with a uniformly distributed load of 25 kN/m as shown below. Furthermore, a uniformly distributed load of 10 kN/m was applied to the roof beams.

Loading on the model — **Figure 9:** Typical loading and dimensions of the model

The analysis comparison is going to check the effect of introducing a diagonal column on the deflection, bending, and shear force on the structure.

Introduction of diagonal — **Figure 10:** 3D rendering of the test model with diagonal columns

Analysis Results: Without diagonal column

deflection 1 — **Figure 11**: Deflection profile of the structure (without diagonal columns)

As can be seen above, without the presence of the diagonal columns, the maximum deflection at the free end of the cantilever was 35.261 mm.

bending moment 1 — **Figure 12:** Bending moment diagram of the structure (without diagonal columns)

Without the presence of the diagonal column, the maximum cantilever moment was 372.701 kNm. However, the column sitting on top of the cantilever beam appeared to be tension, assisting in supporting the cantilever beams and sending the load to the roof beams.

shear 1 — **Figure 13:** Shear force diagram of the structure (without diagonal columns)

A maximum shear force value of 158.518 kN was observed at the fixed end of the cantilever without diagonal columns.

Axial 1 — **Figure 14:** Axial force diagram of the structure (without diagonal columns)

The axial force diagram confirms that the column sitting on top of the cantilever beam is in axial tension, as well as the roof beams.

Analysis Result: With Diagonal column

deflection 2 — **Figure 1**5: Deflection profile of the structure (with diagonal columns)

As can be seen in Figure 15, when diagonal columns were introduced, the maximum deflection at the free end of the cantilever reduced to 23.301 mm.

BMD 2 — **Figure 16:** Bending moment diagram of the structure (with diagonal columns)

With the presence of the diagonal column, the maximum cantilever moment was 219.221 kNm (see Figure 16). However, the column sitting on top of the cantilever beam reversed to be a compression column, thereby transferring its loads to the cantilever beams.

sHEAR 2 — **Figure 17:** Shear force diagram of the structure (with diagonal columns)

With the presence of diagonal columns, the shear force at the fixed end of the cantilever is reduced to 106.244 kN as shown in Figure 17.

AXIAL 2 — **Figure 18:** Axial force diagram of the structure (with diagonal columns)

Axial force diagram in Figure 18 shows that the diagonal column is in axial tension.

A table of comparison has been prepared below to show the effect of diagonal tension column support on the deflection behaviour of long span cantilever beams.

	Without diagonal Column	With diagonal column	Percentage decrease
Deflection (mm)	35.261 mm	23.301 mm	33.91%
Bending moment (kNm)	372.701 kNm	219.221 kNm	41.18%
Shear force (kN)	158.518 kN	106.244 kN	32.97%

It can therefore be seen that the diagonal tension columns were effective in reducing the deflection, bending moment, and shear force on the cantilever beams. Where architectural specifications permit, cantilever beams and slabs can be supported using diagonal tension members for more structural efficiency.

Source of images:
Instagram @engnivaldo
Architects: @curvoarquitetos

Finite Difference Solution to Flow of Water Through Soils | Flownets

By

Ubani Obinna

-

May 30, 2022

Manual sketching of flownets is an acceptable way of dealing with two-dimensional groundwater water flow and seepage in soils. However, as the geometry gets more complicated and the flow becomes anisotropic, manual sketching of flownets become more tedious and less accurate. In this case, finite difference solution to flow of water through soils can be adopted.

Many flow problems can be considered as two-dimensional, and in cases where permeability has the same value for all directions, the Laplace’s equation for flow becomes;

(∂²H/∂x²) + (∂²H/∂z²) = 0 ——— (1)

For problems with orthogonal geometry and Laplace’s equation, such as the flow around a pervious barrier shown in Figure 1, the finite difference method provides a quick and accurate solution. This entails replacing the continuous soil cross-section delimited by ABCD with a pattern of discrete points on an orthogonal grid within the cross-section. At each grid point, the governing differential equation can be expressed approximatively in terms of H values at that point and adjacent points.

Finite Difference Solution to Flow of Water Through Soils — **Figure 1:** Finite difference grid and results for seepage flow beneath an impermeable wall (Tomlinson, 2001)

Previously, we solved the elastic analysis of simply supported thin plates using the finite difference method. We will do the same to solve Laplace’s equation to determine two-dimensional confined flow through soils. Let us consider a grid of a flow domain, as shown in Figure 2, where (i, j) is a nodal point.

**Figure 2:** A partial grid of the flow domain (Budhu, 2011)

Using Taylor’s theorem, we have;

k_x(∂²H/∂x²) + k_z(∂²H/∂z²) = k_x/∆x² (h_i+1,j+ h_i-1,j – 2h_i,j) + k_z/∆z² (h_i,j+1+ h_i,j-1 – 2h_i,j) = 0 ——— (2)

Let α = k_x/k_z and ∆x = ∆z (i.e., we subdivide the flow domain into a square grid). Then, solving for h_i,j from Equation (2) gives;

h_i,j= 1/2(1 + α) × (αh_i+1,j + αh_i-1,j + h_i,j+1 + h_i,j-1) ——— (3)

For isotropic conditions, α = 1 (k_x = k_z) and Equation (3) becomes;

h_i,j= 1/4 × (h_i+1,j + h_i-1,j + h_i,j+1 + h_i,j-1) ——— (4)

Since we are considering confined flow, one or more of the boundaries would be impermeable. Flow cannot cross impermeable boundaries and, therefore, for a horizontal impermeable surface;

∂h/∂x = 0 ——— (5)

The finite difference form of Equation (5) is;

∂h/∂x =1/2∆x(h_i,j+1 – h_i,j-1) = 0 ——— (6)

Therefore, h_i,j+1 = h_i,j-1 and, by substitution in Equation (4), we get;

h_i,j= 1/4 × (h_i+1,j + h_i-1,j + 2h_i,j-1) ——— (7)

Various types of geometry of impermeable boundaries are encountered in practice, three of which are shown in Figure 3. For Figure 3a, b, the finite difference equation is;

h_i,j= 1/2 × (h_i+1,j + h_i,j-1) ——— (8)

boundary conditions encountered in practice — **Figure 3:** Three types of boundary encountered in practice (Budhu, 2011)

and, for Figure 3c,

h_i,j= 1/3 × (h_i,_j-1 + h_i+1,j + h_i,j+1 + 0.5h_i-1,j + 0.5h_i,j+1) ——— (9)

The pore water pressure at any node (u_i,j) is;

u_i,j= γ_w(h_i,j – z_i,j) ——— (10)

where z_i,j is the elevation head.

Contours of potential heads can be drawn from discrete values of h_i,j. The finite difference equations for flow lines are analogous to the potential lines; that is, ψ_s replaces h in the above equations and the boundary conditions are specified for ψ_s rather than for h.

The horizontal velocity of flow at any node (v_i,j) is given by Darcy’s law:

v_i,j= k_xi_i,j ——— (11)

Where i_i,j is the hydraulic gradient expressed as;

i_i,j= (h_i+1,j – h_i-1,j)/2∆x ——— (12)

Therefore;

v_i,j= k_x/2∆x × (h_i+1,j – h_i-1,j) ——— (13)

The flow rate, q, is obtained by considering a vertical plane across the flow domain. Let L be the top row and K be the bottom row of a vertical plane defined by column i (Figure 2). Then the expression for q is;

q = k_x/4[h_i+1,L– h_i-1,L+ 2∑(h_i+1,j – h_i-1,j) + h_i+1,K – h_i-1,K] ——— (14)

Procedure of Finite Difference Solution to Flow of Water Through Soils

sheet pile — **Figure 4**: Sheet pile wall (Budhu, 2011)

The procedure to determine the distribution of potential head, flow, and porewater pressure using the finite difference method is as follows (Budhu, 2011):

Divide the flow domain into a square grid. Remember that finer grids give more accurate solutions than coarser grids, but are more tedious to construct and require more computational time. If the problem is symmetrical, you only need to consider one-half of the flow domain.
For example, the sheet pile wall shown in Figure 4 is symmetrical about the wall and only the left half may be considered. The total flow domain should have a width of at least four times the thickness of the soil layer. For example, if D is the thickness of the soil layer (Figure 4), then the minimum width of the left half of the flow domain is 2D.
Identify boundary conditions, for example, impermeable boundaries (flow lines) and permeable boundaries (equipotential lines).
Determine the heads at the permeable or equipotential boundaries. For example, the head along the equipotential boundary AB (Figure 4) is DH. Therefore, all the nodes along this boundary will have a constant head of DH. Because of symmetry, the head along nodes directly under the sheet pile wall (EF) is DH/2.
Apply the known heads to corresponding nodes and assume reasonable initial values for the interior nodes. You can use linear interpolation for the potential heads of the interior nodes.
Apply Equation (4), if the soil is isotropic, to each node except (a) at impermeable boundaries, where you should use Equation (7), (b) at corners, where you should use Equations (8) and (9) for the corners shown in Figure 3a–c, and (c) at nodes where the heads are known.
Repeat item 5 until the new value at a node differs from the old value by a small numerical tolerance, for example, 0.001 m.
Arbitrarily select a sequential set of nodes along a column of nodes and calculate the flow, q, using Equation (14). It is best to calculate q’ = q for a unit permeability value to avoid too many decimal points in the calculations.
Repeat items 1 to 6 to find the flow distribution by replacing heads by flow q’. For example, the flow rate calculated in item 7 is applied to all nodes along AC and CF (Figure 4). The flow rate at nodes along BE is zero.
Calculate the porewater pressure distribution by using Equation (10).

References

[1] Budhu M. (2011): Soil Mechanics and Foundations (3rd Edition). Pearson Education
[2] Tomlinson M. J. (2001): Foundation Design and Construction (7th Edition). John Wiley and Sons, Inc

Seepage of Water into Excavations

By

Ubani Obinna

-

May 29, 2022

Groundwater is typically regarded as one of the most challenging problems encountered during excavation for civil engineering construction works. To control seepage of water into excavations, expensive and continuous pumping may be required, and the constant flow from the surrounding ground may cause settlement of adjacent structures.

The sides of open excavations are susceptible to eroding or collapsing if they are subjected to heavy inflows. Under certain conditions, the base can become unstable due to seepage towards the pumping sump, or if the bottom of excavation in clay is underlain by a pervious layer containing water under artesian pressure.

However, with knowledge of the soil and ground-water conditions and an understanding of the laws of hydraulic flow, it is possible to adopt ground-water control methods that will ensure a safe and cost-effective construction plan under any circumstances. Before beginning work, it is essential to collect all important data, and this aspect should not be overlooked during the site investigation phase.

Sometimes, after an excavation has been begun, more pumps are brought to the site, and with great difficulty the excavation is taken deeper until the inflow is so heavy that the sides start collapsing, endangering adjacent roads and buildings, or ‘boiling’ of the bottom is so extensive that a suitable base for foundation concreting cannot be obtained.

ground water pumping — **Figure 1:** Typical pumping of ground water from excavation

At this point, the contractor may give up and seeks assistance from outside sources to install a wellpointing or bored well system for ground-water lowering, or resorts to underwater construction. The ground-water lowering systems may be effective, but the overall cost of abortive pumping, extra excavation of collapsed material, making good the damage, and standing time of plant and labour would have been significantly higher had the ground-water lowering system been used in the first place.

Due to a lack of understanding of the capabilities of modern groundwater lowering systems, there have been instances in which caissons have been used for foundations in water-bearing soils when conventional construction with the aid of such systems would have been perfectly feasible and significantly less expensive.

Figure 2(a) shows the flow line of groundwater under a relatively low head into an open excavation in permeable soil. The water surface is depressed toward the pumping sump, and as a result of the low head and flat slope, seepage lines do not emerge on the slope and conditions are stable.

Seepage of Water into Excavations — **Figure 2:** Seepage into open excavations (a) Stable conditions (b) Unstable conditions (c) Increasing stability of slope by blanketing (Tomlinson, 2001)

Nevertheless, if the head is increased or the slopes are steepened, the water will flow away from the face, and if the velocity is high enough, it will cause the movement of soil particles and erosion down the face, resulting in undermining as well as the collapse of the upper slopes (Figure 2(b)). The solution is to level the slopes and cover the face with a graded gravelly filter material that allows water to pass but traps soil particles (Figure 2(c)).

Graded filters will need to be properly designed, otherwise erosional instability might occur. This type of erosional instability is most likely to occur in fine or uniformly graded sands. Since well-graded sand gravels act as natural filters and their higher permeability prevents the emergence of flow lines on the excavated face, there is a significantly reduced risk of complications when using these materials.

In the case of close-timbered or sheet-piled excavations, the flow lines descend behind the sheet piling before ascending into the excavation (Figure 3). This condition of upward seepage is especially prone to cause instability, known as “piping” or “boiling,” when the velocity of the upward-flowing water is sufficient to suspend soil particles.

seepage into sheeted — **Figure 3**: Seepage into sheeted excavation (Tomlinson, 2001)

The preceding cases are generally applicable to groundwater flow in permeable soils, such as sands and gravels, or similar materials with relatively low proportions of silt and clay. Little or no difficulty is encountered when excavating in clays. When groundwater is present, it typically seeps through fissures and can be removed by pumping from sump wells.

Typically, the flow velocity is so low that there is no risk of erosion. Silts, however, are extremely problematic. They are permeable enough to allow water to pass through, but their permeability is low enough to make any system of ground-water lowering by wellpoints or bored wells a time-consuming and expensive process. Typically, ground water in rocks seeps from the face as springs or seeps through fissures or permeable layers. There is no risk of instability unless heavy flows pass through a brittle, fractured rock.

Generally, water in rock excavations can be pumped from open sump pits, with the exception of when foundation concrete is intended to be placed against the rock face. If the springs or weeps are strong, the water will wash the cement and fines from the unset concrete and flow out through the concrete’s surface.

The standard procedure is to construct the pumping sump at the lowest point of the excavation and continue pumping from it until the concrete has hardened, while allowing seepage to occur from the face towards the sump through a layer of ‘no fines’ concrete, or behind corrugated sheeting, or bituminous or plastic sheeting attached to a wire mesh frame (Figure 4).

water bearing rock formation — **Figure 4**: Concreting adjacent to water-bearing rock formation (Tomlinson, 2001)

After the concrete work is completed, the area behind the sheeting is grouted using the pipes left for this purpose. Typically, occasional weeping can be remedied by plastering with quick-setting cement mixtures or by applying dry cement to the face prior to pouring concrete.

Calculation of rates of seepage of water into excavations

In large excavations, it is necessary to estimate the amount of water that must be pumped in order to remove the water from below the formation level. This quantity must be known in order to provide the required number and capacity of pumps. In the case of long-relative-to-width trench excavations, the flow calculation can typically be modelled as a condition of gravitational flow to a partially penetrating slot (Figure 5).

ground water flow — **Figure 5:** Gravity flow to trench excavation (partially penetrating slot) – draw-down for gravity flow for sources remote from trench (Tomlinson, 2001)

drawdown flow net — **Figure 6:** Flow net for conditions in Figure 5 (Tomlinson, 2001)

For this flow condition, ground water in the pervious layer is not confined by an impervious layer above it, and the trench does not extend completely through the pervious water-bearing layer. In addition, the source of ground water is distant from the excavation, as there is no large body of water such as a river or the ocean nearby.

For the conditions in Figures 5 and 6, flow to the trench from both sides of the excavation is given by the equation;

q = [0.73 + 0.27(H – h₀)/H] k/L(H² – h₀²) ——— (1)

where;
q = quantity of flow per unit length of trench,
k = coefficient of permeability of the pervious layer,
H, h₀, and L are dimensions as shown in Figure 5.

The dimension L can be obtained by drawing a flow net as shown typically in Figure 6 or approximately by substituting L for R₀ in equation (5) (see below). Where the draw-down (H — h) has been measured in a trial excavation at a distance y from the slot, L can be obtained from the equation;

H² — h²= (L – y)/L (H² – h_c²) ——— (2)

Where the trench penetrates fully the water-bearing layer, the flow to the trench from both sides of the excavation is given by the equation;

q = k/L(H² – h_c²) ——— (3)

If there is a large body of water such as the sea or a river close to one side of the excavation, the flow to the opposite side from a remote source of water will be small in comparison with that to the side close to the line source and it can be neglected. The flow from the nearby source, assuming this to be a line source of infinite length compared to the length of the trench, is equal to half the quantity as calculated by either equations (1) or (3), and the distance L is the distance from the trench to the nearby source.

Where a pumping test has been made in the excavation and the draw-down (H — h) is measured at a radius r from the well, the dimension R₀ for gravity flow can be calculated from the equation;

H² — h² = [(H² — h_w²)/log_e(R₀/r_w)] × log_e(R₀/r) ——— (4)

Alternatively R₀can be obtained by drawing a flow net or it can be obtained approximately from the empirical equation.

R₀= CH√k ——— (5)

where;
C = a factor equal to 3000 for radial flow to pumped wells and between 1500 and 2000 for line flow to trenches or to a line of well points,
H = total draw-down in metres,
k = coefficient of permeability in metres per second

Shape of Draw Down Curve

It is often necessary to determine the shape of the drawdown curve to a well, for example to assess the risk of settlement of existing structures near the excavation. The draw-down in uniform soils can be obtained by means of Figure 7.

ghy — **Figure 7:** Determination of shape of draw-down curve (Tomlinson, 2001)

In the case of large rectangular or irregularly shaped excavations, the flow can be calculated by drawing a plan flow net of the type shown in Figure 8, when for gravity flow to a fully penetrating excavation the flow per unit thickness of the pervious layer is given by the equation;

q = k(H – h_e) × (N_f / N_e) ——— (6)

or for a thickness D of pervious water-bearing soil, the total flow to the excavation is given by;

Q = k(H – h_e) × D(N_f / N_e) ——— (7)

where;
k = coefficient of permeability,
N_f = number of flow lines,
N_e = number of equipotential lines

Worked Example

The plan flow net for a gravity flow to fully penetrating excavation for dry dock is shown below. For the example, H = 12.0 m, h_c= 0, k = 8 × 10^-4 m/s, D = 12.0 m, N_f = 14, N_e = 5

Therefore
Q = 8 × 10^-4 × (12 — 0) × 12 × (14/5) = 0.32 m³/s or 19200 litres/min

It is important to note that the equations generated above are for the quantity of flow when stead state conditions have been obtained. A higher pumping capacity will be required in large and deep excavations if the time required to achieve the necessary draw down is not to be unduly protracted. The volume of water to be pumped out from standing water level to full draw down conditions in the excavation should be calculated and divided by the time required by the construction programme. This will give the initial pumping capacity to achieve the required draw down.

References
Tomlinson M. J. (2001): Foundation Design and Construction (7th Edition). Pearson Education

Flownets: Two-Dimensional Flow of Water Through Soils

By

Ubani Obinna

-

May 30, 2022

Table of Contents

Laplace’s equation is used to describe the movement of water through soils. By comparison, the flow of water through soils is analogous to the steady-state heat flow and steady-state current flow in homogeneous conductors. Flownets can be used to calculate the flow of water through soils based on the Laplace’s equation. The common form of Laplace’s equation for the flow of water through two-dimensional soils is:

k_x(∂²H/∂x²) + k_z(∂²H/∂z²) = 0 ——– (1)

where H is the total head and k_x and k_z are the hydraulic conductivities in the X and Z directions. The condition that the changes in hydraulic gradient in one direction are balanced by changes in the other directions is expressed by Laplace’s equation.

The assumptions in Laplace’s equation are:

• Darcy’s law is valid.
• Irrotational flow (vorticity) is negligible. This assumption leads to the following two-dimensional relationship in velocity gradients.

∂v_z / ∂Z = ∂v_x / ∂X

where v_z and v_x are the velocities in the Z and X directions, respectively. This relationship is satisfied for a uniform flow field and not a general flow field. Therefore, we will assume all flows in this chapter are uniform, i.e., v_z = v_x = constant.
• There is inviscid flow. This assumption means that the shear stresses are neglected.
• The soil is homogeneous and saturated.
• The soil and water are incompressible (no volume change occurs).

Laplace’s equation is also called the potential flow equation because the velocity head is neglected. If the soil is an isotropic material, then k_x = k_z and Laplace’s equation becomes;

(∂²H/∂x²) + (∂²H/∂z²) = 0 ——– (2)

Any differential equation requires knowledge of the boundary conditions in order to be solved. Since the boundary conditions of the majority of “real” structures are complex, an analytical or closed-form solution cannot be obtained for these structures. Using numerical techniques such as finite difference, finite element, and boundary element, it is possible to obtain approximate solutions.

We can also attempt to replicate the flow through the actual structure using physical models. There are two major techniques for solving Laplace’s equation. The first is an approximation known as flownet sketching, and the second is the finite difference method. In this article, we are going to focus on flownet sketching.

Flownet Sketching

The flownet sketching technique is straightforward and adaptable, and it represents the flow regime. It is the preferred method of analysing flow through soils for geotechnical engineers. Before delving into these solution techniques, however, we will establish a few key conditions necessary to comprehend two-dimensional flow.

The solution of Equation (1) is solely dependent on the total head values within the flow field in the XZ plane. Let us introduce a velocity potential (ξ) that describes the variation of total head in a soil mass as follows:

ξ = kH ——– (3)

where k is a generic hydraulic conductivity. The velocities of flow in the X and Z directions are;

v_x = k_x(∂H/∂x) = ∂ξ/∂x ——– (4a)
v_z = k_z(∂H/∂z) = ∂ξ/∂z ——– (4b)

illustration of flow terms — **Figure 1**: Illustration of flow terms.

The inference from Equations (4a) and (4b) is that the velocity of flow (v) is normal to lines of constant total head, as illustrated in Figure 1 The direction of v is in the direction of decreasing total head. The head difference between two equipotential lines is called a potential drop or head loss.

If we draw lines that are tangent to the flow velocity at each point in the flow field in the XZ plane, we will obtain a series of lines that are normal to the equipotential lines. These lines are known as streamlines or flow lines (Figure 1). A flow line represents the expected path of a particle of water in a steady-state flow. ψ_s is a stream function that represents a streamline family (x, z). According to the stream function, the components of velocity in the X and Z directions are as follows:

v_x = ∂ψ_s / ∂z ——– (5a)
v_z = ∂ψ_s / ∂x ——– (5b)

Since flow lines are normal to equipotential lines, there can be no flow across flow lines. The rate of flow between any two flow lines is constant. The area between two flow lines is called a flow channel (Figure 1). Therefore, the rate of flow is constant in a flow channel.

Criteria for Sketching Flownets

A flownet is a graphical representation of a flow field that satisfies Laplace’s equation and comprises a family of flow lines and equipotential lines. A flownet must meet the following criteria (Budhu, 2011):

The boundary conditions must be satisfied.
Flow lines must intersect equipotential lines at right angles.
The area between flow lines and equipotential lines must be curvilinear squares. A curvilinear square has the property that an inscribed circle can be drawn to touch each side of the square and continuous bisection results, in the limit, in a point.
The quantity of flow through each flow channel is constant.
The head loss between each consecutive equipotential line is constant.
A flow line cannot intersect another flow line.
An equipotential line cannot intersect another equipotential line.

An infinite number of flow lines and equipotential lines can be drawn to satisfy Laplace’s equation. However, only a few are required to obtain an accurate solution. The procedure for constructing a flownet is described next.

Flownet for Isotropic Soils

According to Budhu (2011), the procedure for constructing the flownet of isotropic soils are as follows;

Draw the structure and soil mass to a suitable scale.
Identify impermeable and permeable boundaries. The soil–impermeable boundary interfaces are flow lines because water can flow along these interfaces. The soil–permeable boundary interfaces are equipotential lines because the total head is constant along these interfaces.
Sketch a series of flow lines (four or five) and then sketch an appropriate number of equipotential lines such that the area between a pair of flow lines and a pair of equipotential lines (cell) is approximately a curvilinear square. You would have to adjust the flow lines and equipotential lines to make curvilinear squares. You should check that the average width and the average length of a cell are approximately equal by drawing an inscribed circle. You should also sketch the entire flownet before making adjustments.

Flownet of a sheet pile wall — **Figure 2:** Flownet for a sheet pile (Budhu, 2011)

The flownet in confined areas between parallel boundaries typically consists of elliptical and symmetrical flow lines and equipotential lines (Figure 2). Avoid abrupt changes between straight and curved flow and equipotential lines. Transitions should be smooth and gradual. For certain problems, portions of the flownet are enlarged, are not curvilinear squares, and do not satisfy Laplace’s equation.

For instance, the portion of the flownet beneath the base of the sheet pile in Figure 2 is not composed of curvilinear squares. Check these sections to ensure that repeated bisection results in a point for a precise flownet.

Another example of flownet are shown in Figures 3. Figure 2 shows a flownet for a sheet pile wall, and Figure 3 shows a flownet beneath a dam. In the case of the retaining wall, the vertical drainage blanket of coarse-grained soil is used to transport excess porewater pressure from the backfill to prevent the imposition of a hydrostatic force on the wall. The interface boundary, is neither an equipotential line or a flow line. The total head along the boundary is equal to the elevation head.

flownet for a dam — **Figure 3:** Flownet under a dam with a cutoff curtain (sheet pile) on the upstream end (Budhu, 2011)

Flow Rate

Let the total head loss across the flow domain be ΔH, that is, the difference between upstream and downstream water level elevation. Then the head loss (Δh) between each consecutive pair of equipotential lines is;

Δh = ΔH/N_d ——– (6)

where N_d is the number of equipotential drops, that is, the number of equipotential lines minus one. In Figure 1, ΔH = H = 8 m and N_d = 18. From Darcy’s law, the flow through each flow channel for an isotropic soil is;

q = Aki = (b × 1)k(Δh/L) = kΔh(b/L) = k(ΔH/N_d)(b/L) ——– (7a)

where b and L are defined as shown in Figure 14.3. By construction, b/L = 1, and therefore the total flow is;

q = kΔH(N_f/N_d) ——– (7b)

where N_f is the number of flow channels (number of flow lines minus one). In Figure 1, N_f= 9. The ratio N_f/N_d is called the shape factor. Finer discretization of the flownet by drawing more flow lines and equipotential lines does not significantly change the shape factor. Both N_fand N_d can be fractional. In the case of anisotropic soils, the quantity of flow is;

q = ΔH(N_f/N_d)√(k_xk_z) ——– (8)

Summarily, Flow nets are typically designed for homogeneous, isotropic porous media undergoing saturated flow to known boundaries. There are extensions to the basic method that make it possible to solve the following cases:

inhomogeneous aquifer: matching conditions at property boundaries
anisotropic aquifer: drawing the flownet in a transformed domain and scaling the results differently in the principal hydraulic conductivity directions before returning the solution
one boundary is a seepage face: iteratively solving for both the boundary condition and the solution throughout the domain

The method is typically applied to these types of groundwater flow problems, but it can be applied to any problem described by the Laplace equation, such as the flow of electric current through the earth.

References
Budhu M. (2011): Soil Mechanics and Foundations (3rd Edition). John Wiley & Sons, Inc.

Fire Resistance Design of Steel Columns

Buckling Resistance

Shear Resistance

Design Example (Franssen and Real, 2015)

Spacing of Piles in a Group

Ultimate Bearing Capacity of Pile Group in Clay

Ultimate Bearing Capacity of Pile Group in Sand

Grades of Basements

Types of Water-resisting Construction/Protection

Type A – Waterproofing Barrier Protection

Type B – Structurally Integral Protection

Type C – Drained Protection

Rule of Thumb for Structural Design of Water Resistant Basements

Materials for Waterproofing of Basements

Bracing Systems

Vertical bracing

Types of Vertical Bracing

Single Diagonal Bracing

Cross Bracing (X-Braced Frames)

V-Bracing (Chevron Bracing)

Horizontal bracing

Horizontal Diaphragm

Discrete Triangulated Bracing

Identification of Expansive Soils Based on Physical Properties

Methods Based on Plasticity

Free Swell Test

Potential Volume Change (PVC)

Expansion Test (EI) Test

Coefficient of Linear Extensibility (COLE)

Standard Absorption Moisture Content (SAMC)

Mineralogical Methods

Chemical Methods

Cation Exchange Capacity (CEC)

Specific Surface Area (SSA)

Total Potassium (TP)

References

Functions of Rafters

Types of Rafters

Steel Rafters

Timber Rafters

Principal rafter

Common rafter

Hip rafter

Valley rafter

Jack rafter

Barge rafter

Rafter Design and Spacing

Definitions

Rafter Design Tables

Installation of Rafters

Why did the Model Succeed?

How Efficient is the Model?

Analysis Results: Without diagonal column

Analysis Result: With Diagonal column

Procedure of Finite Difference Solution to Flow of Water Through Soils

References

Calculation of rates of seepage of water into excavations

Shape of Draw Down Curve

Worked Example

Flownet Sketching

Criteria for Sketching Flownets

Flownet for Isotropic Soils

Flow Rate

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