Theory of structures is a field of knowledge that is concerned with the determination of the effect of loads (actions) on structures. A structure in this context is generally regarded to be a system of connected members that can resist a load. Therefore in some programs, theory of structures is also referred to as structural analysis.

In application, theory of structures is usually interested in the computation of the deformations (displacements), internal forces, stresses, stability, support reactions, velocity, and accelerations of structures under load. The principles from applied mathematics, applied sciences (physics and mechanics), and materials science are usually employed in achieving this. The results of the analysis are used to evaluate the behaviour of a structure under the load, with the sole aim of verifying the integrity of the structure when in use. Therefore, theory of structures (structural analysis) is a key part of the engineering design of structures.

Theory of Structures and Structural Design

Buildings, bridges, and towers are prominent examples of the application of theory of structures in civil engineering. Ship and aircraft frames, tanks, pressure vessels, mechanical systems, and electrical supporting structures are examples of other engineering areas.

In the design of civil engineering structures for public use, safety, stability, serviceability, functionality, aesthetics, as well as economic and environmental constraints must be considered by the engineer. Thus, structural designs are expected to satisfy two basic limit states which are;

Ultimate Limit State (ULS) and
Serviceability Limit State (SLS)

According to limit state design, ULS requirements have to deal with issues like structural failure, overturning, buckling, instability, etc while SLS requirements have to deal with issues like deflection, vibration, cracking, etc, which affects the appearance and user experience of the building.

Usually, numerous independent evaluations of various solutions are required before a final decision on which structural scheme is best can be made. This approach to the design of structures, therefore, necessitates a basic understanding of material properties and the principles of mechanics that control material behaviour.

Following the proposal of a preliminary structural design, the structure must be examined to ensure that it has the requisite stiffness and strength. To effectively examine a structure, specific idealizations about how the components are supported and connected must be made. The loadings on a structure are calculated using codes and local requirements, and the forces and displacements in the members are calculated using theory of structures (structural analysis), which is the focus of this article.

The results of the structural analysis can then be utilised to redesign the structure, taking into consideration a more precise calculation of the self-weight and size of the members. As a result, structural design is based on a sequence of successive approximations, each of which necessitates a structural analysis.

Types of Structures

When modelling a real-life structure, it is necessary to represent the form of the structure in terms of idealized structural members, e.g. in the case of plane frames as beam elements, in which the beams, columns, etc. are indicated by line diagrams. The lines normally coincide with the centre lines of the members.

The decision as to which type of structural system to use rests with the structural designer whose choice will depend on the use of the structure, the materials to be used, and the original form of the structure as indicated by the architect. It is possible that more than one form of structural systems may be required to satisfy the requirements of the problem and the designer must then rely on experience and skill to choose the best solution. Examples of structural elements/systems are;

Beams
Columns
Trusses
Frames (portal frames, gable frames)
Arches
Cables
Shear walls
Continuum structures (shells, plates, domes, etc)

Beams

Beams are the commonest of all structural systems. They are usually made up of straight prismatic members that span in between supports that may be of any form. Beams are usually important in resisting transverse loads and achieve so by developing mainly bending moments and shear forces. Typically in structural design, beams can be made of reinforced concrete, steel, or timber.

See;
Design of reinforced concrete beams
Design of steel beams according to Eurocode 3
Design of steel beams according to BS 5950
Design of timber beam according to Eurocode 5

When a beam is fixed at one end and free at the other end, it is called a cantilever beam. It is also possible to have a beam extend beyond an external support and in this case, it is called a beam with an overhang (see Figure 3b and 3c). A beam that has intermediate supports is called a continuous beam, and when internal hinges are introduced to make it statically determinate, it is called a compound beam.

simply supported beam — **Figure 2:** Simply supported beam system for bridge

different types of beam systems — **Figure 3**: Different types of beam systems

Cantilever beams and compund beams — **Figure 4:** (a) Cantilever beam (b) Compound beam

Columns

Vertical members that are used to resist axial compressive loads are referred to as columns. Hollow sections, H-sections, and I-sections are often used in the design of steel columns. Rectangular, circular, and square cross-sections with reinforcing rods are often used in the design of reinforced concrete columns. Occasionally, columns are subjected to both axial load and a bending moment. When a column is subjected to a bending moment in one direction, it is called a uniaxially loaded column, and when it is loaded in two directions, it is called a biaxially loaded column.

Trusses

Trusses are arrangements of straight members connected at their ends. They resist loads by developing axial forces in their members but this is only true if the ends of the members are pinned together. The members in a truss are arranged to form a triangulated system so as to make them geometrically unchangeable, and also so that they will not form a mechanism. In trusses, loads are applied only at the joints.

When trusses have their ends fixed or welded together, secondary stresses develop and it is a form of analysis that favours the use of computer programs. Trusses provide practical and economical solutions to engineering problems. They can efficiently span greater lengths than beams, and hence can be found in roofs of buildings, bridges, etc. See design of steel roof trusses.

**Figure 6**: Typical Fink Truss and loading

**Figure 7:** Typical Howe Truss and loading

Pratt truss — **Figure 8:** Typical Pratt Truss and loading

Rigid Frames

Unlike trusses which are pin-jointed frames that transmit axial forces only, rigid frames are designed to transfer both axial forces, shear forces, and bending moment across the members. Rigid frames are comparatively easier to erect since their construction usually involves the connection of steel beams and columns by bolting or welding.

In reinforced concrete design, the beams and columns are usually cast monolithically and it is relatively easy to construct the formwork. Rigid frames can be statically determinate or indeterminate and may involve multi-storey or multi-bay configuration as the case may be.

Types of frames — **Figure 9:** Different forms of rigid frames

Arches

The arches are widely used in modern engineering due to their ability to cover large spans and their attractiveness from an aesthetic point of view. The greater the span, the more an arch becomes more economical than a truss. Materials of the modern arches are concrete, steel, and timber. Arches are mainly classified as three-hinged, two-hinged, and arch with fixed supports.

Arches carry most of their loads by developing compressive stresses within the arch itself and therefore in the past were frequently constructed using materials of high compressive strength and low tensile strength such as stones and masonry. Arches may be constructed in a variety of geometries; they may be semicircular, parabolic, or even linear where the members comprising the arch are straight.

types of arch structures — **Figure 10:** Different forms of arch structures

Cables

Cables are made from high-strength steel wires twisted together and present a flexible system, which can resist loads only by axial tension. The use of cables allows engineers to cover very large spans, especially in bridge design. In theory of structures, cables are deemed extremely efficient because they make the most effective use of structural material in that their loads are carried solely through tension through the wire. Therefore, there is no tendency for buckling to occur either from bending or from compressive axial loads.

A cable as a load-bearing structure has several features. One of them is the vertical load that gives rise to horizontal reactions, which, as in the case of an arch, is called a thrust. To accommodate the thrust it is necessary to have a supporting structure. It may be a pillar of a bridge or a tower or pylon. Cables are utilized in the form of suspension bridges, cable-stayed bridges, tower guy wires, etc.

suspension bridge — **Figure 11:** Suspension bridge

cable stayed bridge — **Figure 12:** Cable-stayed Bridge

Shear Walls

In tall buildings, there is a need to resist lateral forces which may arise from wind load or seismic forces, shear walls have been found to be a veritable solution. Shear walls may be planar solid or coupled and are usually provided alongside the frames of the structure. They are usually provided along the elevator shafts or stairwells. Apart from the use of computer-aided software, shear walls have been analysed using methods such as the continuous connection medium method, frame analogy method, etc.

shear core — **Figure 14:** Plan of a tall building with a shear wall in the elevator shaft

Continuum structures

Continuum structures are made from a material having a very small thickness compared to its other dimensions. Examples of continuum structures are folded plate roofs, shells, floor slabs, etc. An arch dam is a three-dimensional continuum structure as are domed roofs, aircraft fuselages, and wings. Continuum mechanics is concerned with structures that are continuous in space. The simplest elements of this type are surface elements which have a thickness of a different order from the other two dimensions. Surface elements are termed plates if they have a plane form and shells if they form a general surface.

floor slab — **Figure 15**: Reinforced concrete slab

Folded plate and shell structure — **Figure 16:** Shell structure

Loading of Structures

In theory of structures, we have different types of loading configuration and the commonest of them are;

Concentrated loads (point loads)
Uniformly distributed line loads (UDL)
Uniformly distributed area loads (full pressure loads)
Non-uniformly distributed load (e.g. trapezoidal and triangular loads)
Indirect actions

Concentrated loads

These are loads that act on a point in a structural system, and theoretically, they are like knife-edge loads acting at a précised point on a beam. However, in practice, these loads are usually distributed over a small area.

concentrated load on structures — **Figure 17:** Typical idealisation of concentrated loads

For example, a column that is supported on a suspended beam may be represented as a point load on the beam. Furthermore, the effect of secondary beams on primary beams is also treated as concentrated loads (see Figure 18). In Figure 18, Beam No 4 (axis 2-2) is a secondary beam that is supported on beams Nos 1 and 2 (axis A-A and B-B) which are primary beams that are directly supported on columns.

floor slab arrangement — **Figure 18:** Typical floor slab general arrangement

In the analysis, the loading on Beam No 4 is evaluated first, and the resultant support reactions are obtained. The reactions from Beam No 4 are applied as concentrated loads at the mid-span of beams Number 1 and 2. Note that secondary beams are usually shallower than primary beams. The equivalent loading on beams No 1 and 2 can be represented as shown in Figure 19.

loaded simply supported beams — **Figure 19:** Loading on beam No 2 (neglecting horizontal reaction)

Where;
P = Support reaction from beam No 4
q = load from the slab, self-weight of the beam, blockwork loading, and finishes, etc

Also in trusses, all loads are converted to concentrated loads acting on the nodes. It is also very usual for wind loads to be converted to concentrated loads acting at the nodes of a structure (where the nodes are at the floor slab level).

Uniformly Distributed Line Loads (UDL)

UDLs are loads that are spread with the same intensity over a span of a structural element. They are usually distributed per unit length of the element and hence their units are of the form kN/m or N/m as the case may be. For example, the load from a slab that is simply supported or clamped at all sides to a beam that is supporting it may be converted to equivalent UDL using the following relationships (British Codes) given in Table 1. However, it is important to note that the actual load distribution from slab to beams is either triangular or trapezoidal.

Table 1: Equivalent loading from slab to beams according to British Codes

Type of slab / Direction of span	Load to beam on long span (kN/m)	Load to beam on short Span (kN/m)
One-way slab	(nl_x)/5	(nl_x)/2
Two-way slab	(0.5nl_x) × (1 – 0.333k²)	(nl_x)/3

In Table 1 above, n stands for the uniformly distributed load on the slab at the ultimate limit state and is usually given by 1.4g_k + 1.6q_k (BS 8110-1:1997) or 1.35g_k + 1.5q_k (Eurocode 2). g_k stands for the dead load while q_k stands for the live load. k is the ratio between the length of the longer side (L_y) and the shorter side (L_x). This aspect ratio oftentimes is also used to define whether a slab is spanning in one direction or in two directions. Conventionally, if;

L_y/L_x > 2.0 (One way spanning slab)
L_y/L_x < 2.0 (Two way spanning slab).

You can refer to standard design textbooks for more complete information, but this is just to give us a little idea of how these loads come about.

Uniformly Distributed Area Loads (UDL)

Another variation of UDLs is pressure loads that are uniformly distributed over a unit area. These are typically the type of loads on slabs and plates, and even retaining walls. In fact, all walls are usually subjected to pressure loads, including earth intensity loads on foundations. The unit of pressure loads is kN/m². The self-weight of slabs and finishes are typical examples of uniformly distributed loads. The live loads (imposed loads) specified on codes of practice are also uniformly distributed pressure loads.

In theory of structures, the analysis of plates subjected to uniformly distributed pressure load is usually carried out per unit strip and hence, internal forces such as moment and shear are expressed per metre strip (e.g. kNm/m, kN/m). Other examples of uniformly distributed loads are the surcharge loads that act at the back of retaining walls etc.

Non-uniformly distributed loads

These are loads that have varying intensity across the element on which they act. Examples of such loads are trapezoidal loads or triangular loads. Also, beams that do not have their full length subjected to full UDL can also be brought under this category (figure 1.3a). Triangular loads can come from block work on lintels and beams, and other types of loads and trapezoidal loads can come from earth pressure or hydrostatic force that is exerted on an element that is fully immersed in a fluid at rest.

Figure 20(c) shows a gravity retaining wall that is retaining earth at the backface. The pressure due to the retained material varies with depth in a triangular manner as shown in the figure. In the figure;

q = earth pressure (kN/m²)
γ = Density of retained earth material (kN/m³)
K = Coefficient of active pressure (which may be computed using Coulomb’s or Rankine’s theory)
H = Height of the earth fill (m)

Indirect Actions (Loads)

Sometimes, other indirect actions on structures induce internal forces in the structure other than externally applied loads. The effects of these actions must be adequately assessed if the structure is to perform satisfactorily well. Some of these indirect loads are;

Relative movement of supports (differential settlement)
Temperature difference
Imperfections

It is however worth knowing that problems such as differential settlement do not affect statically determinate structures. Such problems are only critical in statically indeterminate structures.

Types of supports in Theory of Structures

In theory of structures, a lot of support systems are available and the one that is adopted is dependent on the one that significantly represents the actual physical behaviour when constructed. The common types of support conditions are;

Pinned supports
Roller supports
Fixed supports
Tension or compression springs
Elastic foundations

The first three are the commonest and they are the ones that are usually considered in theory of structures textbooks. In the selection of the types of support that will be used for analysis, the following may be used as guidelines;

Pinned Supports

These supports appear in the form of a hinge and is characterized by possessing vertical and horizontal reaction components. The joints of the support are free to rotate and as a result, pinned supports are not capable of resisting bending moment. In actual practice, situations that may be idealised as pinned supports are;

Stanchions (steel columns) that are supported on base plates (Figure 21)
Double angle cleat connection for beams and columns (Figure 22)

Column base plate — **Figure 21:** Stanchion on a base plate

double cleat supports — **Figure 22:** Double angle cleat for beams and columns

Roller supports

A single pinned support is sufficient to maintain the horizontal equilibrium of a beam and hence may not necessarily be provided at the other end. It may be advantageous to allow horizontal movement of the other end so that, for example, expansion and contraction caused by temperature variations do not cause additional stresses.

Such support may take the form of a composite steel and rubber bearing as shown in (Figure 23a) or consist of a roller sandwiched between steel plates (Figure 23b). This type of support is called roller support. It is assumed that such support allows horizontal movement and rotation but prevents movement vertically, up, or down. Roller supports resist only vertical loads.

Examples of roller support — **Figure 23:** (a) Steel laminated bearing on a bridge (b) Rocker supporting a bridge beam

Fixed Support

As the name implies, when the support of a structural member is built-in (fixed) such that no rotation or translation occurs, it is referred to as a fixed support. Fixed support resists vertical loads, horizontal loads, and moment and hence there are three reaction components. A beam-column connection that is welded with additional stiffeners can be idealised as fixed support for the beam (Figure 24a). Reinforced concrete columns and footing can also be idealised as fixed support if the base and the supporting soil are very stiff and rigid. (Figure 24b).

In theory of structures, these support conditions and their reaction components are shown in the Table below;

Types of supports in theory of structures

Tension or compression springs

Springs are also used in the idealisation of supports in theory of structures. When a structure is supported on a deformable body, springs can be used idealise the support condition. The stiffness of the spring is modelled to represent the strength of the deformable body. As a result, the interaction between the deformation of the support and the structural response of the supported member is captured in the analysis. A typical example is the use of Winkler’s springs in the idealisation of soils.

Elastic Foundations

When a structure is supported on a continuously deformable medium, it can be modelled as a structure on an elastic foundation. Beams on elastic foundations and plates on elastic foundations are popular topics in geotechnical and structural engineering. The can be used in the idealisation of raft foundations when using flexible approach. Elastic foundations are important in the modelling of soil-structure interaction.

Types of Structural Analysis

After the modelling of the structure and the supports, and assessment of the applied loads, the next step is to carry out an analysis to determine the internal forces and deformations induced in the structural members due to the applied load. Structural analysis usually encompasses one or more types of the following analytical methods.

Static linear analysis

This is the analysis that is carried out to determine the internal forces and displacements due to static loads (time-independent loads) while assuming that the structure is within the elastic state (Hooke’s law is being obeyed).

Non-Linear Static analysis

This is the analysis that is carried out to determine the internal forces and displacements in a structure when it is subjected to static loads and when we assume non-linear conditions. Non-linear analysis can be in the following forms;

Physical – The material has exceeded the yield point and Hooke’s law is no longer being obeyed.
Geometrical – Structure is subjected to large displacement, and analysis is based on the deformed structure
Structural – Analysis for structures with gaps, unbalanced constraints etc
Mixed non-linearity – A combination of the above-mentioned non-linearities

Buckling Analysis

This is the analysis that is carried out to determine the critical load (or critical load factor) including the corresponding buckling mode shapes of structures subjected to compressive loads. It is important is assessing the stability of structures.

P-delta Analysis

This analysis takes into account the additional moments due to compressive loads on the displacement caused by the lateral loads. This happens in tall and flexible structures. Thus, the analysis is also based on the deformed system (non-linear).

Free Vibration Dynamic Analysis

The purpose of this vibration analysis is to obtain the natural frequencies (eigenvalues) and the corresponding mode shapes (eigenfunctions).

Time-history Analysis

This analysis is carried out to determine the response of a structure that is subjected to arbitrarily time-varying loads.

Pushover Analysis

Pushover analysis is a non-linear approach used to determine structural capacity under static horizontal loads that grow until the structure collapses. It is a static process that estimates seismic structural deformations using a reduced nonlinear technique. During earthquakes, structures redesign themselves.

The dynamic forces on a structure are moved to other components as individual components yield or fail. A pushover analysis replicates this phenomenon by applying loads until the weak link in the structure is discovered, then changing the model to account for the structural changes generated by the weak link.

The loads are redistributed in the second iteration. The structure is pushed once more until the second weak link is found. This approach is repeated until a yield pattern for the entire structure is found under seismic loading. Pushover analysis yields several capacity curves based on the variation of base shear as a function of the displacement of a control point on the structure.

Static Determinacy of Structures

A structure is stable when it maintains balance in force and moment. As a result, we know that from statics, if a structure is to be at equilibrium,

∑F_y = 0; ∑F_x = 0; ∑M_i = 0 ——— (1)

Where;

∑F_y = Summation of the vertical forces
∑F_x = Summation of the horizontal forces
∑M_i= Summation of the moment of force components acting in the x-y plane passing through point i.

When the number of constraints in a structure permits the use of the equation of statics (equation 1) to analyse the structure, the structure is said to be statically determinate. Otherwise, it is statically indeterminate, and additional equation(s) which are derived from the load-deformation relationships are used for analysis. For the records, there are two well-known approaches to the analysis of indeterminate structures and they are;

Flexibility Methods – When the structure is analysed with respect to unknown forces
Stiffness Methods – when the structure is analysed with respect to unknown displacements

A structure may be indeterminate due to redundant components of reaction and/or redundant members. Note that a redundant reaction or member is one which is not really necessary to satisfy the minimum requirements of stability and static equilibrium. However, redundancy is desirable in structures because it can be a cheaper alternative to statically determinate structures. The degree-of-indeterminacy is equal to the number of unknown member forces/external reactions which are in excess of the equations of equilibrium available to solve for them.

Webinar Invitation: Design of Pile and Raft Foundation

Ubani Obinna

May 8, 2022

Structville Integrated Services Limited is pleased to announce the third edition of the webinar on the ‘Design of Raft and Pile Foundation’. The details are as follows;

Topic: Structural Analysis and Design of Raft and Pile Foundation
Date: Saturday, 14th of May, 2022
Time: 12:00 noon – 02:00 pm (WAT)
Fee: NGN 4000 only (USD $10.00)
Platform: Zoom Live
Facilitator: Engr. Ubani Obinna (MNSE, R.Engr)

To register for the event click HERE

Features:
(1) Theories and philosophies in the design of shallow and deep foundations
(2) Practical design of pile foundations and pile caps using real-life data
(3) Rigid and flexible approach to the design of raft foundation
(4) Structural design of raft foundation
(5) Full design material (mini textbook) with detailed drawings covering the above topics

To register for the event, click HERE

Temperature Actions on Bridge Decks | Thermal Loads on Bridges

Ubani Obinna

May 23, 2022

Table of Contents

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Temperature Changes on Bridge Decks
- Uniform Temperature Component
- Vertical temperature varying component
Worked Example on Calculation of Temperature Stresses on Bridge Decks

The temperature of a bridge structure and its surroundings varies daily and seasonally, affecting the overall movement of the bridge deck as well as the stresses within it. The temperature actions on bridge decks have an impact on the design of bridge bearings and expansion joints, while the overall movement of the bridge deck and the stresses within it have an impact on the quantity and placement of structural materials such as steel.

The daily effects cause temperature variations within the depth of the superstructure which vary depending on whether it is heating or cooling. Guidance is typically provided in the form of idealized linear temperature gradients to be expected when the bridge is heating or cooling for various types of construction (concrete slab, composite deck, etc.) and blacktop surface thickness.

Internal stresses in a bridge cross-section self-equilibrate as a result of temperature gradients. There are two forms of internal stresses induced as a result of this: primary and secondary. The former is caused by temperature changes across the superstructure (whether simply supported or continuous), while the latter is caused by continuity. Both must be considered and accommodated in the design.

The temperature of a bridge deck varies throughout its mass. The variation is caused by:

the position of the sun
the intensity of the sun’s rays
thermal conductivity of the concrete and surfacing
wind
the cross-sectional make-up of the structure.

The effects of these variations on a bridge deck can be quite complex. On a daily (short-term) and annual (long-term) basis, changes occur. Daytime heat gain and nighttime heat loss occur on a daily basis. The ambient (surrounding) temperature varies from year to year.

On a daily basis, incident sun radiation controls temperatures towards the top, whereas shade air temperature controls temperatures near the bottom. In Figure 1, the whole distribution is shown. Positive indicates a rapid rise in deck slab temperature due to direct sunshine (solar radiation). Negative indicates that the ambient temperature is dropping due to heat loss (re-radiation) from the structure.

Temperature Actions on Bridge Decks — **Figure 1:** Typical temperature distributions

According to research, the distributions (or thermal gradients) for different ‘groups’ of structure as stated in Figure 9 of BD 37/01 Clause 5.4 can be idealized for analysis purposes. The four groups depend on the type of construction and are listed below;

(1) Steel deck on steel box girders
(2) Steel deck on steel truss or plate girders
(3) Concrete deck on steel truss, steel box, or plate girders
(4) Concrete deck on concrete beams or box girders

The thickness of the surface, the thickness of the deck slab, and the type of the beam are all important criteria. The curvature of the deck is caused by temperature changes, which result in internal primary and secondary stresses within the structure.

Temperature Changes on Bridge Decks

Temperature changes on bridges are expressed in terms of a uniform temperature component, a vertical difference component that contains non-linear components, and, when applicable, a horizontal difference component that can be considered to vary linearly. The temperature of the bridge is affected not only by the shade air temperature and solar radiation, but also by the scheme, cross-section, mass, and material used.

As a result, bridges can be categorised into the following categories and subcategories:

Steel bridge: steel box girder, steel truss, or plate girder;
Composite bridge
Concrete bridge: concrete slab, concrete beam, concrete box girder.

Temperature actions on bridge decks are discussed in EN 1991-1-5:2003.

Uniform Temperature Component

The maximum and minimum temperatures, T_e,max and T_e,min, that the bridge can reach over its working life determine the uniform temperature component. The uniform temperature components T_e,max and T_e,min can be calculated using the diagrams in Figure 2, where T_e,max and T_e,min, in °C, are stated in terms of T_max and T_min, in °C, for each bridge type previously mentioned.

Over the years, weather stations have recorded the maximum and minimum shade air temperatures. The maximum and minimum design temperatures that the bridge deck may encounter during its design life are predicted using these records. These temperatures are represented as isotherm maps in the codes.

temperature in bridges — **Figure 2:** Correlation between shade air temperature (T_min, T_max) and uniform
components of the bridge temperature (T_e,min, T_e,max)

To calculate how much the bridge deck will expand and contract, the maximum and minimum shade air temperatures are transformed into ‘effective’ bridge temperatures T_e,max and T_e,min and multiplied by the coefficient of thermal expansion and the deck length. The deck can be accommodated by providing joints and sliding bearings, or by restricting the movement and engineering the structure to resist the forces created. For truss or plated steel bridges (category 1), T_e,max values can be decreased by 3 °C.

A base reference temperature T₀ is used to represent the effective bridge temperature at specific stages of construction. The deck will expand from T₀ to T_e,max, and contract from T₀ to T_e,min.

In the case of a free moving deck, T₀ is used to calibrate the gap for the expansion joint and to set the sliding bearing positions when these units are installed. However, in the case of a restrained deck, it is used to determine the magnitude of movement that the supporting structure has to accommodate after it has been made integral with the deck.

If T₀ is the initial bridge temperature, i.e. the temperature of the bridge at the time when it is restrained, the variation of the uniform bridge temperature ∆T_u is given by;

∆T_u= T_e,max – T_e,min = ∆T_N,esp + ∆T_N,con

where

∆T_N,exp=T_e,max – T₀ and ∆T_N,con = T₀ – T_e,min

are the temperature variations to be considered when the bridge expands or contracts, respectively.
Assessing bearing displacements it can be assumed ∆T_N,exp = T_e,max – T₀+ 20°C and ∆T_N,con = T₀ – T_e,min + 20°C.

Vertical temperature varying component

Vertical temperature variations can arise as a result of the top and bottom surfaces of the bridge being heated and cooled differently. These differences correlate to maximum heating and cooling, respectively, when the top surface is warmer than the bottom surface and when the bottom surface is warmer than the top surface. The vertical temperature profiles can be defined under two different hypotheses, depending on whether the non-linear temperature profile ∆T_E is ignored or not: in the first case, a simplified equivalent linear profile can be considered, whereas in the second case, a non-linear profile, including ∆T_E, is considered.

Equivalent linear vertical temperature profile

Table 1 shows the maximum temperature differences corresponding to maximum heating or maximum cooling, designated as ∆T_M,heat and ∆T_M,cool, for the various bridge categories when an equivalent linear vertical temperature profile is used. The temperature differences for road and railway bridges with a 50 mm surface are represented by the upper bound values in Table 1. Table 1 values should be multiplied by the correction factors k_sur in Table 2 for varying surface thicknesses.

Table 1: Equivalent linear vertical temperature variations for bridges

Type of deck	Top warmer than bottom ∆_TM,heat [°C]	Bottom warmer than top ∆T_M,cool [°C]
Type 1: Steel deck	18	13
Type 2: Composite deck	15	18
Type 3: Concrete deck concrete box girder concrete beam concrete slab	10 15 15	5 8 8

Table 2: Adjustment factors k_sur for road, foot and railway bridges

Non-linear vertical temperature profiles

When more detailed calculations are required, the vertical temperature profile can be considered to be non-linear by using the temperature profiles shown in Figures 3, 4, and 5 for steel, concrete, and composite bridges, respectively, in the heating and cooling conditions. Tables 5.a and 5.b show two alternative profiles for composite bridges: normal and simplified.

In general, the simpler profile is safe. The temperature differences ∆T listed in tables 3 to 5 comprise both the vertical temperature component ∆T_M and the non-linear temperature component ∆T_E, as well as a small portion of the uniform component ∆T_N. The temperatures for other surfacing depths of bridge decks of type 1 to 3 are given in Tables B.1 to B.3 of EN 1991-1-5, Annex B.

Figure 3: Non-linear vertical temperature differences for steel bridges

concrete bridges temperture — **Figure 4:** Non-linear vertical temperature differences for concrete bridges

Non linear vertical temperature differences for composite bridges normal — **Figure 5.a**: Non-linear vertical temperature differences for composite bridges, normal profile

Non linear vertical temperature differences for composite bridges simplified — **Figure 5.b**: Non-linear vertical temperature differences for composite bridges, simplified profile

Horizontal temperature varying component

Horizontal temperature differences in bridges can be generally disregarded, except in special cases, for example when one side of the bridge is much more exposed to the sunlight of the other one. When the horizontal component must be taken into account, a linear variation of 5°C can be assumed.

Worked Example on Calculation of Temperature Stresses on Bridge Decks

Determine the stresses induced by both the positive and reverse temperature differences for the concrete box girder bridge shown in Figure 6 (A = 940000 mm², I =102534 × 10⁶ mm⁴, depth to NA = 409 mm, T = 12 × 10^-6, E = 34 kN/mm²). This example is adapted from Ryall (2008).

temperature stresses on a box girder bridge — **Figure 6:** Box girder dimensions and temperature distribution (Ryall, 2008)

Calculate critical depths of temperature distribution
From BD 37/88 Figure 9 this is a Group 4 section, therefore:

h₁ = 0.3h = 0.3 × 1000 = 300 > 150; thus h₁ = 150mm
h₂ = 0.3h = 0.3 × 1000 = 300 > 250; thus h₂ = 250mm
h₃ = 0.3h = 0.3 × 1000 = 300 > 170; thus h₃ = 170mm

Calculate temperature distribution
Basic values are given in Figure 9 of BD 37/01 which are modified for depth of section and surface thickness by interpolating from Table 24 of BD 37/01.

T₁ = 17.8 + (17.8 – 13.5)20/50 = 16.1°C
T₁= 4.0 + (4.0 – 3.0)20/50 = 3.6 °C
T₁ = 2.1 +(2.5 – 2.1)20/50 = 2.26 °C

Calculate restraint forces at critical points
This is accomplished by dividing the depth into convenient elements corresponding to changes in the distribution diagram and/or changes in the section (see Figure 3.2 of BD 37/01):

F = E_cβ_TT_iA_i
F₁ = 34000 × 12 × 10^-6 × (16.1 – 3.6) × 2000 × 150/1000 = 765 kN
F₂= 34000 × 12 × 10^-6 × (3.6) × 2000 × 150/1000 = 441 kN
F₃ = 34000 × 12 × 10^-6 × [(3.6 + 2.6)/2] × 2000 × (220 – 150)/1000 = 177 kN
F₄ = 34000 × 12 × 10^-6 × (2.6/2) × 2 × (250 – 70) × 250/1000 = 48 kN
F₅ = 34000 × 12 × 10^-6 × (2.26/2) × 1000 × 170/1000 = 78 kN

Total F = 1509 kN (tensile)

Element Forces — **Figure 7:** Element forces (Ryall, 2008)

Calculate restraint moment about the neutral axis
M = [765(409 – 50) + 441(409 – 75) + 177(409 – 185) + 28(409 – 270) – 78(591 – 170 × 2/3)]/1000 = 431 kNm (hogging)

Calculate restraint stresses
f = E_cβ_TT_i

f₀₁ = -34000 × 12 × 10^-6 × 16.1 = 6.56N/mm²
f₀₂ = -34000 × 12 × 10^-6 × 3.6 = 1.47 N/mm²
f₀₃ = -34000 × 12 × 10^-6 × 2.6 = 1.06 N/mm²
f₀₄ = -34000 × 12 × 10^-6 × 0 = 0.00 N/mm²
f₀₅ = -34000 × 12 × 10^-6 × 0 = 0.00 N/mm²
f₀₆ = -34000 × 12 × 10^-6 × 2.26 = 0.92 N/mm²

Calculate balancing stresses
Direct stress f₁₀= (1509 × 10³)/940000 = 1.61 N/mm²
Bending stresses f_2i = My/I:

f₂₁ = [(431 × 10⁶)/(102534 × 10⁶)] × 409 = 1.71 N/mm²
f₂₂= [(431 × 10⁶)/(102534 × 10⁶)] × 259 = 1.08 N/mm²
f₂₃ = [(431 × 10⁶)/(102534 × 10⁶)] × 180 = 0.75 N/mm²
f₂₄ = [(431 × 10⁶)/(102534 × 10⁶)] × 9 = 0.06 N/mm²
f₂₅ = [(431 × 10⁶)/(102534 × 10⁶)] × 421 = -1.76 N/mm²
f₂₆ = [(431 × 10⁶)/(102534 × 10⁶)] × 591 = -2.47 N/mm²

Calculate final stresses
The final stress distribution is shown in Figure 8. Similar calculations for the cooling (reverse) situation are shown in Figure 9.

final stress distribution — **Figure 8:** Final stress distribution (positive)(Ryall, 2008)

negative stress distribution — **Figure 9:** Final stress distribution (negative)(Ryall, 2008)

References
[1] Rryall M. J. (2008): Loads and load distribution, in ICE Manual of Bridge Engineering (2nd Edition). Edited by Parke G. and Hewson N. doi: 10.1680/mobe.34525.0023

Load Combinations for Highway Bridges

Ubani Obinna

May 2, 2022

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Load Combinations for Highway Bridges in the UK
Load Combinations for Highway Bridges (Eurocode)

Highway bridges are subjected to a myriad of direct and indirect forces. For the purpose of obtaining the action effects on bridges, bridge designers must obtain the adequate load combination for highway bridges that will give the worst effect on any part of the bridge. The predominant action on highway bridges is gravity loads due to self-weight and the mass/dynamic effects of moving traffic. Other actions that are frequently considered are temperature loads, construction loads, snow loads, earthquake loads, and possible differential settlement of the foundation.

In this article, we are going to consider the load combinations for highway bridges according to UK standards (BD 37/01) and Eurocodes (EN 1990:2002 and EN 1991-2).

Load Combinations for Highway Bridges in the UK

In the UK, five combinations of loading are considered for the purposes of design: three principal and two secondary. These are defined in Clause 4.4 and Table 1 of BD 37/01. It is usual in practice to design for Combination 1 and to check other combinations if necessary.

Combination 1. For highway and foot/cycle track bridges, the loads to be considered are the permanent loads, together with the appropriate primary live loads, and, for railway bridges, the permanent loads, together with the appropriate primary and secondary live loads.

Combination 2. For all bridges, the loads to be considered are the loads in combination 1, together with those due to wind and, where erection is being considered, temporary erection loads.

Combination 3. For all bridges, the loads to be considered are the loads in combination 1, together with those arising from restraint due to the effects of temperature range and difference, and, where erection is being considered, temporary erection loads.

Combination 4. Combination 4 does not apply to railway bridges except for vehicle collision loading on bridge supports. For highway bridges, the loads to be considered are the permanent loads and the secondary live loads, together with the appropriate primary live loads associated with them. Secondary live loads shall be considered separately and are not required to be combined. Each shall be taken with its appropriate associated primary live load.

Combination 5. For all bridges, the loads to be considered are the permanent loads, together with the loads due to friction at bearings.

The summary of UK load combinations (BD 37/01) is shown in Table 1 below;

Table 1: UK load combinations

Combination	Description
1	Permanent + Primary Live
2	Permanent + Primary live + wind + (temporary erection loads)
3	Permanent + Primary live + temperature restraint + (temporary erection loads)
4	Permanent + Secondary live + associated primary live
5	Permanent + bearing restraint

The partial factors for load (γ_Fl) when carrying out load combination are provided in the Tables below;

Table 2: Partial factors for dead and superimposed loads

Load	Limit State	Comb. 1	Comb. 2	Comb. 3	Comb. 4	Comb. 5
Dead: Steel	ULS SLS	1.05 1.00	1.05 1.00	1.05 1.00	1.05 1.00	1.05 1.00
Dead: Concrete	ULS SLS	1.15 1.00	1.15 1.00	1.15 1.00	1.15 1.00	1.15 1.00
Superimposed: deck surfacing	ULS SLS	1.75 1.20	1.75 1.20	1.75 1.20	1.75 1.20	1.75 1.20
Superimposed dead: other loads	ULS SLS	1.20 1.00	1.20 1.00	1.20 1.00	1.20 1.00	1.20 1.00
Reduced load factor for dead and superimposed load where this has a more severe effect	ULS	1.00	1.00	1.00	1.00	1.00

The partial factors for highway live load combination are shown in Table 3 below;

Table 3: Partial factors for live load combination

Highway Live Loading	Limit State	Comb. 1	Comb. 2	Comb. 3	Comb. 4	Comb. 5
HA alone	ULS SLS	1.50 1.20	1.25 1.00	1.25 1.00
HA with HB or HB alone	ULS SLS	1.30 1.10	1.10 1.00	1.10 1.00
Footway and cycle track loading	ULS SLS	1.50 1.00	1.25 1.00	1.25 1.00
Accidental wheel loading	ULS SLS	1.50 1.20

The partial factors for wind load (on bridges) combination are shown in Table 4;

Table 4: Partial factors for wind load combination

Wind Load	Limit State	Comb. 1	Comb. 2	Comb. 3	Comb. 4	Comb. 5
During erection	ULS SLS		1.10 1.00
With dead load plus superimposed dead load only, and for members primarily resisting wind load	ULS SLS		1.40 1.00
With dead load plus superimposed dead load plus other appropriate Combination 2 loads	ULS SLS		1.10 1.00
Relieving effect of wind load	ULS SLS		1.00 1.00

The partial factors for combination involving temperature loads are given in Table 5;

Table 5: Partial factors for temperature load combination

Temperature Load	Limit State	Comb. 1	Comb. 2	Comb. 3	Comb. 4	Comb. 5
Restraint to movement, except frictional	ULS SLS			1.30 1.00
Frictional bearing restraint	ULS SLS					1.30 1.00
Effect of temperature difference	ULS SLS			1.00 0.80

Load combination factors involving earth pressures are shown in Table 6, while the partial load factors for differential settlement and erection loads are shown in Table 7.

Table 6: Partial factors for earth pressure loads

Earth Pressure Load (retained fill and/or live load)	Limit State	Comb. 1	Comb. 2	Comb. 3	Comb. 4	Comb. 5
Vertical Loads	ULS SLS	1.20 1.00	1.20 1.00	1.20 1.00	1.20 1.00	1.20 1.00
Non-vertical loads	ULS SLS	1.50 1.00	1.50 1.00	1.50 1.00	1.50 1.00	1.30 1.00
Relieving effect	ULS	1.00	1.00	1.00	1.00	1.00

Table 7: Partial factors for differential settlement and erection loads

Load	Limit State	Comb. 1	Comb. 2	Comb. 3	Comb. 4	Comb. 5
Differential settlement	ULS SLS	1.20 1.00	1.20 1.00	1.20 1.00	1.20 1.00	1.20 1.00
Erection load (temporary loads)	ULS SLS		1.15 1.00	1.15 1.00

Load Combinations for Highway Bridges (Eurocode)

Annex A2 to EN 1990:2002 specifies the rules and methods for establishing combinations of actions for serviceability and ultimate limit state verifications (except fatigue verifications), as well as the recommended design values for permanent, variable, and accidental actions and ψ factors to be used in the design of road, footbridge, and railway bridges. It also applies to actions during the execution of bridges.

General guidelines for load combination on bridges (Annex 2 EN 1990)

The effects of actions that cannot occur concurrently owing to physical or functional reasons are not required to be included jointly in combinations of actions, according to Annex 2 of EN 1990. Combinations including activities not covered by EN 1991 (for example, mining subsidence, specific wind impacts, water, floating debris, flooding, mudslides, avalanches, fire, and ice pressure) should be characterized in accordance with EN 1990, 1.1. (3).

The combinations of actions given in expressions 6.9a to 6.12b of EN 1990 should be used when verifying ultimate limit states. Expressions 6.9a is as given below;

γ_SdE{γ_g,jG_k,j; γ_pP; γ_q,1Q_k,1; γ_q,iψ_0,iQ_k,i} j ≥; i > 1

The combination of effects of actions to be considered should be based on the design value of the leading variable action, and the design combination values of accompanying variable actions. The combinations of actions given in expressions 6.14a to 6.16b of EN 1990 should be used when verifying serviceability limit states. Additional rules are given in A2A for verifications regarding deformations and vibrations.

The necessary design situations must be considered throughout execution. Specific construction loads should be taken into account in the appropriate combinations of operations where applicable. Construction loads that cannot occur at the same time due to the implementation of control measures do not need to be considered in the appropriate combinations of actions. When a bridge is put into service in phases, the appropriate design situations must be considered.

typical construction loads on a bridge deck — **Figure 1:** Typical construction load during the execution of bridges

Snow loads and wind actions do not have to be considered at the same time as construction-related loads Q_co (i.e. loads due to working personnel). During some temporary design circumstances, however, it may be required to agree on the requirements for snow loads and wind actions to be taken into account simultaneously with other construction loads (e.g., actions due to heavy equipment or cranes) for a specific project.

Thermal and water actions should be evaluated simultaneously with construction loads where possible. When determining optimal combinations with construction loads, the various parameters governing water actions and components of thermal actions should be taken into account where applicable.

In accordance with the relevant parts of EN 1991-2, variable traffic measures should be taken into consideration simultaneously where applicable. Any group of loads, as described in EN 1991-2, shall be taken into account as one variable action for any combination of variable traffic actions with other variable actions stated in other parts of EN 1991.

If the impacts of uneven settlements are regarded as considerable in comparison to the effects of direct interventions, they should be evaluated. Total and differential settlement restrictions may be specified by the specific project. Uncertainty in the assessment of these settlements should be taken into account where the structure is extremely sensitive to uneven settlements.

Uneven settlements on the structure caused by soil subsidence should be defined as a permanent action and included in the structure’s ultimate and serviceability limit state verifications as a combination of actions. G_set should be represented as a set of values relating to differences in settlements between individual foundations or segments of foundations (relative to a reference level), d_set,i (i is the number of the individual foundation or part of foundation).

Permanent loads and backfill are the main causes of settlements. For some individual projects, variable actions may need to be considered. Settlements change monotonically (in the same direction) throughout time and must be taken into account from the moment they cause structural impacts (i.e. after the structure or a part of it, becomes statically indeterminate).

Furthermore, there may be a connection between the development of settlements and the creep of concrete members in the case of a concrete structure or a structure with concrete elements. Individual foundation or pair of foundation settlement differences, d_set,i, shall be taken into consideration as best-estimate projected values in line with EN 1997, taking into account the structure’s building process.

Groups of Traffic Loads on Highway Bridges

When the simultaneous presence of traffic and non-traffic actions is considerable, the characteristic values of the traffic actions can be calculated using the five separate, mutually exclusive groups of loads listed in Table 4.4a of BS EN 1991-2:2003 (reproduced below), with the dominant component action highlighted. Each load category in the table should be seen as establishing a characteristic action for usage with non-traffic loads, but they can also be used to assess infrequent and frequent values.

group of loads for load combinations in highway bridges

To obtain infrequent combination values it is sufficient to replace in Table 4.4(a) of EN 1991-2 characteristic values with the infrequent ones, leaving unchanged the others, while frequent combination values are obtained by replacing characteristic values with the frequent ones and setting to zero all the others. The ψ-factors for traffic loads on road bridges are reported in the Table below.

Combination Rules for Highway Bridges

(1) The infrequent values of variable actions may be used for certain serviceability limit states of concrete bridges.
(2) Load Model 2 (or associated group of loads gr1b) and the concentrated load Q_fwk (see 5.3.2.2 in EN 1991-2) on footways need not be combined with any other variable non-traffic action.
(3) Neither snow loads nor wind actions need be combined with: braking and acceleration forces or the centrifugal forces or the associated group of loads gr2, loads on footways and cycle tracks or with the associated group of loads gr3, crowd loading (Load Model 4) or the associated group of loads gr4. The combination rules for special vehicles (see EN 1991-2, Annex A, Infonnative) with nominal traffic (covered LM1 and LM2) and other variable actions may be referenced as appropriate in the National Annex or agreed for the individual project.
(4) Snow loads need not be combined with Load Models 1 and 2 or with the associated groups of loads gr1a and gr1b unless otherwise specified for particular geographical areas. However, geographical areas where snow loads may have to be combined with groups of loads gr1a and gr1b in combinations of actions may be specified in the National Annex.
(5) No wind action greater than the smaller of F_w* and ψ₀F_wk should be combined with Load Model 1 or with the associated group of loads gr1a.
(6) Wind actions and thermal actions need not be taken into account simultaneously unless otherwise specified for local climatic conditions.

Partial Factors for Actions in the Limit State

(1) Design values of actions (EQU) (Set A)

For persistent design situations, the recommended set of values for γ are:
γ_G,sup = 1.05
γ_G,inf = 0.95
γ_Q= 1.35 for road and pedestrian traffic actions, where unfavourable (0 where favourable)
γ_Q=1.45 for rail traffic actions, where unfavourable (0 where favourable)
γ_Q= 1.50 for all other variable actions for persistent design situations, where unfavourable (0 where favourable).
γ_P = recommended values defined in the relevant design Eurocode.

For transient design situations during which there is a risk of loss of static equilibrium, Q_k,1 represents the dominant destabilising variable action and Q_k.irepresents the relevant accompanying destabilising variable actions.

During execution, if the construction process is adequately controlled, the recommended set of values for γ are:
γ_G,sup = 1.05
γ_G,inf = 0.95
γ_Q= 1.35 for construction loads where unfavourable (0 where favourable)
γ_Q= 1.50 for all other variable actions, where unfavourable (0 where favourable)

(2) Design values of actions (STR/GEO) (Set B)

γ_G,sup = 1.35 (This value covers the self-weight of structural and non-structural elements, ballast, soil, groundwater and free water, removable loads)
γ_G,inf = 1.00
γ_Q= 1.35 for road and pedestrian traffic actions, where unfavourable (0 where favourable)
γ_Q=1.45 for rail traffic actions, where unfavourable (0 where favourable)
γ_Q= 1.50 for all other variable actions for persistent design situations, where unfavourable (0 where favourable). This value covers variable horizontal earth pressure from groundwater, free water, and ballast, traffic load surcharge earth pressure, traffic aerodynamic actions, wind, and thermal actions, etc.
γ_P = recommended values defined in the relevant design Eurocode.
γ_Gset(partial factor for settlement) = 1.20 in the case of linear elastic analysis, and γ_Gset = 1.35 in the case of non-linear analysis, for design situations where actions due to uneven settlements may have unfavourable effects.
ξ = 0.85 (so that ξγ_G,sup = 0.85 x 1.35 = 1.15)

(3) Design values of actions (STR/GEO) (Set C)

γ_G,sup = 1.00
γ_G,inf = 1.00
γ_Gset = 1.00
γ_Q= 1.15 for road and pedestrian traffic actions where unfavourable (0 where favourable)
γ_Q = 1.25 for rail traffic actions where unfavourable (0 where favourable)
γ_Q = 1.30 for the variable part of horizontal earth pressure from soil, groundwater, free water and ballast, for traffic load surcharge horizontal earth pressure, where unfavourable (0 where favourable)
Y_Q = 1.30 for all other variable actions where unfavourable (0 where favourable)
γ_Gset= 1.00 in the case of linear elastic or non linear analysis, for design situations where actions due to uneven settlements may have unfavourable effects. For design situations where actions due to uneven settlements may have favourable effects, these actions are not to be taken into account.

Thermal Actions | How to Apply Fire Loading on Steel Structures

Ubani Obinna

May 8, 2022

The action of fire on a structure is represented by thermal actions, and EN 1991-1-2:2002 (Eurocode 1, Part 2) provides several options for considering thermal action on steel structures. Time-temperature relationships are one of the numerous ways of representing fire actions on structures. These are relationships that show the evolution of a temperature that represents the environment around the structure as a function of time. The heat flux transported from the environment to the structure can be calculated using this temperature and the relevant boundary conditions.

Relationships that directly give the heat flux impinging on the structure are another option. The temperature evolution in the structure is then determined by combining the impinging heat flux with the flux emitted by the structure. Eurocode 1 distinguishes between nominal temperature-time curves, which include the standard temperature-time curve, the hydrocarbon curve, the external fire curve, and on the other hand, the natural fire models. The thermal action to be employed is usually a legal requirement set by the country or region in which the structure is located, and it is determined by the building’s size, usage, and occupancy (Franssen and Real, 2015).

Some countries impose prescriptive standards that specify the time-temperature curve as well as the time (referred to as fire resistance) that the structure must withstand when subjected to this curve. For example, a hotel in Country A must have a 60-minute resistance to the standard curve, whereas a railway station in country B must have a 30-minute resistance to the hydrocarbon curve. In such circumstances, the designer must guarantee that the construction meets the criterion and must use the time-temperature curve that has been provided (Franssen and Real, 2015).

1117 sd 1 — **Figure 1**: Fire action in a steel building

The regulation in other countries or areas may be more flexible, allowing the designer to create a performance-based design. Although the Eurocode provides some assistance in the form of restrictions of application to some of the proposed natural fire models, it is the designer’s obligation to adopt an acceptable representation of the fire in this scenario.

Such natural fire models should ideally be utilized in conjunction with performance-based requirements, such as the time required for evacuation or intervention. Before beginning any performance-based design, it is recommended to obtain consent from the authority with jurisdiction over the design fire and design scenario.

Nominal Temperature-Time Curves for Thermal Actions

Temperature-time curves are time-dependent analytical functions that yield a temperature. Because these functions are continuous, they can be used to create a curve in a time-temperature plane. Because they aren’t supposed to represent a genuine fire, they’re called nominal. They must be regarded as standard, or arbitrary, functions (Franssen and Real, 2015).

This is why the phrase “fire curve” is a misnomer because it implies that the temperature is the same as the temperature of a fire. In fact, the temperature is on par with what is found in fires. Such relationships are to be used in a prescriptive regulatory context because they are quite conventional. As a result, any requirement defined in terms of a nominal curve is both prescriptive and, in some ways, arbitrary (Franssen and Real, 2015).

The time it takes to evacuate or intervene should not be equated to the resistance of a structure to a nominal fire. Three distinct nominal temperature-time curves are proposed by Eurocode 1. The standard temperature-time curve is the one that has been used in standard fire tests to grade structural and separating elements in the past, and it is still used today. It’s used to symbolize a compartment with a fully formed fire. Because the formula was derived from the ISO 834 standard, it is commonly referred to as the ISO curve. Equation (1) gives us this standard curve;

θ_g = 20 + 345log₁₀ (8t +1) ——– (1)

where θ_g is the gas temperature in °C and t is the time in minutes.

When a regulatory requirement is defined as Rxx, with xx equal to 30 or 60 minutes, for example, it implies that the standard fire curve must be used to evaluate the structural parts’ duration fire resistance.

The external time-temperature curve is used to describe the exterior surface of separating external walls of a building that are exposed to a fire that starts outside the building or flames that come in through the windows of a compartment below or adjacent to the external wall.

Note: This curve should not be used to calculate the effects of a fire on an exterior load-bearing structure outside the building envelope, such as steel beams and columns. Annex B of Eurocode 1 describes the thermal attack on external structural steel parts.

The external curve is given by Equation (2);

θ_g= 20 + 660(1− 0.687e^−0.32t− 0.313e^−3.8t) ——– (2)

The hydrocarbon time-temperature curve is used for representing the effects of a hydrocarbon type fire. It is given by Equation (3);
θ_g = 20 + 1080 (1 − 0.325e^−0.167t− 0.675e^−2.5t ) ——–(3)

Fire curves for thermal actions — **Figure 2:** Different fire curves for thermal actions

In Fig. 2, the standard and hydrocarbon curves are compared. It can be seen that the hydrocarbon curve rises rapidly and achieves a constant temperature of 1100 °C within half an hour, whereas the standard curve rises more slowly but steadily over time. Equation (4) should be used to simulate the heat flow at the surface of a steel element when the environment is represented by a gas temperature, as is the case for nominal curves.

h_net = α_c(θ_g − θ_m) + Φε_mε_f σ[(θ_r + 273)⁴ − (θ_m + 273)⁴] ——– (4)

where;
α_c is the coefficient of convection which is taken as 25 W/m²K for the standard or the external fire curve and 50 W/m²K for the hydrocarbon curve,
θ_g is the gas temperature in the vicinity of the surface either calculated from Eqs. (1), (2) or (3) or taken as 20 °C
θ_m is the surface temperature of the steel member (the evolution of which has to be calculated)
Φ is a configuration factor that is usually taken equal to 1.0 but can also be calculated using Annex G of Eurocode 1 when so-called position or shadow effects have to be taken into account
ε_m is the surface emissivity of the member taken as 0.7 for carbon steel, 0.4 for stainless steel, and 0.8 for other materials
εf is the emissivity of the fire, in general, taken as 1.0
σ is the Stephan Boltzmann constant equal to 5.67 × 10^-8 W/m²K⁴
θ_r is the radiation temperature of the fire environment taken as equal to θ_g in the case of fully engulfed members.

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

Application of Waste Rubber Tyre in Concrete Production: A Brief Review

Ubani Obinna

April 27, 2022

Waste rubber tyre is a serious environmental issue that is becoming increasingly important. Currently, huge amounts of waste rubber tyres are being stockpiled (whole tyres) or landfilled (shredded tyres), with 3 billion in the EU and 1 billion in the US (Sofi, 2018). By the year 2030, the number of tyres used in automobiles is expected to exceed 1.2 billion, equating to about 5 billion waste tyres thrown away on a yearly basis.

Landfilling of tyres poses a severe environmental risk. Waste tyre disposal regions, in particular, contribute to biodiversity loss, as tyres contain hazardous and soluble components (Thomas et al, 2015). Recycling is one of the most significant waste reduction methods; nevertheless, recycling discarded tyres is especially difficult due to their high creation rates and non-biodegradability. Incorporating waste tyres as a partial substitute for coarse aggregate in the most extensively used building material, concrete, is one strategy to reduce the volume of waste tyres in the environment (Muyen et al, 2019).

In the majority of studies and research works, three general categories of waste tyre rubber are usually investigated, namely chipped, crumb, and ground rubber:

(1) Chipped Rubber:
Chipped rubber is waste tyre rubber that has been shredded or chopped to replace the gravel (coarse aggregate in concrete). Two phases of tyre shredding are required to generate this rubber. The rubber should be 300 – 430 mm long and 100–230 mm wide by the end of stage one. Cutting reduces the size to 100–150 mm in the second stage. If the shredding process is prolonged, shredded particles with a size of 13–76 mm are formed, which are referred to as “shredded particles.”

(2) Crumb Rubber:
Crumb rubber, which is used to substitute sand (fine aggregate), is made in special mills where large rubbers are broken down into smaller ripped pieces. Depending on the type of mills used and the temperature generated, different sizes of rubber particles may be produced. Particles with a high irregularity in the range of 0.425–4.75 mm are created using a simple approach.

(3) Ground Rubber:
Ground rubber that could be used to replace cement is dependent on the equipment used for the size reduction. Magnetic separation and screening are commonly performed in two steps on waste rubber tyres. In increasingly complicated techniques, different size percentages of rubber are obtained. The particles produced in the micro-milling process range in size from 0.075 to 0.475 mm.

Processing of Waste Rubber Tyre — **Figure 1:** Processing of waste rubber tyre for concrete production

Generically, rubber aggregates are made from waste tyres using one of two methods: mechanical grinding at room temperature or cryogenic grinding below the glass transition temperature. To substitute coarse aggregates, the first process yields chipped rubber. The second process, on the other hand, frequently results in crumb rubber being used to substitute fine aggregates (Kotresh and Belachew, 2014).

Effect of Waste Rubber Tyre on the Mechanical Properties of Concrete

The inclusion of tyre rubber particles in concrete has been proven to affect the performance of concrete in both positive and negative ways, according to research. According to some research works cited by Sofi (2018), waste rubber tyre in concrete is especially recommended for concrete constructions located in locations with a high risk of earthquakes, as well as applications subjected to high dynamic forces, such as railway sleepers. One of the most significant disadvantages of employing tyre rubber in concrete is that it has a significantly lower mechanical performance than normal concrete.

According to Muyen et al. (2019), a 5% replacement of conventional aggregates with waste tyre chips leads to a 5% loss in compressive strength, a 10% replacement results in a 26% drop, and a 15% replacement results in a 47% reduction in compressive strength. As a result, when waste tyre rubber aggregates are utilized in concrete production, a reduction in compressive strength is predicted. This is consistent with the findings of Ganjian et al (2009) and Su et al (2014).

Variation of crumb rubber with compressive strength of concrete — **Figure 2:** Variation of compressive strength of concrete with rubber content (Sofi, 2018)

On the other hand, when 20% fine aggregate was replaced with rubber aggregate, Su et al. (2015) observed a 12.8%drop in the flexural strength. When the rubber particles were smaller, there was less loss of strength. The tensile strength of concrete reduces with increase in the percentage of rubber replacement in concrete. The tensile strength of concrete containing chipped rubber (replacement for aggregates) is lower than that of concrete containing powdered rubber (for cement replacement) (Sofi, 2018).

Rubberized concrete’s strength and elastic modulus were found to be significantly reduced, according to researchers (Zheng et al, 2008; Elchalakani, 2015). The decreased stiffness (5 MPa) of the rubber, the uneven distribution of the rubber aggregate due to its lightweight, and the poor bonding between the rubber aggregate and the cement paste have all been blamed for the reduced mechanical performance (Panda et al, 2012).

Chipped rubber causes larger strength losses than crumb rubber, according to several researchers (Khatib and Bayomy, 1999; Topçu and Avcular, 1997). The lowering of stress and strain concentrations in the concrete, according to Huang et al. (2013), is what causes the strength improvement when smaller sizes are employed.

Li et al. (2011) investigated the effects of tyre rubber content and rubber particle size on the mechanical performance of concrete. High rubber content and smaller tyre rubber aggregate size were shown to reduce rubberised concrete’s compressive strength and static young’s modulus. The final strain of the rubberised concrete, on the other hand, increased as the rubber content and particle size reduced.

The explanations for the rubberized concrete’s decreased compressive and flexural strength (Ganjian et al, 2009);

(a) The aggregate in the concrete mix would be surrounded by the cement paste containing rubber particles. This leads to a cement paste that will be softer compared to cement paste without rubber particles. When loaded, there will be rapid development of cracks around the rubber particles and this leads to quick failure of specimens.
(b) There will be no proper bonding between rubber particles and cement paste, as compared to cement paste and natural aggregate. This can lead to cracks due to non-uniform distribution of applied stresses.
(c) The compressive strength depends on the physical and mechanical properties of the constituent materials. If part of the materials is replaced by rubber, a reduction in strength will occur.
(d) Due to the low specific gravity of rubber and lack of bonding of rubber with other concrete ingredients, there is a tendency for rubber to move upwards during vibration leading to higher rubber concentration at the top layer. Such a non-homogeneous concrete sample leads to reduced strengths.

Summarily, when tyre rubber aggregate content and rubber aggregate size increased, the mechanical properties of concrete reduced. Different studies by several researchers corroborated these findings. As a result, most studies recommend a maximum rubber content of no more than 20% total aggregate volume and a size no larger than crumb rubber size in terms of rubber content and size. The soft rubber was thought to act as air gaps inside the concrete matrix, providing minimal resistance to loads and causing the particles to become weak spots within the concrete matrix.

References

[1] Elchalakani M. (2015): High strength rubberized concrete containing silica fume for the construction of sustainable road side barriers, Structures (1):20–38
[2] Ganjian E, Morteza K, Ali AM. (2009): Scrap-tire-rubber replacement for aggregate and filler in concrete. Constr Build Mater 2009(23):1828–36.
[3] Khatib Z. K., Bayomy F. M. (1999): Rubberized Portland Cement Concrete, J. Mater. Civ. Eng., 11(3):206–213
[4] Kotresh K.M., and Mesfin G. B. (2014): Study on Waste Tyre Rubber as Concrete Aggregates. International Journal of Scientific Engineering and Technology 3(4): 433-436
[5] Li L. J., Xie W. F., Liu F., Guo Y. C., Deng J. (2011): Fire performance of high-strength concrete reinforced with recycled rubber particles, Mag. Concr. Res., 63(3):187–195
[6] Muyen Z., Mahmud F., and Hoque M. N. (2019): Application of waste tyre rubber chips as coarse aggregate in concrete. Progressive Agriculture 30 (3): 328-334
[7] Panda K. C., Parhi P. S., Jena T. (2012): Scrap-Tyre Rubber Replacement for Aggregate in Cement Concrete: Experimental Study, Int. J. Earth Sci. Eng., 5(6):1692–1701
[8] Sofi A. (2018): Effect of waste tyre rubber on mechanical and durability properties of concrete – A review. Shams Engineering Journal 9 (2018):2691–2700 https://doi.org/10.1016/j.asej.2017.08.007
[9] Su H, Yang J, Ling TC, Ghataora GS, Dirar S. (2015): Properties of concrete prepared with waste tyre rubber particles of uniform and varying sizes. J. Clean. Prod. 2015(91):288–96.
[10] Thomas BS, Gupta RC, Mehra P, Kumar S. (2015): Performance of high strength rubberized concrete in aggressive environment. Constr. Build. Mater. 2015(83):320–6.
[11] Topçu I. B., Avcular N. (1997): Analysis of rubberized concrete as a composite material, Cem. Concr. Res., 27(8):1135–1139
[12] Zheng L., Huo X. S., Yuan Y. (2008): Strength, Modulus of Elasticity, and Brittleness Index of Rubberised Concrete, J. Mater. Civ. Eng., 20, 11 (2008)

Geotextiles: Design, and Applications in Civil Engineering

Ubani Obinna

April 29, 2022

Geotextiles are permeable geosynthetic fabrics (textiles) that can separate, filter, reinforce, protect, or drain when used in conjunction with soil. As the use of geotextile fabrics has grown, so has the creation of geotextile composites and products like geogrids and meshes. Geotextiles and associated materials/products are the umbrella term for these materials. Roads, airfields, railroads, embankments, retaining structures, reservoirs, canals, dams, bank protection, and coastal engineering are among the many civil engineering applications where they are beneficially used.

Geotextiles are employed as an integral part of a human-made project, structure, or system with foundation soil, rock, earth, or any other geotechnical engineering-related material. AASHTO (M288-96) specifies geotextile strength standards in the United States of America. In Europe, EN 13249:2016 specifies the relevant characteristics of geotextiles and geotextile-related products used in the construction of roads and other trafficked areas (excluding railways and asphaltic inclusion), and the appropriate test methods to determine these characteristics. According to EN 13249:2016, the intended use of these geotextiles or geotextile-related products is to fulfill one or more of the following functions: filtration, separation, and reinforcement. The separation function will always occur in conjunction with filtration or reinforcement, and hence will not be specified alone.

Geotextiles in erosion control works and embankment protection — **Figure 1:** Erosion protection using geotextile (Van Baars, 2016)

geotextile in road construction — **Figure 2**: Geotextiles in highway construction

Geotextile tensile strength varies depending on the geotextile designation and the design requirements. A woven slitfilm polypropylene (weighing 240 g/m²), for example, has a strength range of 30 to 50 kN/m. The angle of friction between soil and geotextiles varies depending on the geotextile and the soil. For design purposes, it is usually necessary to apply reduction factors to the laboratory tensile strength of geotextiles in order to suit site conditions.

The different functions of geotextiles in soils are:

Erosion protection
Sealing
Filtering
Reinforcing
Drainage
Separation

different ways of using — **Figure 3:** Six different functions of geotextiles (Van Baars, 2016)

Geomembranes are impervious membranes that are frequently utilized as cut-offs and liners. Geomembranes were mostly utilized as canal and pond liners until recently; nevertheless, one of the most common current uses is the containment of hazardous or municipal wastes and their leachates. Geotextile or mesh underliners are used in many of these applications to support or protect the more flexible geomembrane while also functioning as an escape route for gases and leachates generated in specific wastes.

Construction of Reinforced Earth Structures using Geotextiles

Reinforced earth is a construction material made up of soil fill that has been strengthened by the addition of rods, bars, fibers, or nets that provide frictional resistance to the soil. The idea of using rods or fibers to reinforce soil is not new. Thin metal strips, geotextiles, and geogrids are currently used as reinforcing materials in the construction of reinforced earth retaining walls. The three components of a mechanically stabilised earth wall are the facing unit, the backfill, and the reinforcing material. Modular concrete blocks, currently called segmental retaining walls are most common as facing units.

The type of facing unit and reinforcing material employed in the system usually determine the process of constructing a mechanically stabilised earth wall. The skin, also known as the facing unit, can be flexible or rigid, but it must be robust enough to hold the backfill in place and allow fastenings for the reinforcement to be connected. The facing units only need a small foundation to be built on, which usually consists of a trench filled with mass concrete that provides a footing similar to that seen in domestic housing.

Construction procedure using — **Figure 4:** General construction procedures for using geotextiles in fabric wall construction (Murthy, 2009)

The construction procedure with the use of geotextiles is explained in Figure 4. Here, the geotextile serves both as a reinforcement and also as a facing unit. The procedure is described below as given by Murthy (2009) with reference to Figure 4.

Start with an adequate working surface and staging area (Fig. 4(a)).
Lay a geotextile sheet of proper width on the ground surface with 4 to 7 ft at the wall face draped over a temporary wooden form (b).
Backfill over this sheet with soil. Granular soils or soils containing a maximum of 30% silt and /or 5% clay are customary (c).
Construction equipment must work from the soil backfill and be kept off the unprotected geotextile. The spreading equipment should be a wide-tracked bulldozer that exerts little pressure against the ground on which it rests. Rolling equipment likewise should be relatively lightweight.
When the first layer has been folded over the process should be repeated for the second layer with the temporary facing formwork being extended from the original ground surface or the wall being stepped back about 6 inches so that the form can be supported from the first layer. In the latter case, the support stakes must penetrate the fabric.
This process is continued until the wall reaches its intended height.
For protection against ultraviolet light and safety against vandalism, the faces of such walls must be protected. Both shotcrete and gunite have been used for this purpose.

Design Considerations for Mechanically Stabilised Earth Walls using Geotextiles

The design of a mechanically stabilised earth wall involves the following steps (Murthy, 2009):

Check for internal stability, addressing reinforcement spacing and length.
Check for external stability of the wall against overturning, sliding, and foundation failure.

The general considerations for the design are:

Selection of backfill material: granular, freely draining material is normally specified. However, with the advent of geogrids, the use of cohesive soil is gaining ground.
Backfill should be compacted with care in order to avoid damage to the reinforcing material.
Rankine’s theory for the active state is assumed to be valid.
The wall should be sufficiently flexible for the development of active conditions.
Tension stresses are considered for the reinforcement outside the assumed failure zone.
Wall failure will occur in one of three ways
a. tension in reinforcements
b. bearing capacity failure
c. sliding of the whole wall soil system.
Surcharges are allowed on the backfill. The surcharges may be permanent (such as a roadway) or temporary.
a. Temporary surcharges within the reinforcement zone will increase the lateral pressure on the facing unit which in turn increases the tension in the reinforcements but does not contribute to reinforcement stability.
b. Permanent surcharges within the reinforcement zone will increase the lateral pressure
and tension in the reinforcement and will contribute additional vertical pressure for the reinforcement friction.
c. Temporary or permanent surcharges outside the reinforcement zone contribute to lateral pressure which tends to overturn the wall.
The total length L of the reinforcement goes beyond the failure plane by a length L_g. Only length L_g (effective length) is considered for computing frictional resistance. The length L_R lying within the failure zone will not contribute to frictional resistance
For the purpose of design, the total length L remains the same for the entire height of wall H. Designers, however, may use their discretion to curtail the length at lower levels.

References
[1] Van Baars Stefan (2016): Advanced Soil Mechanics. Edited and published by Stefan Van Baars. Edition May 2016
[2] Murthy V. N. S. (2009): Textbook of Soil Mechanics and Foundation Engineering: Geotechnical Engineering series. CBS PUBLISHERS AND DISTRIBUTORS PVT LTD

Differential Settlement of Foundations

Ubani Obinna

September 29, 2023

cracking of a building due to differential settlement

Table of Contents

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Causes of Differential Settlement
Deformation of Structures and their Supporting Foundations
Methods of Avoiding or Accommodating Excessive Differential Settlement
- Foundations on Sand
- Foundations on Clay
References

When there is relative movement or differential settlement between various parts of a foundation, internal stresses are developed in the structure. Differential settlement occurs when one part of a foundation settles relative to the other. When the settlement of a foundation is uniform, there are usually no structural implications. However, serious cracking, and even collapse of the structure, may occur if the differential movements are excessive.

Causes of Differential Settlement

The differential settlement between parts of a structure may occur as a result of the following;

(a) Variation in soil properties
Highly compressible soil may be used to support one part of a structure and an incompressible material for the other. Such differences are typical, especially in glacial deposits, where clay lenses might be found in primarily sandy material or vice versa.

Furthermore, some parts of a structure may be built on shallow rock and others on soil or compressible weathered rock in places with uneven bedrock surfaces. Sand and gravel deposits thrown down by the wind or water can vary greatly in density both vertically and horizontally. In such cases, differential settlement may occur in the foundations of structures built on such soil deposits.

differential settlement due to non uniform soil property — **Figure 1:** Differential settlement due to variation in soil properties

(b) Variation in foundation loading
When the magnitude of loads coming from superstructure columns or walls vary significantly, differential settlement may occur unless special design considerations are made to prevent it. For example, in a building with a tower and wings, a differential settlement between the tower and the wings would be predicted unless special foundation design procedures were used to prevent it. Furthermore, a light superstructure might surround a very large piece of machinery in a factory building, and the area supporting the machinery may settle relative to the factory building.

(c) Large loaded area on flexible foundations
When built directly on compressible soil, the settlement of large flexible raft foundations, or big loaded regions consisting of independent foundations of a number of columns, takes on a characteristic bowl form, with the largest settlement in the centre and the minimum at the corners.

In most cases, the maximum differential settlement is around half of the entire settlement. Even while the maximum differential settlement between the centre and corners may be significant in a building made up of a large number of closely spaced equally loaded columns, the relative settlement between columns may be only a fraction of the maximum.

However, where the large loaded region is founded on a relatively incompressible stratum (e.g. dense gravel) overlying a compressible layer, settlement of the structure will occur due to consolidation of the deeper compressible layer, but it will not take the form of the bowl-shaped depression. If the dense layer is thick enough, it will produce a rigid raft, which will eliminate differential settlement to a considerable extent.

(d) Differences in time of construction
This problem happens when an extension is added to a structure several years after the original structure was completed. Although the latter’s long-term consolidation settlements may be nearly complete, the new structure (assuming the same foundation loading as the original) will eventually settle an equivalent amount. To prevent distortion and cracking between the old and new structures, special precautions in the form of vertical joints are required.

Cracking of partitions — **Figure 2:** Crack between two adjoining structures

(e) Variation in site condition
On a sloping site, it may be necessary to remove a significant thickness of overburden to produce a level site, or one portion of a building area may have been occupied by a heavy structure that had been demolished. Different stress conditions both before and after loading emerge from these variances, resulting in differential settlement or swelling. One part of a site may be normally consolidated, and another part overconsolidated. This will result in variation in the settlement behaviour.

Deformation of Structures and their Supporting Foundations

In a conference held in Tokyo in the year 1977, Burland et al (1977) highlighted the basic conditions that must be met when considering the limiting movements of a structure due to soil-structure interaction. The criteria stated are still very much the basis for the design of structures and foundations today. The basic criteria that must be satisfied when considering the limiting movements of a structure are;

(a) The visual appearance of the structure
(b) Serviceability or functionality of the structure
(c) Stability of the structure

It is necessary to describe settlements and distortions in line with the established terminology presented in Figure 3 when considering the criteria above in connection to limiting movements. When looking at the visual appearance of a building, a tilt or rotation of more than 1 in 250 is likely to be visible to the human eye. A deflection ratio of more than 1 in 250 or a local rotation of horizontal components greater than 1 in 100 is likely to be very visible. The appearance of framed buildings is affected when load-bearing walls or claddings crack. At eye level, crack widths of more exceeding 3—5 mm are ugly and require repair.

Differential settlement — **Figure 3:** Definitions of differential settlement and distortion for framed and load-bearing wall structures (Tomlinson, 2001)

Cracking in structures can lead to loss of weather/water tightness, fire resistance, and thermal and sound insulation characteristics, thereby affecting the serviceability or functionality of the structure. Total settlement can be important to serviceability when connecting to exterior drains or other piping, while deformations can interfere with the proper operation of overhead cranes and precision gear. Relative deflections and rotations may be important for structural stability because they can produce excessive bending strains in members. Excessive tilting might cause a structure to completely collapse.

The amount of damage produced by settlement is partly determined by the order and timing of construction operations. For example, if a tall building is built on a deep clay basement, the excavation’s base will first heave to a convex shape. The foundation soil will consolidate and finally deform to a concave (bowl) shape when the superstructure is built, resulting in a full reversal of curvature of the basement and lowermost stories.

The structural frame of a multi-story housing or office building bears the major portion of the overall dead load. As a result, by the time the frame is finished, the majority of the building’s settlement will have occurred (see Figure 4). Then, at a later time, claddings or finishes will add to the structure’s rigidity and suffer far less deformation than that which has already occurred in the structural frame.

On the other hand, this will not be the case for structures like silos. The majority of the settlement in silos does not occur until the compartments are filled for the first time. The contents of the silo can weigh significantly more than the confining structure.

**Figure 4:** Typical foundation of a framed structure

Empirical standards for restricting the movement of structures have been established in order to prevent or reduce cracking and other forms of structural damage. Tables 1 and 2 show some of the criteria. Skempton and Macdonald’s (1956) criteria are consistent with the guidelines for acceptable limits in EN 1997-1:2004 (EC 7) Clause 2.4.9. Clause 2.4.8(5)P stipulates that the limiting values must be agreed upon with the structure’s designer during the design of the building. The relative rotation (or angular distortion) is the key factor for framed buildings and reinforced load-bearing walls, but the deflection ratio is the requirement for unreinforced load-bearing walls that fail by sagging or hogging, as illustrated in Figure 3.

Table 1: Criteria for limiting values for relative rotation (Tomlinson, 2001)

Type of damage	Limiting values for relative rotation (angular distortion) Skempton and MacDonald (1956)	Limiting values for relative rotation (angular distortion) Meyerhof (1947)
Structural Damage	1/150	1/250
Cracking in walls and partitions	1/300 (but 1/500 recommended)	1/500

Table 2: Criteria for limiting values for deflection ratio (∆/L) (Tomlinson, 2001)

Type of damage	Limiting values for deflection ratio (∆/L) Meyerhof (1947)	Limiting values for deflection ratio (∆/L) Burland and Wroth (1974)
Cracking by sagging	0.4 × 10^-3	At L/H = 1: 0.4 × 10^-3At L/H = 5: 0.8 × 10^-3
Cracking by hogging	–	At L/H = 1: 0.2 × 10^-3At L/H = 5: 0.4 × 10^-3

Methods of Avoiding or Accommodating Excessive Differential Settlement

Differential settlement does not have to be taken into account only when structures are to be built on relatively incompressible bedrock. When structures are built on weathered rocks or soils, an estimate of total and differential settlements must be made to determine whether the movements are likely to be tolerated by the structure’s design, or whether they are large enough to necessitate special measures to avoid or accommodate them. The Institution of Structural Engineers (1989), in a report, provides general recommendations on how to approach this study.

Foundation on rock — **Figure 5:** Differential settlement may be ignored for foundations on rocks

It is impractical to design foundations to be completely free of cracks caused by differential settlement. This is because temperature and moisture movements in the structure also cause cracking in walls and ceilings in most buildings with internal plaster finishes. Therefore a certain degree of readily repairable cracking owing to differential settlement should be permitted (Tomlinson, 2001). The risks of damage due to settlement can be calculated using empirical principles based on experience in the case of simple structures on generally uniform compressible soils.

Foundations on Sand

The differential settlement for foundations on sand is unlikely to exceed 75% of maximum movement, and since most conventional structures can withstand 20 mm of settlement between adjacent columns, a limiting maximum settlement of 25 mm was proposed by Tezarghi and Peck (1967).

The maximum settlement limit for raft foundations on sand is increased to 50 mm. Skempton and MacDonald (1956) concluded from a study of the movement of 11 buildings that the limiting maximum differential settlement is roughly 25 mm for a limiting angle of distortion (β) of 1 in 500, the limiting total settlement is 40 mm for pad foundations, and 40—65 mm for raft foundations.

Buildings on sands seldom settle by more than 50 mm, according to studies, and in the vast majority of cases, settlement is on the order of 25 mm or less (Sutherland, 1974). These guidelines should not be applied to sands that contain silt or clay, as these materials increase the compressibility of the sand.

Foundations on Clay

Skempton and MacDonald (1956) proposed a design limit for maximum differential settlement of 40 mm for foundations on clays, as well as design limitations for a total settlement of 65 mm for isolated foundations and 65—100 mm for rafts. If the total and differential settlements exceed the serviceability limit state as a result of applying the above empirical rules or conducting a settlement analysis of the structure based on the assumption of complete flexibility in the foundations and superstructure, the engineer has the option of either avoiding settlement or accommodating the movement through appropriate structural design measures.

If the structures themselves are not rigid enough to prevent excessive differential movement with regular spread foundations, one or more of the procedures listed below may be used to limit total and differential settlements to a tolerable level.

(a) Provision of a rigid raft foundation in two or three directions
(b) Provision of deep basements to reduce net bearing pressure on the soil
(c) Transference of foundation loading to deeper and less compressible soil via basements, piers, or piles
(d) Provision of jacking pockets, or brackets, in columns to relevel the superstructure
(e) Provision of additional loading on lightly loaded areas in the form of kentledge or embankments to even out soil pressure distribution

Method (b) is effective in minimizing excessive differential settlement between components of a structure with differing foundation loads, as well as reducing maximum settlements owing to the relief of overburden pressure and excavating for deep basements. As a result, the deepest basements can be given under the structure’s heaviest components, while shallower or no basements can be provided in places with lighter loading.

References

[1] Burland J. B., Broms B. B. and De Mello V. (1977): Behaviour of foundations and structures, in Proceedings of the 9th International Conference on Soil Mechanics, Tokyo, Session 2, 1977
[2] Burland J. B. and Wroth C. P. (1974): Review paper Settlement of buildings and associated damage, in Proceedings of the Conference on Settlement of Structures, Pentech Press, Cambridge, pp 611—654, 1974
[3] Institution of Structural Engineers (1989): Structure—Soil Interaction — The Real Behaviour of Structures, Institution of Structural Engineers, London, 1989
[4] Meyerhof G. G. (1947): The settlement analysis of building frames, Structural Engineer, 25, 309,
[5] Skempton A. W. and MacDonald D. H. (1956):, The allowable settlement of buildings, Proceedings of the Institution of Civil Engineers, 3(5):727—784
[6] Sutherland H. B. (1974) Review paper Granular materials, in Proceedings of the Conference on Settlement of Structures, Pentech Press, Cambridge, pp 473—499, 1974
[7] Terzaghi K. and Peck R. B. (1967): Soil Mechanics in Engineering Practice, 2nd edn, John Wiley, New York, 1967
Tomlinson M. J. (2001): Foundation Design and Construction (7th Edition). Pearson Education Ltd UK

Partial Replacement of Sand with Waste Glass in Concrete Production

Ubani Obinna

April 22, 2022

Glass is a transparent or translucent material that is used in the production of materials like sheet glass and container glass. It is manufactured by rapidly cooling molten components like silica sand to prevent the creation of visible crystals. Glass is an excellent material for recycling, and its applications, including concrete manufacturing, reduce the embodied energy in concrete production (Gautam et al, 2012). The use of waste glass in concrete production is still uncommon due to the alkali-silica reaction (ASR), which reduces concrete durability and strength (Lui, 2011).

Waste glass aggregate is angular in shape and has a smooth feel. It is tough, but also fragile and brittle (Chen et al, 2006; Taha and Nounu, 2008). Recycling waste glass and turning it into fine aggregate for concrete production reduces landfill space and the demand for natural raw materials in the construction sector (Rakshvir and Barai, 2006). In a study by Ibrahim (2017), it was discovered that waste glass may replace sand up to 40% by weight without affecting the tensile and compressive strengths when compared to control concrete. He observed that 15% partial replacement was the optimum dosage.

To address the cement/concrete industry’s environmental and economic challenges, Malik et al. (2013) used waste glass as a partial replacement for fine aggregates (sand) in concrete. Several samples were made by replacing sand with glass contents of 10%, 20%, 30%, and 40% by weight in M-25 grade concrete. The samples were tested for compressibility, splitting tensile strength, and density 28 days after curing. The experiment’s findings were compared to those of conventional concrete. Specimens containing crushed waste glass had higher compressive strength for particle sizes of 0.1 – 1.18 mm, with up to 30% weight replacement of small aggregates. Specimens made of glass were also shown to be more cost-effective and environmentally friendly.

Ramana and Samdani (2013) studied the effects of replacing fine sand aggregates with waste glass in the ranges of 0%, 5%, 10%, 15%, 20%, 25%, and 30%. The research work investigated compressive strength, split tensile strength, and flexural strength, among other mechanical properties. The results of the laboratory tests were recorded and compared to traditional concrete results. The results showed that mechanical properties improved when fine aggregates were replaced with crushed glass at 15% but reduced when fine aggregates were replaced at a rate of 30%.

Dabiri et al. (2018) assessed the effects on compressive strength and, more importantly, the effects on the weight of the concrete by replacing fine aggregates with waste glass particles. 27 cube samples were produced to achieve the objectives, with 6 specimens produced of normal concrete and the rest incorporating glass particles mixed in varying amounts. Micro-silica was added to the glass cubes to prevent the Alkali-Silica reaction (ASR). According to the results of the testing, replacing aggregates with glass particles increased the compressive strength by more than 30%. The weight of the concrete was observed to be nearly constant for the most part. The optimum proportion for replacing aggregates with waste glass particles, according to the research, is 50%.

Ganiron et al. (2014) conducted an experimental study to discover a substitute for coarse aggregates in concrete mixtures. In the study, crushed glass bottles were utilised in place of coarse aggregates, and the influence on the mixture’s physical and mechanical properties was observed. The results of the testing showed that replacing coarse aggregates with recycled glass bottles up to 10% by weight and adding 5% by weight to the concrete mix produced acceptable compressive strength values. In the experiment, it was shown that recycled glass bottles may effectively replace coarse aggregates.

Turgut and Yahlizde (2009) studied and compared the physical and mechanical properties of concrete cubes by substituting varying degrees of fine glass (FG) and coarse glass (CG) in the concrete mixture. The values of several parameters including compressive strength, flexural strength, splitting tensile strength, and abrasion resistance of the samples were measured and observed at a 20% FG replacement. The results showed that compressive strength, flexural strength, splitting tensile strength, and abrasion resistance were 69%, 90%, 47%, and 15% respectively greater than the typical concrete sample. According to the findings, at a 20% replacement level by weight of FG, the alkali-silica reaction (ASR) in concrete is reduced.

waste glass into concrete — Waste glass in concrete

Kavyateja et al. (2016) studied the use of crushed glass as a substitute for fine aggregates in concrete production. The control mix proportion of 1:1.5:3 was batched by volume with a water/cement ratio of 0.5. The replacement rates in the samples ranged from 0% to 40%, with a 10% difference between the two. To study the variation in their compressive strength, concrete cube samples of 150mm x 150mm x 150mm were cast and tested after 3 days, 7 days, 28 days, 56 days, and 90 days. According to the experimental data, the compressive strength increases up to a 20% substitution dosage, then drops at 30% and 40% substitution dosage. The split tensile strength test also demonstrated that the split tensile strength reduces as the glass content increases.

Compressive strength for different ages by using different percentages of crushed glass as sand in concrete (Kavyateja et al., 2016).

Jain et al (2020) investigated the possibility of utilizing solid waste, such as granite powder from the granite industry, and wasted soda-lime glass powder from waste glass bottles, in concrete production. The durability of blended concrete mixes using waste glass powder and granite powder at various replacement amounts was investigated in the research work. Glass powder (GP) was added to the concrete mixes in amounts of 5%, 10%, 15%, 20%, 25%, and granite powder (GrP) in amounts of 10%, 20%, 30%, 40%, and 50%, respectively, as a partial replacement for cement and sand.

Water absorption, water permeability, acid attack, sulphate attack, and a quick chloride penetration test (RCPT) were used to assess the durability of a series of blended mixes. Microstructure investigation was done using a scanning electron microscope (SEM) and X-ray diffraction (XRD). The durability properties of concrete containing 15% GP and 30% GrP in place of cement and sand, respectively, showed a considerable increase. The results show that glass granite blended concrete has improved water permeability and absorption. The response of the blended mix to sulphate and acid attack was significantly better than that of the control concrete mix.

Ibrahim (2020) utilized waste glass as a partial replacement for coarse aggregate, with ratios of 0%, 5%, 10%, 15%, 20%, 25%, 30%, 35%, 40%, 45%, and 50% by weight. At both the hardened and fresh stages, some mechanical and other properties of concrete were investigated. The results from the study showed that when waste glass was used as a partial replacement for coarse aggregate, it caused a reduction in the slump, density, and water absorption of the concrete. However, it improved the concrete’s tensile and compressive strengths until a 25% weight substitution ratio was reached. According to the results of the tests, when the waste content increases, the strengths gradually increase up to a certain point, after which they gradually decline. The highest level of influence was a 25% substitution ratio.

According to studies published in multiple papers, waste glass and glass powder have all been effectively employed as partial replacements for fine and coarse aggregates in concrete. Waste glass particles can improve concrete’s compressive strength, flexural strength, workability, and tensile strength, according to the findings. By replacing fine aggregates with glass powder in a 20% ratio by weight, the highest compressive strength can be achieved. The addition of waste glass particles in concrete has also shown to be more cost-effective and environmentally friendly than ordinary concrete.

References

[1] Chen C. H., Huang R, Wu J. K., and Yang C. C. (2006): Waste E-glass particles used in cementitious mixtures. Cement and Concrete Res 36(3):449–56
[2] Dabiri H., Sharbatdar M. K., Kavyani A. and Baghdadi M. (2018): The Influence of Replacing Sand with Waste Glass Particle on the Physical and Mechanical Parameters of Concrete. Civil Engineering Journal 2018 (4):1646-1652
[3] Ganiron T. U. (2014): The Effect of Waste Glass Bottles as an Alternative Coarse Aggregates in Concrete Mixture. International Journal of ICT-aided Architecture and Civil Engineering 2014 (02):1-10
[4] Gautam S. P., Srivastava V. and V.C. Agarwal V. C. (2012): Use of glass wastes as fine aggregate in concrete. J. Acad. Indus. Res. 1(6):320-322
[5] Liu M. (2011): Incorporating ground glass in self-compacting concrete. Construction and Building Materials 25 (2): 919-925 https://doi.org/10.1016/j.conbuildmat.2010.06.092
[6] Ibrahim K. I. M (2017): The Effect of Using Waste Glass [WG] as Partial Replacement of sand on Concrete. IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE) 14(2):41-45
[7] Ibrahim KIM (2020): Recycled Waste Glass [WG] in Concrete. Glob J Eng Sci. 6(1): 2020 http://dx.doi.org/10.33552/GJES.2020.06.000627
[8] Jain K. L.,Sancheti G., Lalit Kumar Gupta L. K. (2020): Durability performance of waste granite and glass powder added concrete, Construction and Building Materials, Volume 252, 2020, 119075, https://doi.org/10.1016/j.conbuildmat.2020.119075
[9] Kavyateja B. V., Reddy P. N., Mohan U. V. (2016): Study of Strength Characteristics of Crushed Glass used as Fine Aggregates in Concrete. International Journal of Research in Engineering and Technology (05):157- 160
[10] Malik M. I., Bashir M., Ahmad S., Taruq T. and Chowdhary U. (2013): Study Of Concrete Involving Use of Waste Glass as Partial Replacement of Fine Aggregates. International Organization of Scientific Research Journal of Engineering 2013 (3):08-13
[11] Taha B., and Nounu G. (2008): Properties of concrete contains mixed color waste recycled glass as sand and cement replacement. Construction and Building Materials 22(5):713–720
[12] Rakshvir M. and Barai S.V. (2006): Studies on recycled aggregates-based concrete. Waste Management & Research 24(3):225–233
[13] Ramana K.V. and Samdani S. S. (2013): Study on Influence of Crushed Waste Glass on Properties of Concrete. International Journal of Science and Research 2013 (4):1034-1039
[14] Turgut P. and Yahlizade E. S. (2009): Research into Concrete Blocks with Waste Glass. International Journal of Civil and Environmental Engineering 3(3):186-92

Settlement of Foundations

Ubani Obinna

April 15, 2022

Foundation soils undergo settlement and deformation when loaded. When permissible stress methods are utilized in design, settlement due to foundation soil consolidation is usually the most essential consideration when calculating the serviceability limit state or analyzing acceptable bearing pressures. Even though ultimate limit state calculations or the application of an arbitrary safety factor to the calculated ultimate bearing capacity protects foundations from sinking due to soil shear failure, it is still necessary to investigate the likelihood of settlements before the allowable bearing pressures can be determined.

The causes of settlement of foundations, the implications of total and differential movements on the structure, techniques of calculating settlement, and foundation design to eliminate or minimize settlement will all be discussed in this article. Where (EN 1997-2:2007) EC 7 recommendations are used to compute the serviceability limit state, the partial factor γ_Ffor actions is unity for unfavorable loads and zero for favorable loads. When determining the characteristic deformation modulus or the coefficient of compressibility, the material factor γ_M is equal to one.

There are three parts to the settlement of a structural foundation.

The ‘immediate settlement‘ (S_e) occurs as a result of elastic deformation of the soil without a change in water content during the application of the loading.
The ‘consolidation settlement‘ (S_c) occurs when the volume of the soil is reduced due to the ejection of some pore water from the soil.
‘Creep‘ or ‘secondary settlement‘ (S_s) occurs over a very long period of years after completing the extrusion of excess pore water. It is caused by the soil particles’ viscous resistance to compression adjustment.

The ‘immediate’ and ‘consolidation’ settlements are of a small magnitude in the case of foundations on medium-dense to thick sands and gravels. By the time the foundations are fully loaded, a large amount of the whole settlement has been accomplished. Similarly, a large share of foundation settlement on loose sands occurs as the load is applied, whereas settlement on compressible clays is a mixture of immediate and long-term movements. The latter is more likely to account for the majority of the movement and could take many years to complete.

EC 7 defines immediate settlement as “settlement without drainage for fully saturated soil due to shear deformation at constant volume.” EC 7 emphasizes the importance of considering foundation settlement due to reasons other than normal soil compression and consolidation. Some of them are groundwater level variations, effects of animals and vegetation, earthquakes, subsidence, etc.

Settlement of foundations is not only limited to very massive and heavy constructions. Under light loadings, considerable settlement can occur in soft and compressible silts and clays. According to Tomlinson (2001), settlement and cracking happened in two-storey homes built on soft silty clay in Scotland. The foundation loading was probably not more than 32 kN/m run of wall in the dwellings, which were made of precast concrete blocks. Differential settlement and cracking of the blocks of houses were so severe in less than three years after completion that a number of the dwellings had to be evacuated. A relative displacement of 100 mm along the wall was observed in one block.

The amount of the differential or relative settlement between one portion of a structure and another is more important to the superstructure’s stability than the magnitude of the total settlement. Only in relation to adjacent works is the latter significant. A flood wall to a flyer, for example, might be built to a crest level at a set height above the maximum flood level. Excessive wall settlement over a lengthy period of time could result in the wall overtopping during flood events. There is no influence on the superstructure if the entire foundation area of a structure settles to the same degree.

If there is relative movement between distinct elements of the foundation, however, stresses in the structure are created. If the differential movements are extreme, serious cracking and even collapse of the structure may ensue.

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

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