The 2225 mm x 2225 mm pad footing shown above is meant to support a 230 x 230 mm column from the superstructure of a building. The characteristic dead and imposed loads on the column are 600 kN and 210 kN respectively, excluding the weight of the column stub. Making use of your personal engineering judgment, kindly answer the following questions using the information provided;
(1) Will you accept the size of the pad footing provided if the verified allowable bearing capacity of the soil is 175 kN/m2?
(2) What is your calculated value of the total service load for determining the footing dimensions?
(3) What informed your decision? Did you make any assumptions?
Helical staircases provide alternatives for vertical circulation in buildings. Due to their curved and elegant nature, they inspire awe and admiration whenever they are properly designed and constructed. The analysis and design of a helical staircase can be complex due to an inherent interaction of vertical moments, horizontal moments, vertical shear, lateral shear, axial force, and torsion. The complexity of helical staircases is due to their geometry.
Geometrically, a helical surface is a three-dimensional structure in space consisting of a warped surface that is generated by moving a straight line touching a helix so that the moving line is always perpendicular to the axis of the helix. In an oblique helix, the generating line always maintains a fixed angle with the helix. Since a helical staircase is a space member, all the six internal actions come into play at any section, having varying directions and lines of actions.
Fig 1: Internal stresses in a helical staircase
Despite the existence of the six internal stresses at any section in a helical staircase, they are usually designed for the horizontal bending moment only. Furthermore, helical staircases can sometimes be idealised as a fixed ended curved beam for the purposes of design due to their complexity.
Bergman (1956) proposed an approximate solution to helical staircases called the in-plane beam solution. The method reduces the problem to that of a horizontal bow girder, which fails to take into account the inherent structural strength of a helical beam.
Morgan (1960) and Scordelis (1960) presented the analysis of the longitudinal elastic axis of the helical beam as a three-dimensional structure indeterminate to the sixth degree. By selecting the redundant at mid-span and using the principle of symmetry, all but two of the redundants become zero. The two redundants at mid-span are the horizontal shear H and the vertical moment Mo.
Cusens and Trirojna (1964) carried out a test on half scaled fixed ended helical staircase and confirmed the equilibrium equations derived by Morgan. Santathadaporn and Cusens (1966) worked further on helicoidal stair slabs to simplify the design process. They proposed some simple equations for design purpose with the provision of using coefficients in the equations. They used computer program to solve those complicated equations and constructed a series of design charts to provide the values of the coefficients.
Reynolds and Steedman (1988) attempted further modifications of the charts provided by Santathadaporn and Cusens. The analytical approach they chose was the one made by Morgan and Scordelis. The proopsed equations and coefficients for determination of the internal stresses are as follows;
Fig 2: Geometry and internal stresses in helical staircase
Horizontal moment Mn = Mo sinθsinϕ – HR2θtanϕcosθsinϕ – HR2sinθcosϕ + nR1sinϕ(R1sinθ – R2θ) —— (1)
Vertical moment My = Mocosθ + HR2θtanϕsinθ – nR12(1 – cosθ) —— (2)
Where; Mo = Redundant moment acting tangentially at the midspan = k1nR22 H = Horizontal redundant force at midspan = k2nR2 Mvs = Vertical moment at supports= k3nR22 n = total design load on the staircase (kN/m2) R1 = Radius of centreline of loading = 2(Ro3 – Ri3)/3(Ro2 – Ri2) R2 = Radius of centreline of steps = 0.5(Ri + Ro) θ = angle subtended in plan between point considered and mid-point of stair ϕ = slope of tangent to helix centreline measured from horizontal β = Total angle formed by the helical staircase in plan k1 = GC/EI1 k2 = GC/EI2 I1, I2 = Second moment of area of stair section about horizontal axis and axis normal to slope respectively.
The 36 design charts presented by Santathadaporn and Cusens (1966) covered ranges of β of 60° to 720°, ϕ of 20° to 50°, b/h of 0.5 to 16 and R1/R2 of 1.0 to 1.1, based on a ratio of G/E of 3/7.
Four design charts are provided in Table 177 of (Reynolds and Steedman 10th Edition) and (Tables 2.90 and 2.91 of 11th Edition) for a ratio of G/E of 0.4 and by taking C to be one-half of the St. Venant value for plain concrete. These charts cover ranges of β of 30° to 360° and ϕ of 20° to 40°, with values of b/h of 5 and 10 and R1/R2 of 1.0 and 1.1, these being the ranges most frequently met in helical stair design. According to Reynolds and Steedman (1988), interpolation between the various curves and charts on the Tables will be sufficiently accurate for preliminary design purposes.
According to a 2015 research from the Department of Civil Engineering, Addis Ababa University, Ethiopia, the solution provided in Reynolds and Steedman is the best so far, even though it has its limitations. The charts presented in the book give good information about internal actions but only qualitatively, and indicate set of constants to put in his equations then solved. And, as Reynolds himself stated in his book, his results are only satisfactory for preliminary analysis only as lots of interpolation is involved.
Design Example
Design a helicoidal stair having an angle of inclination ϕ of 25° to the horizontal plane to support a uniform imposed load of 3 kN/m2. The stair is to have a width of 1.5 m and the minimum thickness of the waist is 150 mm, the radius to the inside of the stair Ri is 1000 mm, and the angle β turned through by the stair is 180°. (Weight of finishes = 1.2 kN/m2; weight of riser/thread = 1.875 kN/m2)
Solution Permanent actions; Self-weight of waist (150 mm thick) = 0.15 × 25 = 3.75 kN/m2 Weight of stepped area = 1.875 kN/m2 Weight of finishes = 1.2 kN/m2 Total dead load (characteristic permanent action) gk = 6.825 kN/m2
The calculation should be carried out for different angles, and the maximum values used for the structural design.
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References [1] Bergman V.R. (1956): Helicoidal staircases of reinforced concrete, A.C.I. Journal Proceedings 53(4):403‐412 [2] Cusens A.R., Trirojna S. (1964): Helicoidal Staircase Study. ACI Journal, Proceedings, 61(1):85‐101.[2] [3] Morgan V.A. (1960): Comparison of Analysis of Helical Staircases. Concrete and Construction Engineering, (London) 55(3):127‐132. [4] Reynolds C.E., Steedman J.C. (1988): Reinforced Concrete Designers Handbook, Tenth Edition, E & FN Spon Ltd, London [5] Santathadaporn, S. and Cusens, A R. (1966): Charts for the Design of Helical Stairs with Fixed Supports. Concrete and Construction Engineering, 61(2):46-54. [6] Scordelis A. C. (1960): Closure to Discussion of Internal Forces in Uniformly Loaded Helicoidal Girder. ACI Journal Proceedings, 56(6):1491‐1502.
As we celebrate the awesomeness of this special season, may your life, home, and career be filled with peace, love, joy, laughter, and happiness. Merry Christmas and Happy New Year in Advance.
The frame loaded as shown above is hinged at point D, simply supported at point F, and fixed at point C. Provide the following solutions based on the diagram.
(1) What is the vertical support reaction at point F? (A) 2.5 kN (B) 0.5 kN (C) 0.75 kN (D) 1.0 kN
(2) What is the bending moment just to the right of point B? (A) 4 kNm (B) -8 kNm (C) -5 kNm (D) 6 kNm
For a tall building to be successful, at a minimum, the structure should employ systems and materials appropriate to the building’s height and configuration. The structural system for a tall building must perform well and lend itself to efficient construction [1]. According to [1], a successful tall building should have the following features;
Create a friendly and inviting image that has positive values to building owners, users, and observers.
Fit the site, providing proper approaches with a congenial layout for people to live, work, and play.
Be energy-efficient, providing interior space with a controllable climate.
Allow flexibility in office layout with easily divisible space.
Offer space oriented to provide the best views.
Most of all, the building must make economic sense, without which none of the development would be a reality.
In the year 1969, Fazlur Khan classified structural systems for tall buildings relating to their heights with considerations for efficiency in the form of “Heights for Structural Systems” diagrams [2]. This marked the beginning of a new era of skyscraper revolution in terms of multiple structural systems [3].
Later, Khan upgraded these diagrams by way of modifications [4, 5]. He developed these schemes for both steel and concrete as can be seen from Figures 1 and 2 respectively [6]. Khan argued that the rigid frame that had dominated tall building design and construction so long was not the only system fitting for tall buildings.
Fig 1: Classification of tall building structural systems for steel [1]
Fig 2: Classification of tall building structural systems for concrete [1]
Bungale [1] also presented a table to show the appropriate structural system for reinforced concrete tall buildings. This is shown in Table 1.
Table 1: Appropriate structural systems for concrete tall building [1]
We are going to give a brief description of the various and popular structural systems for tall buildings in the sub-sections below. The structural system to be adopted in any design should be able to carry different types of loads, such as gravity, lateral, temperature, blast and impact loads. The drift of the tower should be kept within limits, such as H/500.
The structural systems usually adopted for high rise buildings are as follows;
Column and slab systems
Concrete floors in tall buildings often consist of a two-way floor system such as a flat plate, flat slab, or a waffle system which can resist lateral loads (see Figure 3). In a flat plate system, the floor consists of a concrete slab of uniform thickness which frames directly into the columns. Two way flat slabs make use of either capitals in columns or drop panels in slab or both, requiring less than a flat plate because extra concrete is provided only at columns where the shears and moments are the greatest.
Fig 3: Response of flat slab-frame to lateral load [1]
The waffle system is obtained using rows of joists at right angles to each other; the joists are commonly formed by square domes (see Figure 4). The domes are omitted around the columns to increase the moment and shear capacity of the slab [7]. Any of the three systems can be used to function as an integral part of the wind-resisting systems for buildings in the 10 to 20 storey range. The concept of an “effective width” is usually used in the analysis of such buildings subjected to lateral forces.
Fig 4: Typical floor systems for flat slab-frames: (a) flat plate, (b) flat slab with drop panels, and (c) two-way waffle system [1]
Rigid Frames
A rigid frame is characterized by the flexure of beams and columns and rotation at the joints. Interior rigid frames for office buildings are generally inefficient because;
(1) The number of columns in any given frame is limited due to leasing considerations and (2) The beam depths are often limited by the floor-to-floor height.
However, frames located at the building exterior do not necessarily have these limitations. An efficient frame action can thus be developed by providing closely spaced columns and deep spandrels at the building exterior. A rigid-frame high-rise structure typically comprises of parallel or orthogonally arranged bents consisting of columns and girders with moment-resistant joints [8].
The continuity of the frame also increases resistance to gravity loading by reducing the positive moments in the girders [1]. The advantages of a rigid frame are the simplicity and convenience of its rectangular form. Its unobstructed arrangement, clear of structural walls, allows freedom internally for the layout and externally for the fenestration.
Rigid frames are considered economical for buildings of up to about 25 stories, above which their drift resistance is costly to control. If, however, a rigid frame is combined with shear walls, the resulting structure is very much stiffer so that its height potential may extend up to 50 stories or more [1].
The horizontal stiffness of a rigid frame is governed mainly by the bending resistance of the girders, the columns, and their connections, and in a tall frame, also by the axial rigidity of the columns (see Figure 5). The accumulated horizontal shear above any story of a rigid frame is resisted by shear in the columns of that story.
Fig 5: Rigid frame – Forces and deformations [1]
Rigid frame systems are not efficient for buildings over 30 storeys in height because the shear racking component of deflection caused by the bending of columns and girders causes the building to sway excessively [3].
Braced frames are a type of moment-resisting frames that have single diagonal x-braces and k-braces. Lattice and knee bracing are also used. Concrete braced frames are often not used, since shear walls are superior for construction and lateral resistance. Lattice bracing is used in pre-cast panel construction.
Steel braced frames are used in interior cores, so connections could easily be made with wall panels. Composite braced frames may have steel bracings in concrete bracings in steel frames. Concrete encasement of columns and composite floor beams has also been used.
Shear wall-Frame Systems
In this system, resistance to horizontal loading is provided by a combination of shear walls and rigid frames [9]. The shear walls are often placed around elevator and service cores while the frames with relatively deep spandrels occur at the building perimeter. When a wall–frame structure is loaded laterally, the distinctly different deflected forms of the walls and the frames can be quite effective in reducing the lateral deflections to the extent that buildings of up to 50 stories or more are economical [1].
The potential advantages of a wall–frame structure depend on the intensity of horizontal interaction, which is governed by the relative stiffness of the walls and frames, and the height of the structure. The taller the building and the stiffer the frames, the greater the interaction [10].
The interaction of frame and shear walls has been understood for quite some time, the classical mode of the interaction between a prismatic shear wall and a moment frame is that the frame basically deflects in a so-called shear mode while the shear wall predominantly responds by bending as a cantilever (see Figure 6).
Fig 6: Shear wall-frame interaction [1]
Compatibility of horizontal deflection introduces an interaction between the two systems which tends to impose a reverse curvature in the deflection pattern of the system. The combined structural action, therefore, depends on the relative rigidities of different elements used in the makeup of the lateral-load-resisting system.
The distribution of total wind shear to the individual shear walls and frames as given by the simple interaction diagram is valid only if one of the following two conditions is satisfied.
1. Each shear wall and frame must have constant stiffness properties throughout height of the building. 2. If stiffness properties vary over the height, the relative stiffness of each wall and frame must remain unchanged throughout the height of the building.
Shear Truss-Outrigger Braced Systems
The structural arrangement for this system consists of a main concrete core connected to exterior columns by relatively stiff horizontal members such as one or two-story deep walls commonly referred to as outriggers. The core may be centrally located with outriggers extending on both sides or it may be located on one side of the building with outriggers extending to the building columns on one side [1].
The outrigger system is a development due to the desire to make inner cores and outer columns as one by linking them together at one or more levels with rigid arms – outriggers. It may be formed by any combination of steel, concrete or composite construction and reduce the structure’s internal overturning moment by up to 40 % compared to that of a free cantilever [10].
Multilevel outrigger systems can provide up to five times the moment resistance of a single outrigger system. Outrigger systems have been used for buildings up to 70 stories but the concept should hold for even higher buildings [11].
The basic structural response of the system is quite simple. When subjected to lateral loads, the column-restrained outriggers resist the rotation of the core, causing the lateral deflections and moments in the core to be smaller than if the freestanding core alone resisted the loading as shown in Figure 7 [3].
The external moment is resisted not by bending of the core alone, but also by the axial tension and compression of the exterior columns connected to the outriggers. As a result, the effective depth of the structure for resisting bending is increased when the core flexes as a vertical cantilever, by the development of tension in the windward columns, and by compression in the leeward columns.
Fig 17: Core supported outrigger structure [3]
Framed-Tube System
In its simplest terms, a framed tube can be defined as a three-dimensional system that engages the entire building perimeter to resist lateral loads [1]. A necessary requirement to create a wall-like three-dimensional structure is to place columns on the building exterior relatively close to each other, joined by deep spandrel girders as shown in Figure 8.
Fig. 8: Frame tube building. (a) Schematic plan and (b) isometric view [1]
The system works quite efficiently as a hollow vertical cantilever. However, lateral drift due to the axial displacement of the columns—commonly referred to as chord drift—and web drift, caused by shear and bending deformations of the spandrels and columns, may be quite large depending upon the tube geometry.
In framed tube systems, the “strong” bending direction of the columns is typically aligned along the face of the building, in contrast to a typical transverse rigid frame where it is aligned perpendicular to the face. The frames parallel to the lateral load act as webs of the perforated tube, while the frames normal to the load act as the flanges.
When subjected to bending, the columns on opposite sides of the neutral axis of the tube are subjected to tensile and compressive forces. In addition, the frames parallel to the direction of the lateral load are subjected to the in-plane bending and the shearing forces associated with an independent rigid frame action.
Trussed Tube Systems
A trussed tube system improves the efficiency of the framed tube by increasing its potential for use in taller buildings and allowing greater spacing between the columns. This is achieved by adding diagonal bracing at the faces of the tube as shown in Figure 9 to virtually eliminate the shear lag in both the flange and web frames [1].
Fig. 9: Trussed tube System [1]
The framed tube, as discussed previously, even with its close spacing of columns is somewhat flexible because the high axial stresses in the columns cannot be transferred effectively around the corners. For maximum efficiency, the tube should respond to lateral loads with the purity of a cantilever, with compression and tension forces spread uniformly across the windward and leeward faces.
The framed tube, however, behaves more like a thin-walled tube with openings. The axial forces tend to diminish as they travel around the corners, with the result that the columns in the middle of the windward and leeward faces may not sustain their fair share of compressive and tensile forces. An example of an application is the Onterie Centre building in Chicago (Figure 10).
Fig. 10: Onterie Center Building, Chicago (www.skyscrapercentre.com)
Bundled Tube Systems
The bundled tube structure consists of four parallel rigid frames in each orthogonal direction, interconnected to form nine bundled tubes (see Figure 11). The principle is the same as for the single tube structure where the frames in the horizontal load direction act as webs and the perpendicular frames acts as flanges.
Fig. 11: Bundled tube structure [3]
By introducing the internal webs the shear lag is drastically reduced and as a result the stresses in the columns are more evenly distributed and their contribution to the lateral stiffness is more significant. This allows for the columns to be spaced further apart and to be less striking. In essence, the underlying principle to achieve a bundled tube response is to connect two or more individual tubes into a single bundle. The main purpose is to decrease shear lag effects.
References
[1] Bungale S. T. (2010): Reinforced Concrete Design of Tall Buildings. CRC Press, Taylor and Francis Group [2] Khan, F.R. (1969): Recent structural systems in steel for high-rise buildings. In Proceedings of the British Constructional Steelwork Association Conference on Steel in Architecture. London: British Constructional Steelwork Association. [3] Ali M.M., and Moon K.S. (2007): Structural developments in tall buildings: Current trends and future prospects. Architectural Science Review 50(3):205-223 [4] Khan, F.R. (1972): Influence of design criteria on selection of structural systems for tall buildings, In Proceedings of the Canadian Structural Engineering Conference. Toronto: Canadian Steel Industries Construction Council, 1-15. [5] Khan, F.R. (1973): Evolution of structural systems for high-rise buildings in steel and concrete. In J. Kozak (Ed.), Tall Buildings in the Middle and East Europe: Proceedings of the 10th Regional Conference on Tall Buildings-Planning, Design and Construction. Bratislava: Czechoslovak Scientific and Technical Association [6] Ali, M.M. (2001): Art of the Skyscraper: The Genius of Fazlur Khan. New York: Rizzoli. [7] Reddy S.V.B., and Eadukondalu M. (2018): Study of the lateral structural systems in tall buildings. International Journal of Applied Engineering Research 13(15):11738 – 11754 [8] Zalka K. A. (2013): Structural Analysis of Regular Multi-storey Buildings. CRC Press – Taylor and Francis Group, USA [9] Aginam C.H., Chidolue C.A., and Ubani O.U. (2015): Effect of Planar Solid shear wall-frame arrangement on the deformation behaviour of multi-story frames. IOSR Journal of Mechanical and Civil Engineering 12(1):98-105 [10] Sandelin C. and Bujadev E. (2013): The stabilization of high-rise buildings: An evaluation of the tubed mega frame concept. Dissertation submitted to the Department of Engineering Science, Applied Mechanics, Civil Engineering, Uppsala University [11] Hallebrand E., and Jakobsson W. (2016): Structural design of high-rise buildings. M.Sc thesis presented to the Department of Construction Sciences (Division of structural mechanics), Lund University, Sweden
Masonry is defined as an assemblage of masonry units (blocks, bricks, etc) laid in a specified pattern and joined together with mortar. They are usually used in components subjected to compressive loads such as walls, columns, arches, domes, vaults, etc. On other hand, masonry elements have limited capacity to support horizontal loads and bending moments.
Within the last decades, the efficiency of masonry units has increased due to higher allowable stresses, and refined possibilities in design. This, therefore, calls for more precision in the analysis, construction, and production of masonry to be used as structural members. This brings the design of masonry walls as a task of civil engineers.
Previous researches have shown that the out-of-plane (OOP) two-way bending failure of structural components can be one of the most predominant failure mechanisms in unreinforced masonry (URM) buildings. This is according to a submission from research carried out at the Faculty of Civil Engineering and Geosciences, Delft University of Technology, Netherlands. The study was published in the Elsevier – Structures journal in December 2020.
According to the researchers, extensive analytical formulations have been well developed for one-way vertically spanning walls, while analytical formulations for URM walls in OOP two-way bending require further improvement in accuracy and extension for the application range.
In order to bridge this knowledge gap, the researchers carried out an international testing campaign on a dataset of 46 testing specimens, in order to evaluate current analytical formulations. The current analytical formulations (codes of practice) used for the evaluation of the test specimens are;
two other virtual work formulations related to AS3700 (W2006 and G2019)
The analytical formulations that have been developed in the past decades were incorporated into design standards to assess the wall capacity in engineering practice. The current analytical formulations are mainly based on the yield line method or on the virtual work method.
The underlying assumptions used in the development of the yield line method are:
masonry is simplified as a homogeneous material;
all cracks develop simultaneously;
the force capacity is calculated from the equilibrium between the applied forces and the reaction forces along cracking lines.
One drawback of the yield line method is that some crucial factors such as bonding patterns are neglected since masonry is considered as a homogenous material. This can affect the crack pattern therefore possibly resulting in misevaluation. Another drawback of the yield line method is that all cracks are assumed to develop concurrently, which can lead to inaccuracy for calculating the force capacity since contributions of all cracks are taken into account.
Another category of analytical formulations originates from the virtual work method. The major underlying assumptions of the virtual work method are:
the contributions from horizontal cracks are neglected;
diagonal cracks start right from the wall corners;
the cracking pattern is assumed to follow the mortar joints and is determined by the aspect ratios of the units and of the wall; horizontal and diagonal bending moment capacities are calculated independently;
the virtual work done by external loads is equal to the strain energy along cracking lines in pre-assumed cracking patterns
Figure 1: Classic pre-assumed cracking patterns used in formulations based on the virtual work method [1]
Additionally, the virtual work method provides coefficients and formulas to consider the presence of openings, while the yield line method does not. It is important to note that the afore-mentioned formulations are all force-based methods.
The dataset of the study carried out consists of tests performed mostly on clay brick and calcium silicate (CS) brick masonry walls. 37 out of 46 testing specimens were subjected to quasi-static cyclic loading, while the others to dynamic loading.
Figure 2: The testing configuration of the testing specimens [1]
Eurocode 6 evaluates the force capacity w of a wall by following equation;
w = [(fx1 + σd)Z]/μα2Lw2 —— (1)
where the flexural strength ratio μ is defined as:
μ = (fx1 + σd)/fx2 —— (2)
Where; fx1 and fx2 are the flexural strength of masonry obtained for planes of failure parallel to and perpendicular to the bed joints, respectively; σd is the vertical compressive stress at a specific height of the wall caused by self-weight and pre-compression σ; Z is the section modulus of the wall; α2 is the bending moment coefficient
The formulations AS3700, W2006 and G2019 assess the force capacity w of a wall by the following equation:
w = [2αf /Ld2] (k1Mh + k2Md) —— (3)
with the components of Eq. (3):
G = 2(hu + tj)/(lu + tj) —— (4) α = GLd/Hd ——(5)
where; Hd and Ld are the design height and design length of the wall, respectively G is the assumed slope of the crack line; α is the slope factor that identifies the expected cracking pattern including a vertical central crack in the case α < 1, or a horizontal central crack in the case α ≥ 1; αf, k1 and k2 are coefficients determined by the presence of the openings, the slope factor α and the number of supported vertical edges Mh and Md are the horizontal and the diagonal bending moment capacity of masonry, respectively.
The equations for Md and Mh are provided in the research article accordingly.
To compare the accuracy of the formulations, the tested force capacity from the dataset was predicted according to the equations of the analytical formulations. Lower and upper bounds for each testing specimen were calculated. The lower bound of the force capacity was estimated by considering the wall hinged on all sides in EC6 or assuming Rf = 0 in the other formulations; the upper bound of the force capacity was estimated by considering the wall clamped on all sides in EC6 or assuming Rf = 1 for the other formulations.
The comparison showed that EC6 has an incorrect prediction rate of 58.7% with the highest overestimation rate of 47.8%. W2006 and G2019 have incorrect prediction rates of 71.7% and 65.2%, respectively. Both these two formulations tend to underestimate the force capacity. AS3700 provides the lowest incorrect prediction rate of 56.6%. Also, the incorrect prediction rate on walls without openings of AS3700 is the lowest.
Table 1: Percentage of incorrect predictions for the considered dataset [1]
Nevertheless, the accuracy of AS3700 requires further improvement considering 21.8% and 34.8% of testing specimens are overestimated and underestimated, respectively. The formulations based on the virtual work method provide close incorrect prediction rates for walls with and without openings.
In conclusion, the formulations based on the virtual work method returned the most accurate predictions for the testing specimens evaluated in the study, especially for partially clamped walls and walls with openings. Nevertheless, drawbacks and limitations were revealed when analytical formulations were applied to assess the influence of crucial factors on the force capacity such as precompression of the wall, bonding pattern, boundary conditions, material properties, area of openings, and eccentricity of load.
To improve the accuracy and application range of the analytical formulations, further study is suggested regarding the influence of above-mentioned crucial factors on the wall failure mechanisms and quantifying the relations between the force capacity and the crucial factors.
Disclaimer The contents of this article are culled from [Lang-Zi Chang, Francesco Messali, Rita Esposito (2020): Capacity of unreinforced masonry walls in out-of-plane two-way bending: A review of analytical formulations. Structures 28 (2020) 2341-2477] and does not belong to www.structville.com. It has been presented here in accordance to the requirements of open access articles under the CC BY license (http://creativecommons.org/licenses/by/4.0/)
References [1] Lang-Zi Chang, Francesco Messali, Rita Esposito (2020): Capacity of unreinforced masonry walls in out-of-plane two-way bending: A review of analytical formulations. Structures 28 (2020) 2341-2477 https://doi.org/10.1016/j.istruc.2020.10.060
Deep foundations are employed when the soil stratum beneath the structure is not capable of supporting the load with a tolerable settlement or adequate safety against shear failure. The two common types of deep foundations are well foundations (or caissons) and pile foundations. Piles are relatively long, slender members that are driven into the ground or cast-in-situ. The design of pile foundation involves providing adequate pile type, size, depth, and number to support the superstructure load without excessive settlement and bearing capacity failure. Deep foundations are more expensive and technical than shallow foundations.
Pile Foundations can be used in the following cases;
When the upper soil layer(s) is (are) highly compressible and too weak to support the load transmitted by the superstructure, piles are used to transmit the load to underlying bedrock or a stronger soil layer. When bedrock is not encountered at a reasonable depth below the ground surface, piles are used to transmit the structural load to the soil gradually. The resistance to the applied structural load is derived mainly from the frictional resistance developed at the soil–pile interface.
When subjected to horizontal forces, pile foundations resist by bending while still supporting the vertical load transmitted by the superstructure. This situation is generally encountered in the design and construction of earth-retaining structures and foundations of tall structures that are subjected to strong wind and/or earthquake forces.
In many cases, the soils at the site of a proposed structure may be expansive and collapsible. These soils may extend to a great depth below the ground surface. Expansive soils swell and shrink as the moisture content increases and decreases, and the swelling pressure of such soils can be considerable. If shallow foundations are used, the structure may suffer considerable damage.
The foundations of some structures, such as transmission towers, offshore platforms, and basement mats below the water table, are subjected to uplifting forces. Piles are sometimes used for these foundations to resist the uplifting force.
Bridge abutments and piers are usually constructed over pile foundations to avoid the possible loss of bearing capacity that a shallow foundation might suffer because of soil erosion at the ground surface
Figure 1: Schematic representation of pile foundation
Classification of Piles
Piles may be classified in a number of ways based on different criteria:
(a) Function or action (b) Composition and material (c) Method of installation
Classification Based on Function or Action
Piles may be classified as follows based on the function or action:
End-bearing piles Used to transfer load through the pile tip to a suitable bearing stratum, passing soft soil or water.
Friction piles Used to transfer loads to a depth in a frictional material by means of skin friction along the surface area of the pile.
Tension or uplift piles Uplift piles are used to anchor structures subjected to uplift due to hydrostatic pressure or to overturning moment due to horizontal forces.
Compaction piles Compaction piles are used to compact loose granular soils in order to increase the bearing capacity. Since they are not required to carry any load, the material may not be required to be strong; in fact, sand may be used to form the pile. The pile tube, driven to compact the soil, is gradually taken out and sand is filled in its place thus forming a ‘sand pile’.
Anchor piles These piles are used to provide anchorage against horizontal pull from sheetpiling or water.
Fender piles They are used to protect water-front structures against impact from ships or other floating objects.
Sheet piles Sheet piles are commonly used as bulkheads, or cut-offs to reduce seepage and uplift in hydraulic structures.
Batter piles Used to resist horizontal and inclined forces, especially in water front structures.
Laterally-loaded piles Used to support retaining walls, bridges, dams, and wharves and as fenders for harbour construction.
Classification Based on Material and Composition
Piles may be classified as follows based on material and composition:
Timber piles These are made of timber of sound quality. Length may be up to about 8 m; splicing is adopted for greater lengths. Diameter may be from 30 to 40 cm. Timber piles perform well either in fully dry condition or submerged condition. Alternate wet and dry conditions can reduce the life of a timber pile; to overcome this, creosoting is adopted. Maximum design load is about 250 kN.
Steel piles These are usually H-piles (rolled H-shape), pipe piles, or sheet piles (rolled sections of regular shapes). They may carry loads up to 1000 kN or more.
Figure 2: Steel H-section piles
Concrete piles These may be ‘precast’ or ‘cast-in-situ’. Precast piles are reinforced to withstand handling stresses. They require space for casting and storage, more time to cure and heavy equipment for handling and driving. Cast-in-situ piles are installed by pre-excavation, thus eliminating vibration due to driving and handling.
Figure 3: Precast concrete piles
Composite piles These may be made of either concrete and timber or concrete and steel. These are considered suitable when the upper part of the pile is to project above the water table. The lower portion may be of untreated timber and the upper portion of concrete. Otherwise, the lower portion may be of steel and the upper one of concrete.
Classification Based on Method of Installation
Piles may also be classified as follows based on the method of installation:
Driven piles Timber, steel, or precast concrete piles may be driven into position either vertically or at an inclination. If inclined they are termed ‘batter’ or ‘raking’ piles. Pile hammers and pile-driving equipment are used for driving piles.
Cast-in-situ piles Only concrete piles can be cast-in-situ. Holes are drilled and these are filled with concrete. These may be straight-bored piles or may be ‘under-reamed’ with one or more bulbs at intervals. Reinforcements may be used according to the requirements.
Driven and cast-in-situ piles This is a combination of both types. Casing or shell may be used. The Franki pile falls in this category.
However, the commonest type of pile foundation in Nigeria is bored piles using continuous flight auger (CFA).
Design of Pile Foundation
Section 7 of EN 1997-1:2004 is dedicated to the geotechnical design of pile foundations. There are some design standards that are dedicated to the design and construction of pile foundations. A design standard that is referred to is the part of Eurocode 3 for the structural design of steel piles:
EN 1993-5: Eurocode 3, Part 5: Design of Steel Structures – Piling
Other standards that can be referred to for the execution of piling work are;
EN 1536:1999 – Bored Piles
EN 12063:1999 – Sheet pile walls
EN 12699:2000 – Displacement piles
EN 14199:2005 – Micropiles
Approaches to the design of pile foundations
According to clause 7.4(1)P of EN 1997-1, the design of piles shall be based on one of the following approaches:
The results of static load tests, which have been demonstrated, by means of calculations or otherwise, to be consistent with other relevant experience
Empirical or analytical calculation methods whose validity has been demonstrated by static load tests in comparable situations
The results of dynamic load tests whose validity has been demonstrated by static load tests in comparable situations
The observed performance of a comparable pile foundation provided that this approach is supported by the results of site investigation and ground testing.
Static load test is the best way of verifying the load-carrying capacity of piles, however, it is not very attractive because it is expensive and time-consuming. Traditionally, engineers have designed pile foundations based on calculations from theoretical soil mechanics. The commonest approach is to divide the soil into layers and assign soil properties to each layer. The most important soil parameters given to each layer is cohesion (C) and angle internal friction (ϕ). These two properties will enable the quick determination of the bearing capacity factors for evaluation of the load-carrying capacity of the pile.
From the soil profile, the shaft friction on the pile from different layers is summed up to obtain the total shaft friction resistance of the pile. The base resistance of the pile is also obtained based on the soil properties of the layer receiving the tip of the pile.
Where qs is the unit shaft resistance of the pile and As is the surface area of the pile for which qs is applicable. Ab is the cross-sectional area of the base of the pile while qb is the base resistance.
For pile in cohesionless soil (C = 0) Qs = q0KstanδAs —— (2)
For pile in cohesive soil (ϕ = 0) Qs = αCuAs —— (3)
Where; q0 is the average effective overburden pressure over the embedded depth of the pile for which Kstanδ is applicable. Ks is the lateral earth pressure coefficient δ is the angle of wall friction Cu is the average undrained shear strength of clay along the shaft α is the adhesion factor.
Typical values of δ and Ks are given in the table below;
On the other hand, the typical equations for obtaining the base resistance of a single pile are given below;
Qb = Base resistance = qbAb Where qb is the unit base resistance of the pile and Ab is the area of the pile base.
For pile in cohesionless soil (C = 0) Qb = q0NqAb —— (4)
For pile in cohesive soil (ϕ = 0) Qb = cbNcAb —— (5)
For pile in c-ϕ soil; Qb = (cbNc + q0Nq)Ab —— (6)
Where Nq and Nc are bearing capacity factors.
Therefore for a design to be considered acceptable, the applied load ≤ Ultimate Capacity/Factor of Safety. The factor of safety usually varies between 2.0 and 3.0 and depends on the quality of ground investigation carried out.
Pile Foundation Design to Eurocode 7
EN 1997-1:2004 allows the resistance of individual piles to be determined from;
static pile formulae based on ground parameters
direct formulae based on the results of field test
the results of static pile load test
the results of dynamic impact tests
pile driving formulae, and
wave equation analysis
According to clause 7.6.2.1 (1)P, to demonstrate that the pile foundation will support the design load with adequate safety against compressive failure, the following inequality shall be satisfied for all ultimate limit state load cases and load combinations:
Fc,d ≤ Rc,d —— (7)
Where Fc,d is the design axial load on the pile, while Rc,d is the compressive resistance of the pile. Fc,d should include the weight of the pile itself, and Rc,d should include the overburden pressure of the soil at the foundation base. However, these two items may be disregarded if they cancel approximately. They need not cancel if the downdrag is significant, or when the soil is very light, or when the pile extends above the ground surface.
For piles in group, the design resistance shall be taken as the lesser of the compressive resistance of the piles acting individually, and the compressive resistance of the piles acting as a group (block capacity). According to clause 7.6.2.1(4), the compressive resistance of the pile group acting as a block may be calculated by treating the block as a single pile of large diameter.
Static pile formulae based on ground parameters
Methods for assessing the compressive resistance of a pile foundation from ground test results shall have been established from pile load tests and from comparable experience. Generally, the compressive resistance of the pile shall be derived from;
Rc,d = Rb,d + Rs,d —— (8)
Where; Rb,d = Rb,k/γb Rs,d = Rs,k/γs
The values of the partial factors may be set by the National annex. The recommended values for persistent and transient situations are given in Table A6, A7, and A8 of EN 1997-1:2004 for driven, bored, and CFA piles respectively;
where ξ3 and ξ4 are correlation factors that depend on the number of profiles of tests, n. The values of the correlation factors may be set by the National annex. The recommended values are given in Table A10 of EN 1997-1:2004. For structures with sufficient stiffness and strength to transfer loads from “weak” to “strong” piles, the factors ξ3 and ξ4 may be divided by 1.1, provided that is never less than 1.0.
The characteristic values may be obtained by calculating: Rb,k = Ab qb,k —— (11) Rs,k = ∑As,i qs,i,k —— (12)
where qb,k and qs,i,k are the characteristic values of base resistance and shaft friction in the various strata, obtained from values of ground parameters.
To estimate pile shaft friction and end bearing from ground parameters, the following relationships may be applied;
Cohesive soil or weak rock (mudstone) qs,k = αCu —— (15) qb,k = CuNc —— (16)
Adhesion factor (α) can be read from chart, or determined from the unconfined compression test result (UCS). For piles in clay, Nc is usually taken as 9.0.
Figure 5: Relationship between adhesion factor and undrained conhesion of soil
It is usually recommended that Cu < 40 kPa, α should be taken as 1.0.
Figure 5: Relationship between adhesion factor and unconfined compressive strength of soil
Design of pile foundation using static pile load test
The procedure for determining the compressive resistance of a pile from static load tests is based on analysing the compressive resistance, Rc,m, values measured in static load tests on one or several trial piles. The trial piles must be of the same type as the piles of the foundation, and must be founded in the same stratum.
An important requirement stated in Eurocode 7 is that the interpretation of the results of the pile load tests must take into account the variability of the ground over the site and the variability due to deviation from the normal method of pile installation. In other words, there must be a careful examination of the results of the ground investigation and of the pile load test results. The results of the pile load tests might lead, for example, to different ‘homogeneous’ parts of the site being identified, each with its own particular characteristic pile compressive resistance.
To use static load test result to design pile foundation, determine the characteristic value Rc,k from the measured ground resistance Rc,m using the following equation:
Rc,k = Min{(Rc,m)mean/ξ1; (Rc,m)min/ξ2} —— (17)
where ξ1 and ξ2 are correlation factors related to the number n of piles tested, and are applied to the mean (Rc,m)mean and to the lowest (Rc,m)min of Rc,m, respectively. The recommended values for these correlation factors, given in Annex A, are intended primarily to cover the variability of the ground conditions over the site. However, they may also cover some variability due to the effects of pile installation.
The design pile compressive resistance, Rc,d is obtained by applying the partial factor γt to the total characteristic resistance or the partial factors γs and γb to the characteristic shaft resistance and characteristic base resistance, respectively, in accordance with the following equations:
Rc,d for persistent and transient situations may be obtained from the results of pile load tests using DA-1 and DA-2 and the recommended values for the partial factors γt or γs and γb given in Tables A.6, A.7 and A.8 of EN 1997-1:2004.
Footbridges are susceptible to traffic-induced vibration due to their typical slender nature. Therefore, the serviceability level of pedestrian bridges is influenced by the deformations and vibrations caused by the human traffic on the bridge. In considering the dynamics of footbridges, all types of vibration on the main structure such as vertical and horizontal vibrations, torsional vibrations (which may be alone or coupled with vertical and/or horizontal vibration), should be identified and taken into account.
In studying the dynamics of footbridges, it is important to select and consider the appropriate design situation. This can be influenced by pedestrian traffic admitted on individual footbridges during their design working life, and how access to the bridge will be authorised, regulated, and controlled.
According to [1], the design situations should include:
The simultaneous presence of a group of about 8 to 15 persons walking normally as a persistent design situation;
The simultaneous presence of streams of pedestrians (significantly more than 15 persons), which could be persistent, transient or accidental depending on boundary conditions, like location of the footbridge in urban or rural areas, the possibility of crowding due to the vicinity of railway and bus stations, schools, important building with public admittance, the relevance of the footbridge itself;
Occasional sports, festive or choreographic events, which require specific studies.
Periodic forces exert by a pedestrian normally walking are;
vertical, with a frequency ranging between 1 and 3 Hz, and
horizontal, with a frequency ranging between 0.5 and 1.5 Hz, perfectly synchronised with the vertical ones.
The forces exerted by several persons walking on a footbridge are usually not synchronised and characterised by different frequencies. When the frequency of the forces normally exerted by pedestrians is close to a natural frequency of the deck, it commonly happens that the subjective perception of the bridge oscillation induces the pedestrian to synchronise their steps with the vibrations of the bridge, so that resonance occurs, increasing considerably the response of the bridge.
It is important to note that the number of persons contributing to the resonance of a footbridge is highly random; beyond about 10 persons on the bridge, it is a decreasing function of their number. The resonance is in most cases mainly, but not solely, marked with the fundamental frequency of the bridge. Correlation between forces exerted by pedestrians may increase with movements.
Dynamic models of pedestrian loads
Two separate dynamic models of pedestrian loads for the design of footbridges could be defined:
a concentrated force (Fn), representing the excitation by a limited group of pedestrians, which should be systematically used for the verification of the comfort criteria;
a uniformly distributed load (Fs), representing the excitation by a continuous stream of pedestrians, which should be used also where specified, separately from Fn.
Load model Fn, which should be placed in the most adverse position on the bridge deck, consists of one pulsating force (N) with a vertical component;
Fnv = 280kv(fv) sin(2πfvt) —— (1)
and a horizontal component;
Fnh = 70kh(fh) sin(2πfht) —— (2)
where; fv is the natural vertical frequency of the bridge closest to 2 Hz, fh is the natural horizontal frequency of the bridge closest to 1 Hz, t is the time in seconds and, kv(fv) and kh(fh) are suitable coefficients, depending on the frequency, given in the figure below;
Figure 1: Relationships between coefficients kv(fv), kh(fh) and frequencies fv, fh [1]
The two components Fn,v and Fn,h should be considered separately.
When inertia effects are evaluated as well as for the calculation of fv or fh, Fn should be associated with a static mass equal to 800 kg (if unfavourable), applied at the same location. The uniformly distributed load model Fs, to be applied on the whole deck of the bridge, consists of a uniformly distributed pulsating load (N/m2) with a vertical component;
Fsv = 15kv(fv) sin(2πfvt) —— (3)
and a horizontal component;
Fsh = 4kh(fh) sin(2πfht) —— (4)
to be considered separately.
When inertia effects are evaluated as well as for the calculation of fv or fh, Fs should be associated with a static mass equal to 400 kg/m2 (if unfavourable), applied at the same location.
For a particular project, especially for big footbridges, it may be possible to increase the reliability degree of the assessments, by specifying to apply Fs on limited unfavourable areas (e.g. span by span) or with an opposition of phases on successive spans.
Human Comfort Criteria on Footbridges
In order to ensure pedestrian comfort on footbridges, the maximum acceleration of any part of the deck should not exceed;
0.7 (m/s2) for vertical vibrations; or
0.15 (m/s2) for horizontal vibrations.
The assessment of comfort criteria should be performed when the natural vertical frequency is less than 5 Hz or the horizontal and torsional natural frequencies are less than 2.5 Hz. The assessment of natural frequencies fv or fh should take into account the mass of any permanent load. The mass of pedestrians should be taken into account only for very light decks.
The stiffness parameters of the deck should be based on the short term dynamic elastic properties of the structural material and, if significant, of the parapets. When comfort criteria do not seem to be satisfied with a significant margin, it is recommended to make provision in the design for the possible installation of dampers in the structure after its completion.
Evaluation of accelerations shall take into account the damping of the footbridge, through the damping factor ζ referring to the critical damping, or the logarithmic decrement δ, which is equal to 2πζ.
For rather short spans, when calculations are performed using the groups of pedestrians given before, the acceleration can be reduced by multiplying it by;
knv = 1 – e(-2πnζ) for vertical vibrations or by
knh = 1 – e(-πnζ) for horizontal vibrations, being
n, equal to 12 times the number of steps necessary to cross the span under consideration. For a simple span, the design value of the vertical acceleration (m/s2) due to the group of pedestrians may then be taken as equal to;
ad = 165kv(fv) × (1 – e(-2πnζ))/Mζ —— (5)
M is the total mass of the span, f is the relevant, i.e. the determining, fundamental frequency, and kv(fv) is given in Figure 1.
References [1] Pietro Croce P., Sanpaolesi L. (2005): Bridges – Actions and Load Combinations. In Handbook 4 Design of Bridges (Guide to basis of bridge design related to Eurocodes supplemented by practical examples) Leonardo da Vinci Pilot Project CZ/02/B/F/PP-134007
