Retaining walls are structures used for supporting earth materials at different levels. There are different types of retaining walls such as cantilever retaining walls, gravity retaining walls, counterfort retaining walls, buttressed retaining walls, etc. Counterfort retaining walls have similarities with cantilever retaining walls, with the difference of having triangular or rectangular web panels spaced at regular intervals at the back of the retaining wall.
These web panels are called counterforts, and they serve the purpose of tying the base slab and the wall (stem) together. By so doing, they reduce the internal stresses induced in the structure and increase the weight of the structure for stability. The main characteristic of a counterfort retaining wall is the inclusion of the counterforts.
These counterforts are positioned at regular intervals along the length of the wall, extending from the base to the top. They act as braces or buttresses, helping to distribute the lateral forces exerted by the retained soil.
The counterforts are connected to the main wall, known as the stem, by horizontal slabs or beams called tie beams. These tie beams create a robust structure, distributing the forces evenly and increasing the overall stability of the wall. The toe of the wall is typically thicker and wider than the stem, providing additional resistance against overturning and sliding.
The design of a counterfort retaining wall takes into account factors such as soil properties, anticipated loads, and water pressure. It must be designed to withstand the lateral pressure exerted by the retained soil and any potential surcharges, such as additional loads from adjacent structures or traffic.
One advantage of counterfort retaining walls is their ability to span longer heights compared to other types of retaining walls. The presence of counterforts and tie beams enhances the structural integrity, allowing for the construction of taller walls. This makes them suitable for applications where a high retaining wall is required, such as highway embankments, bridge abutments, and building foundations.
Another advantage is the ease of construction. Counterfort retaining walls can be built using precast concrete elements or cast-in-place methods, depending on the project requirements. The use of precast elements can expedite the construction process and reduce costs.
Counterfort retaining wall
Modelling and Design of Counterfort Retaining Walls
Counterfort retaining walls can be easily modelled on Staad Pro software, and loaded to obtain the internal forces and deformations due to the retained earth. We are going to demonstrate that using the video below.
The data on the retaining wall is shown below;
Loads on counterfort retaining wall
Design Data Height of wall from base = 7 m c/c Length of base = 4.5m Projection of toe = 0.8 m c/c Projection of heel = 3.7 m c/c Thickness of stem wall = 0.3 m Thickness of base = 0.5 m Thickness of counterfort = 0.3 m Spacing of counterfort = 2.5 m c/c Unit weight of concrete = 25 kN/m3 Unit weight of retained earth = 19 kN/m3 Angle of internal friction φ = 30° Surcharge pressure on retaining wall = 10 kN/m2 Modulus of subgrade reaction of supporting soil = 50000 kN/m2/m
We are going to neglect the effect of passive earth pressure on the retaining wall.
Model of counterfort retaining wall on Staad Pro
Coefficient of active earth pressure Ka = (1 – sinφ)/(1 + sinφ) = 0.333 Earth pressure at the back of the wall (triangularly distributed) = 0.333 x 19 kN/m3 x 7m = 44.289 kN/m2 Surcharge pressure at the back of the retaining wall = 0.333 x 10 = 3.33 kN/m2
Earth pressure on the base (heel) = (7m x 19 kN/m3) = 133 kN/m2 Surcharge pressure on the base (heel) = 10 kN/m2
Watch the video for the analysis of counterfort retaining walls on Staad Pro below;
Many engineering vibration problems can be idealised as single degree of freedom systems using mass-spring-dashpot model. In civil engineering, some water tank models and structures can be idealised this way for dynamic analysis such as the water tank shown above.
The ‘dashpot’ is the simplest mathematical element to simulate a viscous damper. The force in the dashpot under dynamic loading is directly proportional to the velocity of the oscillating mass.
Mathematical model for free vibrating system with damping
For such structures under free vibration, the equation of motion is;
M.(d2z/dt2) + c(dz/dt) + kz = 0 ——— (1)
Where M is the mass of the vibrating system c is the coefficient of viscous damping expressed in force per unit velocity k is the stiffness of the system z is the displacement
There are three different type of solutions that can be obtained from equation (1); roots are real and negative, roots are equal, and roots are complex. The solution obtained can be used to describe the nature of damping of the system such as overdamped, underdamped, critically damped etc. For more information consult standard dynamics of structures textbook.
Solved Example For the SDOF system shown below, plot the displacement time history analysis of the system for the initial conditions; z = 0.1m, dz/dt = 0, at t = 0
The equation of motion of the system can therefore be given by;
d2z/dt2 + 40(dz/dt) + 10000z = 0
x2 + 40x + 10000 = 0
The solution to the above equation has complex roots given by; x = -20 ± 97.979i
The general solution to the equation is; z = e-20t(A cos97.979t + B sin97.979t) —– (2)
From the initial conditions; z(0) = 0.1 m 0.1 = e-20(0)[A cos97.979(0) + B sin97.979(0)] 0.1 = 1(A + 0) Therefore A = 0.1
Hence; z = e-20t(0.1cos97.979t + B sin97.979t) —- (2a)
Differentiating the equation (2a) using product rule;
For the first term of equation (2a); u = 0.1e-20t; du/dt = -2e-20t
v = cos(97.979t); dv/dt = -97.979 sin(97.979t)
For the second term of equation (2a); u = Be-20t; du/dt = -20Be-20t
In the design of reinforced concrete structures, floor loads are usually transferred from slabs to beams, and from the beams, the loads are transferred to the columns. Ultimately, the columns transfer the superstructure load to the foundation supporting the structure. Load transfer from slab to beams is one of the most intriguing aspects of reinforced concrete design, especially for beginners.
Usually, slab pressure loads (force per unit area) are transferred to the supporting beams as line loads (force per unit length). The line load can be triangular, trapezoidal, or partially distributed on the beam. Depending on the analytical method employed in the design, some idealisations can be made in order to simulate load transfer from slab to beam. The most popular methods of transferring slab load to beams are;
Finite element analysis
Yield line method
Approximate method using formula
Finite element analysis is suited more to computer calculation since it can be a very lengthy process when done by hand. In this method, the slab is divided into finite element meshes connected by nodes. The reactive forces on each node along the beam are transferred to the beams (which must be broken into finite elements too with nodes connected to the slab).
In the yield line method, the most appropriate yield lines are constructed (usually at 45° angles) on the slab, and the corresponding load on each part of the yield line transferred to the beam adjacent to it. For two-way slabs, this method usually leads to trapezoidal and triangular loads on the beams.
In the manual design of structures, some formulas can be used to idealise slab loads on beams as uniformly distributed loads. The main reason for this is to simplify manual analysis since it is not a very accurate method. The results obtained from the method are usually very conservative.
Some of the formulas can be obtained from Reynolds and Steedman (2005) for transfer of load from two-way slab to beams. The formulas are presented below;
Two-way slab (ly/lx < 2) Long span: p = nlx/2(1 – 1/3k2) Short span: p = nlx/3
One-way slab (ly/lx > 2) Long span: p = nlx/2 Short span: p = nlx/5
Where; n = load from slab ly = length of long side of the slab lx = length of short side pf the slab k = aspect ratio = ly/lx
In this article, we are going to review load transfer from slab to beams using the three approaches;
(1) Full finite element analysis of beams and slabs using Staad Pro (2) Yield line method of load transfer using Staad Pro (3) Manual method using formula
CASE 1: Two way slab of dimensions (5m x 6m) simply supported by beams on all sides and subjected to a pressure load of 10 kN/m2
(a) Finite element analysis
Long span beam: Maximum span moment = 73.063 kNm Support moment = -2.71 kNm End shear = 37.6 kN
Short span beam: Maximum span moment = 54.495 kNm Support moment =-0.814 kNm End shear = 31.9 kN
(b) Yield line method
Long span beam: Maximum span moment = 76.562 kNm Support moment = -9.897 kNm End shear = 39.4 kN
Short span beam: Maximum span moment = 46.987 kNm Support moment =-5.096 kNm End shear = 30.151 kN
(c) Manual analysis using formula k = ly/lx = 6/5 = 1.2 Load on long span beam = nlx/2(1 – 1/3k2) = [(10 x 5)/2] x [1 – 1/(3 x 1.22)] = 19.212 kN/m Maximum span moment = ql2/8 = (19.212 x 62)/8 = 86.454 kNm End shear = ql/2 = (19.212 x 6)/2 = 57.636 kN
Load on the short span beam = nlx/3 = (10 x 5)/3 = 16.667 kN/m Maximum span moment = ql2/8 = (16.667 x 52)/8 = 52.084 kNm End shear = ql/2 = (16.667 x 5)/2 = 41.6675 kN
Summary Table for Two-Way Slab
Analytical Method
Ly – Span Moment (kNm)
Ly – Support Moment (kNm)
Ly – End shear (kN)
Lx – Span Moment (kNm)
Lx – Support Moment (kNm)
Lx – End shear (kN)
Finite Element Analysis
73.063
2.71
37.6
54.495
0.814
31.9
Yield line method
76.562
9.897
39.4
46.987
5.096
30.151
Formula
86.454
0.00
57.636
52.084
0.00
41.66
CASE 2: One-way slab of dimensions (2.5 m x 7 m) simply supported by beams on all sides and subjected to a pressure load of 10 kN/m2
k = ly/lx = 7/2.5 = 2.8
(a) Finite Element Analysis
Long span beam: Maximum span moment = 60.689 kNm Support moment = -6.337 kNm End shear = 29.7 kN
Short span beam: Maximum span moment = 12.091 kNm Support moment = +2.81 kNm End shear = 11.6 kN
(b) Yield line method
Long span beam: Maximum span moment = 63.4 kNm Support moment = -9.9 kNm End shear = 35.9 kN
Short span beam: Maximum span moment = 6.16 kNm Support moment = -0.346 kNm End shear = 7.81 kN
(c) Manual analysis using formula Load on long span beam = nlx/2 = (10 x 2.5)/2 = 12.5 kN/m Maximum span moment = ql2/8 = (12.5 x 72)/8 = 76.56 kNm End shear = ql/2 = (12.5 x 7)/2 = 43.75 kN
Load on the short span beam = nlx/5 = (10 x 2.5)/5 = 5 kN/m Maximum span moment = ql2/8 = (5 x 2.52)/8 = 3.906 kNm End shear = ql/2 = (5 x 2.5)/2 = 6.25 kN
Summary Table for One-Way Slab
Analytical Method
Ly – Span Moment (kNm)
Ly – Support Moment (kNm)
Ly – End shear (kN)
Lx – Span Moment (kNm)
Lx – Support Moment (kNm)
Lx – End shear (kN)
Finite Element Analysis
60.689
6.337
29.7
12.091
2.81
11.6
Yield line method
63.4
9.9
35.9
6.16
0.346
7.81
Formula
76.56
0.00
43.75
3.906
0.00
6.25
Discussion of results
(a) Two-way slab systems (1) In the long span direction, finite element analysis and yield line method gave very close results for bending moment and shear forces. Manual analysis overestimated the load transferred. (2) In the short span direction, the yield line method underestimated the load transferred to the short span beams when compared with finite element analysis. The formula method gave results that are close to finite element analysis. (3) Manual analysis using formula gave bending moment values that can be used for design purposes but generally overestimated the shear forces. In the long span direction, the sagging moment can be redistributed to the supports by 10% (reduce the span moments by 10% and take the value taken away as support moment).
(b) One-way slab systems (1) As with two-way slabs, finite element analysis and yield line method gave very close results for bending moment and shear forces in the long span beams. Manual analysis overestimated the load transferred. (2) In the short span direction, the yield line method underestimated the load transferred to the short span beams when compared with finite element analysis. Manual analysis using formula underestimated the load transferred. (3) As with two-way slabs, manual analysis using formula gave bending moment values that can be used for design purposes, but overestimated the shear forces in the long span beams. The shear force and bending moment in the short-span beam were underestimated when the formula method was used. (4) In the long span direction, the sagging moment can be redistributed to the supports by 10% (reduce the span moments by 10% and take the value taken away as support moment) when using formula method.
Conclusion and Recommendation
(1) In a strict technical sense, there is nothing like a one-way action for a slab supported by beams on all the edges. There is always a two-way action even though it is greater in the long span. (2) Formula should not be applied when assessing the shear force induced in beams supporting floor loads. (3) Yield line method of load transfer from slab to beams should be used for manual design of structures, despite the more onerous computational effort.
The cost of building a duplex in Nigeria varies, and it is generally influenced by the size of the building, the price of construction materials, the design specifications, the expertise and machinery required, and the environment/location. Depending on the soil condition of the area, a special foundation may be needed for the building, which will affect the overall cost of the project. For instance, a raft foundation will be more expensive than a pad foundation, while deep foundations such as piles will be more expensive than raft foundation.
The construction cost of a building can also be influenced by the nature of the contract, and it will be in the best interest of the client to hire a professional consultant or project manager who will represent his interest throughout the duration of the project.
Excavation of trench for construction of a duplex by Structville Integrated Services Limited
Intending homeowners must engage professionals during the design stage, in order to get their project right from the scratch. A complete construction drawing in Nigeria should include;
Full sets of architectural drawing
Structural drawings
Electrical drawings, and
Mechanical drawings
The client representative or project manager is expected to advise the client on how to get the drawings approved for construction depending on the jurisdiction. Requirements for approval varies from state to state and from local government to local government.
Construction of foundation The construction of the substructure of a building (foundation) is very critical because any mistake in a foundation is very difficult and expensive to correct. A poorly constructed foundation can compromise the integrity of the entire building. Foundation construction has very little to do with the specifications of the architect, but he has to inspect the setting out of the building to ensure that his design has been followed. The major costs and activities involved in the foundation of a building such as residential duplexes are basically functions of engineering design.
Setting out of column starter bars in a substructure
A good design will minimise cost, identify possible challenges in the construction of the building, guarantee the integrity and structural stability of the building, and subsequently lead to fewer difficulties during construction. Now that you are here, it is important that you play your part as an intending homeowner and engage registered professionals in your projects. This can help stop the problem of building collapse in Nigeria.
In this article, let us briefly review the cost of constructing a simple duplex from the foundation to the DPC (ground floor slab). The building is to be constructed in a semi-urban area in South-Eastern Nigeria. As stated earlier, the cost is dependent on the drawing provided by the structural engineer and not by guesswork. The actual price of materials in the locality, delivery to site, and labour will also influence the cost. Therefore, the cost provided in this article may not reflect the cost of materials in your locality.
The plan of the building is shown below;
Foundation layout of a duplex
From the foundation layout, it can be seen that the structural engineer provided three types of pad foundations (BT1, BT2 and BT3). The size of any foundation is determined by the strength of the soil, and the load coming from the column. The details of the pad bases are given below;
Base Type 1 and Base Type 2 structural details
Base Type 3 structural details
The activities that will take place in the construction of the foundation are;
(1) Setting out works (2) Excavation works (3) Reinforcement works (4) Formwork (5) Concrete works (6) Blockwork (7) Backfilling and compaction (8) Casting of ground floor slab
(1) Setting out Width of building = 12.275 m Length of building = 15.7 m
If we make a setback of 1.2 m from all sides of the building line for the profile board, the total perimeter of the profile board will be 65.55 m. At 1.5m spacing, we will need 45 pegs, and 20 pieces of 2″ x 3″ softwood. Let us assume that the equipment needed for setting out is available except lines.
(a) 2” x 3” soft wood – 25 pcs @ ₦400 = ₦12,500 (b) 2” x 2” pegs – (3 bundles @ 20 pieces per bundle) @ ₦1200 = ₦3,600 (c) Nails – 1 bag of 2 inches nail, and 1 bag of 3 inches nail = ₦26,000 (d) 6 rolls of lines = ₦2,000
Labour and supervision cost (say) = ₦30,000
Total cost of setting out = ₦74,100
(2) Excavation works (a) Excavation of 19 column bases according to structural drawings to a depth not less than 1200 mm to receive blinding for pad foundation – Total volume = 42.42 m3 19 column bases @ ₦1000 = ₦19,000
(b) Excavation of strip footing 690 mm wide and 950 mm deep to receive mass concrete strip footing – Total volume = 70.13 m3 Labour cost for 30 partitions @ ₦2000 = ₦60,000 Supervision cost (say) = ₦20,000
Sub-total for excavation = ₦99,000
Excavation and column setting out works
(3) Concrete works (a) Provision of 50 mm thick weak concrete blinding (1:3:6) on column bases to receive footing reinforcement – Total volume = 1.8 m3 Cement – 8 bags @ ₦4,100 per bag = ₦32,800 Sand – 1.98 tonnes @ ₦3500 per tonne = ₦6,930 Granite – 2.52 tonnes @ ₦16000 per tonne = ₦40,320
(b) Provision of concrete with strength not less than 25 MPa after 28 days for the column bases – Total volume = 10.7 m3 Cement – 65 bags @ ₦4,100 per bag = ₦226,500 Sand – 12 tonnes @ ₦3500 per tonne = ₦42,000 Granite – 15 tonnes @ ₦16000 per tonne = ₦240,000
(c) Provision of concrete with strength not less than 20 MPa after 28 days for the mass concrete strip footing – Total volume = 8.5 m3 Cement – 51 bags @ ₦4,100 per bag = ₦209,100 Sand – 9.35 tonnes @ ₦3,500 per tonne = ₦32,725 Granite – 12 tonnes @ ₦16,000 per tonne = ₦192,000
(d) Casting of Column Stubs (1.5 m3) Cement – 8 bags @ ₦4,100 per bag = ₦32,800 Sand – 1.65 tonnes @ ₦3500 per tonne = ₦5,775 Granite – 2.1 tonnes @ ₦16000 per tonne = ₦33,600
Labour cost for mixing, pouring and consolidation of concrete = ₦159,600 Supervision cost = ₦50,000
Cost of concrete works = ₦1,304,150
(4) Reinforcement Works (a) Column base mat reinforcement 50 lengths of Y12 mm @ ₦3,700 per length = ₦185,000
(b) Column starter bars 20 lengths of Y16 mm @ ₦8,000 per length = ₦160,000
(c) Column links 13 lengths of Y8mm @ ₦2,100 per length = ₦27,300
(d) Binding wire 20 kg roll of binding wire @ ₦14,000 per roll = ₦14,000
Labour cost for cutting, bending, and placement of reinforcement = ₦40,000
Cost of reinforcement works = ₦426,300
(5) Blockwork (a) Total number of 9 inches blocks required to raise the building to DPC = 1600 blocks 1600 pieces of 9” blocks @ ₦350 per block = ₦560,000
Labour cost for laying of blocks = ₦144,000 Cement for mortar = 32 bags @ ₦4100 per bag = ₦131,200 Sand = 10 tonnes @ ₦22,000 = ₦22,000 Supervision = ₦20,000
Cost of blockwork = ₦877,200
Blockwork in substructure
(6) Formwork (a) Provide formwork for sides of columns up to a height not less than 1225 mm. 20 pieces of 1” x 12” x 12 plank @ ₦1,350 per plank = ₦27,000
Labour cost for formwork preparation and placement = ₦15,000
Cost of formwork = ₦42,000
(7) Backfilling and compaction (a) Backfill and compact substructure to a height not less than 550 mm above ground level with selected backfill material. Total volume = 100 m3 33 trips (165 tonnes) of laterite @ ₦12,000 per trip = ₦396,000 Labour cost for filling and compaction = ₦50,000
Total Cost of filling and compaction = ₦446,000
(8) Damp proof membrane (a) Provide and install damp-proof membrane over an area not less than 181 m2 181 m2 of high density polythene sheet @ ₦385 per m2 = ₦69,685
Damp proof membrane = ₦69,685
(9) BRC mesh (a) Provide and Install A142 BRC MESH (TOP) over an area not less than 181 m2 181 m2 of A142 BRC Mesh @ ₦1,200 per m2 = ₦217,200
Labour cost for installation = ₦5,000
Total Cost of BRC mesh = ₦222,200
(10) Ground floor Slab (a) Cast ground floor slab over an area not less than 181 m2 and concrete of volume = 27.15 m3 Cement – 163 bags @ ₦4,100 per bag = ₦668,300 Sand – 30 tonnes @ ₦3500 per tonne = ₦105,000 Granite – 40 tonnes @ ₦16000 per tonne = ₦640,000
Labour cost for mixing, pouring and consolidation of concrete = ₦170,000 Supervision cost = ₦50,000 Casting of ground floor slab = ₦1,633,300
Therefore, the tentative cost of raising the building from foundation to DPC is ₦5,193,935 without the contractor’s profit and overhead.
Completed substructure of a duplex by the author
For design, construction, and professional management of your building project, contact;
Earth materials are extensively utilised for highway construction in Nigeria. It has been recognised that high quality lateritic soils which are abundant in Nigeria can be used as fill, sub-base, and base course materials in highway construction. In this article, we are going to show the recommended standards for highway materials based on Federal Ministry of Works Specifications for Roads and Bridges (1997).
Generally, the material to be used for highway construction shall not be excavated from swamps, marshes or bogs. Furthermore, they shall be free from peat, logs, stumps, roots, and other perishable or combustible materials. Top soils and highly organic clays and silt shall not to be used for constructiont. All clays having liquid limit exceeding 80% or plasticity index exceeding 55 should be rejected.
The basic recommendations given for base course materials are for crushed stone base in clauses 6250, 6251, and 6252. Therefore, the recommendations for sub-base course materials shall be deemed to apply to base course earth materials too. Recommendations for crushed stone base will not be covered in this this article.
List of Tests for selection of highway earth materials The lists of test that shall be conducted for highway materials are;
Materials for sub-base course (Type 1) – Heavy Traffic (1) The percent by weight passing No 75μm sieve shall not be greater than 35% (2) The material passing 425μm sieve shall have a liquid limit not more than 35% and plasticity index of nit more than 12% (3) The material shall have unsoaked CBR value of 80% using Modified AASHTO or West African Standard Compaction and minimum CBR of 30% after 24 hours soaking.
Materials for sub-base course (Type 2) – Light Traffic (1) The percent by weight passing No 75μm sieve shall not be greater than 35% (2) The material passing 425μm sieve shall have a liquid limit not more than 35% and plasticity index of nit more than 12% (3) The material shall have unsoaked CBR value of 80% using Modified AASHTO or West African Standard Compaction and minimum CBR of 20% after 24 hours soaking.
Materials for sub-base course (Type 3) – Substandard materials When the site engineer recognises that suitable materials are not available for use, and the materials slightly fall short of the required standard, the following measures can be adopted;
(1) Compacting the material to a lesser density at the wet side of the optimum to contain the tendency of the material to shrink or swell. (2) Mechanical stabilisation of the material with sand (if available) to reduce the fines content
Growing up as a Nigerian from one of the least developed communities, I have identified that our continent Africa and Nigeria in particular has a problem in infrastructure that is yet to be addressed.
Engineering as a discipline is male dominated both globally and locally. For us to retain the women we have, we need to create more opportunities for them whether as colleagues, wives, sisters or friends.
In the construction industry generally, it is perceived that only men work on site. Everyday at work, I hear a lot of people say things like, “this is my first time of seeing a female engineer on site”. What this literarily means is that we have left the work for men alone, allowing them to decide a significant part of our lives – building homes.
There is need for us to begin to rethink the way we build infrastructure and who builds them. This will begin from design to procurement, construction, finishes, and even commissioning. On the long run, this will help us to recognize talent inclusively, bridge the gender gap, increase the retention of women, and also work towards achieving global goals.
For example, on a construction site, you will discover that most safety wears, boots and signs are designed to be more compatible with men. Some read – ‘Men at Work’. Some are with visual signals that denotes men.
As a starting point to inclusion on construction sites, access to construction areas including walk-ways, stairs, and temporary platforms should include women in the design. Also, initial site planning and management should include restrooms for women as well as men.
While some women are working hard and pushing to be outstanding in the profession whether as technical leaders, engineers and project managers, there are some reasons why many people feel women should not be on site.
Firstly engineering and construction is male oriented, as workers on site are already used to taking instructions from men, which has been a norm for centuries.
Secondly women are believed to have poor leadership skills and as such receive bais from both the society and those above them who ought to be an excellent support to enhance their productivity and performance.
Thirdly women are believed to be too sensitive amidst a few others which are not true. Perhaps on site when they delegate responsibilities and follow up to ensure that it gets done, managers may conclude that those are small things. One way to help is by constructively critising them when necessary, evaluating their performance for the sole purpose of providing useful feedback that could lead to self-improvement.
Overall, we recognize that over 90% of workers in construction are men. There is need for managers, leaders, engineers and decision makers to begin to shape the future through inclusion, shared opportunities, equity and promoting a culture of respect and empathy.
EN 1991-1-7 discussed and gave guidance on accidental actions and their applications in the design calculations for bridge structures. It is obvious that there is a possibility of vehicles impacting on the substructure and/or superstructure of bridges. In order to reduce the negative effects in accidental actions, priority should also be given to reducing the risk of accidental impacts. Some risk reducing measures are;
(1) Preventing the accidental action from occurring or reducing the probability and/or magnitude of the action to a reasonable level. (2) Protecting the structure against accidental actions (for example by using traffic bollards) (3) Designing the structure in a such a way that neither the whole structure nor an important part of it will collapse if local failure should occur. (4) Designing key elements of the structure with special care (5) Applying prescriptive design and detailing rules which will lead to a robust structure.
Fig 1: Typical vehicle collision with bridge pier
Fig 2: Failure of bridge pier due to vehicle collision
Traffic Impact on Bridge Piers Impacts on the substructure of bridges (such as piers) by road vehicles are a relatively frequent occurrence and may have considerable consequences (see Figure 1 and 2). For soft impacts (when the impacting body consumes most of the available kinetic energy), the design values for the horizontal actions due to impact on vertical structural elements (e.g columns, walls) are shown in Table 1.
Table 1: Equivalent horizontal static action for traffic impact of bridge substructures (Vrounwenvelder and Diamantidis, 2010)
The forces Fdx and Fdy denote respectively the forces in the driving direction and perpendicular to it. There is no need to consider them simultaneously. The collision forces are supposed to act at 1.25 m above the level of the driving surface (0.5 m for cars). The force application area may be taken as 0.25 m (height) by 1.50 m (width) or the member width, whichever is the smallest.
Design Example Consider a circular bridge pier with a diameter of 1200 mm. The height of the column is 6 m and is assumed to be hinged to the bridge deck and fixed to the foundation (pile cap) as shown in Figure 3. The main reinforcement consists of 30Y25 (Asprov = 12570 mm2) with a yield strength of 460 Mpa, and concrete strength of 40 Mpa. Let us check the column for truck vehicle collision under motorway conditions.
Fig 3: Truck collision model for a bridge pier
The simple structural idealisation of the system and the analysis result is shown in Figure 4. You can analyse the structure using any means at your disposal.
Fig 4: Internal stresses diagram due to vehicle collision
Note that other loads are not relevant in this case. The self weight of the bridge deck and traffic loads on the bridge will lead to normal force in the pier. Therefore, at the point of impact, the axial force will interact with the bending moment, but for this article, let us ignore the effect of the axial force and confine ourselves to the accidental action only. Note that this accidental action is not supposed to be factored.
Therefore the simplified moment capacity of the section without considering axial force can be obtained from the interaction diagram (d/h = 0.9) given in Figure 5.
Fig 5: Moment-Axial interaction diagram for a circular column (d/h = 0.9)
Design strength of reinforcement fyd = 0.87fyk = 0.87 x 460 = 400.2 MPa Design strength of concrete in compression fcd= αccfck/γc= (0.85 x 40)/1.5 = 22.667 MPa
Asfyd/h2fcd= (12570 x 400.2)/(12002 x 22.667) = 0.154
MEd/h3fcd = (886.5 x 106)/(12003 x 22.667) = 0.0226 At zero axial force (NEd/h2fcd = 0), Asfyd/h2fcd= 0.05 (Figure 4).
Therefore, the area of steel required to resist bending moment due to impact only; As = (0.05 x 12002 x 22.667)/400.2 = 4078 mm2
A little consideration will show that this is less than the area of steel provided (Asprov = 12570 mm2). It is left for the designer to evaluate the axial force coming from the bridge deck to fully verify the adequacy of the reinforcement provided. The shear capacity of the section should also be checked.
References Vrounwenvelder T., and Diamantidis D. (2010): ‘Accidental Actions’ in Guidebook 2 Design of Bridges (Pietro Croce Ed). Faculty of Engineering University of Pisa and Leonardo De Vinci Project.
Vibration (or Oscillation) is a time-dependent, repeated motion which a body undergoes when it is excited in its natural state or by an external force. Frequency is the number of cycles of vibration the system undergoes in a unit of time which is expressed in Hertz (Hz) or (cycles per second). When a structural system is undergoing undamped vibration in its natural state (under self excitation), it is said to be undergoing free vibration. A structure has as many natural frequencies as its degree of freedom, but the frequency with the highest mass participation is often regarded as the natural frequency.
In the video shown above, a ten storey frame of total height of 30 m (each storey height = 3m) was analysed to determine the natural frequency under a floor load of 40 kN/m at each level. All the columns are 400 mm x 400 mm in dimension while the beams are 600 mm x 400 mm. The support conditions were treated as fixed. When analysed using the steps described in the video, the results below were obtained;
The horizontal natural frequency was observed to be 0.587 Hz, with a period of 1.705 seconds. The mass participation factor for this mode vibration was found to be 81.89%. The implication of this is that when carrying out the wind load analysis, the frequency of the wind action should not be close to 0.587 to avoid resonance.
The vertical frequency of the structure was observed to be 5.854 Hz with a period of 0.171 seconds, and mass participation factor of 70.981%. This can be important when evaluating human-structure interaction if the building will be subjected to crowd action.
Here are some textbooks you can purchase from Amazon that will help your understanding of the subject:
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In the design of bridges, environmental actions such as wind, snow, and temperature should also be considered alongside traffic actions. In this article, we are going to show how to apply wind action on bridge decks according to the procedures described in EN 1991-1-4. The specifications in this code apply to bridges of constant cross-section with one or more spans. Different bridge deck sections are permitted such as mono-cell box sections, closed box sections, beam and slab deck systems, etc (see Fig 8.1 of EN 1991-1-4).
In EN 1991-1-4, wind load on bridge decks are considered to be coming from the longitudinal (y-direction) or from transverse (x-direction) axes, and these actions generate stresses in the x,y,z directions of the bridge deck (see Figure 1). During analysis, you can only consider the wind coming in one direction only (either x or y direction) for each load combination.
Fig 1: Wind direction on bridge deck (Fig 8.2 EN 1991-1-4)
Wind forces acting on a bridge deck Wind forces acting in the x-direction of a bridge deck is given by the simplified equation (1);
Fwk= 0.5ρVb2C.Aref,x —– (1)
Where; ρ = density of air = 1.25 kg/m3 Vb = basic wind speed of the site C = Wind load factor for the bridge Aref,x = Reference area
In the absence of traffic, the reference area Aref,x should take into account the total height d of projection on a vertical plane of all beams, including the part of one cornice or footway or ballasted track projecting over the front main girder, plus the sum d1 of solid parapets, noise barriers, windshields, and open safety barriers installed on the bridge. In the case of truss girders, the total height d of the projection on a vertical plane of all truss members should be considered.
In the presence of traffic, the reference area Aref,x should be assumed as the larger between the area evaluated considering the absence of traffic, and the area obtained taking into account the presence f traffic. For road bridges, the lateral surface of vehicles exposed to wind is represented by a rectangular area 2m in height starting from the carriageway level.
The wind load factor C is given by equation (2);
C = cecf,x —–(2)
Where ce is the exposure coefficient for kinetic pressure and cf,x is the force coefficient which is the drag coefficient without free end flow. The exposure coefficient can be evaluated by considering a reference height ze given by the distance from the lowest point of the ground and the center of the bridge deck disregarding additional parts, parapets, barriers, etc.
For bridges with solid parapets and/or solid barriers and/or traffic, the force coefficient cf,x can be determined using equation (3);
cf,x = min {2.4; max[2.5 – 0.3(b/dtot); 1.3]} —- (3)
Where b is the total width of the bridge and dtot is the height considered in the evaluation of Aref,x = dtot.L
From the above considerations, equation (1) can also be given as equation (4);
Fwk= qp(ze)Cf,x.Aref,x —– (4)
Analysis Example Evaluate the wind load on the bridge deck with the profile shown in Figure 2. The bottom of the bridge deck is 7m above the ground (see Figure 3), and it is located in a category III area.
Fig 2: Bridge deck profile
Fig 3: Height of the bridge deck above ground level
We will therefore take our reference height ze = 7.0 + 1.25 = 8.25 m
For the area under consideration, let the basic wind velocity Vb,0 = 40 m/s. Therefore; Vb = Cdir . Cseason . Vb,0 = 1.0 × 1.0 × 30 = 40 m/s
The mean wind velocity Vm(z) at a height z above the terrain depends on the terrain roughness and orography, and on the basic wind velocity, Vb, and should be determined using the expression below;
Vm(z) = cr(z). co(z).Vb
Where; cr(z) is the roughness factor (defined below) co(z) is the orography factor often taken as 1.0
The terrain roughness factor accounts for the variability of the mean wind velocity at the site of the structure due to the height above the ground level and the ground roughness of the terrain upwind of the structure in the wind direction considered. Terrain categories and parameters are shown in Table 1. We will assume that the tank support we are designing is located in an area that can be described as Category III.
Table 1: Terrain Categories and parameters (Table 4.1 EN 1991-1-4:2005)
cr(z) = kr. In (z/z0) for zmin ≤ z ≤ zmax cr(z) = cr.(zmin) for z ≤ zmin
Where: Z0 is the roughness length Kr is the terrain factor depending on the roughness length Z0 calculated using Kr = 0.19 (Z0/Z0,II)0.07
Where: Z0,II = 0.05 m (terrain category II) Zmin is the minimum height Zmax is to be taken as 200 m kr = 0.19 (0.3/0.05)0.07= 0.215
cr(z) = kr. In (z/z0) cr(8.5 m) = 0.215 × In(8.5/0.3) = 0.605
Therefore; Vm(8.5 m) = cr(z). co(z).Vb = 0.605 × 1.0 × 40 = 24.2 m/s
Wind turbulence The turbulence intensity Iv(z) at height z is defined as the standard deviation of the turbulence divided by the mean wind velocity. The recommended rules for the determination of Iv(z) are given in the expressions below;
Iv(z) = σv/Vm = k1/[c0(z).In (z/z0)] for zmin ≤ z ≤ zmax Iv(z) = Iv.(zmin) for z ≤ zmin
Where: k1 is the turbulence factor of which the value is provided in the National Annex but the recommended value is 1.0 Co is the orography factor described above Z0 is the roughness length described above.
From Figure 2, we can verify that dtot = 2.5 m + 1.0 m = 3.5 m b = 10.5 m
cf,x = min {2.4; max[2.5 – 0.3(b/dtot); 1.3]} = min {2.4; max[2.5 – 0.3(10.5/3.5)]; 1.3} = min {2.4; max[1.6; 1.3]} The minimum value of 1.3 recommended is always deemed unsafe sided, therefore take cf,x = 1.6
Fwk= 1.132 x 1.6 x 3.5 = 6.3392 kN/m
Fig 4: Application of wind load on the deck without vehicle
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Concreting or casting days are usually big days for site engineers. It is very typical to see engineers and project managers work very hard to ensure that nothing goes wrong on such days. Thinking ahead is an important skill in construction site management as it gives room for the elimination of all factors that may cause glitches during construction. In this article, we are going to provide some important checklists to help you know whether you are fully ready to cast concrete on site.
A lot of activities precede concrete casting on site such as formwork installation, reinforcement installation and fixing, quality control checks etc. Having gone through these processes to get everything right, the final stage is concreting. It is important to make a checklist in order to ensure that you have done everything properly. Omitting any of these checklists might cause a delay you will not expect on your day of casting.
Here are some important checklists before concreting in a low scale – low-cost construction project:
(1) Formwork The formwork installation must be checked and approved by a third party. Checks should include dimensions and tolerances, bracings, location of props, tightness of formwork to prevent excessive loss of cement slurry etc.
(2) Reinforcement The reinforcement works must be checked and approved by the structural engineer and other relevant agencies. Checks should ensure that the correct grade and sizes of bars have been used, the rebar spacings are according to the drawings, lap lengths and positioning of bars are appropriate etc. The inspector should also check the concrete covers, and certify them as adequate.
(3) Levels The levels for concreting should be established and checked using available instruments. The levels can be established using nails, markers, pegs etc. It is not ideal to start establishing casting levels on the day of concreting. It will lead to delays.
(4) Scaffolds, walkways, and platforms Sometimes site engineers may forget to make provisions for walkways and platforms and start running around on a casting day. If you are casting at a height, make sure you provide safe benches, walkways, scaffolds, and platforms to enable the casters to walk freely and pour concrete in a safe manner.
(5) Personal Protective Equipment(PPE) All PPEs must be available on-site prior to the day of casting and given to workers before the commencement of casting. Safety officers should stop the casting operation if safety precautions are not taken seriously.
(6) Materials All the materials needed for casting should be available before the day of casting. These include sand, granite, cement, admixtures, and water. As practically as possible, ensure that all the materials you need to complete the casting are on the ground a day prior to the casting date.
Suppliers may disappoint you on the casting day or there might be a breakdown of vehicles or unforeseen interruptions. This will completely ruin your big day and you will not meet your target. Also, make sure that you have calculated the quantity of materials you need properly, and verify that the suppliers did not undersupply. Lack of water on site can ruin your casting day too. Therefore, you must pay careful attention to the materials you need.
(7) Equipment/Machinery Ensure that all tools, equipment, and machinery you need for the job are on stand-by prior to the day of casting. You should have at least two vibrators and two concrete mixers on-site, depending on the size of the job to be done. Breakdown of equipment can completely ruin your day. Also make sure that ancillary equipment such as your concrete cube moulds, buckets, headpans, shovels, trowels, etc are all available.
(8) Personnel Make calls and confirm the availability of all personnel you will need for the job at least 24 hours before the casting day. This includes all supervisors, safety officers, operators, foremen, labourers, etc. Also, make sure that at least one iron bender, one carpenter, and one mechanic (technician) are available on your casting day for quick fixes just in case something goes wrong.
(9) Casting sequence/planning Plan your casting operations very well before the actual casting date. You may want to look at the areas that you will cast before the others. Factors that may influence these are the location of materials, location of concrete mixers, concrete thermal cracking considerations, ease of access and pouring, construction methodology etc.
A site manager must work out these details properly and discuss them with the foremen and supervisors. Their inputs will be helpful for a successful casting operation. Also based on the size of the job, you can request two or more gangs (two or more concrete mixers with different casting teams) working simultaneously so that you can finish on time. Make provisions for adequate lighting on site if the casting must finish in one day, and you anticipate it might creep into dusk.
With all these checklists certified, you can be sure that your casting can progress without many problems.
