It is more common for reinforced concrete columns in a building to be in compression. This makes sense because, in a conventional load path of a building, gravity loads are transmitted through the columns to the foundation, which exerts an equal and opposite reaction on the column, thereby placing it in compression. For tension columns, the load is probably going somewhere else before being transmitted to the foundation. This is usually deliberate!
The most significant internal force associated with columns is the axial force, even though more often than not, columns are also subjected to bending moment and shear forces. Design equations and charts exist in almost all codes of practice for reinforced concrete columns subjected to compressive axial force, bending moment, and shear. However, when a concrete column is in tension, the approach is usually very cautious. It is advisable to avoid placing concrete columns in tension unless you cannot help it.
Concrete is very good in compression but weak in tension. For instance, for a block of concrete with a 28 days cylinder compressive strength of 25 N/mm2, we will expect the tensile strength to be about 2.6 N/mm2 (See Table 3.1 EN 1992-1-1:2004). In reinforced concrete design, we normally assume the tensile resistance of concrete to be zero unless we are dealing with serviceability issues such as cracking.
If a reinforced concrete column is to be in axial tension, the entire axial stress will be carried by the steel reinforcements, unlike when it is compression. The area of steel required to resist the tensile axial force will be given by;
Ast,req = Nt,Ed/0.87fyk
Where; Ast,req = Area of steel required Nt,Ed = Ultimate axial compressive force fyk = characteristic yield strength of the reinforcement
However, this is not as simple and straightforward as it looks. For a concrete section subjected to a significant axial tensile force, cracking is going to be a major problem. As a result, additional reinforcements will be required to control the cracking, aside from the main reinforcements resisting the axial tension. The additional reinforcements will have to be caged as in the case of a wall reinforcement using smaller diameter bars. The column might have two layers of reinforcements; the inner layer for resisting the tensile force, and the outer layer for controlling the cracking (skin reinforcement).
The maximum bar diameter for the skin reinforcement can be estimated from equation 7.7N of EN 1992-1-1:2004 for members subjected to uniform axial tension;
φs = φ∗s(fct,eff/2.9)hcr/(8(h-d))
where: φs is the adjusted maximum bar diameter φ∗s is the maximum bar size given in Table 7.2N of EN 1992-1-1:2004 h is the overall depth of the section hcris the depth of the tensile zone immediately prior to cracking, considering the characteristic values of prestress and axial forces under the quasi-permanent combination of actions d is the effective depth to the centroid of the outer layer of reinforcement
The discussion above has been based on the assumption that the tension column is subjected to pure axial force. However, what happens when there is bending and shear in the section? In that case, the interaction of axial force and bending will have to be considered, and this is expected to have an effect on the size of the tension column and the quantity of the reinforcements.
From the ongoing, it can be seen that structural steel or composite sections are the best for tension columns, provided the connection details are well designed by the structural engineer. The problem of cracking due to low tensile strength is the major challenge of constructing tension columns using reinforced concrete. However, if this can be overcome with the use of additional reinforcements, the problem can be solved at the expense of additional costs.
Furthermore, the reinforcements in tension columns must not be lapped and must be detailed in such a way that the reinforcement hooks and carries the load below it. For reinforcement continuity, mechanical couplers are expected to perform better than lapping.
Practical Applications of Tension Columns
Tension columns are usually floating columns, and transmit the load they are supporting to the member above them. This depends on the structural scheme adopted by the structural engineer. However, when tension columns are to be supported on the ground, uplift forces will have to be properly checked.
An example of the application of tension columns is in the United States Court House in Downtown Los Angeles. The building’s structural system is depicted in the diagram below. All of the columns are set around the perimeter of the structure, but they do not convey the load to the ground; instead, they take the load up. As a result, there is no compressive force applied to the columns.
3D Model of United States Court House (Source; Faizal Manzoor, 2020)
Structural scheme and load path of the United States Court House (Source; Faizal Manzoor, 2020)
As can be seen above, the slab is not supporting the column’s weight; rather, it is hanging on to it. Therefore, the columns are subjected to axial tension due to the pull from the downward weight of the slab. The load is transferred from the slab to the column, which then passes through the roof truss, which is supported by shear walls. It is however important to note that steel columns were utilised in the construction described above.
In civil engineering practice, two methods for modelling structure-soil interaction are commonly used. The beam/plate sitting on an elastic foundation is one way, while the continuum method, which employs finite element analysis (FEA), is another. These approaches take into account the soil and structure deformation.
The Winkler foundation model is connected to the beam/plate resting on an elastic foundation. The Finite Element Analysis (FEA) is a sophisticated method of calculating mechanical problems where the constitutive equation (stress – deformation relationship) determines the outcome of the finite element (FE) computation. Although the FEA is a more advanced method of modeling the interaction, it is still a complicated method that may require a large number of input parameters depending on the model. On the one hand, these input parameters are not always known ahead of time and can be difficult to determine. The Winkler model, on the other hand, just requires one parameter to model the structure-soil interaction.
Soil-structure foundation models can be created in one dimension, two dimensions, or three dimensions. Each of these models has its own set of boundary conditions, calculation method, and, ultimately, advantages and disadvantages when applied to civil engineering calculations.
Models in One Dimension
Analytical or numerical procedures can be used to calculate one-dimensional (1D) models. To characterize the system’s behaviour, an ordinary differential equation can be used which can be solved using the boundary conditions of the system (see analysis beams on elastic foundation). For example, suppose the classical beam theory is used to represent the plate and the Winkler model is used to model the soil (assuming that the soil behavior is entirely linear-elastic). When analyzing slender structures that sit on an elastic foundation, this method is frequently used.
This model has the advantage of taking less time, but it has the disadvantage of being overly basic when it comes to superstructures, particularly slabs. The stiffness parameters for the structure and soil are especially crucial because they are the only parameters in the model that describe the structure and soil medium.
Models in Two Dimensions
This 2D model gives the soil-structure interaction system another dimension and can be used to represent plate structures (see plates on elastic foundation). When using 2D elements in structural software, the soil-structure interaction must be taken into account.
Because of the additional dimension, this model gets near to matching reality. The output results are easy to understand. However, the interaction between the soil and the structure is still difficult to simulate in 2D spaces, which is a disadvantage.
Models in Three Dimensions
These models are the most accurate representations of reality. All of the dimensions are taken into account. Both the structure, foundation, and soil are modelled using three-dimensional elements.
These 3D models have the following advantages: (1) More dispersal of the loads, which could result in material savings. (2) Load interactions from several directions are considered. (3) The model is more accurate.
The disadvantages are as follows: (1) It is time-consuming (modelling, analysis, and interpretation). (2) During the design process, not all parameters may be known. (3) In comparison to a 2D model, the outcomes are more difficult to control. (4) There is a possibility of apparent accuracy.
Modelling of Soils
For computational analysis, soils are modelled by making some idealisations and following some well-documented approaches which are discussed in the sub-sections below.
Winkler’s Model
The Winkler foundation concept idealizes the soil as a set of springs that displace as a result of the load applied to it. The model has a flaw in that it does not account for the interaction between the springs. Furthermore, a linear stress-strain behavior of the soil is also assumed in the model. This linear relationship simplifies calculations, but the truth is that soil does not exhibit linear elastic behaviour when loaded.
The Wikler model does not portray the settlement in a particularly accurate way, but it does provide an indication of what will happen in reality. The benefit of this model is that it only has one parameter to represent the soil (the modulus of sub-grade reaction, also known as the “k” parameter). This is why it is commonly referred to as a one-parameter model.
For Winklers Model,
p = wk
p = pressure w = settlement k = modulus of sub-grade reaction
Pasternak’s Model
Improved versions of the Winkler model have been developed to address the model’s weaknesses. The Pasternak foundation model is one of these variations. The major difference is that the springs in Pasternak’s model are connected to one another. To account for this, it includes an additional factor in addition to the ‘k’ factor, hence this model is also known as a two-parameter model. The word for this is “the Gp parameter.” This parameter physically describes the contact between the spring elements due to shear action. When compared to a one-parameter model, the displacement of the model with this extra parameter can be more realistic.
The differential equation is:
p = wk – (Gp)d2w/dx2 p = pressure k = modulus of sub-grade reaction Gp = shear modulus of the shear layer
The shear modulus (G) is related to the Gp value, however they are not the same. In a three-dimensional space, their dimensions, G is kN/m2 and Gp is kN/m, show that they are not the same. Gp is G times the effective depth of shearing in the soil. There isn’t a lot of literature or theory on this Gp parameter. Gp is an interaction parameter, according to the available and consulted publications on the Pasternak foundation model. The interaction of the springs is taken into account with this parameter. With this additional parameter, the Winkler model’s flaw is improved. This parameter physically describes the contact between the spring parts owing to shear action.
Soil-Structure Interaction
The difference in stiffness between the structure and the soil can be used to explain the soil-structure interaction. A flexible slab foundation, for example, has the largest settlement in the centre and uniformly distributed contact stresses and low moments. On the other hand, the length of a rigid slab foundation settles uniformly. The contact stresses are higher at the edge because the soil acts more stiffly there since the load can spread there. As a result, the slab’s contact stress has a parabolic shape, with maximum stresses at the edge and minimal values in the centre. The stiff slab’s bending moment is substantially greater than that of a flexible foundation slab.
The interaction between soil and foundation is caused by the linear elastic behavior of the foundation slab and the non-linear elastic behavior of the soil. The difference in stiffness, on the other hand, is the cause of the interaction between structure and soil and can be utilized to explain it.
A method for determining a system’s stiffness category is offered in the literature. The stiffness ratio (kr) can be used for this purpose;
kr = Et3/12EsL3
kr = stiffness ratio E = young’s modulus of the slab Es = young’s modulus of the soil t = thickness of the slab L = length of the slab
The terms t and L can be straightforwardly determined by the designer, while E and Es can be determined experimentally or using correlations.
For a kr ≤ 0.01 the structure may be defined as flexible and for kr > 0.1 the structure may be defined as stiff.
According to Annex G of EN 1992-1-1:2004, the column forces and the contact pressure distribution on the foundations are both influenced by relative settlements. For general design purposes, the problem can be solved by ensuring that the soil and structure’s displacements and accompanying reactions are compatible.
If the superstructure is considered flexible, then the transmitted loads do not depend on the relative settlements, because the structure has no rigidity. In this case, the loads are no longer unknown, and the problem is reduced to the analysis of a foundation on a deforming ground. If the superstructure is considered rigid, then the unknown foundation loads can be obtained by the condition that settlements should lie on a plane. It should be checked that this rigidity exists until the ultimate limit state is reached.
An analysis comparing the combined stiffness of the foundation, superstructure framing components, and shear walls with the stiffness of the soil can be used to calculate the approximate rigidity of the structural system. Depending on the relative stiffness KR, the foundation or structural system will be considered rigid or flexible. For building structures, expression (G1) of EN 1992-1-1:2004 can be used:
KR = (EJ)S / (EL3)
where: (EJ)S is the approximate value of the flexural rigidity per unit width of the building structure under consideration, obtained by summing the flexural rigidity of the foundation, of each framed member and any shear wall E is the deformation modulus of the ground L is the length of the foundation
Relative stiffnesses higher than 0.5 indicate rigid structural systems.
To reduce soil heave under a foundation or subgrade, expansive soils can be overexcavated and replaced with nonexpansive or treated soils. In this approach, the expansive soil is excavated to an adequate depth to reduce heave, and then replaced with properly treated and compacted fill up to grade.
The required depth of removal, as well as the volume, location, and cost of the fill, must all be considered. The depth of soil that must be removed is determined by the overall soil profile, the nature of the fill material, and the amount of heave that may be tolerated. A stiff layer of compacted low to nonexpansive fill has the added benefit of tending to even out variations in the heave of the underlying native soil, thereby eliminating differential heave.
In the absence of suitable nonexpansive fill around the area, moisture-conditioning and compaction control can be used to change the swell properties of the expansive soils on-site. Compacting the material to a lower density at the wet side of the optimum moisture content will minimize the expansion potential, but care must be taken to ensure that the recompacted soil is densified well enough to avoid settlement. Chemical additives can be utilized in conjunction with moisture-conditioning in some cases.
Typical behaviour of an expansive soil
However, if the expansive soil layer extends to a depth that makes total removal and replacement prohibitively expensive, appropriate soil tests and studies should be carried out to design the overexcavation and assess the projected potential heave after the overexcavation and recompaction procedure. The expected heave must be factored into the depth of overexcavation design.
Chen (1988) suggested a maximum overexcavation depth of 3 to 4 feet (1 to 1.3 meters), but these depths have been oberved to be ineffective for sites with highly expansive soils. Thompson (1992a and 1992b) studied some insurance claims and found that if there was 10 feet (3 meters) or more of nonexpansive soil beneath the footings, the frequency of claims was lower than at shallower depths. Overexcavation depths of 10 ft (3 m) or more have been specified regularly in Thompson’s works. In certain situations, the top 20 feet (6 meters) of soil have been excavated, moisture-conditioned, and recompacted in place.
Water content changes in the underlying expansive soil layers can be controlled by overexcavation and replacement. The majority of the seasonal water content variation will occur in the top few feet of soil. However, if the underlying soil has a high potential for expansion, the overexcavated zone may not be enough to prevent surface heave or shrinkage. If the underlying expansive soil gets wet, it might cause uncontrollable movement. Some of the potential water sources are impossible to forecast or control. As a result, the design engineer must consider that such events might occur during the structure’s lifetime and make appropriate design decisions.
Overexcavation and Replacement of an expansive soil
Furthermore, water collection in the overexcavation zone must be avoided at all costs. The use of permeable granular fill as a replacement fill is not suggested. Highly permeable fill will allow water to flow freely and create a reservoir for it to collect in. The ‘bathtub effect‘ is a term used to describe this phenomenon. Seepage into expansive subgrades or foundation soils will occur as a result of this situation. Any fill material that is impermeable and nonexpansive is therefore more preferable. If granular material is required, permanent, positive drainage and moisture barriers, such as geomembranes, should be installed to prevent moisture from infiltrating this zone.
The removed and recompacted material is expected to have a higher hydraulic conductivity than the underlying in situ soils and bedrock, even without granular soil. Groundwater can be intercepted using an underdrain system installed at the bottom of the overexcavation zone. Care must be taken to ensure that the drain has positive drainage and that it does not just concentrate water in an area where it would cause increased soil wetting and heave.
Advantages of Overexcavation and Replacement
The following are some of the benefits of overexcavation and replacement treatment:
• Because soil replacement does not require special construction equipment, it might be less expensive than other treatment options. • Soil treatment additives can be mixed in a more equal manner, resulting in some soil improvement. • Overexcavation and replacement may cause construction to be delayed less than other processes that need a curing period.
Disadvatantages of Overexcavation and Replacement
The following are some of the disadvantages of overexcavation and replacement methods:
• The expense of nonexpansive fill with low permeability can be high if the fill must be imported. • If the recompacted on-site soils demonstrate intolerable expansion potential, removing and recompacting the on-site expansive soils may not be enough to limit the danger of foundation movement. • The recompacted backfill material’s needed thickness may be too considerable to be practicable or cost-effective. • If the backfill material is overly permeable, the overexcavation zone could act as a reservoir, storing water for the foundation soils and bedrock over time.
If overexcavation and replacement are ineffective on their own, they can be combined with other foundation options. It may be conceivable to employ a rigid mat foundation instead of a more expensive deep foundation if the potential heave can be suitably mitigated. The needed length of the piers may also be lowered when used in conjunction with a deep foundation.
References [1] Chen, F. H. 1988. Foundations on Expansive Soils. New York: Elsevier Science. [2] Thompson, R. W. 1992a. “Swell Testing as a Predictor of Structural Performance.” Proceedings of the 7th International Conference on Expansive Soils, Dallas, TX, 1, 84–88. [3] Thompson, R. W. 1992b. “Performance of Foundations on Steeply Dipping Claystone.” Proceedings of the 7th International Conference on Expansive Soils, Dallas, TX, 1, 438–442
Recent research carried out at the Zhengzhou University of Technology, China has offered more insight into the improvement of interlayer mechanical properties of mass concrete. The study was published in the International Journal of Concrete Structures and Materials.
During the construction of mass concrete structures such as gravity dams, concrete is poured in layers. This can be as a result of a lapse in mixing and placement time, ease of construction, possible re-use of formworks, etc. As a result, the interlayer of the concrete (joint between the new and old concrete) becomes a potential weak point that is very susceptible to cracking. A structure’s durability and stability will be severely affected if interlayer bonding characteristics deteriorate. Hence, to maintain the structure’s safety, the interlayer bonding quality of mass concrete must be closely controlled.
The chemical bonding force of cementitious materials and the degree of mutual embedding of aggregates determine the interlayer bonding strength of concrete. According to research, the interlayer bonding strength of concrete can be ensured by pouring the upper layer of concrete before the initial setting time of the lower layer (substrate).
The interlayer bonding characteristics of mass concrete are therefore heavily influenced by the interval time between the placement of new concrete on old concrete. Parameters such as compressive strength, interlayer splitting tensile strength, shear strength, and impermeability of concrete reduce with an increase in interval time according to many research works. Temperature, relative humidity, and wind speed are additional important parameters that influence the quality of mass concrete construction.
As a result, researchers (Song, Wang, and Lui, 2022), carried out research focusing on the effect of harsh environmental conditions on the quality of mass concrete construction, with emphasis on the interlayer properties and cracking. The concrete layer condition and interlayer splitting tensile strength were tested in harsh situations (high temperature, strong wind, steep temperature decline, and short-term heavy rainfall).
Fig. 1: Cracks in a concrete wall
Secondly, the cracking risks of concrete in extreme weather were evaluated (coupling of high winds and dry heat, strong winds and cold waves, and short-term heavy rainfall). Finally, effective strategies to deal with construction risks during harsh weather conditions were proposed by the authors. The strategies considered were covering the concrete with an insulation quilt, artificial introduction of grooves on the old concrete, and addition of Polyvinyl alcohol (PVA) fibres.
From the study, it was observed that under harsh weather conditions, the interlayer mechanical characteristics of concrete reduced significantly (high temperature, strong wind, a steep descent in temperature, and short-time heavy rainfall).
For instance, under high temperatures (40 deg celsius), the water content of the cement mortar decreased gradually with time, while the penetration resistance increased continuously. However, when the concrete sample was covered with an insulation quilt, the water content was closer to the designed water content of 133 kg/m3. Generally, the results showed that in a high-temperature setting, covering an insulating quilt can reduce mortar water loss and lower penetration resistance of concrete specimens to a degree.
The study, therefore, showed that the interlayer bonding strength of concrete can be improved by covering it with an insulation quilt. The reason for this is that an insulation quilt can lessen the impact of the external environment on concrete, resulting in less water evaporation and a slower setting rate. Artificial grooves can also help to strengthen interlayer bonding. This is due to the artificial grooves increasing the roughness of the lower layer of concrete and improving the mutual embedding degree of the upper and lower layers.
Fig. 2: Cracking of concrete surface under the coupled conditions of wind, dryness, and heat (Song, Wang, and Lui, 2022).
Furthermore, under harsh weather conditions (coupling of strong winds and dry-heat, strong winds and cold waves, and short-time heavy rainfall), mass concrete has an increased risk of cracking. Concrete cracking can be efficiently prevented by using an insulation quilt (see Figure 2). It is mostly due to the insulation quilt’s ability to prevent water evaporation, resulting in a significant reduction in water loss shrinkage stress. Furthermore, the high water content fully hydrates the cement and enhances early tensile strength, which is beneficial to the anti-cracking properties of concrete at an early stage.
Finally, the addition of Polyvinyl alcohol (PVA) fibers to the concrete can help to prevent the formation and propagation of microcracks. This is due to the fact that PVA fibers can resist some of the tensile stress induced by moisture loss and shrinkage, as well as play a role in crack resistance and bridging. As a result, PVA fiber concrete can be put into important portions of concrete dams to improve the dam’s crack resistance.
References
Song H., Wang D. and Liu WJ (2022): Research on Construction Risks and Countermeasures of Concrete. Int J Concr Struct Mater (2022) 16:13 https://doi.org/10.1186/s40069-022-00501-3
Cantilever beams are beams that are free at one end and rigidly fixed at the other end. Beams that are free at one end and continuous through the other support (beams with overhangs) are also treated as cantilever beams. The primary design of cantilever beams involves the selection of an adequate cross-section and reinforcements to resist the internal stresses due to the applied loads and to limit the deflection to an acceptable minimum.
Due to the structural system of cantilever beams, they are very sensitive to deflection and vibration. When loaded, the maximum shear force and bending moment occur at the fixed support, while the maximum deflection occurs at the free end. As a result, under normal circumstances in reinforced concrete design, the length of cantilevers is usually kept to a minimum in order not to have bulky sections and heavy reinforcements. In order to save material and to reduce the load due to self-weight, cantilever beams can be tapered, increasing linearly from the free end to the fixed support.
For cantilever beams, the tensile moment occurs at the top, therefore the main reinforcements are provided at the top. At the bottom, standard beam detailing requirements recommend that at least 50% of the reinforcement provided at the top be provided at the bottom. The anchorage length of the top reinforcement is expected to enter at least 0.25 times the effective span of the backspan or 1.25 times the effective length of the cantilever (whichever is greater).
A cantilever beam relies on the backspan or an alternative counterweight for equilibrium or structural stability.
Design Example of a Cantilever Beam
Design a two-span cantilever beam (beam with overhang with the following information provided). The beam is to support a rendered 230 mm hollow block wall up to a height of 2.7m, in addition to the load transferred by the floor slab. The ultimate design load on the slab is 12 kN/m2. fck = 25 N/mm2; fyk = 500 N/mm2; Concrete cover = 35mm; Unit weight of concrete = 25 kN/m3; Unit of block wall = 3.5 kN/m2
Load Analysis
Aspect ratio of the slab k = Ly/Lx = 6/5 = 1.2 Factored load transferred from slab to beam B1 = 0.5(12 × 5) × [1 – 0.333(1.2)2] = 15.61 kN/m Factored load transferred from slab to beam B2 (factored) = γGnlx/4 = 1.35 × (12 × 2.5)/4 = 10.125 kN/m Factored self-weight of the beam (considering the 300 mm drop) = 1.35(25 × 0.3 × 0.23) = 2.33 kN/m Factored weight of block wall = 1.35(3.5 × 2.7) = 12.76 kN/m
Nominal cover to top reinforcement; cnom_t = 35 mm Nominal cover to bottom reinforcement; cnom_b = 35 mm Nominal cover to side reinforcement; cnom_s = 35 mm
Fire resistance
Standard fire resistance period; R = 60 min Number of sides exposed to fire; 3 Minimum width of beam – EN1992-1-2 Table 5.5; bmin = 120 mm
FlexuralDesign of the Cantilever Section
Design bending moment; MEd = 78.8 kNm
Distance between points of zero moment; L0 = (0.15 × Lm1_s1) + Lm1_s2 = (0.15 × 6000) + 2500 = 3400 mm Maximum flange outstand; b1 = bf – b = 720 mm Effective flange outstand; beff,1 = min(0.2 × b1 + 0.1 × L0; 0.2 × L0; b1) = 484 mm Effective flange width; beff = beff,1 + b = 714 mm Effective depth of tension reinforcement; d = 399 mm
K = M / (beff × d2 × fck) = 0.028
K’ = (2 × η × αcc / γC) × (1 – λ × (δ – k1) / (2 × k2)) × (λ × (δ – k1) / (2 × k2)) = 0.207 Lever arm; z = min(0.5 × d × [1 + (1 – 2 × K / (η × αcc / γC)0.5], 0.95 × d) = 379 mm Depth of neutral axis; x = 2 × (d – z) / λ = 50 mm
λx < hf – Compression block wholly within the depth of flange K’ > K – No compression reinforcement is required
Area of tension reinforcement required; As,req = max(M / (fyd × z), As,min) = 478 mm2
Tension reinforcement provided;3H16 Area of tension reinforcement provided; As,prov = 603 mm2
Minimum area of reinforcement – exp.9.1N; As,min = max(0.26 × fctm / fyk, 0.0013) × b × d = 122 mm2 Maximum area of reinforcement – cl.9.2.1.1(3); As,max = 0.04 × b × h = 4140 mm2 PASS – Area of reinforcement provided is greater than area of reinforcement required
Minimum area of shear reinforcement – exp.9.5N; Asv,min = 0.08 N/mm2 × b × (fck )0.5 / fyk = 184 mm2/m
Design shear force at support ; VEd,max = 63 kN Min lever arm in shear zone; z = 379 mm Maximum design shear resistance – exp.6.9; VRd,max = αcw × b × z × v1 × fcwd / (cot(θmax) + tan(θmax)) = 392 kN PASS – Design shear force at support is less than maximum design shear resistance
Design shear force at 399mm from support; VEd = 53 kN
Design shear stress; vEd = VEd / (b × z) = 0.608 N/mm2 Angle of concrete compression strut – cl.6.2.3; θ = min(max(0.5 × Asin(min(2 × vEd / (αcw × fcwd × v1),1)), 21.8 deg), 45deg) = 21.8 deg
Area of shear reinforcement required – exp.6.8; Asv,des = vEd × b / (fyd × cot(θ)) = 129 mm2/m Area of shear reinforcement required; Asv,req = max(Asv,min, Asv,des) = 184 mm2/m
Shear reinforcement provided; 2 × 8 legs @ 200 c/c Area of shear reinforcement provided; Asv,prov = 503 mm2/m PASS – Area of shear reinforcement provided exceeds minimum required
Maximum longitudinal spacing – exp.9.6N; svl,max = 0.75 × d = 299 mm PASS – Longitudinal spacing of shear reinforcement provided is less than maximum
Foundations can be subjected to dynamic loads in addition to static loads in engineering practice. Dynamics loads can be found in pile foundations supporting machines, oil and gas facilities, buildings under seismic effect, wind turbines, etc. The design of foundations under dynamic loading is complex and involves the inputs of structural, mechanical, and geotechnical engineering, as well as the theory of vibration.
In some cases, using deep foundations rather than shallow foundations may be required for foundations subjected to dynamic loads. Many factors influence whether a structure should be supported on a shallow or deep foundation system, including subsurface conditions and induced dynamic and static stresses.
Pile foundations are utilized to prevent bearing capacity failure, improve the system’s dynamic stiffness, and reduce dynamic oscillations. However, when a complete understanding of the dynamic interaction between the pile and the soil (pile-soil interaction) and between adjacent piles (pile-soil-pile interaction) is necessary, calculations become more difficult.
Fig 1: Construction of pile foundation
In general, applying dynamic loads to piles in cohesive soils reduces their skin friction and end-bearing value, i.e., reduces their ultimate carrying capacity, whereas applying dynamic loads to piles in granular soils reduces their skin friction but increases their end-bearing resistance at the expense of increased settlement under working load (Tomlinson, 1994).
The reduction in skin friction and end-bearing resistance of piles in cohesive soils is due to cyclic loading reducing the shearing strength of these soils. The ratio of the applied stress to the ultimate stress of the soil determines the amount of decrease for an infinite number of load repetitions.
Novak (1974) proposed an approximate method for simulating the dynamic interaction between soil and single piles. His method presupposed that the soil is made up of a series of infinitesimally thin horizontal strata that extend indefinitely. As a result, it can be seen as a generic Winkler medium with inertia and the ability to dissipate energy. Novak’s work was able to establish the value of geometric damping, in addition to being more precise than earlier attempts.
Novak and AboulElla (1978) modified this approach to include the effect of having a soil profile that changes with depth. Novak and Sheta (1982) also incorporated the effect of the weak zone around the pile, which can be used to represent either soil-pile interface slippage or the real weak zone generated around the pile during construction.
All of these investigations reveal that frequency has a considerable impact on single pile dynamic impedance characteristics. Field tests (Manna and Baidya, 2009; Elkasabgy et al, 2010) have also corroborated this effect. As a result of soil non-linearity, field investigations on large-scale piles have also revealed a non-linear dynamic pile reaction.
Dynamic Behaviour of Single Piles
Khalil et al (2019) carried out numerical and experimental modelling on the dynamic behaviour of piles. In the study, a range of excitation frequencies ranging from 10Hz to 60Hz was considered in order to verify their effects on the dynamic behaviour of single piles. Under vertical and horizontal vibrations, the finite element model from the study reveals that as the excitation frequency increases, the stiffness Ks increases and the damping Cs reduces (see Figure 2).
The study found out that stiffness increased by 64% (under vertical vibrations) and by 120% (under horizontal vibrations) when the frequency is increased from 10 Hz to 60 Hz (a 500% increase). Damping, on the other hand, is reduced by 40% under vertical vibrations and by 26% under horizontal vibrations. These findings show that the excitation frequency affects the dynamic pile-soil interaction.
Fig 2: Vertical dynamic behavior of single pile: (a) stiffness, (b) damping, (c) peak displacement (L/D = 20) (Khalil et al, 2019)
The effect of varying soil stiffness (Es) was also investigated in the study. When the value of soil stiffness was reduced by 50%, the numerical model revealed a corresponding drop in the impedance parameters of up to 33% for the stiffness and 12% for the damping (under vertical vibrations). For horizontal vibrations, this was up to 43% for stiffness and 27% for damping.
As a result, peak displacements can rise by 30 to 45% under vertical vibrations and 54 to 79% under lateral vibrations. This is to be expected, because lowering the soil stiffness lowers the soil resistance around the pile, resulting in a reduction in system stiffness and damping.
The slenderness ratio (L/D) of single piles subjected to vertical or lateral vibrations has no significant effect on their dynamic behavior. Peak displacements are reduced by less than 10% (under vertical vibrations) and 3% (under horizontal vibrations) when the pile slenderness ratio is increased from 20 to 30. This is consistent with the findings of Novat (1974) which show that raising the slenderness ratio has little effect on long flexible piles, especially when subjected to lateral motion. This happens because, regardless of the pile’s entire length, the soil mass contributing to the system’s dynamic resistance is confined to a specific depth.
Dynamic Behaviour of Piles in Group
Khalil et al. (2019) investigated the dynamic behavior of pile groups using a 3D finite element model. According to the findings, the group stiffness Kg increases at a variable rate as the frequency increases (see Figure 3). However, the group damping Cg increased significantly till it reaches f = 30 Hz to 40 Hz and decreased again after this point. Furthermore, the peak displacement reduced until it is nearly constant beyond 45 Hz. The study found that a pile group’s response is more sensitive to frequency than a single pile’s response.
The finite element model from the study showed a non-uniform drop in Kg as a result of lowering the soil stiffness (Es) by 50%. The reduction varies from as low as 4% (between 25 and 35 Hz) to as high as 39% at f = 60 Hz under vertical vibrations. Meanwhile, between 10 and 27 Hz, Cg slightly increased (by less than 5%). It however dropped by up to 39% for frequencies greater than 27 Hz.
. Fig 3: Effect of group size on vertical dynamic behavior: a) stiffness group efficiency, b) damping stiffness group efficiency, c) peak displacement (100%Es, L/D = 20, S/D = 5) (Khalil et al, 2019)
An overall rise in peak displacements was observed for the pile groups. However, this increase is inconsistent over the frequency range investigated, ranging from 8% to 52%. The reduction under lateral vibrations varies from as low as 24% at f = 35 Hz to as high as 42% at f = 60 Hz. In the meantime, the Cg drops by up to 35%. As a result, there was an overall rise in peak displacements.
Under vertical vibrations, the effect of modifying the dimensionless spacing ratio (S/D = distance between piles/pile diameter) was investigated using values of 3, 5, and 10 for a pile group of four. The phase at which the stress waves reach the adjacent vibrating piles changes as the spacing between piles increases. The stress waves become in-phase or out-of-phase with the vibrating nearby piles as a result of this change, which might cause the impedance parameters to reduce or rise.
The pile slenderness ratio (L/D) has a minor effect on the dynamic behavior of a pile group subjected to vertical or lateral vibrations, according to the research. Peak displacements are reduced by less than 5% (under vertical vibrations) and less than 7% (under horizontal vibrations) when the pile slenderness ratio is increased from 20 to 30. (under lateral vibrations).
Design of Pile Foundation under Dynamic Loading
To accommodate for dynamic load application on piles, it is common practice to double the safety factor on the combined skin friction and end bearing. The lateral loading of supporting piles can be caused by the torque of rotating machinery. The approaches can be used to determine the deflection under lateral loading in accordance with established methods. The deflections computed for the comparable static load should be doubled to account for dynamic loading.
The type of pile used, whether driven, driven-and-cast-in-place, or bored-and-cast-in-place, has no impact on the behavior of piles founded entirely in cohesive soils. Because of the development of an enlarged hole around the upper part of the shaft, lateral movements of piles with driven pre-formed shafts (e.g. precast concrete or steel H-piles) may be greater than those of cast-in-place piles (Tomlinson, 1994).
In granular soil, a pile’s skin-frictional resistance to static compressive force is quite low. When the pile is subjected to vibratory stress, this resistance is further reduced, and it is best to discard all frictional resistance on piles carrying high-frequency vibrating loads. If such piles are ended in loose to medium-dense soils, the settlement will continue to an unsatisfactory level for most machinery installations.
As a result, piles must be driven to a dense or very dense granular soil stratum, and even then, settlements can be large, especially if large end-bearing pressures are used. This is due to the soil grains’ increasing attrition at their places of contact. The slow but steady settlement of the piles is caused by the continued deterioration of the soil particles. Piles supporting vibrating machinery should, if possible, be driven completely through a granular soil stratum and terminate on bedrock or within a firm clay.
References
Elkasabgy M, El Naggar MH., and Sakr M. (2010): Full-scale vertical and horizontal dynamic testing of a double helix screw pile. Proc. of the 63rd Canadian Geotech; Conf., Calgary, Canada; 2010, pp. 352–359.
Khalil M. M., Hassan A. M., and Elmamlouk H. H. (2019): Dynamic behavior of pile foundations under vertical and lateral vibrations. HBRC Journal, 15(1):55-71, DOI: 10.1080/16874048.2019.1676022
Novak M. (1974): Dynamic stiffness and damping of piles. Can Geotech J. 11:574–598.
Novak M. and Aboul-Ella F (1978): Impedance functions for piles embedded in layered medium. J Eng Mech ASCE. 104(3):643–661.
Novak M. and Sheta M. (1982): Dynamic response of piles and pile groups. 2nd International Conference on Numerical Methods in Offshore Piling; Austin, TX; 1982.
Manna B and Baidya DK. (2009): Vertical vibration of full-scale pile—analytical and experimental study. J Geotech Geoenviron Eng. 135(10):1452–1461.
Tomlinson M. J. (1994): Pile Design and Construction Practice. E & FN SPON, London, UK
An orthotropic steel deck bridge is made up of a steel plate with welded stiffeners running in opposite directions. Longitudinal stiffeners are known as ribs, and transversal stiffeners are known as cross beams or floor beams. The entire deck is supported by main girders running longitudinally. The presence of two different members in the two orthogonal directions means that the stiffness of the deck is anisotropic (not the same in every direction). The term ‘orthotropic’ is coined from orthogonal-anisotropic.
Figure 1: Typical components on an orthotropic steel deck bridge
The major structural components of an orthotropic steel deck are;
The wearing surface
The deck plate
The transverse stiffeners
The longitudinal stiffeners (ribs), and
The main girders
Because the deck serves as a top flange for the longitudinal and transversal stiffeners, as well as the major girders, the orthotropic steel deck is a cost-effective and efficient system. This concept saves material, reduces self-weight, and increases the rigidity of the deck at the same time (Håkansson and Wallerman, 2015).
Structural Behaviour of Orthotropic Steel Deck
Almost every structure that exists is made up of several structural elements such as beams, columns, and slabs. In a sophisticated way, those elements contribute to the overall behavior of the structure, however, it is common for these members to be isolated and designed individually.
The elements of orthotropic steel deck bridges are linked in a more complicated way, and the same structural elements can perform several functions. The plate functions as a load distributer between the ribs as well as a top flange for ribs, crossbeams, and main girders, as previously stated. Due to this complex interaction, individual members should not be designed in isolation from each other. In other words, the structural elements cannot be treated individually if the true response of the bridge must be known (Håkansson and Wallerman, 2015).
The diagram below depicts the transfer of a concentrated load to the major girders. The load is applied to the deck plate, which distributes it among the ribs. The load is transferred from the ribs to the cross beams, which are then distributed between the main longitudinal girders.
Figure 2: Typical load path in an orthoptropic steel deck bridge (Karlsson and Wesley, 2015)
Subsystems for Analysis of Orthotropic Steel Deck Bridges
It has been proposed that the entire bridge deck be broken into subsystems in order to make hand computations and characterize the complex structural behavior of orthotropic steel decks. Because these subsystems are believed to work independently of one another, the impacts of the several subsystems can be combined using superposition (US Department of Transportation, 2012).
The proposed subsystems are described below;
Subsystem 1: Local Plate Deformation
In this subsystem, the deck plate should only transfer the imparted wheel load to the adjacent rib walls (US Department of Transportation, 2012). Deck plate bending is used to transfer the load. When a concentrated force is applied over a rib, the deck plate deforms as shown in Figure 3.
Figure 3: Local deformation of deck plate (Karlsson and Wesley, 2015)
Subsystem 2: Panel Deformation
Due to the fact that ribs share the same top flange, they are unable to operate independently, resulting in panel deformation. As explained in the preceding section, a concentrated force applied to the deck plate will be transferred to the neighboring ribs, but because of the shared top flange, even ribs that are not loaded will deflect. This action reduces the stresses in loaded ribs while increasing stresses in unloaded ribs (US Department of Transportation, 2012). Figure 4 shows how the panel deforms as a whole, with all ribs deflecting at the same time.
Figure 4:Panel deformation of deck plate (Karlsson and Wesley, 2015)
Subsystem 3: Longitudinal Flexure of the Ribs
Ribs are constructed in a continuous pattern over cross beams, and the fact that cross beams deflect under load must be taken into account. The ribs are treated as continuous across discrete flexible supports to account for this flexure. Cross-beams are simply supported between rigid main girders in this concept, and they deflect when loaded.
Subsystem 4: Cross Beam In-plane Bending
The ribs are built in a continuous pattern over the cross beams, as previously stated. This will result in cut-outs in the cross-section of the cross beam where the rib passes through. As a result, the geometry of the cross beam will change, making hand computations of in-plane stresses from bending and shear more difficult. According to the US Department of Transportation (2012), FE-analysis should be used to simulate the entire cross beam. The deformed shape of a cross beam subjected to in-plane forces is seen in Figure 5.
Figure 5: In-plane bending of cross beam (Karlsson and Wesley, 2015)
Subsystem 5: Cross beam distortion
Three separate effects affect the local stresses in the cross beam at the cross beam and rib intersection. Out-of-plane distortion from rib bending, distortion of rib walls due to shear stresses, and distortion of ribs due to unequal deflection are the local mechanisms at these intersections.
Subsystem 6: Rib Distortion
The rib will twist about its rotating centre if a concentrated force is applied in the mid-span between two cross beams and is eccentric about the axis of the rib (US Department of transportation, 2012). Because the rib-cross beam junction will be a fixed or partially fixed barrier, depending on how large cut-outs are employed, there will be substantial stress concentrations in the welds where twisting is inhibited. When the ribs are loaded, they distort as shown in Figure 6.
Figure 6: Distortion of ribs when loaded (Karlsson and Wesley, 2015)
Subsystem 7: Global Behaviour
When there is no consideration for local effects, the global system explains the displacement of the main girders as well as the overall behavior. It is possible to determine stresses and strains in the structure using traditional methods in this system (US Department of Transportation, 2012). Figure 7 depicts the orthotropic steel deck bridge’s global bending.
Figure 7: Global bending of the OSD (Karlsson and Wesley, 2015)
References
(1) Håkansson J. and Wallerman H. (2015): Finite Element Design of Orthotropic Steel Bridge Decks. Masters Thesis submitted toChalmers University of Technology, Göteborg, Sweden (2) Karlsson A. and Wesley C. (2015): Necessity of Advanced Fatigue Analysis for Orthotropic Steel Deck Bridges. Chalmers University of Technology, Göteborg, Sweden (3) US Department of Transportation (2012): Manual for Design, Construction, and Maintenance of Orthotropic Steel Deck Bridge. US Department of Transportation, Federal Highway Administration, Publication no. FHWA-IF-12-027. USA
In highway engineering, pavements should be constructed on subgrades with a known strength. The subgrade is the foundation or existing soil surface where the new highway will be built. Subgrade strength is normally described in terms of the CBR. The design idea for heavily trafficked roads is to improve the subgrade as needed and to establish a stable platform with the sub-base for the construction of the bound structural layer and surfacing above. In the long run, the capping and/or sub-base keep water out of the bound layers and offer a platform for compacting the bound layers.
The theory for both composite and flexible pavements is that the thickness of the sub-base/capping layer does not fluctuate with traffic, but rather with the strength of the subgrade. The thickness is the same in both types of roads.
Subgrade of a highway development
A pre-construction geotechnical site investigation shall be carried out for all sites prior to the start of any highway design/construction in order to assess a number of design issues, including the stiffness (CBR) of the material, its moisture sensitivity, and, if necessary, its suitability for earthworks and stabilisation to form a capping layer, sub-base, or road base material.
Despite problems in measurement, particularly on mixed fine and coarse graded soils and when moisture effects are considered, the California Bearing Ratio (CBR) remains the greatest predictor of soil strength. A method based on the Portable Dynamic Plate is available for measuring the stiffness of the sub-grade under dynamic loading. This is especially useful when a road is being widened such that the foundation is at or near equilibrium, while for new roads, additional approaches will be required to calculate equilibrium CBR values.
Regardless, any site investigation can only sample the soils at discrete areas inside the site. Variability is unavoidable, and it should be factored into the work’s design. A competent Geotechnical Engineer can provide guidance on the expected range of CBR values if necessary.
Typical laboratory CBR test
Aside from determining design CBR values for both short and long-term characterization of subgrade performance, several other elements might affect subgrade performance and must be considered during the design stage. The following are examples of typical issues to be addressed:
a) Water table depth/perched water tables b) Chemical contamination risk assessment c) Control of fine-grained soil piping d) The possibility of encountering loose Made Ground. a) The requirement for foundation soils to be improved on the ground (e.g. soft Alluvium, loose made ground etc.) f) The risk of collapse settlement of dry engineered fills g) The possibility of landslides. h) The possibility of subsurface caves, deneholes, and so forth. I The effect of nearby developments on locations with soft alluvium. j) The occurrence and handling of subgrade solution characteristics. k) Solution characteristics below drainage runs are treated. l) Treatment and frequency of other subgrade soft spots. m) Frost sensitivity of the subgrade. n) Risks of differential settlement/need for ground improvement. o) Subgrade soil chemistry if in-situ lime/cement stabilisation is considered. p) The ability of over-consolidated clays to shrink and expand. (Especially where trees have been destroyed) q) The possibility of open cracks in the underlying rock. r) The possibility of soft clay layers in granular soil.
The strength of a subgrade may be defined in terms of the soaked CBR. Subgrade strength classification in terms of CBR is given in the Table below;
Soaked CBR
Strength Classification
Comments
< 1%
Extremely weak
Geotextile reinforcement and separation layer with a working platform typically required.
1% – 2%
Very weak
Geotextile reinforcement and/or separation layer and/or a working platform typically required.
2% – 3%
Weak
Geotextile separation layer and/or a working platform typically required.
3%–10%
Medium
Geotextile separation layer and/or a working platform typically required.
10% – 30%
Strong
Good subgrade to Sub – base quality material
> 30%
Very Strong
Sub – base to base quality material
Selection of Method for Determination of CBR
The method for determining CBR should be chosen depending on the scale of the scheme, the precision required, and the likely soils encountered. Special precautions are required to provide adequate foundation support for CBR less than 2%.
The strength of most soils is highly dependent on moisture content. Some soils experience a rapid loss of strength as the moisture content increases. On such soils, structure protection as described in the Specification for Highway Works is very critical, as is a conservative approach when considering the effect of subgrade drainage.
For design purposes, two scenarios must be considered: the likely CBR at the time of construction and the long-term equilibrium value. Both of these are essential for design purposes. If the ‘as found’ CBR at the time of construction is less than that determined during the site investigation, a change in foundation layer thickness may be required.
The plasticity index should be computed for cohesive soils, and the description of the soil type from a grading examination of a bulk sample, as well as some knowledge of the probability of saturation in the future, should be examined for other soils. The table below can be used to approximate the CBR based on this information.
Methods of Determination of CBR
CBR’s performed upon pot samples
Advantages (i)It enables the evaluation of a soil’s CBR at various levels of saturation (CBRs, soaked/unsoaked)
Disadvantages (i) Considerable disturbance is created during the sample operation, which has a significant impact on the test results. (ii) Requires a large trial pit.
Plastic and Liquid Limits
Advantages (i) It enables a lower bound estimate of the CBR under recompacted circumstances for a wide range of effective stresses (i.e. construction conditions)
Disadvantages (i) Unless standard graphics are utilized, the analytical approach is costly. (ii) The typical graphs are based on the worst-case scenario. (iii) Can only be utilized on soils containing cohesive material (iv) Can be difficult to evaluate on granular and cohesive material mixtures
Soil Assessment Cone Penetrometer (MEXE Probe)
Advantages (i) It is simple and affordable to carry out.
Disadvantages (i) Correlation dependant (ii) Only offers the current CBR value (iii) Insensitive to the influence of the soil’s microstructure (iv) It is not suitable for usage in stony soils.
Measurement of Shear strength (hand vane/Triaxial tests)
Advantages (i) The hand vane is a quick and low-cost method of measuring undrained shear strength. (ii) Triaxial measurements of undrained shear strength take into consideration the soil microstructure (iii) For recompacted subgrades, remolded tests provide a lower bound.
Disadvantages (i) Only offers the existing CBR value when using the hand vane (ii) Triaxial measurements are somewhat expensive to perform, and sample and testing time might be lengthy (iii) It is dependent on correlation although this has a theoretical basis (iv) can only be applied for cohesive soils.
In-situ CBR
Advantages (i) Realistic CBR measurement (ii) Takes into account the macrostructure of the soil account (iii) can be used to determine the present chalk value in chalks
Disadvantages (i) Expensive to complete (ii) Only offers the present CBR value (iii) It is difficult to perform beneath the existing ground surface (iv) Not recommended for coarse granular soils.
Laboratory compaction test
Laboratory Compaction
Advantages (i) Calculates the CBR of remoulded soils at various moisture levels.
Disadvantages (i) Insensitive to soil macrostructure impacts (ii) Expensive to do (iii) Requires a big sample (iv) variable outcomes with coarse granular soils
Conclusion
CBR measurements are used to determine the current subgrade strength and to forecast the worst-case scenario for future service inside the pavement. In actuality, the current value may underestimate the subgrade’s strength at the time of construction because pessimistic conditions may not be present at the time of building, such as during summer construction. The current value, on the other hand, could have been taken in the summer and the plan built in the winter, when the subgrade may be weaker due to the moisture accumulating at that time of year.
As a result, it’s critical to think about the implications of the site investigation results in relation to those at the time of building. Unfortunately, there are no “hard and fast” guidelines, therefore field data may need to be interpreted by a professional. To develop an interpretative statement on the equilibrium CBR for design purposes, local knowledge of the effect of moisture on the relevant soil and the compaction/moisture content vs CBR connection should be combined with laboratory experiments. It will be necessary to make a request for this. Other elements impacting the performance of the subgrade, such as drainage and the possibility of foundation material settlement, must also be considered in the design.
Beams that have V-shapes (in the plan view) are commonly found in the corners of residential and commercial buildings. According to Hassoun and Al-Manaseer (2008), such beams can be analysed and designed using strain-energy principles especially when the beam is fixed at both ends.
In some scenarios, engineers may be tempted to design v-shaped beams as two cantilevers meeting at a point, but this is not strictly the case. There will be a need to assess other internal stresses such as torsion, and to understand the actual nature of the distribution of internal forces.
For a v-shaped beam subjected to a uniformly distributed load w (kN/m), the internal forces in the members can be obtained as follows;
(The bending moment at the centre of the beam Mc is given by; Mc = (wL2)/6 × [sin2θ/(sin2θ + λcos2θ)
Where; λ = EI/GJ L = half the total length of the beam AC θ = Half the angle between the two sides of the v-shape beam.
The torsional moment at the centreline of the section is given by; TC = (MC/sinθ) × cosθ = MCCotθ
At any section N along the length of the beam at a distance x from the centreline C,
A v-shaped beam at the corner of a building has a depth of 400mm and a width of 225 mm. The plan view of the beam is shown below. It is to support an ultimate uniformly distributed load of 30 kN/m inclusive of the factored self-weight. Design the beam according to the requirements of EC2. fck = 30 MPa, fyk = 500 MPa, Concrete cover = 35 mm
Solution
For fck = 30 MPa, Modulus of elasticity Ecm = 31476 MPa Shear modulus G = Ecm/2(1 + v) = 31476/2(1 + 0.2) = 13115 MPa Moment of inertia I = bd3/12 = (225 × 4003)/12 = 12 × 108 mm4 Polar moment of inertia J = 985033091.649413 mm4 λ = EI/GJ = (31476 × 12 × 108)/(13115 × 985033091.649413) = 2.9237
Mc = [wl2sin2θ/6(sin2θ + λcos2θ) = (30 × 2.52 × 0.75)/[6 × (0.75 + 2.9237 × 0.25)] = 140.625/8.88555 = 15.755 kNm MA = MB = MC – wl2/2 = 15.755 – (30 x 2.52)/2 = -77.995 kNm Torsional moment = TA = MCcotθ = 15.755 x 0.5773 = 9.095 kNm Shear at support = VA = VB = 30 × 2.5 = 75 kN
Let us check these answers using Staad Pro software;
By implication, one can confirm that the analysis method adopted is accurate. The beam can now be designed for torsion, bending, and shear using the guidelines provided in EN 1992-1-1:2004.
References Hassoun N. M. and Al-Manaseer A. (2008): Structural Concrete Theory and Design. Wiley and Sons Inc, New Jersey, USA
