This report paper focuses on the deep beam through analysis of its failure analysis, failure modes, deep beam nodal zones, shear value results, implementation cost of the deep beam as well as examples of the deep beam. Some of the examples of deep beam include the Strut tie representing combined mechanism, Strut tie representing vertical truss mechanism, and Strut tie representing arch mechanism. Previous studies have shown that deep beams made of reinforced concrete are an important subject in the sector of structural engineering especially in foundations, offshore structures, and tall buildings.
For the purposes of design, the transition beam behaviour that is ordinary to behaviours of the deep beam is vague, it is normally considered to happen at a ratio of span and depth which is approximated to be 2.5. However, the ratio of the span of span and depth is a parameter that is quoted frequently in the behaviour of the deep beam. The significance of the ratio of the shear span and depth was stressed numerous years ago and for instability and bucking, the ratio of depth and thickness and thickness and bucking ratio are both important (ACI 318, 2011).
Some of the constructions which use the deep beam include foundations of most of the buildings and structures, offshore structures which include those buildings constructed near the seashore and there is the need for them to be raised as well as tall buildings (Clarke, 2011).
The early investigation of deep beams majorly focused on elastic behaviour. The studies of the elastic behaviour can be done easily by the use of finite elements techniques and standard finite difference. Nevertheless, a shortcoming of the studies if the elasticity behaviour is the normal assumption of materials that are obeying Hooke’s Law hence do not provide enough guidance for the theoretical design (Cognini, 2015). The intention of this project is to analyse the deep beams and find out alternative methods of reinforcing steel so as to minimize sizes of cracks in concrete. A deep beam is a beam with a short span of shear as shown in figure 1 below:
This shows that a deep beam has a small span which is donated by ‘a’ and depth donated by b ration of a/d that is less than two. Specifically, the right section of the figure 1 above would be considered a deep beam because of the way the beam would crack after load P has been loaded to on it. The cracks have the shape of bottles since the distance between the support and load is near. This distance is donated by span ‘a’ of the beam which is compared to its depth which is donated by during the process of identification of the deep span. The deep beams are utilized in the construction of the bridges in specifically their supports to hold the bridge span over a column (Cranston, 2010).
The deep beams should possess a minimum quantity of reinforcement which is equal to 0.3 percent of the area of cross-section in both the horizontal and vertical directions. It has been proved that this quality of the deep beam does not need any requirement of strength. However, this deep beam quality is needed in controlling the cracks or limiting the extreme crack width to be less than 0.016 inches when the services load is applied (Desayi. P. and Krishnan, 2012).
The analysis of beam failure is done to evaluate the lowest quantity of shear strength of the beam that is weak. The beam without any additional conventional reinforcement in the region to be tested is the best sample to be analysed on its failure analysis. The flexural analysis of the bean is done before the beginning of the process of gathering the values required to perform the shear analysis (Dipti R. Sahoo, 2013). The seep beam failure analysis is done with an aim of determining the beam’s strength before flexural or bending failure is provided with the dimensions. This analysis can be done by the use of the diagram in figure 2 below:
The left section of the image above shows the view of the cross-section of the deep beam and the right section of the beam shows the graph of the stress of the deep beam. The compression produced in the beam possess an equivalent value to the beams tension leading to the formation of the equilibrium. The compression forces are denoted by the arrows Cc and Cs as indicated in figure 2 above (Gerardo Aguilar, 2014). By setting the compression to be equivalent to the tension in the deep beam, the following equation can be derived:
C * β * b * f’c * 0.85 = fy * As = ?s * Es * As’
Where fy = reinforcement’s specified yield strength (psi)
As = area of longitudinal tension reinforcement that is prestressed
As’ = Compressional reinforcement area (in2)
Es = modulus of elasticity of structural steel and reinforcement (psi)
C = distance from neutral axis to extreme compression fibre (in.)
β = ratio of short to long dimensions; clear spans for slabs that are two-way
b = compression face width of the member (in.)
f’c = the compressive strength of the concrete which is specified (psi)
After solving the variables above, the value of c is found to be equal to 3.52 in. (Gerardo Aguilar, 2014)
The value of ‘a’ which is a parameter utilized in describing the rectangular stress block depth is found to be equivalent to c * β resulting in a value of ‘a’ to be 2.98. After the determination of ‘a’, this value was utilized in finding the values of ?s2 and ?s1 which denotes the longer node sections depicted in figure 3 below (Institution, 2014). The ?s2 and ?s1 were used in numerous equations in finding the values of the shear strength at every face of the node of the compression-compression-tension (CCT) and the node of compression-compression-compression (CCC) as shown in figure 3 below:
The CCT node and CCC node are close up views of the zones of the node of the deep beam. The figure above shows the connection between the nodes by a section of cracks that are bottle-shaped (John W. Vallace, 2013).
The modes of failure are critically determined by the concrete’s compression strength, web reinforcement, tension percentage, and sheer depth to span ratio. The failure modes of the deep beam can be categorised into flexural failure mode and shear failure mode (Kong, 2010). The shear failure mode can further be subdivided into the following groups:
Compressional or shear proper failure: This failure mode is normally witnessed in minute beams with minute shear depth to span ratio. If the span to depth ratio is small, the thrust line will be very steep and the action of the arch will efficiently maintain the force required and also reserve the capacity of the flex in most instances. The arch is normally witnessed in the failure of beams as a result of compression crush in the ordinary direction of the axes of the strut or unexpected formation of tensile crack parallel to the axes of the strut. (Précontrainte, 2013)
Shear compression failure: In this failure mode, the RC reinforced beam fails as a result of the diagonal crack development into the zone of compression and minimizing excessively the area of the resisting region and the deep beam crashes when the compressive stress generated surpasses the strength of compression of the deep beam.
Diagonal tension failure: In this mode of failure, the thrust line becomes unusual and lead to the failure of the flex in the zone of comprehension. It is critical to note that this failure mode is caused by extension of the tensile crack in the zone of comprehension because of flexural weight (Tuchschere, 2014).
The results gotten from the beam failure analysis are used in numerous equation in determining the shear strength of different faces of the node of compression-compression-tension which is abbreviated as CCT and node of compression-compression-compression which is abbreviated as CCC. There are two different types of nodal zones in the deep beam, namely CCC nodal zone and CCT nodal zone as shown in the figures below:
The node zones, CCT and CCC nodal zones are close up views as shown in figure 3 above which represents half a beam and portrays the connection of the nodes by a section of cracks which are bottle-shaped (Cognini, 2015). The section between the compression-compression-tension which is abbreviated as CCT and node of compression-compression-compression which is abbreviated as CCC is commonly referred to as test region of the deep beam. The thickness of the nodal zone should be equivalent to the member. The figure above shows a nodal zone. The horizontal and vertical forces equal to the forces in the strut inclined. In case the stresses are equivalent in all the three struts, then the hydrostatic zone of the node can be utilized and the strut’s width will be proportional to the stud’s forces (Gerardo Aguilar, 2014).
The table below shows a sample of the sheer results gotten forms the calculations done with an aim of determining the least value of shear of the beams mentioned above:
From the table above, it is clear that the least value of shear strength evaluated was 106.97 kips. This means that the beam that is weakest will have the least shear of approximately 107 kips, however, this does not mean that the beam will automatically fail at this point. This analysis of shear value was done with an aim of predicting the least shear strength of the nine beams used. The definite position of the strengths and cracks at which they will happen cannot be evaluated through analysis of the shear and will have to be evaluated after the occurrence of the tests (Gerardo Aguilar, 2014).
The cost of implementing deep beam which was carried out from the beginning to the end can be evaluated through the consideration of the costs of the following materials that will be used in the construction of the deep beam wood, steel, insulating blankets, concrete, steel fibres, bolsters, lifting inserts, and travel expenses. The cost of the implementation of the deep beam is as shown in the table below with an inclusion of the concrete despite not being considered as a part of the cost of the deep beam (Clarke, 2011).
Table 2: Cost Implementation of the Deep Beam
The strut and tie models are effective in designing and detailing various structural types of elements in the deep beam. This model is perfect in the representation of the structural mechanisms and behaviour. The following are some of the examples of the strut tie model representing arch mechanism, strut tie model representing vertical truss mechanism, and strut tie model representing combined mechanism (Desayi. P. and Krishnan, 2012). These examples of the deep beam are explained below:
In this model, the resultant topology improves with the increased iteration. To find out the ultimate topology which may be translated to the strut and tie model that is optimum, an index of performance that is considered as the objective function, may be derived based on the concept of the scaling design where the real design flexible like the thickness of the element is scaled in terms of the constraint of the design. The optimization of the topology of the deep beam can be illustrated as shown below:
Where V is the volume of the entire domain of the design and that of the element. It is critical to note that the current research that reduces the volume is equal to reducing the weight due to a single material type such as deep beam and plain concrete is presumed to occupy the whole domain of the design (Gerardo Aguilar, 2014).
This model of Strut tie was developed by Morsch and Ritter related to reinforced concrete beam in an equal structure of the truss. The elements that are discrete represent the tensile field and the stresses of compression which take place inside the structure of the element as the effect of bending. The model has been improved and is still being utilized by the standard that is technical in the reinforced concrete beams design in shear and flexural force and developing numerous methods for determination of the limits that is safe in its processes. With the indication of the topology gotten from the current formulation, a proposal for the model can be presented directly (Cranston, 2010).
In this model, a uniaxial stresses ties and struts possessing finite dimension are utilized in representing the real tensile and compressive field of stress. The pin connection combining the ties and struts together agree to the triaxially or biaxially zones of nodal stress. This model is normally implemented in design practice with an aim of predicting the strength and examining the equilibrium of the loads applied, internal forces and reactions for regions of the structures disturbed with frequent geometry where the internal movement of force is not known well. This model is able to capture the monotonic reaction of the deep beam beyond the elastic range. The interior force request established in the structural component may be efficiently assessed by the use of this model (Institution, 2014).
Conclusion
This report paper is about the deep beam through analysis of its failure analysis, failure modes, deep beam nodal zones, shear value results, implementation cost of the deep beam as well as examples of the deep beam. Some of the examples of deep beam include the Strut tie representing combined mechanism, Strut tie representing vertical truss mechanism, and Strut tie representing arch mechanism. The analysis of beam failure is done to evaluate the lowest quantity of shear strength of the beam that is weak. The cost of implementing deep beam which was carried out from the beginning to the end can be evaluated through the consideration of the costs of the following materials that will be used in the construction of the deep beam wood, steel, insulating blankets, concrete, steel fibres, bolsters, lifting inserts, and travel expenses.
References
ACI 318, 2. (2011). Building Code Requirements for Structural Concrete. Michigan: American Concrete Institute.
Clarke, J. a. (2011). Concrete opportunities for the structural engineer. Colorado: Stru.
Cognini, S. D. (2015). Viscoelastic moment-curvature. Michigan: Am. Con. Inst. Mat. J.
Cranston, W. (2010). Analysis and design of reinforced concrete columns. New York: Cement and Concrete Association.
Desayi. P. and Krishnan, S. (2012). The equation for the stress-strain curve for concrete. Toledo: Proc. Am. Conc. Inst.
Dipti R. Sahoo, C. A. (2013). The behaviour of Steel Fiber-Reinforced Concrete Deep Beams. London: ACI Structural Journal.
Gerardo Aguilar, A. B.-M. (2014). Experimental evaluation of design procedures for shear strength of deep reinforced concrete beams. Chicago: ACI Structural Journal.
The institution, C. a. (2014). Bibliography on deep beams. Michigan: Cement and Concrete Association.
John W. Vallace, S. W. (2013). Use of Headed Reinforcement in Beam-Column Joints Subjected to Earthquake Loads. London: ACI Structural Journal.
Kong, F. (2010). Reinforced concrete deep beams. Perth: Lecture delivered at Ove Arup.
Précontrainte, C. E. (2013). The design of deep beams in reinforced concrete. Paris: CIRIA Guide 2. Ove Arup & Partners and CIRIA.
Tuchschere, R. (2014). Minimum Web Reinforcement in Deep Beam. Moscow: ACI Structural Journal.
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