5.1 EN 1998-1:2004
According to EN 1992-1-1:2004, Clause 5.3.1(3) has definition “A beam is a member for which the span is not less than 3 times the overall section depth. Otherwise it should be considered as a deep beam.”
Deep beams should normally be provided with an orthogonal reinforcement mesh near each face, with a minimum of reinforcement ratio. The distance between two adjacent bars of the mesh should not exceed the lesser of twice deep beams thickness or 300 mm. Reinforcement, corresponding to the ties considered in the design model, should be fully anchored for equilibrium in the node in the design model, by bending the bars, by using U-hoops or by anchorage devices, unless a sufficient length is available between the node and the end of the beam permitting an anchorage length of lbd (Clause 9.7 in EN 1992-1-1:2004).
Where a non-linear strain distribution (discontinuity regions or D-region) exists (e.g. supports, near concentrated loads or plain stress) strut-and-tie models (STM) may be used (Clause 6.5.1(P) EN 1-1:2004). Strut-and-tie model is given in EN 1992-1-1:2004: Clause 5.6.4-Analysis with struts and tie models, and Clause 6.5-Design with strut and tie models.
5.2 ACI 318-08
The deep beams will be designed and detailed as member subject to shear load, flexural or axial load or to combined flexure and axial loads.
Clause 10.7.1 requires “Deep beams are members loaded on one face and supported on the opposite face so that compression struts can develop between the loads and the supports, and have either:
(a) clear spans, ln, equal to or less than four times the overall member depth; or (b) regions with concentrated loads within twice the member depth from the face of the support.”
The deep beams shall be designed either taking into account nonlinear distribution of strain, or by strut-and-tie models. Lateral buckling shall be considered.
Vn for deep beams shall not exceed: 10bwd(f’c)1/2.
For detail requirements of the deep beams, such as minimum area of flexural tension reinforcement, minimum horizontal and vertical reinforcement in the side faces of deep beams, are satisfied Clause 10.7.3, 10.7.4, respectively. The anchorage of positive and negative moment tension reinforcement in the deep beams subject to flexural load shall be designed in accordance with Clause 12.10.6 (Adequate anchorage shall be provided for tension reinforcement in flexural members), and 12.11.4 (Anchorage at simple support, interior support), 12.12.4 (Anchorage at interior support).
The longitudinal reinforcement in the deep beams should be extended to the supports and adequately anchored by embedment, hooks, or welding to special devices; and bent-up bars are not recommended. And Appendix A-Strut-and-Tie Model (ACI 318-08) shall be applied for other design and detail requirements for the deep beams.
5.3 Discussion
Firstly, as regards to analysis model for the deep beams, the EN 1992-1-1:2004 (not EN 1998-1:2004) and Chapter 10, ACI 318-08 introduce on STM method for determining the shear strength of reinforced concrete deep beams. The main concepts of
STM are struts (compression member), ties (tension member), and nodes (joint member) (Nagarajan et al. [52]). The STM for the deep beams (Figure 5.1) consists of two concrete compressive struts, longitudinal reinforcement serving as a tension tie, and joints referred to as nodes. The concrete around a node is called a nodal zone. The nodal zones transfer the forces from the inclined struts to other struts, to ties and to the reactions (Bower [53]). Nodes are described by the type of the members that intersect at the nodes. For example, a CCT node is one, which is bounded by two struts (C) and one tie (T). Nodes are classified as CCC, CCT, CTT or TTT (Nagarajan et al. [51]).
Principle of STM method is equilibrium condition only and STM is applied for design of local regions (Hsu [54]). The STM shall be in equilibrium with the applied loads and the reactions.
For inclined angle between the strut and the tie, ACI 318-08 requires the smallest angle between the strut and the tie in a D-region is arctan(1/2)=26.5 degrees, rounded through 25 degrees, and inclined angle is recommended : 25o65o. While for EN 1992-1-1:2004, =31o though 59o. The inclined angle is not allowed less than
min in preventing too long tie and too short strut. ACI 318-08 does not contain detailed requirements for designing deep beams for flexure except that nonlinearity of strain distribution and lateral buckling is to be considered. Suggestions for the design of deep beams for flexure are given in References 10.22, 10.23, and 10.24. (ACI 318-08, R10.7).
Clause 11.8 (ACI 318-08) applies only to single span deep beams. Continuous deep beams can be applied STM. About reinforcement layout in the deep beams, longitudinal bars should be arranged along the height of the deep beams to support the transverse reinforcement along the deep beam axis, not only arranged at the bottom of the deep beams to subject bending moment as normal beams (Nawy [55], Tuan [56]). In
design, shear force in the deep beams is major consideration. The ratio and space of both the vertical and horizontal shear reinforcement differ considerably from those used in the normal beams, as well as the expressions that have to be used for their design (Nawy [55]). According to Kong [57], the deep beams can be designed as normal beams in flexural strength. While for shear strength, can not use formulas of the normal beams to calculate the deep beams, and STM will be applied. According to ACI 318-08, the design principle is based on:
Vu Vn (5-1)
Vn=Vc+Vs (5-2)
where: Vu is the design shear force at the critical section;
Vn is the nominal shear strength;
Vc is the shear strength provided by concrete;
Vs is the shear strength provided by steel;
is the capacity reduction factor for shear, taken as 0.75.
In simply supported deep beams, design of shear strength is carried out for the critical section. But for continuous deep beams, shear strength design are not based on the design shear force at the critical section. Instead, the shear reinforcement at any section is calculated from the design shear force Vu at that section. And for continuous deep beams, the concrete nominal shear strength Vc is to be taken not the same with simply supported deep beams (Kong [57]).
Following the increased interest in STM regarding complex load states in high rise buildings, general methods for the application of STM began to appear. This method provided basic concepts and tools that could be applied to complex structures for designs based on behavior models. STM began appearing in North American codes for general design use. The Canadian CSA A23.312 was the first to adopt STM in 1984.
STM have been applied in the EN code before the ACI code. ACI introduced STM provisions in the 2002 edition (ACI 318-02). The STM provisions of ACI 318-05 were written largely by compiling information and provisions from European codes (Brown and Bayrak [58]). The ACI 02 is only for reference and introduction, and ACI 318-08 will be officially applied. The STM method uses the principle of lower bound which gives a conservative result. Simplicity of STM in modeling and analysis makes it a valuable tool that may be used by almost students and structural engineers for design of complex or unusual structural concrete members. The STM also has many difficulties;
one of them is that the sketch of trajectories of the principle stress distribution in reinforced concrete components. This should have experience in the process of selecting a model for specific components. For one component may have many different models for calculating and will give different results. The combined with finite element method to determine accurately the trajectories of principle stress distribution and based on model that is selected model to reflect the proper working of the discontinuous regions (Tuan [56], Ley et al. [59]). The greatest difficulty of STM is to build a good model for the considered element or region. One of the most important issues here is the correct selection of inclination of compressive struts (Baczkowsk [51]).
Continuous deep beams occur as transfer girders in reinforced concrete frames in high rise buildings, as pile caps and as foundation wall structures, etc. The definition of continuous deep beams is not formally in EN 1998-1:2004 and ACI 318-08. Frederick and Jonathan [60] shown that ACI 318 defines deep beams as flexural members with clear span-depth ratios less than 2.5 for continuous spans and 1.25 for simple spans.
According to Kong [57], the ACI deep beams definition is based on shear behavior while EN definition is based on flexural behavior. When considering the design recommendations in two codes will recognize the different definitions. Continuous deep
beams behave differently from either simply supported deep beams or continuous shallow beams. Continuous deep beams develop a distinct ‘tied arch’ or ‘truss’ behavior not found in shallow continuous beams. And the result of this is that detailing rules of conventional reinforcement, based on shallow beams or simply span deep beams, are not necessarily appropriate for continuous deep beams. In simple span beams, the region of high shear coincides with a region of low bending moment. In continuous beams, the locations of maximum negative moment and shear coincide, and the point of inflection may be very near the critical section for shear. At an interior support in a continuous beam, the zone of high shear and high negative bending moment coincide.
These differences cast further doubt on the usefulness of empirical equations based on simple span experimental test data. The usual design practice for continuous deep beams has been to employ empirical equations, which are invariably based on simple span deep beams experimental data tests.
For design of the deep beams, especially for practicing engineer, it is recommended that the equilibrium truss model to be use, Zhang and Tan [61] presented STM method for two span continuous deep beams using truss model as shown in Figure 5.2.
The experimental tests by Rogowsky et al. [62] on continuous deep beams with column stub showed that the load capacity of continuous deep beams would not be properly estimated by formulas developed for simple deep beams. Therefore, proper design of continuous deep beams would require further investigations to understand the influence of various parameters on their load capacity. Brown and Bayrak [58, 63]
checked results of 596 experimental tests of beams with shear span-depth ratios less than two were compiled from the technical literature. Laboratory tests of beams with small shear span-depth ratios are among the simplest types of structural members for
which strut-and-tie is appropriate. Typically, laboratory specimens are subjected to one or two concentrated loads and have simple supports. In fact, however, shapes of deep beams or transfer beams are complex, they are not simple as experimental tests, examples and analysis data in papers and text book. Yang and Ashour [64] investigated total of 75 two spans with top loaded reinforced concrete deep beams were compiled from different sources, such as Rogowsky et al [62]-1986, Ashour-1997, Subedi-1998, Asin-1999, and Yang et al.-2007. Figure 5.3 illustrates STM for continuous deep beams on ACI 318 code. All beams were reported to fail in shear due to a major diagonal crack within interior shear spans, joining the edges of load and intermediate support plates.
Figure 5.4 showed crack patterns in simple and two span deep beams tested by Rogowsky et al. [62].
Many experimental tests, design examples; results of analysis, checking and evaluation; proposals and recommendations, for a span and two span deep beams, deep beams with opening, under one and two point loadings, have been done by researchers, such as Untrauer and Siess [65], Rogowsky et al. [66], Ove Arup [67], Nawy [55], Rogowsky et al. [62], Hwang and Lee [68], Kong [57], Reineck [69], Aguilar et al. [70], Wight and Parra-Montesinos [71], Nilson et al. [33], Dirar and Morley [72], Singh et al.
[73], Ong [74], Zhang and Tan [61, 75], Yang and Ashour [64], and Arabzadeh et al.
[76]. Since, it is difficult to practicing structural engineers in the fact. Some design recommendations for ACI 318-05 has introduced by Brown and Bayrak [63] after they checked results of 596 experimental tests. Particularly, experimental tests have been done by Ley et al. [59], Untrauer and Siess [65], etc. with bearing plates at loading point and support locations as shown in Figure 5.5, and other ones have been carried out by Rogowsky et al. [62, 66] with column stubs as shown in Figure 5.6, column stubs are more clearly similar to practice model than bearing plates. Reineck [69] collected and
presented many examples for the structural concrete design with STM according to ACI 318 code, in which bearing plates are provided at all loading and support locations as shown in Figure 5.5.
There are some the methods and its discussions for deep beams analysis, such as elastic analysis, finite element analysis, ACI 318 (1977), Kong, Robins and Sharp (1975), truss models (Kong 2002), nonlinear finite element analysis (Dirar and Morley [70]), Strut-and-tie design methodology for three-dimensional reinforced concrete structures (Leu et al. [77]), 2D finite element analysis (Ong [74]), micro truss model (Nagarajan et al. [52]). Quangfeng and Hoogenboom [78] also have been presented other method, namely “stringer-panel model”. Method of “stringer-panel model” is intermediate method between strut-and-tie models and finite element method (Tuan [56]). There are some other methods based on STM: Softened strut-and-tie models of Hwang and Lee [68], modified strut-and-tie models of Zhang and Tan [61], simple strut-and-tie models of Arabzadeh et al. [76]. According to Arabzadeh et al. [76], some existing methods are simplified softened truss model of Mau Su (1989), combined softened STM of Matamoros et al. (1986), formula proposed by Foster-Gilbert based on plastic truss model (1996). There are three methods for formulating strut-and-tie models: Elastic analysis based on stress trajectories, load path approach, experimentally test. Table 5.3 shows existing problems for the deep beams in practice.
Basic design procedure for a structures using STM according to ACI 318-08 Appendix A is as following below:
(i) Definition of structural system, determination of loads and reactions, estimate dimensions and sizes of members.
(ii) Definition of B and D-regions.
(iii) Design for B-regions as normal beams in flexural strength (by other methods).
(iv) Design for D-regions: Design of struts, ties, and nodal zones shall be based on:
The effective compressive strength of the concrete in a strut:
fce = 0.85sf’c (5-5)
- The nominal strength of a longitudinally reinforced strut:
Fns = fceAcs + A’sf’s (5-6)
(b) For ties:
- The nominal strength of a tie shall be taken as
Fnt = Atsfy + Atp(fse + fp) (5-7) where (fse + Δfp) shall not exceed fpy, and Atp is zero for nonprestressed members.
(c) For nodal zones:
- The nominal compression strength of a nodal zone shall be
Fnn = fceAnz (5-8)
The effective compressive strength of the concrete in the nodal zone:
fce = 0.85nf’c (5-9)
(v) Arrangement and detail of reinforcements.
Appendixes A shows calculation for examples of simple deep beams using data tested by Rogowsky et al. [62] according to ACI 318-08, Appendix A. Model of specimens was similar to practice shape about column stubs. Results are quite conservative, it may be due to data of experimental test had been done for long time ago.
Almost experimental tests are listed above subjected to monotonic loadings.
Since, current provisions of STM are based on theoretical and/or experimental tests that depart from assuming monotonic loadings. Structural members, however, are often subjected to cyclic demands, such as earthquake loading. There is little evidence on the behaviour of members designed using STM and subjected to reversed cyclic loads. The doubts therefore have been cast upon STM being appropriate for seismic design. The adequacy of using STM for seismic design was assessed by Alcocer and Uribe [79]. In four tests, due to loading equipment availability, upward loading (negative direction) was limited to approximately half of the maximum load that could be applied in the positive direction. Figure 5.7 demonstrate final crack patterns for all deep beams, first two beams (MT, MR) were tested under monotonic loads, and last two beams (CT, CR) were tested under reversed cyclic loads. Hysteretic loops of beam CT and CR are shown in Figure 5.8. Hysteretic loops show considerable pinching, especially at deflections of 20 mm, and severe stiffness degradation. The hysteretic loops observed are typical of structural members failing in shear. Average rotations at beam strength under positive loading were 2.3% and 1.6% for beams CT and CR, respectively. First cracking, first yielding, and last yielding recorded before failure are also indicated in Figure 5.9. The curves show a nearly elastic behavior up to a deflection of approximately 9 mm. At this deflection, the first yielding occurred in the transverse reinforcement (beams MT, CT, and CR) or in the longitudinal reinforcement (beam MR). In terms of strength, stiffness,
and deformation capacity, the performance of all four beams tested exceeded expectations, thus verifying the reliability of STM for structural members subjected to either monotonic or reversed cyclic loads. The STM may be used for seismic design of structural members subjected to reversed cyclic shear demands up to 0.42√f’c (MPa) (5.0√f’c [psi]) and to inelastic deformation demands up to 2.3%. For these cyclic deformations and shear demands, code provisions on strut and node strengths need not be altered. This conclusion departs from an assumed failure mode controlled by yielding of the tie reinforcement. The predicted strengths applying the provisions of Appendix A of ACI 318-05 were smaller than the actual strengths obtained in the laboratory by an average of 27%. One of the most important from this experimental test is that current provisions, ACI 318-05, for STM may be used in seismic design.
Regardless of analysis problems of transfer structures as transfer beams, according to Puvvala [80], unlike normal deep beams, there is no particular span to depth ratio for estimating structural behavior and failure mechanism of transfer beams.
Transfer beams behave either as full tension, deep beams or as normal beam in flexural moment depending on type of upper structure form as well as relevant parameters such as span to depth ratio of transfer beams, stiffness of support columns, span of shear wall and degree of coupling on the coupled shear wall… There are 3 major problems are introduced by Puvvala [80]: Single-span and two-span continuous transfer beams supporting in-plane loaded shear walls (Figure 5.10, 5.11), transfer beams supporting in-plane loaded equal and unequal coupled shear walls (Figure 5.12), and frame structures supported on transfer beams (Figure 5.13). Method of analysis, modeling by finite element method, structural behavior, relevant parameters, and stiffness of the coupling beams in case of coupled shear walls also presented. Transfer beam should be
designed as flexural-tension members, but not ordinary beams in bending or normal deep beams. Li [81] used non-linear finite element analysis to investigate failure modes, failure loads and load transfer mechanism, and introduced some formulas for problem of transfer beams support an in-plane loaded shear wall. This mode of failure changes from shear failure to flexural-shear failure then turning into flexural failure according to different span to depth ratio or width of beam. When the stiffness of column is enough, the transfer beams likes fixed ends beam. Otherwise, it likes simple supporter beam.
With proposed formulas, flexural moment and axial force occurred at mid-span of transfer beams may be calculated simply, quickly, effectively. Table 5.4 and Figure 5.14 indicated that, with the width of beam is fixed at 2 times of the width of shear wall, and span to depth ratio of transfer beams range from 2 to 12, cracking load decrease slowly with increase in the span to depth ratio, so that change of ratio for span-depth has little effect to cracking loads. Figure 5.15 demonstrated that failure load decreases very little when span to depth ratio change from 2 to 3. Nevertheless, failure load then decreases quickly with span to depth ratio from 3 to 12. It means that the depth of transfer beams is larger than 1/3 of total span, it may not be useful to increase the depth for getting larger failure load. Regardless of changes for the width of transfer beams, with span to depth ratio of the beam fixed to 6, the width of transfer beams ranges from 0.5 to 5
With proposed formulas, flexural moment and axial force occurred at mid-span of transfer beams may be calculated simply, quickly, effectively. Table 5.4 and Figure 5.14 indicated that, with the width of beam is fixed at 2 times of the width of shear wall, and span to depth ratio of transfer beams range from 2 to 12, cracking load decrease slowly with increase in the span to depth ratio, so that change of ratio for span-depth has little effect to cracking loads. Figure 5.15 demonstrated that failure load decreases very little when span to depth ratio change from 2 to 3. Nevertheless, failure load then decreases quickly with span to depth ratio from 3 to 12. It means that the depth of transfer beams is larger than 1/3 of total span, it may not be useful to increase the depth for getting larger failure load. Regardless of changes for the width of transfer beams, with span to depth ratio of the beam fixed to 6, the width of transfer beams ranges from 0.5 to 5