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1st Civil and Environmental Engineering Student Conference

25-26 June 2012 Imperial College London

Strut and Tie Modelling of Reinforced Concrete Short Span

Beams

W.T. Yap ABSTRACT Strut-and-tie modelling of reinforced concrete structures is used in the design of discontinuous or D-region structural elements which includes short span beams of clear shear span to depth ratio of less than 2. For those beams, shear is the critical mode of failure and Eurocode 2 allows for its design using strut-and-tie models (STM). However Eurocode 2 provides very little guidance in using strut-and-tie models, which covers mainly the effective concrete strength provisions for the various strut-and-tie elements. Comparing different simple strut-and-tie models to Eurocode 2 sectional model with regards to shear capacity predictions shows that the traditional sectional method gives conservative estimates. Among the various chosen strut-and-tie model methods, the model according to Eurocode 2 with concrete strength estimates based on the Modified Compression Field Theory (MCFT) yields the most accurate predictions. This paper discusses the applicability of the different design methods for shear and provides some design recommendations in designing short span beams without shear reinforcement using strut-and-tie models. 1. INTRODUCTION

The shear capacity of a reinforced concrete beam without shear reinforcement has always been the focus of research due to the importance of accurately predicting the shear failure. Due to the sudden and brittle failure nature of shear, it is required that structures are designed to ensure the shear capacities are far above the structures flexural strength. This ensures that the failure mode by flexure will give significant ductility to the structure and be gradual, therefore giving ample warning to the occupants.

Shear failure can only occur once the concrete in beam develops diagonal tensile cracks. The cracked state of concrete is difficult to analyse as the internal forces are redistributed and conventional elastic stress analysis will not apply. In order to analyse the behaviour for a cracked concrete state, the truss analogy is used. This was first introduced by Ritter (1899) and Mrsch (1909) and then later developed as a design method for shear and torsion by Collins & Mitchell (1980). The method however only applies for regions of a structure where Bernoullis hypothesis, which states that plane sections remain plane after deformation, is applicable. This region is known as Bernoulli or B-region.

Regions where the beam theory does not apply are known as discontinuity or D-regions. These occur at regions of geometrical or static discontinuity, such as openings, changes in cross section or near concentrated loads and reactions. The strain distributions for this section will not be linear and the length is usually governed by St.

Venants principle. Examples of structural elements that fall into this category are short span beams, corbels, beam to column connections and beam openings.

According to Schlaich, Schafer & Jennewein (1987), the standard practice for D-region design is based on experimental data, rule of thumb and past design experiences. The behaviour has not been well understood until Schlaich et al (1987, 1991) developed the truss analogy further to include D-region of a structure. This method is known as strut-and-tie models.

Using strut and tie models as a basis to design D-region concrete structures eliminates uncertainty by having a much more systematic approach, since it is based on physical models which are more understandable. The method also gives flexibility to the designer in terms of going for either the cheapest or safest solution (Schlaich & Schafer, 1991).

Short span beams are considered D-region structures and therefore can be analysed using strut-and-tie models. This paper will focus on modelling simply supported short span beams, where the clear shear span to depth ratio / is less than two, using different strut-and-tie-models to predict shear failure. 2. OBJECTIVES

The main objective of this paper is to review different simple strut-and-tie model methods that can be used to design short span beams without shear reinforcement for shear in accordance to

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Eurocode 2 (BSI, 2004). The models will be based on different failure mode assumptions and strength parameters which can be easily implemented in practical applications for short span beams. The effectiveness in predicting shear failure loads will be compared with existing sectional design provisions for shear according to Eurocode 2. The details of the differences are looked at in the form of a parametric study where varying parameters can show in detail how the models perform and their sensitivity in the predicted shear failure loads. The models will also be compared to experimental findings based on tests conducted by Rogowsky, MacGregor & Ong (1983) on short span beams. The outcome is coming up with suitable design recommendations when it comes to designing short span beams using strut-and-tie models. This paper will also look at the shear failure behaviour for beams without shear reinforcement and it implication in design, along with the current state of the art in strut-and-tie modelling of reinforced concrete structures. 3. LITERATURE REVIEW 3.1. Shear Failure Behaviour

A reinforced concrete beam without shear reinforcement can only be subjected to shear failure once the beam exhibits inclined tension cracking, which follows vertical flexural cracking. This occurs when the principal tensile stresses exceed the concretes tensile strength at the final stages of the elastic stage. The behaviour of this type of beam after cracking and way it fails in shear depends mainly on the shear span to depth ratio /. Taking an example of a simply supported beam with symmetrical two point loading without shear reinforcement, the shear at flexure failure, shear failure and for inclined cracking is plotted against varying shear span with constant depth .

Figure 1. Shear at cracking and failure - adapted from (Wight &

Macgregor, 2012)

With reference to Figure 1, for very slender beams of / greater than 6.5, the beam will fail by flexure before any inclined cracking is developed. For short beams of / between 2.5 and 1, once the loading has reached the inclined cracking load, internal forces are reoriented to create a tied arch, changing from beam to arching action. This

explains why for these type of beams, additional loading can still be carried once the inclined cracks are formed. As for the slender beam region of / between 2.5 and 6.5, failure loads will be when the inclined cracks start to form. This is due to the compression strut, which is required for arching action, not forming due to the long shear span distances. Forces are unable to be transferred from the point of loading to the supports and therefore will fail once beam action can no longer exist. Deep beams on the other hand, with / less than 1, exhibit cracking that runs from the loading to the support. This eliminates the shear flow completely and the beam has arching action behaviour. 3.2. Discontinuity Regions

The reason for short span beam behaviour can be explained by St. Venants principle. This states that the localised effect due to a disturbance, like a point load or reaction, will dissipate at a distance of one beam depth away from the load in question. The localised disturbance is what causes the beam to behave differently, to behave as arching action. This region of one beam depth from each point of disturbance, be it a point load or reaction, is classified as a D-region. Therefore short span beams with shear spans that are equal to or shorter than two beam depths dictates that the whole beam will behave like a D-region, which in this case is arching action. 3.3. Strut-and-Tie Models

Once the concrete has cracked in the D-regions, there will be a redistribution of internal forces to maintain equilibrium. In order for the region to be analysed and designed effectively, the flow of forces can be modelled using a strut-and-tie model Figure 2. This is a truss visualisation of the forces which consists of compression struts and tension ties, which are carried by the concrete and tensile steel reinforcement respectively. These members are interconnected at joints known as nodes and are surrounded by nodal zones that transfer forces between struts to ties and reactions (ACI, 2002). It is essentially a model of the tied arch that is formed by arching action. The strut-and-tie model that represents the structure should satisfy the condition of adhering to statics, such that the internal forces are in equilibrium with the applied loading. The second condition is the failure criteria, where the factored design forces in the elements such as struts, ties and nodes must be within limits of the design strength of these elements as the basis for limit analysis. Thirdly, the concrete must be able to withstand sufficient rotation and not yield before the assumed stresses develop in the struts and ties. In other words, the concrete must not crush before the tensile reinforcement yields (Muttoni, Schwartz & Thurlimann, 1997). Upon meeting the

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above three conditions, the model will follow a lower bound theorem of plasticity in which solution is a safe estimate below the actual collapse load. The difference between the truss analogy and strut-and-tie model is therefore clear, given that the latter is simply a set of forces in equilibrium and do not form a stable truss system (Chen & EI-Metwally, 2011). Hence, the truss analogy can be said to be a special case of the strut-and-tie model.

Figure 2. Strut-and-tie model of a short span beam adapted

from (ACI, 2002)

3.4. Modified Compression Field Theory

Predicting the shear strength of a concrete beam has always been done through the use of simple beam tests to form a theoretical mode in for design. Collins, Mitchell & Bentz (2008) feels that although the tests are simple in nature, such as just testing beams with single or double point loading, the results are difficult to use and apply to different beam and load cases. This is when Vecchio and Collins (1986) came up with the modified compression field theory which was derived through testing reinforced concrete elements in pure shear. It was found that diagonally cracked concrete has a softening effect compared to concrete that goes through a standard cylinder test. Moreover, the differences in the principal compressive stress and strain in the concrete, which gives an idea of the concrete strength, are influenced by the level of principle tensile strain. This can be visualised with the understanding that the concrete crushing strength is derived from a uniaxial compressing load, and with concrete being loaded biaxially with an additional lateral tension force, there is a pulling out effect which decreases the compressive strength. The results give rise to new sets of stress-strain relationships for diagonally cracked concrete using equilibrium and compatibility of strains equations. 3.5. Eurocode 2 STM Provisions

The code provides guidance for establishing the effective concrete strength values to use in the struts and nodes for a specific internal force

Table 1. EC2 design concrete compressive strength provisions for STM elements

STM Element

Effective concrete strength

Strut

Without transverse tension

With transverse tension

0.6

Node

Compression node without ties (C-C-C)

Compression node with ties in one direction

(C-C-T)

0.85

where is the design concrete strength; = 1 /250;

is the concrete cylinder strength.

condition and arrangement, as summarised in Table 1.

3.6. Eurocode 2 Sectional Model

The shear strength of slender beams without shear reinforcement is determined from Equation 1.

, =0.18

(100. 1 )

1/3 (1 + 200/) (1)

where is the concrete partial safety factor; 1 = /(); is effective depth and is member width. To account for the increased strength in short span beams due to arching action, the code specifies the reduction in shear force by a factor of /. 4. PROPOSED STRUT-AND-TIE MODEL

The proposed STM for a simply supported beam without shear reinforcement can be seen in Figure 3, while its geometry is seen in Figure 4. The nodal stresses and the relevant concrete strength at the nodes are given in Figure 5.

Figure 3. Simple strut-and-tie model arrangement

Figure 4. Strut-and-tie model geometry, with being the area

of tensile reinforcement

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Figure 5. Nodal stresses: (a) Top node (C-C-C); (b) Bottom

node (C-C-T)

5. DERIVATION OF MODEL FOR FAILURE LOAD PREDCTION

There are three major failure classification modes in a strut-and-tie mode: direct strut crushing, bearing failure and flexural failure. The failure load can be calculated by first determining the critical failure mode.

5.1. Eurocode 2 STM (1)

According to Sagaseta & Vollum (2010), the majority of beams that have been tested through experiments seem to fail by crushing of the direct strut. The critical strut location will be at the bottom node as the strength is more critical than due to the presence of transverse tension and cracking, as per EC2. The stress at the rear face of the top node is also assumed to reach the concrete strength. Therefore setting 2 = and 3 = through geometry and using EC2 provisions gives Equations 2 and 3 respectively.

= 2 ( sin2 + sin 2) (2)

= 4 tan [ ( 4 ) tan ] (3)

The failure load is is found by solving Equations 2 and 3 simultaneously. 5.2. Eurocode 2 STM (2)

Failure is again assumed as crushing at the bottom strut but this time limiting the top strut strength to be the same as the bottom, instead of a higher capacity according to EC2, the strut will be an idealised prismatic one. The limited strut capacity can account for cracking in the direct strut. Taking 2 = , Equation 4 is formed.

= ( sin2 + sin 2) (4)

The failure load is is found by solving Equations 2 and 4 simultaneously with = . 5.3. Eurocode 2 STM (3)

Here, the strength of the bottom strut is derived using Modified Compression Theory. As recommended by Collins and Mitchell (1991), from the compressive stress-strain equations developed for diagonally cracked concrete and assuming the maximum principal concrete compressive strain at

failure as 0.002, the concrete strength of the strut can be expressed as

/(0.8 + 170 1) (5)

where 1 is the principle tensile strain in cracked concrete calculated from Mohrs circle of strain given as

1 = + ( + 0.002) cot2 (6)

where is the tensile strain in the tie passing through the strut and is the angle of the strut from the tie. Taking as Equation in Equation 2, the failure load is can be found by iterating different values of such that Equation 3 is satisfied. A thing to note is that the area of tensile reinforcement for the purpose of calculating tensile strain, is taken as the minimum steel required from the tensile force as derived in EC2 STM (1). 6. PARAMETRIC STUDY

In order to evaluate the competence of the different analysis methods, a parametric study is conducted. By changing different particular beam parameters, the sensitivity of how each method performs in terms of shear strength will become known and give a broad view of its variability. The suitable ranges for application can become clear. Arbitrary beam dimensions are taken to Table 2. For the calculation of STM (3), the area of steel from STM (1) is used instead of STM (2) to calculate the tensile strains in the tie which are then used to calculate the strength of the diagonal strut. This ensures that the calculated shear strength is the critical one and only the minimum amount of steel required for equilibrium is provided. Table 2. Beam dimensions. (All dimensions in mm)

1400 200 500 450 50 700 500 200 200

6.1. Top Node Strength

With regards to STM (1) and STM (3), the strength of the top node is taken as

as per Eurocode 2 guidance. The shear capacity of the beam with top node strength set with different coefficients such as 1, 0.85 and 0.6 is seen in Figure 6. The shear capacity will be proportional to the strength of the top node and a larger top node height is required for a reducing top nodal strength to maintain equilibrium. STM (2) is governed by the strength in the direct strut and not the top node, therefore not compared. 6.2. Bearing Length

The bearing length is an important parameter such that it can vary substantially depending on the particular beam configuration. The bearing length is influenced by the size of the columns or beams in which the loading is applied. (Figure 7)

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Figure 6. Shear capacity of STM (1) & STM (3) with varying top

node strength and corresponding top node height

Figure 7. Shear capacity with varying bearing length

6.3. Maximum Tensile Reinforcement Strain

The maximum tensile strain in the tie is plotted against the shear strength using STM (3) in Figure 8. The marked point A shows the steel strain obtained from /( ), where is the area

of steel calculated in STM (1) required for equilibrium in the tie and is the tensile force. Point B is the point where the steel strain is calculated by / assuming the tensile bars have reached yield point and maximum possible strain is achieved. 6.4. / Ratio

The applicability of STM (1), (2), (3) and Sectional Model for shear failure load prediction can be looked at for different general cases by varying the clear shear span to depth ratio / of the beam to include short span beams (/ 2) and slender beams (/ > 2), with 2% flexural reinforcement, while keeping constant (Figure 9). 7. EXPERIMENTAL VALIDATION

STM (1), STM (3) and EC2 Sectional model will be used to model actual simply supported short span beams sans shear reinforcement that have been tested by Rogowsky, MacGregor & Ong (1983). STM (2) will not be used as the failure predictions are very close to that of STM (1). The failure loads

Figure 8. Shear strength as a function of maximum tensile

reinforcement strain for STM (3)

Figure 9. Shear capacity with varying clear shear span to depth

ratio with 2% flexural reinforcement ratio

due to shear as calculated from the three models is compared with the tested failure loads . The results are plotted in Figure 10.

Figure 10. / versus / for STM (1), STM (3) and

Sectional Model (COV Coefficient of Variation)

8. CONCLUSIONS

The presence of a tied arch developing after cracks have been formed is the concept behind the representation of a beam with a strut-and-tie model. A truss model is created for a simple supported beam that satisfies equilibrium with different concrete strength criterions according to Eurocode 2 provisions and the Modified Compression Field Theory. The EC2 model is

80 100 120 140 160 180 200

2.6

2.8

3

3.2

3.4

3.6

3.8

x (mm)

V/b

d (

MP

a)

0.6 v fck

0.85 v fck

v fck

STM (1)

STM (3)

0 0.2 0.4 0.6 0.8 11.5

2

2.5

3

3.5

4

4.5

5

5.5

l/av

V/b

d (

MP

a)

STM Model (1)

STM Model (2)

STM Model (3)

0 0.5 1 1.5 2 2.5 3

x 10-3

2.6

2.8

3

3.2

3.4

3.6

3.8

4

t

V/b

d (

MP

a)

(A) T/(Asreq

E)

(B) fy/E

0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

av/d

V/b

d (

MP

a)

STM (1)

STM (2)

STM (3)

EC2 Sectional Model

EC2 Flexure

STM (1) Mean: 1.15 COV: 45% STM (3) Mean: 0.93 COV: 20% Sectional Model Mean: 0.58 COV: 12%

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further divided into two separate models based on the different governing failure modes. It was found that the assumption of a prismatic strut being developed is more conservative than only just assuming the strength of the direct strut is reached at the bottom for failure. STM using MCFT is found to be much more accurate in strength predictions as it is based on more parameters such as the orientation of the direct strut with the ties and the tie strains, governed by the amount of tensile steel reinforcement provided. Parametric studies have shown that strut-and-tie models are dependent on parameters such as concrete cover and bearing length, parameters which are not normally accounted for in the EC2 sectional theory. The EC2 strut-and-tie-model provisions have shown to have an undefined / ratio region where the model is applicable due to a discontinuity in the transition from slender to short span beams. This case is not seen in strut-and-tie model which incorporates modified compression field theory. This region has the potential to provide unsafe shear strength predictions. The experimental observations confirm the presence of arching action seen from constant steel strains in the longitudinal reinforcement. Test result comparisons enforces the inference that strut-and-tie models using modified compression field theory gives more accurate predictions while the Eurocode 2 strut-and-tie model can be unsafe when / ratio is in between 1 and 2. 9. DESIGN RECOMMENDATIONS

Under the guidance of design using EC 2 STM provisions, the procedure for estimating the concrete direct strut strength should be done in accordance to Modified Compression Field Theory. This will achieve not only accurate shear strength predictions but also give consistent results for beams with a clear shear span to depth ratio of less than 2. The maximum strains in the tensile reinforcement can be assumed to be strains at the point of yielding. The Eurocode 2 Sectional model provides very conservative shear strength predictions and the maximum possible shear strength obtained with this method can be used as an upper bound limit for the shear strengths calculated using strut-and-tie models. Despite of the fact that the beam still has the additional shear capacity, according to the strut-and-tie models, the limits should be taken for safety to account for the variability in shear failure in the randomness of diagonal shear crack formation. Adequate anchorage should also be provided for the tensile steel reinforcement at the supports to prevent premature reinforcement slip failure. If design is to be made purely on EC2 STM provisions, it should be done for beams that have a clear shear span to depth ratio / of less than 1, which are considered as deep beams. This is to

avoid any unsafe predictions in the shear strength. For 1 / 2, the effective concrete strengths of the direct strut from Modified Compression Field Theory should be used. Anything above the / range of 2 ought to be designed with the Eurocode 2 Sectional model. ACKNOWLEDGEMENTS

I would like to thank my supervisor, Dr. Robert Vollum for the invaluable guidance and advice given for this project. REFERENCES

American Concrete Institute. (2002) Examples for the Design of Structural Concrete with Strut-and-Tie Models. Report number: SP-208.

British Standards Institution (2004) EN 1992-1-1:2004. Eurocode 2: Design of Concrete Structures. Part 1-1: General Rules and Rules for Buildings. London, BSI.

Chen, W. F. & El-Metwally, S. E. (2011) Understanding Structural Engineering: From Theory to Practice. Florida, CRC Press.

Collins, M. P. & Mitchell, D. (2008) Shear design of concrete structures. The Structural Engineer. (May), 32-39.

Collins, M. P. & Mitchell, D. (1991) Prestressed Concrete Structures. New Jersey, Prentice Hall.

Collins, M. P. & Mitchell, D. (1980) Shear and Torsion Design of Prestressed and Nonprestressed Concrete Beams. PCI Journal. 25 (5), 32-100.

Collins, M. P. & Vecchio, F. J. (1986) The Modified Compression Field Theory for Reinforced Concrete Elements Subjected to Shear. ACI Journal. (March-April), 219-231.

Morsch, E. (1909) Concrete-Steel Construction: Der Eisenbetonbau. Trans.Goodrich, E. P. , Kessinger Publishing.

Muttoni, A., Schwartz, J. & Thurlimann, B. (1997) Design of Concrete Structures with Stress Fields. Basel, Birkhuser.

Ritter, W. (1899) Die Bauweise Hennebique (Construction Techniques of Hennebique. Schweizerische Bauzeitung. 33 (7).

Rogowsky, D. M., MacGregor, J. G. & Ong, S. Y. (1983) Tests of Reinforced Concrete Deep Beams. University of Alberta, Edmonton. Report number: 109.

Sagaseta, J. & Vollum, R. L. (2010) Shear design of short-span beams. Magazine of Concrete Research. 62 (4), 267-282.

Schlaich, J. & Schafer, K. (1991) Design and detailing of structural concrete using strut-and-tie models. The Structural Engineer. 69 (6), 113-125.

Schlaich, J., Schafer, K. & Jennewein, M. (1987) Toward a Consistent Design of Structural Concrete. PCI Journal. 32 (May-June), 74-150.

Wight, J. K. & Macgregor, J. G. (2012) Reinforced Concrete Mechanics & Design. New Jersey, Pearson Education.