# Deep beam

### Introduction

A deep beam is a beam having a depth comparable to the span length. Deep beams are usually characterised as being short and deep, and having a thickness that is small relative to their depth or span. They have become more popular in recent years and more useful in structures. Such examples are tall buildings, where they appear as girders, foundation walls and offshore structures. Other common examples are pilecaps, tanks, bins, shear walls and brackets or corbels. There are several modes of failure for reinforced concrete (RC) deep beams. The most frequent mode of failure is shear failure, where compression and tension occurs from shear action. The heavy concentrated loads causes' disturbance to the internal stresses due to the degeneration in the load capacity carrying members, thus causing shear failure.

One approach to analyse shear failure is the Strut and Tie Model (STM). The model can be created within a deep beam to show how they can resist shear after cracking occurs. The model considers moments, shear, compression and tension simultaneously. With deep beams in all different sizes there isn't a particular STM to be used within a RC deep beam. However, there are techniques to follow which helps you to develop the correct method for a particular RC deep beam. Their guidelines are in codes such as Eurocodes 2 and the ACI codes.

The shear strength of RC deep beams is closely related to the section size. In most design purposes, deep beams have been considered to have a span/depth (l/h) of up to 2.5, which was discovered by Kong 1986. Although, this is a very common parameter which has been quoted for many years to this day, it's still a relevant term. Numerous of studies trying to predict shear strength on different types of RC deep beams with test variables (span/depth l/h) have been conducted. However, till this present day, there isn't a procedure to predict shear strength of RC deep beams due to their dimensions. Large amounts of research have been conducted but nothing conclusive (Kong, 1990 and P.E Regan 1993).

This paper will look into how different sized deep beams failed and what caused them to fail. Past experimental results will be put into a database and created in Microsoft Excel. The results will be from different sized RC deep beams, and the database will be examined to see how much concentrated load caused a RC deep beam of a particular size to fail. The STM will then be implemented into the database to predict shear failure for the experimental results.

### Aims and Objectives

In this paper, there will be a number of aims and objectives that will be addressed. The first part of the project will cover the strength and behaviour of reinforced concrete deep beams, which is one of the main aims of the project description. This will cover the understanding of what a reinforced concrete deep beam actually are some of the different types of deep beams and their fundamental behaviour in structures. Flexural capacity and shear capacity (the concepts) will be covered about reinforced deep beams. This will be the tension and compression. Behaviour at Ultimate Limit State will be discussed; this section will cover the causes of failure, and will be linked to the third part of the project. How deep beams are tested will also be covered.

The second part of the project will discover techniques for analysing shear strength or failure of reinforced concrete deep beams. The Strut and Tie Model (or Truss Model) is an example for doing this. The Strut and Tie model will be used in the project and how it's incorporated into a deep beam. This method will study the compression of the struts and the tension of ties in a deep beam where the Strut and Tie Model is applied. Where the struts and tie meet, the nodes will be covered to see how many different types there are and which ones are suitable for the appropriate situation. The Strut and Tie Model will show how they strengthen deep beams and increase the amount of load to cause the deep beams to fail. The main objective from this section would be to predict the shear capacity of the failure when the STM is incorporated into the RC deep beam. Calculations and an example will be shown to explain how it can be applied within a deep beam. The material and coefficients will be used from the ACI codes and the National Annex.

The final main aim of the project will be creating a database using experimental results of deep beams which were tested in the past. The experimental results will be taken from an American report completed in November 1983, (INSERT DOCUMENT RESULT NAMES). The database will be created in Microsoft Excel and include data from the articles such as, the heights and depths of the deep beams, the amount of main top and bottom longitudinal reinforcement, the amount of web reinforcement which is the vertical and horizontal stirrups, as well as the failure load of the deep beams. The purpose of the database will then show a summary of results for the deep beams at failure, and will help in identifying the effect of the span and web reinforcement on the deep beam strength. Then using the STM, the final aim would be to use this approach in the experimental results to predict the shear strength for the RC deep beams. The objective would be able to correctly predict the failure load when a STM is used within a RC deep beam.

### Strength and Behaviour of Deep Beams

### Introduction

A study (M.D Kotsovos,1990) shown that the introduction of the concept of the compressive force path which is due to the multiaxial effects has been shown to provide a realistic description of the causes of failure for reinforced concrete, but also the basis for design models for providing safe design solutions.

Deep beams are often in a state of plate stress in which is shear is a common feature; this is called two dimensional members. Beams can't be determined by ordinary beam theory and procedures for determining strength because of their structure size, span and depth. They can hold a lot of load which is transferred to the supports by the compression thrust. The following paragraphs will have details on RC beams for ultimate limit state and the concept in RC deep beam design.

### Ideas for Current Beam Designs

The first stage in beam design is to design a RC beam for flexural capacity and then ensure that any type of failure is prevented (other than flexural because that would occur when the flexural capacity is acquired). Flexural capacity is realistic deformation in a RC beam and a design tool noted for simplicity (M.D Kotsovos, 1990).

Before flexural capacity can occur an RC beam can expose a number on different types of failure. The most general type of failure is shear types of failure; shear capacity of the beam is not used up before the flexural capacity is obtained. Other types of failure are an anchorage failure or a bearing failure, which is occurs in regions acted by intent loads, are prevented by proper detailing.

There isn't a proper theory to describe shear failure in a deep beam. However, M.D Kotsovos (1990) study showed that there are a number of concepts, based on design methods for shear design. His concepts are:

a) When the shear capacity of a cross-section is surpassed, then this would cause shear failure to take place, and the structure would fail.

b) The segment of the cross section below the neutral axis is the main cause to shear resistance. This then provides “aggregate interlock” and “dowel action” which are methods of shear in a RC beam.

c) An RC beam can treat itself as a truss once inclined cracking occurs. The concrete between the cracks and shear reinforcement acting on the struts and ties of the truss can cause compression failure in the struts and tension failure in the ties. This section is covered in more detail in the “Strut and Tie Model”

For reinforced deep beams the effect of transverse stresses needs to be considered, these are Flexural and Shear capacity.

### Flexural Capacity

M.D Kotsovos, 1990 study found that Flexural Capacity is approached on the basis of the plane section theory, which is a relationship between the flexural capacity and geometric characteristics by considering the equilibrium conditions at the cross sections. Transverse stresses are not considered to affect flexural capacity and are therefore ignored.

An RC structure main purpose to sustain compressive forces because concrete is very weak in tension but very strong in compression. The main purpose in a RC member is to support the compressive forces, therefore steel reinforcement is provided to make the RC structure stronger in tension and provide protection. Reinforced concrete maintains the compressive forces in the member, so it's important that the strength and deformational response are familiar.

Figure 2.1 - (a) Design details for a RC Beam and (b) Measured average stress in compressive zone

Figure 2.1 above shows how the deformational responds to the compressive zone under two-point loading and measuring strains at the top face of a beam (typical RC beam). Experiments like Figure 2.1 have been carried out by M.D Kotsovos, 1982, also experiments to measure strain in the deepest flexural cracks have been carried out.

In conclusion from these experiments, M.D Kotsovos stated that the post-ultimate uniaxial stress-strain characteristics cannot describe the behaviour of an element of concrete in the compressive zone of an RC beam in flexure. This means that the descending branch doesn't behave like normal material behaviour due to the interaction between the specimen and the testing machine (M.D Kotsovos, 1983). For the ascending branch however, M.D Kotsovos came to the conclusion that the data was inadequate to describe the behaviour.

### Shear Capacity

As talked about in the section “Ideas for Current Beam Designs”, M.D Kotsovos, 1990 said that RC beam with shear capacity is the maximum shear force which can be sustained by a critical cross section. When shear force that cannot be sustained by concrete by itself, shear reinforcement is introduced to carry that section. The amount of reinforcement depends on the truss idea, where the shear reinforcement lies; it's believed to behave like a truss once failure starts to occur, which is cracking.

In design, “aggregate interlock” is one of the main contributors to shear resistance (Fenwick and Paulay, 1968, Taylor, 1968, Regan, 1969 citied M.D Kotsovos, 1990). The cracked web of an RC beam is only cause to the shear resistance, from aggregate interlock of an RC T-Beam (British Standards Institution, 1985). The idea of shear capacity of critical section is itself, from the truss concept because of the loss of shear capacity underneath the neutral axis.

The concepts currently used to describe the causes of shear failure have been shown in a test programme by M.D Kotsovos in 1987). It was based on an investigation of the behaviour of RC beams, with different arrangements of shear reinforcement. There were tests on RC beams with different span to depth ratios. This is shown in figure 2.3 below. Figure 2.4 shows the main results of this experiment programme graphically.

M.D Kotsovos study on the concept of shear capacity was that all beams lack shear reinforcement. They can lack a large share of its length or over their entire shear span. However, M.C Kotsovos experiment programme shown that beams C and D [in figure 2.3] could carry a higher capacity of load compared to beams A, which didn't have any shear reinforcement. Beams D had ductile behaviour which was counted as flexural failure. This experiment shown that aggregate interlock makes a huge contribution to shear resistance, because mainly beams C and D deflections of the inclined crack led to a large increase in it which meant it reduced aggregate interlock. M.C Kotsovos then concluded that in the lack of shear reinforcement, the main cause of shear resistance of an RC beam is the compressive zone, which is the middle of the beam.

### Behaviour at Ultimate Limit State

An RC beam can be divided into four types of behaviour depending on the ad, beams subjected to two point loading, or ld It can only be divided if it is a rectangular cross section and without shear reinforcement (Kani, 1964 citied by M.D Kotsovos, 1990). M.D Kotsovos conducted a compressive force path concept with 4 beams, varying in different a/d ratio and while under maximum load.

Figure 2.5 above, shows the four beams behaviour for the concept and shows the characteristics for the mode of failure on each of the beams. In following sub-section, ‘Causes of Failure' types III and IV will be discussed because these portray deep beam behaviour, unlike types I and II.

### Causes of Failure

Type III behaviour; Figure 2.6 below shows a typical mode of failure for this case of a deep beam. Which this deep beam doesn't have web reinforcement and was subjected to a two point loading with ad = 1.5 (Walther and Leonhardt, 1962 citied by M.D Kotsovos, 1990). The figure shows how it's failed by an inclined crack which has formed within the shear span. The inclined crack started from the bottom of the deep beam, near the support, and has prolonged towards the top face near the load point, then caused failure of the compression zone.

There isn't a clear indication what causes the failure, it's associated with shear capacity of a critical section within the shear span appears to be distant. As talked about in ‘Shear Capacity' M.C Kotsovos experiment programme (Beam D in Figure 2.3), delaying of the extension of the inclined crack can be done by using stirrup reinforcement in the middle section, this then permits the beam to act in a ductile manner. Therefore, M.C Kotsovos, 1990 concluded, the causes of failure should therefore be sought within the middle, rather than the shear, span of the beams. The description of failure is based on the concept of compression force path. It appears that the type III behaviour the most critical conditions develop within the horizontal portion of the path.

Type IV behaviour, Figure 2.7 above is another typical mode of failure in this case for a deep beam without shear reinforcement and under a two point loading again with ad = 1.0 (Walther and Leonhardt, 1962 citied by M.D Kotsovos, 1990). The above failure has an inclined crack which has formed within the shear span independently of the flexural cracks. Comparing Type III and IV together, the crack in type IV coincides with the line joining the load point and support. This starts about half way through the load being applied where increasing load causes the expansion of cracking towards to the top and bottom face. This particular type of failure is referred to “diagonal-splitting” and its causes are to be the shear span of the beam. The shape forms a barrel-shaped region with its larger cross section placed half way between the load point and the support.

The mode of failure for Figure 2.7 can be characterised by a fair few cracks, which are flexural and inclined. With incline cracks in both types III and IV, they both extend towards the compressive zone but they both differ in depth which is alike flexural cracks. A different mode of failure can be characterised in type IV by the failure in the middle span of the beam (compressive zone). It is very similar to type III failure around the compressive zone, but for type IV they are inclined cracks around certain regions under compressive and tensile forces which could cause the beam to fail. However, in general, this only happens when flexural capacity is obtained.

### Different types of Deep Beams

There are many different types of reinforced concrete deep beams. Some are used in particular design methods and constructions, and have better factors than each one.

Deep Beams with Web Openings

Figure 3 above shows a deep beam with a web opening in a building. Openings in the web region of deep beams are sometimes provided for essential services and accessibility. The main factors affecting behaviour and performance of deep beams with web openings are:

- The span to depth ratio
- Amount of longitudinal reinforcement (and location)
- Amount of web reinforcement (and location)
- Cross sectional properties
- Size, shape and location of web opening

General behaviour in shear failure is that the strain in concrete at mid span section indicates that before first cracking, the beam then behaves elastically and shows the distribution of strain and more than one natural axis. Neutral axes decreases with loads and at the ultimate stage, only one natural axis is present.

### Continuous Deep Beams

Continuous deep beams, they're fairly common structural elements which occur as transfer girders, pile caps and foundation walls. Continuous deep beams behave differently from either simply supported deep beams or continuous shallow beams. A tied arch or truss behaviour are developed in continuous deep beams, but not found in shallow deep beams. The result of this is the amount of reinforcement detailing rules, based on shall or simple beams. As other deep beams, they exhibit the same trend of increased shear strength with a decrease in shear-span/depth ratio as found in simply supported deep beams.

### Flanged Deep Beams

Flanged deep beams are usually deep and consist of a thin web; figure 4 above shows a typical flanged deep beam. Flanged beams are a major structural component on the foundation of offshore gravity type structures, an example would be wind forces in tall buildings. There are a couple modes of failures for flanged deep beams, the main differences between a conventional deep beam and a flanged deep beam are:

- Overall depth/web thickness ratio h/t - Conventional deep beams are designed to have a h/t to about 10 or less. Flanged deep beams are much larger and tend to be up to 60.
- Accommodation of main tensile reinforcement - The main reinforcement is accommodated within the plane of the web. Flanged deep beams have wider flanges to accommodate them.

The usual mode of failure of slender reinforcement concrete flanged beams involves the diagonal splitting of web between the edge of the loading plate and the support. The modes of failure are:

- Flexural shear failure - This occurs in a beam with a very small amount of main tensile reinforcement.
- Flexural shear compression failure - This is similar to the first mode of failure except crushing the compression flange will occur before the tensile capacity.
- Diagonal splitting failure - This occurs by diagonal splitting and excessive cracks in web in beams with thin webs and minimum reinforcement in top and bottom of flanges.
- Splitting with compression failure - This is similar to the third mode of failure but the crushing of the compression flange will occur.

How Deep Beams are tested

There are a few certain ways deep beams are tested; a common method is that they are pressured at the support loads until failure.

Loading frame - This is where hydraulic rams are used for loading deep beams and pressurised with “air and oil” system which maintains a constant load. The pressurised load is kept constant so the strain results can be taken equally. After a couple of minutes under pressure, the cracking and deflections start to show

### Procedure

To begin with, a set of displacement readings were first data taken. After this, concrete and steel strains were measured; each gage read twice for accurate readings and waits a couple of minutes between readings). Cracks and marks begin to show and then should be photographed for data and the final stage a set of displacement readings should be taken before increasing the load to the next increment. The testing measurements look for mid span deflections, concrete and steel strains, loads and reactions and flexural cracking from reinforcement.

### The Strut and Tie Model

### Introduction

In the past, there were two approaches used to analyse shear problems in reinforced concrete, one has been the Truss Model (or the Strut and Tie Model), the other approach was the Shear Strength Plastic Method. Agreed over the last few years by researchers, the truss model theory provides a better way to treat shear, and have also increased popularity. This is due to the fact that a designer can design complex regions in reinforced concrete and can model the flow of forces in the struts and ties. Thus, rather using experimental based concepts, the designer can use the Strut and Tie Model which elucidates the flow of forces and the resisting components. Figure # below shows a typical example of a Strut and Tie model in a deep beam.

Since the beginning of reinforced concrete design, the Strut and Tie models have been a worthy design tool. The original truss model prototype was first presented by Ritter (1899) and Morsch (1909) to treat shear problems. Their theory was that a concrete concept reinforced with diagonal steel bars and subjected to shear stresses would cause cracking at the same angle as the steel bars. The cracks were caused by axial compression; which led to the cracks being separated into concrete struts. With the steel bars together, they formed a truss form to resist applied shear stresses. The concrete struts were assumed to be at 45° angle to the steel bars, their theories are known as the 45° truss model. Their equations were derived from equilibrium conditions.

The Strut and Tie model (STM) is a unified approach that considers all load factors, like moments, shear, compression and tension simultaneously. For shear critical structures or other disturbed regions in concrete structures, the Strut and Tie model has been one of the most useful design methods for them, the model can clearly show how a reinforced concrete deep beam can resist shear after cracking. The model provides an approach by representing a structural member with an appropriate simplified truss models. Like most common concrete structures there all different in size and shape, so there isn't a single unique STM for design encountered. However, there are rules and techniques which can help develop the correct model for a concrete structure, they serve the formation of basic design codes in practice. Over this chapter, these rules and techniques will be talked about more.

### Components of a Strut and Tie Model

B-Regions and D-Regions

James G. MacGregor defined that concrete structures can be split into beam regions where assumptions of the flexure theory apply and disturbed regions. Disturbed regions are changes in loading at reactions or changes in the geometry, like holes or the cross section. These different portions are referred to B-regions and D-regions.

The B (Bernoulli or Beam) region is understood by the flexural behaviour can be predicted by calculation. Dr C.C. Fu (2001) states that Bernoulli hypothesis states that “Plane section remains plane after bending, which means the flexural behaviour of concrete structures allow a linear strain distribution for loading stages. Basically, the B-region is a design approach to shear. The D (Disturbed or Discontinuity) regions (such as deep beams or corbels) can be modelled using trusses consisting of struts, ties and nodes. The concrete struts are stressed in compression, the steel ties stressed in tension while joined together at the joints by the nodes. They are then referred to as STM's. Figure # above showed an STM of a single deep beam, which has two concrete inclined struts, horizontal tie joined together at three nodes. In D regions, Dr C.C. Fu (2001) states that a majority of the portion load are transferred to the supports by the in-plane compressive forces in the concrete.

Yielding at the ties, crushing of the struts or failure at the nodes are all forms of assumed failure for a STM. The struts are assumed to reach their capacities when the compressive stresses reach effective compressive strength fcu.

Figure # below compares experimental shear strengths of simply supported beams with various shear spans to depth ratios ad, from 1 -7 (Figure # originally from “Prestressed Concrete Structures” [Mitchell and Collins, 1991] and citied by James G. MacGregor (2002). As shown by the horizontal line to the right of ad = 2.5, The B region behaviour shows the strengths of the beams greater with ad greater than 2.5. For the D region, the behaviour controlled the strengths of beams with ad ratios less than about 2.5 as shown by the steeply sloping line to the left of ad = 2.5.

Decisions needed to develop rules for STM's

As mentioned earlier in the Introduction section, there are certain rules and techniques which can help develop a correct Strut and Tie model for a particular concrete structure. According to James G. MacGregor (1997), the certain items to be defined and specified are:

a) The Layout of Strut and Tie models, geometrically.

b) The effective concrete strengths and ɸ factors be used

c) The strength and shape of struts

d) The layout and strength of the ties and nodes

e) The Detailed requirements

The definitions are similar, but different in various codes and other design documents.

Geometric layout of Strut and Tie models

According to James G. MacGregor, the Strut and Tie model is a hypothetical truss that transmits forces from loading points to supports. Minimising the amount of reinforcement approaching the ideal model, is a strut and tie model. For two dimensional structures some researchers (Schlaich, Shafer and Jennewein, 1987) recommended using an analysis to determine stress points for a given example. They stated the struts then can be aligned with 15° of the resultant compression forces from such an analysis, and the ties within 15° of the resultant tensile forces.

James G. MacGregor method for selecting a Strut and Tie model was to select initial trail location for the nodes and use initial cycle of calculation for member forces. The location of Struts and Ties can then be rearranged if a cracking pattern occurs within the structure. Example of this would be if the struts fall between the cracks in a given structure, which would be failure. In any case, struts should not cross cracked regions.

The American Concrete Institute (ACI) has major guidelines that must be satisfied by a Strut and Tie model (located Section A.2 of 2002), these are:

- The Strut and Tie model must be in equilibrium with the applied and dead loads. Calculations of the reactions and strut and tie forces satisfies statics
- The forces of the struts, ties and nodal zones must equal or exceed the strengths of these members. If the strengths equals or exceed its said to have a safe distribution in the members of the Strut and Tie.
- Only consider the axes of the struts and ties when laying out a STM in the early stages of a D-region. It is general to consider the width of the struts, ties and nodal zones and support regions when laying out a Strut and Tie model.
- The struts must never cross each other; however, ties are allowed to cross struts or other ties. The widths of struts are chosen to carry the forces in the struts using effective strength of the concrete on the struts. If overlapping occurred, the struts would be overstressed.
- The smallest angle between the strut and tie that are joined at the node is set at 25°.

### Forces in Struts and Ties

Reactions to the applied loads and self weight loads are computed once the initial Strut and Tie model has been selected. Once the reactions have been calculated, the forces in the struts, ties and nodal zones loads are computed using the truss analysis. The struts, ties and nodal zones are then based on the following equation can be applied:

ɸfn ≥ fu

Where fu is the force in the member (strut, tie or nodal zone) due to the loads, fn is the nominal strength in the struts, ties and nodal zones which are notated as fns, fnt and fnn , and ɸ is a strength reduction factor (depends on the condition of the STM's). To simplify, the ACI 199 code states that ɸ = 0.9 in flexural terms and ɸ = 0.85 for shear in beams, corbels and deep beams. In most cases, ɸ will always be 0.85. The strength would always have to be greater than the force in the strut, tie or nodal zone, otherwise this would be failure.

### Struts

Generally, struts very in shape and size. A STM in a deep beam, similar to figure #, the concrete of a beam webs adjacent to a strut is stressed by the lateral spread of the strut stresses into the adjacent concrete in the shear spans. If there is spread to occur, the struts are referred to bottle-shaped struts. Most struts in two-dimensional STM's will be bottle-shaped. Compression struts fulfil 2 functions in the STM. They serve as the compression chord of the truss mechanism which resists moment and they also serve as the diagonal struts which transfer shear to the supports. Diagonal struts are generally oriented parallel to the expected axis of cracking.

There are three types of struts:

- Prism = The simplest type which has a constant width
- Bottle = Where the strut expands or contracts along its length
- Fan = Array of struts with varying inclination meet at or radiate from a single node

Struts are designed to satisfy the following equations:

Fns = fcuAc

fcu=vfc'

ɸfcu= ɸα1βsfc'

Where, fcu is the effective compressive strength of the concrete in a strut, Ac is the end area of the strut. v is the effectiveness factor and ɸ Is a coefficient factor depending on the struts, ties and nodal zones. α1 Is a 0.85 coefficient from the ACI codes, βs is the effectiveness factor for a strut, table 1 shows the different values for βs. If fcu is different at the two ends of a strut, the strut is uniformly tapered.

Case |
1999 Load Factors (ɸ = 0.85, α1 = 0.85) |
2002 Load Factors (ɸ = 0.75, α1 = 0.85) | ||

Values |
Chosen |
Values |
Chosen | |

Prism |
0.76-0.83 |
1 |
0.77-0.86 |
1 |

Bottle-shaped (Reinforcement) |
0.61-0.66 |
0.75 |
0.61-0.69 |
0.75 |

Bottle-shaped (No reinforcement) |
0.46-0.50 |
0.6 |
0.46-0.52 |
0.6 |

Table 1 - Values for βs for Struts in STM's (ACI)

Effective Compressive Strength of Struts

It is assumed that the stress at the cross sectional area at the ends of the struts are constant. There are three factors that influence the effectiveness of concrete struts. These three factors are the Load Effects, Cracking at the Struts and the Restriction from Surrounding Concrete.

Load Effects

The following equations give the effective strength of the struts:

ɸfn ≥ fu ɸfcu= ɸα1βsfc'

The coefficient α1 is accounting for the load duration effects or it can account for the flexural stress blocks. This coefficient is taken from the ACI section, there have been several relationships put forward that the coefficient should change relating to the force in the strut, but nothing has been set into the codes. In the equation, βs the subscript s stands for strut.

Cracking in the Struts

Struts are subjected to a lot of compression, which causes struts to develop certain cracks, like axial, diagonal and transverse cracks. An example of a strut which this can happen is a Bottle-shaped strut. These are struts which are wider at mid-length because the concrete can spread greater at mid-length than the ends when the strut stresses. Figure # below shows can example of this.

The reduction in compressive strength in the struts can cause forces to separate along the length of the strut which can cause longitudinal splitting near the ends of the struts, Figure # below shows this. With the lack of reinforcement in the strut to support it, the cracks weaken the strut. A study by Schlaich et al (1987) has investigated this type of cracking and can predict when it will occur along the strut. When the compressive stress at the end of the strut, it exceeds near 0.55fc'

Restriction from Surrounding Concrete

Having a large volume of concrete confined around the strut can result in an increase in the compressive strength. This method is usually used in three-dimensional concrete structures, example are pile caps. Researchers, Adebar and Zhou (1993) studied this method and proposed equation for pile caps to predict the compressive strength increase.

### Nodes and Nodal Zones

The difference between nodes and nodal zones are that nodal zones are the regions around the joint areas in which the members are connected and nodes are the points where axial forces in the struts and ties intersect. Another way to describe a node is where the forces are relocated within the STM. There must be at least three forces acting on the node in a structure like a deep beam for when the node is in a vertical and horizontal equilibrium state.

Types of forces that meet at the node are referred to as the nodes. Thus, a C-C-C anchors three struts because it's three compressions or the three struts are fixed. A C-C-T node anchors one tie and two struts and a C-T-T node anchors one strut and two ties. In rare cases there can be a T-T-T node where three ties are anchored and there are no struts connected. Figure # below illustrates these types of nodes. It is assumed that when nodal zones are in compression, their width remains the same at the end of the struts. Width of the faces anchoring the ties will be looked into more detail later on.

Types of Nodal Zones in STM's

In Strut and Tie Models, there are two different nodal zones:

### a) Hydrostatic Nodal Zones

To begin with, researchers believed that nodal zones were assumed to have equal stress on all in-plane sides. This is because Mohr's circle for the in-plane stresses would be able to plot points; this is where the name hydrostatic nodal zones come from. If the stresses on all sides on the nodal zones were equal, the ratio of the length of the nodal zone would be equal to the forces acting on the sides.

Hydrostatic nodal zones were extended to C-C-T or C-T-T nodes by assuming the ties extended through the nodal zones to be anchored on the far side by tie reinforcement beyond nodal zone. A hypothetical anchor plate is located behind the joint. The bearing pressures on the plate being equal to the stresses acting on the other sides of the nodal zone, an area can be chosen for the hypothetical anchor plate. The tie force divided by the bearing stress for the struts at the node is equal to the effective area of the tie.

### a) Extended Nodal Zones

Extended nodal zones are bounded by the outlines of the compressed zones at the intersection of the reactions, the struts and the assumed width of the ties. Transfer of forces from strut to strut or strut to tie is due to the compressive stresses which aids the transfer. The figure below shows a typical extended nodal zone, the ACI guidelines tends to use extended over hydrostatic nodal zones.

As shown in figure #, the total shaded zone is the extended nodal zone and the darker side of the node is the hydrostatic nodal zone. The area where it's stressed in compression is due to the reactions and the struts would be the extended nodal zone.

Relationship between the Sizes of a Nodal Zone

Relating the widths of the struts, widths of ties and the bearing areas, equations can be derived for a relationship for the dimensions. For the following equation, it's assumed that the stresses are equal and the members meet in a C-C-T node.

ws= wtcosϴ+ lbsinϴ

Where ws the width of the strut, wt is the effective width of the tie, lb is the length of the bearing plate and ϴ is the angle between the strut and the tie. This equation is useful for regulating the size of the nodal zones in a strut and tie model. The strut width can be changed by using the width of the strut or the length of the bearing plate. As the stresses become more and more unequal, this equation would then loose accuracy. This equation comes from the ACI codes.

Effective Compressive Strength of Nodal Zones

Only a few tests are available for the effective compressive strength of nodal zones. Jirsa et al (1991) did tests on ten C-C-T and nine C-T-T nodal zones. For the C-C-T test, the results for the calculated strengths were 1.17 with a SD of 0.14. For the C-T-T test, the calculated strengths were 1.02. He also came to the conclusion that fcu=0.80fc' can be calculated for nodal zones if properly detailed.

The following equations are taken from the Canadian Code (1994) which can be used for the effective compressive strength of nodal zones, ɸfcu as:

- For nodal zones loaded by compressive struts: 0.85 ɸ fc'
- For nodal zones anchored for one tie: 0.75 ɸ fc'
- For nodal zones anchored for more than one tie: 0.65 ɸ fc'

These equations will be used later on to work out the effective compressive strength of nodal zones. In this case,ɸ is equal to 0.60 due to the Canadian code.

The values of the effective compressive strength for nodal zones are summarised in the table below:

Case |
Range of βn |
1999 Load Factors |
Range of βn |
2002 Load Factors |

C-C-C Nodal Zones |
0.79-0.80 |
1 |
0.76-0.83 |
1 |

C-C-T Nodal Zones |
0.70-0.71 |
0.8 |
0.67-0.73 |
0.8 |

C-T-T Nodal Zones |
0.61 |
0.6 |
0.58-0.63 |
0.6 |

Table # - Recommended values of βn for nodal zones in STM's

The 1999 load factors were recommended values for βn assuming α1 = 0.85 and ɸ = 0.85. For the 2002 load factors, assume α1 =0.85 and ɸ = 0.75. These values are taken from the 2002 ACI Code. Using this table, we can find a suitable value for βn (effective strength of a nodal zone).

### Ties

### Ties in Strut and Tie Models

Tension ties include stirrups, longitudinal (tension chord) reinforcement, and any special detail reinforcement. The anchorage for the reinforcement is a critical part of detailing for the strut and tie model. If a unique development was provided incorrectly, there would be a small anchorage failure at the load below the ultimate capacity.

Struts and Ties should be dimensioned so that stresses within nodes are hydrostatic. Based on hydrostatic nodal zones in Strut and Tie Models, the tie reinforcement is spread over the height of the tie calculated as:

wt= Fuɸfcubw

Where wt is the effective width of the tie, Fu is the force in the strut or tie, ɸ is a coefficient factor, fcu is the effective compression strength of struts and bw is the effective width of the deep beam.

The tie is assumed to consist of the reinforcement and a hypothetical prism. The ACI code states that the tie reinforcement should be distributed equally over the effective width of the tie, wt. Figure #b - distributed steel image shows an example of this, putting the steel reinforcement into many layers. Figure # - the one layer of steel shows the reinforcement concentrated near the face of the beam.

Figure #a also shows what the a tie would be like if extended nodal zones were used. The height of the tie would correspond to the steel being placed in one layer. The effective width of the tie would equal the diameter of the bars plus double the cover bars within the tie.

### Strength of Ties

A tie is assumed to reach capacity when the force in the tie reaches:

Tn=Asfy

Where Tn the strength of the tie, As is the amount of longitudinal reinforcement in the structure or deep beam and fy is the strength of the concrete.

### Anchorage of Ties

Anchorage of a tie in a nodal zone is generally a major problem in the designing of ties. In serviceability checks reduced strain can also reduce the elongation of the tie, which the members with less deflection. The ACI codes also states that by the time the geometric layout of the bars in the tie leaves the extended nodal zones, the anchorage of the tie should be in place. This reduces any potential failure in the tie. Figure # shows an example of this. Where diagonal struts are anchored by stirrups at nodal zones in a beam structure, the forces at the truss node changes occur within the width of the nodal zone.