L2-A1 Strut-And-Tie Modelling 6p RobertVollum TheStructuralEngineer Apr2015

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TheStructuralEngineer April 2015

Technical Concrete design

Concrete Design Guide No. 4: An introduction to strut-and-tie modelling This series is produced by The Concrete Centre to enable designers to realise the potential of concrete. The Concrete Centre, part of the Mineral Products Association (MPA), is a team of qualified professionals with expertise in concrete construction, engineering and architecture. www.concretecentre.com

Dr Robert L Vollum Department of Civil Engineering, Imperial College London Strut-and-tie modelling is a simple method of modelling complex stress patterns in reinforced concrete as triangulated models. It is based on the same truss analogy as the design for shear in Eurocode 21 and can be applied to many elements, but is particularly useful where normal beam theory does not apply, i.e. where plane sections do not remain plane, e.g. in deep beams, corbels and pile caps. EC2 provides information about the use of strut-and-tie modelling and this article is an introduction for engineers who want to take advantage of this useful analysis method. Regions where normal beam theory cannot be used arise at geometrical discontinuities, supports and concentrated loads and are known as D or ‘disturbed regions’. D regions are dimensioned using Bernoulli’s principle (Figure 1). Regions where normal beam theory can be used are called B or ‘beam’ or ‘Bernoulli’ regions. In strut-and-tie modelling, the structure can be split into D and B regions and a strut-and-tie model (STM) prepared for the D regions. The method uses the truss analogy where the actual flow of stresses is represented by a series of struts and ties (Figure 2). The method is an application of the lower bound theory of plasticity. Consequently, there is no unique STM for any given problem. The main requirements are to ensure that equilibrium is satisfied and to proportion the cross-sectional areas of struts and ties such that their resistances are everywhere greater than or equal to the internal forces. Adequate ductility is typically ensured through the provision of code-specified minimum reinforcement areas.

Figure 1 Dimensioning of D regions

The scope of this article is restricted to strut-and-tie modelling of planar structures. The reader is referred to specialised texts2–7 for more in-depth treatment of the method.

Development of STM The first step is to determine the basic STM geometry within the element. The serviceability limit state (SLS) of cracking is usually satisfied if the ultimate limit state (ULS) STM geometry is based on the elastic stress field, with members positioned at the centroids of the compressive and tensile elastic stress distributions (Fig. 2a). An elastically based STM can lead to overly conservative designs, since such models do not recognise the redistribution in stress that occurs when the concrete cracks, putting more stress into the reinforcement. Therefore, some deviation from the elastic solution is permissible, as in the design of continuous beams. Consideration of Fig. 2a

and Fig. 2b shows that the STM geometry changes as the load is increased to failure, due to the internal redistribution of forces that occurs on cracking and subsequent yielding of reinforcement, which in the ultimate case lies within the plastic limit. The STM in Fig. 2b neglects the contribution of the minimal web reinforcement required by EC21 to control cracking, which further increases load resistance.

Load path method The load path method2 is a useful tool for developing the STM geometry within elements. The starting point is to determine the stress distribution at the boundaries of the D region after which the STM can be developed using the following rules of Schlaich and Schäfer2: 1. Load paths do not cross each other 2. Load paths take the shortest streamlined route between the centres of gravity of stress diagrams 3. Curvatures concentrate near stress concentrations, reactions and concentrated loads 4. In the absence of elastic analysis, position struts from experience and the application of simple rules

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37 Figure 2 Development of STM for uniformly loaded deep beam a) based on elastic stress field b) at ULS

4. The orientation of the model may be allowed to depart from the elastic stress field at the ULS if the SLS is checked Figure 5 shows how rule 2 can be used to distinguish between ‘good’ and ‘bad’ STMs for a deep beam. The STM in Fig. 5a is ‘bad’ because insufficient tensile reinforcement is provided to control cracking at the SLS. Fig. 5b shows that there is an area of concrete in the top section of the STM which is in tension and therefore this section should be reinforced for the tensile stress. It can be shown from finite-element models that this tensile zone is created. Normally the tensile stress is taken by the nominal reinforcement provided in the deep beam.

Compressive stress fields

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Figure 3 Application of load path method (adapted from EC21)

The procedure is illustrated in Figure 3 for the anchorage zone of a post-tensioned beam. A similar loading arrangement arises when a wall is loaded with a concentrated load at its corner. Note that the forces F1 and F2 are equal and opposite. The geometry of the STM is fully defined once the dimensions z1 and z2 are defined. The dimension z2 can be found by assuming that the angle of the o diagonal strut Ʌ is 45 . Alternatively, the strut can be positioned at the centroid of the corresponding elastic compressive stress distribution as done for the deep beam of Fig. 1a. Figure 4 shows two alternative STMs for a deep beam. Both are lower bound solutions but the STM in Fig. 4b is not optimal, or ‘bad’, since the load path is incompatible with the elastic stress field. The STM in Fig. 4a is more appropriate, or ‘good’, because the strut orientation follows the elastic compressive stress trajectories. As noted by Schlaich and Schäfer2, “it is helpful to realise that loads try to use the path with the least forces and deformations”. Consequently, a crude rule of thumb for distinguishing between ‘good’ and ‘bad’ models is to choose the model with shortest ties, since ties are much more deformable than struts.

Choice of limiting angle between struts and ties The angle between struts and ties should be large enough to avoid strain incompatibilities due to ties extending and struts shortening in almost the same direction. Model Code 19908 gives the following useful rules of thumb for developing STMs which broadly follow the elastic stress field: 1. Orientation with the elastic stress field is more important for struts than ties, which can usually be arranged parallel to the edges of the member following practical considerations 2. In highly stressed regions, main struts and ties should meet at angles of about 60° and not less than 45° (this is a guideline and not mandatory) 3. If the arrangement of the model is made in accordance with the elastic stress field, the ULS does not require checking

Compressive stress fields are the way in which the compressive stress in the idealised struts spreads through the concrete. They are categorised as prismatic, fan- or bottle-shaped (Figure 6). Prismatic stress fields typically arise in B regions, whereas fan- and bottleshaped stress fields arise in D regions due to the dispersion of the stress paths radiating out from concentrated loads or reactions. Bottle stress fields are a feature of elastic stress field analysis. The curvature of the principal compressive stress trajectories of the bottle field develops significant transverse tension. The compressive resistance of bottle stress fields is limited by splitting along the axis of the strut, unless transverse reinforcement is provided to maintain equilibrium after cracking. The fan-shaped stress field is an idealisation in which the compressive stress trajectories are assumed to be straight. Consequently, no tension develops at right angles to the stress trajectories of the fan. Fan-shaped stress fields are assumed to develop at supports and concentrated loads in plastic stress field analysis. Failure is assumed to occur at supports and concentrated loads where stresses are greatest. Prismatic or parallel-sided stress fields are a special case of the fan- and bottle-shaped stress fields. Figure 7 shows a deep beam loaded with a concentrated load. In reality, the compressive stress field is bottle-shaped as indicated with the dotted line. However, in practice, struts are often idealised as prismatic (Fig. 7). EC2 requires a minimum amount of horizontal and vertical web reinforcement to be provided in deep beams to control cracking. The code requires an isotropic mesh with a minimum area of 0.001Ac mm2 or 0.001t mm2/ mm (where Ac is the cross-sectional area of the member and t is its thickness) to be provided in each face, but the UK National Annex to EC29 increases this area to 0.002Ac.

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Figure 4 Good and bad model based on minimising length of ties

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Technical Concrete design

" Figure 5 Use of Model Code 19908 rules to distinguish between good and poor STM

Table 1: Eurocode 2 node strengths Code

Type of node

Strength

EC2

&-C-C

1.0ɋǯfcd

C-C-T

0.85ɋǯfcd

C-T-T

0.75ɋǯfcd

Note: ɋǯ = 1Ȃfck/250

Figure 6 Compressive stress fields a) prismatic b) fan-shaped c) bottle-shaped

node dimensions need to be chosen to ensure that stresses on node boundaries are less than or equal to the design concrete strengths given in Table 1. It is necessary to ensure that the design stresses are less than the design strengths in both nodes and at node-strut interfaces where strut resistance is checked. The design strength of the strut depends on the reinforcement provided within the depth of the element and is often less than the node strength.

Dimensioning of nodes Compressive strength of struts EC2 gives design concrete strengths for struts which are based on simple criteria that account for the effects of cracking and confinement. In general: • transverse compression is favourable. Especially confinement from transverse reinforcement or surrounding concrete • transverse tension is detrimental Skew cracks are particularly detrimental as equilibrium requires shear forces to be transferred across cracks by aggregate interlock. Struts are checked at their ends where the cross-sectional area is defined by the node dimensions. EC2 reduces the design concrete strength in prismatic struts with transverse tension to 0.6ɋǯfcd where ɋǯ = 1Ȃfck/250. The strength of 0.6ɋǯfcd also applies to the ends of bottle-shaped struts like the diagonal struts of Fig. 7 if insufficient transverse reinforcement is provided to maintain equilibrium after cracking of concrete.

Classification of nodes Nodes are defined as regions were struts change direction or struts and ties intersect (Fig. 7). Smeared nodes arise in regions where the orientation of struts is diverted by a band of evenly distributed reinforcement bars and is not checked in design. Concentrated nodes are typically highly stressed and need to be designed to resist the incoming forces without concrete failing in compression. EC2 classifies nodes as C-C-C (three compressive struts), C-C-T (two compressive struts and one tie), and C-T-T (one compressive strut and two ties) (Figure 8). The node where a column meets a pile cap would typically be a C-C-C node, and the node where a two-pile pile cap meets the pile would typically be a C-C-T node. Forces are transferred from ties into nodes through a combination of bearing and bond stresses within the node. The dimensions of the nodes in Fig. 7 are governed by the dimensions of the loading and support plates as well as the cover to the flexural reinforcement. The

The dimensions of concentrated nodes are frequently determined by the widths of load and support plates as well as the position of ties (Fig. 8).

Sizing of C-C-C nodes The width of the loading plate of the deep beam in Figure 9 needs to be chosen to limit the bearing stress to a maximum of 1.0ɋǯfcd which is the design strength for C-C-C nodes. The depth x of the C-C-C node in Fig. 9 depends upon the assumed flexural compressive stress which cannot exceed 1.0ɋǯfcd. The choice of flexural compressive stress in Fig. 9 determines the widths of the incoming diagonal struts. Clause 6.5.4 (8) of EC2 states that C-C-C nodes can normally be sized on the basis that stresses are equal on all node boundaries. In this event, the stress distribution is hydrostatic within the node and the incoming strut centrelines are normal to the node boundaries. In practice, the width of the loaded area is normally governed by other considerations than the design concrete strength. Consequently, the bearing stress can be significantly less than ɋǯfcd. It is suggested that in the absence of shear reinforcement, the flexural compressive stress is taken as ɋǯfcd, to avoid the widths of the incoming struts being overestimated, but this is not an EC2 requirement.

Sizing of C-C-T nodes The design bearing stress at C-C-T nodes is limited to a maximum of 0.85ɋǯfcd. The node width normal to the inclined strut centreline in Fig. 8b is given by: (1) where Lb is the length of the support plate, u is the height of the back face of the node and Ʌis the angle of the strut centreline to the horizontal. EC2 does not require stresses to be checked at the back face of C-C-T nodes. It is beneficial to provide the tensile reinforcement in several layers in highly stressed concentrated nodes since this increases u and

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hence the strut width w. The reinforcement should be sufficiently anchored to develop its design tensile force at the node.

Additional recommendations with respect to STM PD6687-1 (2010)10, the background document to the UK National Annex to EC29, gives the design compressive concrete strength as fcd = 0.85fck/ɀc for STM. It also states that “for shear, the shear resistance of sections should be verified using BS EN 1992-1-1:2004, 6.2.2 at all sections where av > 1.5d, where av is the distance of the section from a concentrated load (or support) and d is the effective depth of the section”.

8. If necessary, increase the design shear resistance by increasing the node dimensions or providing shear reinforcement if C > CRd 9. Calculate the area of flexural reinforcement required to resist the design force T1. (As = T1/fyd) Consider the beam in Fig. 9 with L = 5000mm, h = 3000mm, t = 200mm, Lt = 300mm, Lb = 150mm, d’ = 70mm, P = 500kN and fck = 30MPa. The design concrete strengths are as follows:

ɋǯfcd = (1Ȃfck/250)(0.85fck/1.5) = 15.0MPa (bearing strength at top node)

Example 1. Design of deep beam with central point load Fig. 9 shows an STM for a deep beam of thickness t loaded with a central point load P. The inclined strut width at the bottom node is given by Equation 1. At the top node, the inclined strut width is given by:

0.85ɋǯfcd = 0.85(1Ȃfck/250)(0.85fck/1.5) = 12.7MPa (bearing strength at bottom node)

0.6ɋǯfcd = 0.6(1Ȃfck/250)(0.85fck/1.5) = 9.0MPa (strut strength without calculated transverse reinforcement)

(2) The minimum required widths of the bearing plates are: where: (3) in which: (4) where ɐc0 ) ɋǯfcd is the flexural compressive stress.

cotɅ = (0.5LȂ0.25Lt)/(hȂ0.5xȂd’)

(5)

The compressive force in the strut is given by:

C = 0.5P/sinɅ

(6)

In the absence of minimal reinforcement within the depth of the element, the compressive resistance of the strut is given by:

CRd = 0.6ɋǯfcdmin(wt,wb)t

Trial and error gives: x = 70mm, Ʌ = 50ι, C = 326kN, wb = 205mm, wt = 160mm, and T1 = 209kN. Therefore, the required area of flexural reinforcement is T1/fyd = 480mm2. The inclined strut is overstressed, indicating that shear reinforcement is required. The UK National Annex to EC2 requires nominal horizontal and vertical web reinforcement of area 0.002Ac to be provided in each face. The effectiveness of this

Figure 7 STM for simple deep beam

(7)

EC2 does not define how the strut resistance CRd should be calculated. The strut strength is 0.6ɋǯfcd at the C-C-T node-to-strut interface, but EC2 does not define the strut strength at C-C-C nodes. It is suggested that, in the absence of code-prescribed minimal web reinforcement, the strut strength is also taken as 0.6ɋǯfcd at C-C-C nodes. Typical design solution procedure Equations 2–7 can be solved using the following step-by-step procedure: 1. Find the minimum allowable lengths of the top and bottom nodes. (The maximum allowable bearing stress at the top and bottom nodes are ɋǯfcd and 0.85ɋǯfcd respectively) 2. Choose x (e.g. u) 3. Calculate cotɅ with Eq. 5 4. Calculate T1 with Eq. 4 5. Calculate ɐco with Eq. 3. If minimal web reinforcement is provided, the chosen value of x is acceptable if P/(Ltt) [bearing stress at top node] ) ɐco ) ɋǯfcd. Otherwise, adjust x by trial and error until ɐco = ɋǯfcd 6. Calculate the strut force C in terms of the applied load P with Eq. 6 7. Calculate the strut widths wt and wb with Eq. 1 and Eq. 2 and hence the strut capacity CRd with Eq. 7

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Figure 8 Classification of node types a) C-T-T b) C-C-T c) C-C-C

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Figure 10 Bottle stress fields a) partial discontinuity b) full discontinuity

Figure 9 STM of deep beam

reinforcement can be assessed using the EC2 design equation for full discontinuities, stress field models or the EC2 design equations for shear enhancement within 2d of supports.

Transverse reinforcement in bottle stress fields Equations 6.58 and 6.59 of EC2 are used to calculate the area of transverse reinforcement required to equilibrate the transverse tension in bottle-shaped struts with partial and full discontinuity. Partial discontinuity For cases of partial discontinuity the design tensile force for the transverse reinforcement (Figure 10) is given by:

T = F(bȂa)/4b

Technical Concrete design

(Exp 6.58)

reinforcement with minimum area of 0.002Ac = 400mm2/m in each face in each direction. For orthogonal reinforcement, the normal resistance nRd per unit length to the strut centreline is given by:

nRd = (As1sin2Ʌ/s1 + As2cos2Ʌ/s2)fyd

(8)

where s1/s2 = spacing of horizontal bars of cross-sectional area As1 and vertical bars of cross-sectional area As2 and Ʌ is the angle of the strut centreline to the horizontal. For isotropic reinforcement As1 = As2 and nRd = Asfyd/s in all directions. Hence, the required minimum reinforcement grid of 0.002Ac (i.e. 400mm2/m vertically and horizontally) in each face can resist a transverse force:

where F = the force in the strut, b = the available strut width and a = the node width. Full discontinuity For cases of full discontinuity (b > H/2) (Fig. 10b):

Alternatively, the strut capacity corresponding to minimum web reinforcement is given by: (9)

T = F[1.0Ȃ0.7a/H]/4

(Exp 6.59R)

in which H is the strut length and a is the width of the loaded area. (Expression 6.59R replaces the term a/h in Expression 6.59 of EC2 with a/H as directed by BSI paper B525/2 11 0034.) EC2 requires the area of transverse reinforcement resisting T to be provided over the length the compression trajectories are curved, but does not define this distance. Schlaich and Schafer2 show the transverse reinforcement being distributed over the central 0.8H of the partial discontinuity shown in Fig. 10b. However, this paper takes the length of the transverse tension zone as 0.6H as suggested by Hendy and Smith11.

Example 2. Inclined strut with full discontinuity The inclined strut in Fig. 9 is a full discontinuity of length:

Hence, the design tensile force to be resisted by transverse reinforcement over each half of the strut length is:

T = F[1.0Ȃ0.7a/H]/4 = 326[1.0Ȃ0.7×160/3776]/4 = 79kN The tensile force T is provided by a mesh of horizontal and vertical

in which a = wt, H is the strut length and nRd is given by Equation 8. The maximum possible shear force corresponding to crushing of the concrete at the bottom node of the deep beam of Example 1 is Vmax = 0.85ɋǯfcdtLb = 381kN. The shear resistance can be increased further by increasing the length of the bearing and loading plates as well as d’.

Use of EC2 beam equations for design of shear reinforcement Alternatively, shear reinforcement can be designed using a truss model like that of Schlaich and Schäfer12 (Figure 11) or with the EC2 design equations for shear enhancement in beams. In the case of the EC2 shear enhancement equations, no calculated shear reinforcement is required, provided the design shear stress Ⱦ˜Ed ) ˜Rdc where ˜Rdc is given by Equation 6.2a in EC2 and Ⱦ = av/2d where av is the clear shear span. The main advantage of using a truss model over the bottle stress field is to steepen the angle of the inclined strut at the support. For example, the effect of introducing the vertical tie in the simplified truss model of Fig. 11 is to increase the resultant angle of the inclined strut from:

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to:

where Model Code 19908 estimates the tie force F1 as: (10) where z is the flexural lever arm. Clause 6.2.3 (8) of EC2 reduces the design shear force due to loads applied to the top surface of beams within 0.5d ) av ) 2d of supports by Ⱦ = av/2d where av is the clear shear span. Shear reinforcement needs to be provided within the central ¾ of av to resist the reduced design shear force. Consideration of Clause 6.2.3 (8) shows that the reduced design shear force is equivalent to the tie force F1 of Fig. 11. Clause 6.2.3 (8) is typically less onerous than Equation 10, with both being less onerous than Expression 6.59 of EC2 for a full discontinuity. It is also possible to design the shear reinforcement using stress field models7,13.

General application The STM is a very versatile method that can readily be applied to a wide range of structures that would otherwise only be designable with finite-element analysis or empirical design methods. The advantage of an STM over finite-element analysis is that an STM requires load paths to be clearly identified. Hence, structures can readily be checked for equilibrium. An STM also has the advantage of clearly identifying the anchorage requirements of reinforcement, unlike finite-element analysis.

Summary It is often convenient to base the STM geometry at the ULS on the elastic stress field, since this typically ensures satisfactory performance at the SLS. However, some deviation from the elastic solution is permissible. If the elastic stress distribution is not available, the STM geometry can be developed using the load path method of Schlaich and Schäfer2. Model Code 19908 suggests a 2:1 dispersion rule which is useful for establishing the basic STM geometry. The axial resistance of struts in a planar member equals the minimum of wbtfsb and wttfst (in which wb = strut width at bottom

Figure 11 Simplified truss model according to Schlaich and Schäfer13

node, wt = strut width at top node, t = member thickness, fsb = strut strength at bottom node and fst = strut strength at top node). In the absence of shear reinforcement, take fstα fsb α 0.6(1Ȃfck/250)fcd. The strut strength can be increased if necessary by increasing node dimensions, providing shear reinforcement or a combination of the two. The minimum reinforcement required by the UK National Annex within the depth of deep beams is often sufficient to increase the strut strength at C-C-T and C-C-C nodes to the node strengths of 0.85ɋǯfcd and ɋǯfcd respectively.

References and further reading 1) British Standards Institution (2004) EN-1992-1-1:2004. Eurocode 2. Design of concrete structures. Part 1. General rules and rules for buildings, London, UK: BSI 2) Schlaich J. and Schäfer K. (1991) ‘Design and detailing of structural concrete using strut-and-tie models’, The Structural Engineer, 69 (6), pp. 113–125 3) International Federation for Structural Concrete (2011) fib Bulletin No. 61: Design examples for strut-and-tie models, Lausanne, Switzerland: fib 4) Thurlimann B., Muttoni A. and Schwartz J. (1989) Design and detailing of reinforced concrete structures using stress fields, Zurich, Switzerland: Swiss Federal Institute of Technology 5) Goodchild C., Morrison J. and Vollum R. L. (2015) Strut-and-tie Models, London, UK: MPA The Concrete Centre 6) Schlaich J., Schäfer K. and Jennewein M. (1987) ‘Towards a consistent design of structural concrete’, PCI Journal, 32 (3), pp. 74–150 7) Sagaseta J. and Vollum R. L. (2010) ‘Shear design of short-span beams’, Magazine of Concrete Research, 62 (4), pp. 267–282 8) CEB-FIP (1990) Model Code for Concrete Structures, Lausanne, Switzerland: CEB-FIP 9) British Standards Institution (2005) NA to BS EN 1992-1-1:2004 UK National Annex to Eurocode 2. Design of concrete structures. Part 1. General rules and rules for buildings, London, UK: BSI 10) British Standards Institution (2010) PD 6687-1:2010 Background paper to the National Annexes to BS EN 1992-1 and BS EN 1992-3, London, UK: BSI 11) Hendy C. R. and Smith D. A. (2007) Designers’ Guide to EN 1992 Eurocode 2: Design of concrete structures. Part 2: concrete bridges, London, UK: Thomas Telford 12) Schlaich J. and Schäfer K. (2001) ‘Konstruieren im Stahlbetonbau’ (in German), BetonKalender (Vol 2), Berlin, Germany: Ernst & Sohn, pp. 311–492 13) Vollum R. L. and Fang L. (2014) ‘Shear enhancement in RC beams with multiple point loads’, Engineering Structures, 80, pp. 389–405 Further reading The Concrete Centre has recently published a guide to Strut-and-tie Models5, which gives more information on how to construct and use STMs. This is available via the Concrete Centre website: www.concretecentre.com