Strut and Tie lecture

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

Topic A.1: STRUT-AND-TIE MODEL METHOD

TABLE OF CONTENTS

STRUT-AND-TIE MODEL METHOD ........................................................................................ 2 1.

The Structure's B- and D-Regions ................................................................................. 2

2.

Essential Principles of the Strut-and-Tie Model Design ............................................... 5

3.

Modelling D-Regions .................................................................................................... 7

4.

Design Procedures of the Strut-and-Tie Model ............................................................. 7

5.

Dimensioning the Struts, Ties and Nodes ..................................................................... 8 5.1 Ties ............................................................................................................... 8 5.2 Concrete Struts or Compression Stress Fields .............................................. 8 5.3 The Nodal Zones or Nodes ........................................................................... 9 5.4 The Nodal Zone Stress Limit...................................................................... 11

6.

Example Problem: Column Bracket ............................................................................ 12

7.

Selected References ..................................................................................................... 16 WEEK 2 TUTORIAL PROBLEMS: STRUT AND TIE MODEL METHOD ...................................... 17

Department of Civil Engineering, Monash University Edition: 2-2003

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

Strut-and-Tie Model Method

The Strut and Tie Model method (STM) is a powerful tool for the design of what is known as ‘discontinuity’ or ‘disturbed’ regions in reinforced and prestressed concrete structures. These regions are normally referred to as the D-regions. These are regions where a complex state of stress and strain develops. Examples of Dregions include corbels, deep beams, joints, walls with openings, anchorage zones and so on, see Figure 1. The method idealises the D-region by a system of truss members that serve to carry the load to the boundaries of the D-region. The truss model consists of compression struts (concrete) and tension ties (reinforcing bars). The proportioning of the sizes of the compression struts and the truss nodes (joints) are based on satisfying certain stress limits. It is considered as a lower bound plasticity method because it relies on assuming certain distribution of stress and load path that satisfy equilibrium and maximum stress conditions. The load capacity calculated from this state will always be less than or equal to the true ultimate load.

1. The Structure's B- and D-Regions In using the strut-and-tie model approach, the first step is to subdivide the structure into its B- and D-regions. Those regions of a structure in which linear strain distribution (the Bernoulli-Navier hypothesis) is appropriate are referred to as Bregions. These regions of a structure are usually designed with high accuracy. Their internal forces or stresses can be obtained from moment and shear diagrams, analysed by means of the statical system of beams. For uncracked B-regions, these stresses are calculated using bending theory for linear elastic material. For cracked B-regions, the truss models or the standard methods of Codes apply. Department of Civil Engineering, Monash University Edition: 2-2003

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

Figure 1: D-regions (shaded areas) with nonlinear strain distribution due to (a) Geometrical discontinuities. (b) Statical and/or geometrical discontinuities.

However, those regions of a structure where the strain distribution is significantly nonlinear, e.g., near corners, joints, corbels and other discontinuities, and the standard methods of Codes fail to apply in these areas will be called D-regions (Figure 1).

The internal flow of forces in D-regions can be reasonably well

described by strut-and-tie models.

In B-regions, the stresses and stress trajectories are quite smooth as compared to the turbulent pattern near D-regions. Stress intensities decrease rapidly with the distance from the location of the concentrated load, as shown in Figs. 2 and 3. This behaviour helps to identify the separation of B- and D-regions in a structure.

Department of Civil Engineering, Monash University Edition: 2-2003

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

Figure 2: Stress trajectories in a D-region, maximum principal stress

Figure 3: Stress trajectories in a D-region, minimum principal stress Department of Civil Engineering, Monash University Edition: 2-2003

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

2. Essential Principles of the Strut-and-Tie Model Design In a strut-and-tie model, the struts represent concrete stress fields with prevailing compression in the direction of the strut and the ties normally represent one or several layers of tensile reinforcement. Occasionally model ties can also stand for concrete tensile stress fields. Strut-and-tie models provide the designer with considerable insight into the flow of forces in D-regions. It is well understood that cracked reinforced concrete carries load principally by compressive stresses in the concrete and tensile stresses in the reinforcement. After significant cracking has occurred, the principal compressive stress trajectories in the concrete tend towards straight lines.

Hence, those

compressive stress trajectories can be approximated by straight compressive struts.

The internal flow of forces in D-regions can be modelled using concrete struts for principal compression stress fields, ties for the principal tensile reinforcement and nodal zones or nodes for the regions of concrete subjected to multi-directional stresses where the struts and ties meet (Figs. 4 and 5).

When a suitable model of a D-region is known, the forces of the struts and ties will be calculated, thereby satisfying equilibrium between applied loads and inner forces. The struts, ties and their nodes will be dimensioned and checked to carry the inner forces.

Department of Civil Engineering, Monash University Edition: 2-2003

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

a

V* N*

Tension ties

Compression struts

T

C

Figure 4: Strut-and-tie model for corbel of Figure 2.

C 2

T 2

Figure 5: Corner joint subjected to opening moments. (a) Cracking in an improperly designed joint. (b) Strut-and-tie model of joint behaviour. Department of Civil Engineering, Monash University Edition: 2-2003

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

3. Modelling D-Regions All the moments, shear and axial forces, and reactions acting on the D-region must be known before modelling of D-regions can commence (Fig. 6a).

Figure 6: The load path method. (a) The structure and its loads. (b) The load paths through the structure. (c) The corresponding strut-and-tie model. Usually load path method can be used to systematically develop strut-and-tie models by tracing flow of forces through the structure.

4. Design Procedures of the Strut-and-Tie Model

The following design procedure is used to construct the strut-and-tie model: 1. Draw the strut-and-tie models to scale and with the help of the finite element analysis, sketch the flow path of the forces.

2. Develop the strut-and-tie-model as explained in the section 4. The struts and ties condense the real stress fields by resultant straight lines and concentrate their curvature in nodes

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

3. Determine the geometry of the strut-and-tie model. The nodes of the strut-andtie model are located at the points of intersection of the forces at the nodal zones. With the geometry of the strut-and-tie model determined, the forces in the struts and the ties of the model can be found from statics. These are the inner forces generated to resist the external applied loads.

4. Dimension the concrete compressive struts, and the distribution and details of the reinforcement are determined based on consistent equilibrium and ultimate strength considerations.

The cross-sectional area of a compressive strut is

determined by the dimensions of the nodal zones at the ends of the struts. The nodal zones must be chosen large enough to ensure that the nodal zone stresses are less than the limits specified in the Code. But, the resultant dimensions of struts and nodes should be compatible with the geometric constraints of the Dregions.

5. Dimensioning the Struts, Ties and Nodes The nodes should be dimensioned so that the strength of the struts bearing on them can be fully developed and ties which are anchored in them, should prevent anchorage failure. 5.1 Ties Normally tie forces are carried by reinforcement. Its cross section follows from the tie force in the ultimate limit state and the design yield strength of the steel.

5.2 Concrete Struts or Compression Stress Fields To cover all cases of compression stress fields, three typical configurations are sufficient. Department of Civil Engineering, Monash University Edition: 2-2003

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

(a)

The fan-shaped stress field is an idealised stress field with no (negligible)

curvature and it does not develop transverse stresses (Fig. 7a).

σ ≤ f b1

σ ≤ f b1

σ ≤ f b1

Figure 7: The basic compression fields. (a) the ‘fan’, (b) the ‘bottle’, (c) the ‘prism’

(b)

The bottle-shaped stress field with its bulging stress trajectories develops

considerable transverse stresses i.e., compression in the bottle neck and tension further away. The transverse tension can initiate longitudinal cracks and cause an early failure. Therefore, it is essential to reinforce the stress field in the transverse direction. The transverse tension can be determined from a strut-and-tie model of the stress field (Fig. 7b).

(c). The prismatic stress fields is a frequent special case of the two preceding two stress fields in which the transverse stress and curvature is zero (Fig. 7c).

5.3 The Nodal Zones or Nodes The nodes of the model are derived as the intersection points of three or more straight struts or ties, which themselves represent either straight or curved stress fields or reinforcing bars.

In actual reinforced concrete structure, a node is Department of Civil Engineering, Monash University Edition: 2-2003

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

introduced to indicate an abrupt change of direction of forces. In reality, the node usually occurs over a certain length and width.

Fig. 8 shows two typical nodes encountered in the strut-and-tie model. The ‘smeared’ nodes (Node B - consisting of three compressive struts) represent the intersection point where wide concrete stress fields join each other or with closely distributed reinforcing bars. These types of nodes are normally not critical. When sufficient anchorage of the reinforcing bars in the smeared node is ensured, and sufficient reinforcement is provided to ‘catch’ the outermost fibres of the deviated compressive stress field, then the node is considered safe.

On the other hand, the singular nodes (Node A - consisting of two struts and one tie) occur where concentrated forces are applied and the deviation of the forces is locally concentrated. These nodes have to be carefully detailed in order to prevent excessive deformations to the structure.

Figure 8: ‘Singular nodes’ A and ‘smeared nodes’ B of strut-and-tie model, their stress fields, nodes and corresponding reinforcement. These nodes or nodal zones must be chosen large enough to ensure that the nodal zone stresses are less than the nodal zone stress limits. The geometry of the node region and the arrangement of reinforcement in it should be consistent with the Department of Civil Engineering, Monash University Edition: 2-2003

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

model on which the design of the structure is based and with the applied forces. Thereby the equilibrium condition should be satisfied.

5.4 The Nodal Zone Stress Limit The average compressive stresses in the node region boundaries have to be checked to be less than the limit stated. The following simplified strength values, fb1 are proposed for dimensioning all types of struts and nodes and is taken from the CEB-FIP Model Code 1990.

fb1 = αfcd

(1)

where α = 1.0 - for an undisturbed and uniaxial state of compressive stress. - for node regions where only compression struts meet. It could be taken a 1.1 when only compressive struts meet, thus creating a 2- or 3-dimensional state of compressive stresses in the node region. α = 0.8 - for compression fields with cracks parallel to the compressive stresses. - for nodes where main tensile bars are anchored.

Note that fcd denotes the concrete compressive design strength for uniaxial compression, which is related to the specified compressive strength f'c and which in turn depends on the safety factor of the designated Code of practice.

fcd may be

determined by:

Fcd =

0. 85 f ' c

γm

fcd = concrete compressive design strength.

γm = material safety factor for the concrete in compression = 1. (in this case) Coefficient 0.85 accounts for sustained loading. Department of Civil Engineering, Monash University Edition: 2-2003

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

6. Example Problem: Column Bracket

A column bracket having the general features shown in Figure 9 is to be designed to carry the end reaction from a long-span precast girder. Vertical and horizontal reactions V* = 542 kN and N* = 108 kN are applied 130 mm from the column face. A steel bearing plate will be provided for the girder, which will rest directly on a 270 × 100 × 15 mm steel plate anchored at the outer corner of the bracket. Bracket reinforcement will include main steel As welded to the underside of the steel plate, closed hoop stirrups having total area Ah distributed appropriately through the bracket depth, and framing bars in a vertical plane near the outer face. Select appropriate concrete dimensions, and design and detail the column bracket. Material strengths are f'c = 35 MPa and fsy = 400 MPa.

350 mm

180

a

V*

As

270 x100 steel x15 N* angle

180 d

Ah

h

1 60

1 Framing bars Column 300 x 350 mm

Figure 9: Column Corbel

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

Strut-and-Tie Model

Firstly, draw the strut-and-tie model to scale and then, find the orientation of the struts and ties of the model. The actual geometry of the strut-and-tie model should be within geometric constraints of the physical dimensions of column bracket. The forces in the strut and ties can be determined from the static equilibrium at the nodes. The design loads and angles of the struts are given below.

θ1 = 59o, θ2 = 53o

V* = 542 kN, N* = 108 kN,

a

V* N*

2

T1

θ2

θ1

1

C1

C2

T2 T3

3

C3

Figure 10: Strut and Tie model

Department of Civil Engineering, Monash University Edition: 2-2003

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

Node 1 ΣFy = 0

C1Sinθ1 = V* C1Sin59 = 542 kN C1 = 633 kN

ΣFx = 0

C1Cosθ1 + N* = T1 633 Cos59 + 108 = T1 T1 = 434 kN

Node 2 ΣFx = 0

T1 = C2 Cosθ2 C2 = 434 / Cos53 = 721 kN

ΣFy = 0

T2 = C2 Sinθ2 = 721 Sin53 = 576 kN

Node 3 ΣFx = 0

T3 = C 2 Cos53 - C1 Cos59 = 721 Cos53 – 633 Cos59 = 108 kN

ΣFy = 0

C3 = C1 Sin59 + C2 Sin53 = 633 Sin59 + 721 Sin53 = 1118 kN

Department of Civil Engineering, Monash University Edition: 2-2003

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

The main tension tie force, T1 = 434 kN. Take yield strength of reinforcing steel, fsy = 400 MPa.

The total area of main reinforcement, As

T1 = Asfsy As = T1 / fsy = 434×103 / 400 = 1085 mm2

A total of three Y24 bars, providing As = 1357 mm2, will be used.

Additional closed horizontal ties Ah having an area of at least 50 percent of As and distributed within the top two-thirds of the corbel, should be provided in accordance with the ACI Code Method detailing requirement.

The fanning compression strut has its maximum stress in the nodal zone (node 1 of Fig. 11) at the top of the corbel. The top nodal stress under the 100 mm wide by 270 mm long and 15 mm thick bearing plate is = 542 × 1000 / (100 × 270) = 20 MPa. This is less than the nodal stress limit of 0.8(0.85f'c) = 0.8(0.85 × 35) = 23.8 MPa. Anchorage of the main reinforcement bars will be provided by welding to the underside of the steel angle and at the far end by a standard 90o bend.

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

7. Selected References 1. P. Marti, "Basic Tools of Reinforced Concrete Beam Design," Journal of ACI, Vol. 82, No. 1, 1985, pp. 46-56. 2. P. Marti, "Truss Models in Detailing," Concrete International, Vol. 7, No. 12, 1985, pp. 66-73. 3. J. Schlaich, K. Schafer, and M. Jennewein, "Toward a Consistent design of Structural Concrete," Journal of Prestressed Concrete Institute, Vol. 32, No. 3, 1987, pp. 74-150. 4. J. Schlaich and K. Schafer, "Design and detailing of structural concrete using strut and tie model," The Structural Engineer, Vol 69, No.6, March 1991, pp. 113-125. 5. CEB-FIP Model Code for Concrete Structures, Comite Euro-International du Beton (CEB), 1990. 6. Michael Schlaich and Georg Anagnostou, "Stress Fields for Nodes of Strut-andTie Models," Journal of Structural Engineering, Vol 116, No. 1, Jan 1990, pp. 13-23.

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

Week 2 Tutorial Problems: Strut and Tie Model method Deep Beam: The deep reinforced concrete member shown in figure below is to be proportioned to carry a factored uniformly distributed load w* = 500 kN/m. The width of the beam is b= 200 mm. fsy = 400 Mpa, f’c = 32 Mpa. 1. Sketch a feasible strut and tie model indicating the position and orientation of the compression struts and tension ties. 2. From node equilibrium, calculate the forces in the struts and ties. 3. Calculate the area of reinforcement required to provide the tie force. 4. What is the minimum width required for the inclined compression strut. 5. Calculate the minimum width of the support bs. 6. Calculate the maximum value of the tie force T. w* = 500 kN/m

D = 3.2 m

bs

L = 4.8 m

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Unit CIV4235: Advanced Structural Design Topic A.1: Strut-and-Tie Model Method

Corbel: The reinforced concrete corbel shown in figure below is to be proportioned to carry the factored loads V* =750 kN and N* = 140 kN. fsy = 400 Mpa, f’c = 32 Mpa. 1. Sketch a feasible strut and tie model indicating the position and orientation of the compression struts and tension ties. 2. From node equilibrium, calculate the forces in the struts and ties. 3. Calculate the area of reinforcement required to provide the tie force. 4. What is the minimum width required for the inclined compression strut. 5. Check the minimum width required for the base plate.

400 mm

200 mm

V* Base plate 270 x 100 x 15 mm N* 200 mm 1

200 mm

1

Column 400 mm x 300 mm

Department of Civil Engineering, Monash University Edition: 2-2003