5-Strut & Tie Methods

Assoc Prof Tan Kiang HweeDepartment of Civil EngineeringNational University of Singapore

2/16/2004

CE5510 Advanced Structural Concrete Design

- STRUT-AND-TIE METHODS -

DEPARTMENT OF CIVIL ENGINEERING

Tan K H, NUS2/16/2004

In this lecture

We will explore

!the concept of strut-and-tie models!their applications to new construction

(and strengthening works)


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At the end of the lecture

You should be able to

!identify cases where strut-and-tie models are applicable or appropriate

!formulate strut-and-tie models in structural concrete members

!design the reinforcement according to the strut-and-tie models


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!B- and D-regions

!Concept of Strut-and-Tie Models• Geometric Layout • Design of Struts• Nodes and Nodal Zones• Design of Ties• Detailing

Contents


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!Design Examples for New Construction• High Wall • Corbel • Dapped-Beam• Transfer Girder• Deep Beam with Opening • (Stepped (Non-Prismatic) Beams)

! (Examples for Strengthening Works)• Dapped Beams• Beam with Openings or Recesses


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B-region

Main (B-) & Local (D-) regions

D-region


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!regions of relatively uniform stresses!Bernoulli hypothesis of linear strain

distribution applies!internal forces or stresses are derived

from statics!“Standard” methods of Codes apply

Main (B-) regions


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!significantly non-linear strain distribution

!near concentrated loads, corners, bends, openings and other discontinuities

!internal flow of forces well described by strut-and-tie models

!conventionally design by thumb-rule

Local (D-) regions


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! Components! concrete compression

struts! steel tension ties! nodes (nodal zone) where

struts and ties meet! Dual purpose

! describe essential aspects of structural behaviour

! provide tools for structural dimensioning

Concept of Strut-and-Tie Models

steel

concrete


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Boundaryforces/stresses

Load path?

Geometric Layout of strut-and-tie models

follows the flow of internal forces in the structure


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! Major requirements! S-T model must be in equilibrium with applied

loads (statically admissible field)! Strength of struts, ties and nodal zones must

equal or exceed forces in these members (safe)! Sufficient to consider only axes of struts and ties in

the early design stage; need to consider widths in general

! Struts must not overlap each other! Ties may cross struts or other ties! Angle between a strut and a tie joined at a node

should not be less than 25 degrees.


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Basic steps

! Compute internal stresses on boundaries, subdivide boundary and compute force resultants on each sub-length; or

! Compute action effects onboundaries

! Draw truss to transmit forces

! Check stresses in individual truss member

P


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Elasticstresstrajectories

Some rules for

estabilshingstrut-and tie

model


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Minimum steel content

"""" ××××

ΣΣΣΣFilllliεεεεmi=minimum


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Agreement with Crack Pattern

××××


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Superposition of models

""""


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Truss 2 can form only if truss 1 does not fail prematurely


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Exercise 1

! Explore the application of strut-and-tie model in the design of anchorage zone of a post-tensioned beam

Principal compressiveStress trajectories

Stress contours

compression

tensionor


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Exercise 2! A T-beam is post-tensioned with a cable anchored at

the centroid of the section at its end. Given that the area of the flange is one-third of the overall cross-section, explain by sketching in the following figures, how you would obtain the required reinforcement to resist bursting tension in the web due to the prestressing force.

x-section strut-&-tie model reinforcement


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! Compression struts! line along centre-line of strut!strut with width

! Tension ties!band of steel reinforcement!anchorage (hooks, development length)

! Nodes!bounded by compressive forces (CCC)!anchoring one tension tie (CCT)!anchoring more than one tie (CTT, TTT)

Elements of strut-and-tie model


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Forces in struts and ties

In general,φFn ≥ Fu

φ : strength reduction factorFn : nominal strength of the memberFu : force in the member due to factored

loads


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Struts

!Types of struts


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! Design of strutsFns = fcuAc

fcu : effective compressive strengthfcu = ν fc’

ACI Code: φ fcu = φ ν fc’ = φSTM α1 βs fc’

(to ensure same load capacity as FIP Recommend-ations, consistency between AC1 1999 and 2002 Codes, & consistency between B-and D- regions)


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!Factors affecting fcu!Load duration effects (α1 = 0.85)!Cracking of struts

• Bottle-shaped struts• Cracked struts• Transverse tensile strains

!Confinement from surrounding concrete (e.g. pile caps)


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Prismatic strut


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Bottle-shaped strut


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Nodal zones

!Forces must be in equilibrium

!CCC, CCT, CTT, TTT joints

C

CC

C

C

T


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CCC CCT


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Extended Nodal Zones

Ws = wt cos θ + lb sin θ


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Ties

!Spread of tieswt = (Fu/ φ)/(fcu bw)

!Strength of tiesTn = Asfy

!Anchorage of ties


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Reinforcing Requirements

!Minimum reinforcement!To ensure ductility!For crack control

!Bottle-shaped struts:Σ(Asi/bsi)sin γi ≥ 0.003

!Other code requirements

(Asi/bsi)

γγγγi


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Summary

! Isolate D-region and compute force resultants on boundaries

! Draw truss to transmit forces!use of elastic analysis, crack patterns!equilibrium of forces, width of struts,

anchorage of ties! Provide steel reinforcement for ties &

check concrete stresses in struts and nodes where necessary


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Example 1 –Column on wall

534 kN534 kN

187

187

187 187

187187

263565

534534

100

mm

1.80MPa

4.67MPa

267

427 655 678 678

904 226mm

586518203

2T13 each face

2T13 each face

3T13 each face

305 x 305mm column

2438 x 305 mm wall

fc’=20 MPafy=414 MPa

Based onfce=0.66fc’

σσσσ=P/A+M/I

MacGregor


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890 kN

213

222

687687

474 530

745371

785

457

mm

457

mm

292 309

213

763

222

992

890

96

1510

89

1155

785 kN

158

b=406 mmh=508 mmd=457 mm

241 mm

w=1732/(0.61fc’)=200mm

100 mm

Finallayout


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Example 2 -Corbel

Short membercantilevering froma column or wall


305 x 127mm bearing plate

486 x 486mm

MacGregor


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a=241 mm178x102x9.5

angle

279

229

356406

3T25

2T133T13closed

stirrups

4T25

2T13C = 1155 kN; a = C/(0.8νννν2fc’b) = 127mmbef = a+llll/6 = 127+413/6 = 196 mmAsfy ≥≥≥≥ ΣΣΣΣ[(C/4)(1-a/bef)] = 203 kNAs ≥≥≥≥ 490 mm2


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Example 3 - Dapped ends

516

516

37

37553443

443

369

369

57638

5523

381

mm

381

mm

419 mm762 mm deep by381 mm width beamfc’= 20 MPa, fy=414 MPa

686

mm

H=74V=369

914 mm

Bearing area = V/(0.85νννν2fc’)

369

Checkstrut width;

compute steelrequired in ties

MacGregor


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4T13 closed stirrups 4T13 U stirrups

2T13

1T13 U bar

4T20 welded to angle

2T13U bars

2T20U bars

4T25 bars

178 mm


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Example 4 - Transfer girder


3600 700 6850

3600

10450 mm700

b=700 mm11600 kN

140.4 kN/m

MacGregor


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Combined truss and strut action

6543kN6579

kN


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Combined truss and strut action

6543kN6579

kN

Right end:

V=6543 - 11x(140.4x0.6)= 5616 kN

∴∴∴∴ req’d Av=(5616/9)x103

÷÷÷÷ (410x610)=2537 mm2/m

∴∴∴∴ use φφφφ22 U-stirrups @ 300mm c/c (2540 mm2/m)

For K-UU,D= (624+84.2)/sin 280

= 1508 kN∴∴∴∴ req’d width w =D/(bfce)

=D/(0.5bfc’) =123 mmFor S-UU, w=65 mmAverage w = 94 mm→→→→assume all struts to be100 mm and lower tensile tie located at mid-heightof truss node at UU.

Left end:

To ensure ductility, at least 30% ofshear to be transmitted by stirrups;the rest by a major diagonal strut.→→→→try φφφφ22 U-stirrups @ 225 mm c/c

(Avfyv=854 kN per 600 mm spacing)

∴∴∴∴ V transmitted by stirrups= 3x854 = 2562 kN = 39% of 6579 kN

V transmitted by strut H-AA= 6579-2562-6x84.2 =3512 kN

For H-AA, D = 5102 kN; w=416 mm.For E-AA, D = 1174 kN; w=96 mm.


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Example 5 - Deep beam with opening

fcd=17 MPa

fyd=434 MPa

Schlaich


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Right side, complete model


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left side, model 1

left side, model 2


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Check concrete stresses:

Stresses under bearing plates:σσσσp=3000x103/(700x400)

= 10.7 MPa < 1.0 fcd=17 MPaσσσσA=1070x103/(500x400)

= 5.4 MPa < 0.8 fcd=13.6 MPaσσσσB=1930x103/(500x400)

= 9.7 MPa < 0.8 fcd=13.6 MPa

Required depth of compression zone:C=T= 1070 kNd ≥≥≥≥ 1070x103/(400x1.0fcd)

= 135 mm < 400 mm ∴∴∴∴ OK(Nodes taken 200 mm below top surface.)


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Other criticalanchorages

- C, D

Checkanchoragelength of

reinforcingbars

> Anchorage length


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Furtherreinforcement

mesh on eitherface of wall

nominal column reinforcementstirrups


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References

! J.G. MacGregor, “REINFORCED CONCRETE: Mechanics and Design”, 3rd Ed., Prentice-Hall, 1997, Ch. 18.

! A.H. Nilson, D. Darwin and C.W. Dolan, “Design of Concrete Structures”, McGraw-Hill, 2003, pp.

! K.H. Reineck (Ed), “Examples for the Design of Structural Concrete with Strut-and-Tie Models”, ACI SP-208, 2002, 244 pp.

! Strut-and-Tie Resource Web Sitehttp://www.cee.uiuc.edu/kuchma/strut_and_tie/STM/


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Further reading:

! J. Schlaich, et al., “Toward a Consistent Design of Structural Concrete”, J. of Prestressed Concrete Institute, V.32, No. 3, 1987, pp.74-150.

! P. Marti, “Basic Tools of Reinforced Concrete Beam Design”, ACI Journal, V. 82, No. 1, Jan-Feb 1985, pp. 46-56.

! Tan, K.H. and Naaman, A.E., "Strut-and-Tie Model for Externally Prestressed Concrete Beams", ACI Structural Journal, Vol. 90, No. 6, USA, November-December 1993, pp. 683-691.

! Tan, K.H., “Shear Strengthening of Dapped Beams Using FRP Systems", Fifth International Symposium on Fibre Reinforced Plastics for Reinforced Concrete Structures (FRPRCS-5), Cambridge, UK, July 16-18, 2001, Vol. 1, pp. 249-258.

! Mansur, M.A., Tan, K.H. and Weng, W., “Effects of Creating an Opening in Existing Beams”, ACI Structural Journal, Vol. 96, No. 6, USA, November-December 1999, pp. 899-905.

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5-Strut & Tie Methods

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