STRUT & TIE MODELS (S-T-M)

1. STRUT & TIE MODELS (S-T-M) Module 2

2. Topics • Introduction • Development • Design Methodology • IS and ACI provisions • Applications • Deep beams • Corbels • Beam-column joints

3. Hydrostatic state of stress • Nodal zone dimensions proportional to the applied compressive forces • One dimension by the bearing area • Other two, for a constant level of stress ‘p’ • Preselected strut dimensions , non hydrostatic • Extended Nodal zone • Inadequate length of hydrostatic zone for tie anchorage • Intersection of the nodal zone and associated strut • The portion of the overlap region between struts & ties, not already counted as part of a primary node

4. Strut and Tie design Methodology • Steps in design • Define and Isolate D-regions # • Compute the resultant forces on each D-region boundary # • Select a truss model to transfer the forces across a D-region # • Select dimensions for nodal zones # • Verify the capacity of node and strut; for struts at mid-length and nodal interface # • Design the ties and tie anchorage # • Prepare design details and minimum reinforcement requirements # HOME

5. Strut and Tie design Methodology • Strength and serviceability • Strength criteria • ACI A.2.6 • Strength reduction factor 0.75 • Serviceability checks • Spacing of reinforcement within ties

6. Strut and Tie design Methodology HOME • Steps in design • D-regions (ACI A) • Region extending on both sides of a discontinuity by a distance ‘h’

7. Strut and Tie design Methodology • Steps in design • Resultant forces on D-region boundaries • Helps in constructing the geometry of the truss model • Subdividing the boundary into segments • Distributed load • Moments at faces of beam column joints HOME

8. Strut and Tie design Methodology Alternative truss models for a deep beam • Steps in design • The Truss model • Multiple solutions • Axes of truss members to coincide with centroids of stress fields • Struts must intersect only at nodal zones; ties may cross struts • Effective model-minimum energy distribution through D-region • Stiffest load path • Minimum no. of ties • Equilibrium ,structural stiffness • Effectively mobilizes ties -cracking • Points of maximum stresses HOME

9. Strut and Tie design Methodology Alternative truss models for a deep beam • Single tension tie • direct load path Greater number of transfer points & ties More flexible truss • Complex layout • Upper tension tie • Lower tension tie HOME

10. Strut and Tie design Methodology • Steps in design • Selecting dimensions for Struts and Nodal zones • Width on magnitude of forces & dimensions of adjoining elements • External effects bearing plate area on Nodal zone dimensions • Angle between struts and ties at a node>25◦ • Design of nodal zones • Principal stresses within the intersecting struts and ties are parallel to the axes of these truss members • Width of struts and ties α forces in the elements • Width of strut by Geometry of bearing plate / tension tie – non-hydrostatic HOME

11. Strut and Tie design Methodology • Steps in design • Selecting dimensions for Struts and Nodal zones • Thickness of strut, tie and nodal zone typically equal to that of the member • If thickness of bearing plate < thickness of member, reinforcement perpendicular to the plane of the member to be added – confinement, splitting HOME

12. Strut and Tie design Methodology • Steps in design • Capacity of Struts • Based on, the strength of the strut & strength of nodal zone • Insufficient capacity of strut – revising the design • Add compression reinforcement • Increase size of nodal zone • Bearing area of plate and column HOME

13. Strut and Tie design Methodology • Steps in design • Design of Ties and Anchorage • At service loads, stress in reinforcement well below yield stress (crack control) • Geometry of tie – reinforcement fits within tie dimensions, full anchoring • Anchorage – nodal and extended nodal zones + available regions on far side HOME

14. Strut and Tie design Methodology • Length available for anchorage of ties la • Extended nodal zone • Extend beyond or hooks for full development HOME

15. Strut and Tie design Methodology • Steps in design • Design details and minimum reinforcement requirements • Complete design demands the verification • Tie reinforcement can be placed in the section • Nodal zones confined by compressive forces or ties • Minimum reinforcement requirements • Tie details – development length, mechanical anchorage • Shear reinforcement – permissible shear force(code), controlled longitudinal cracking of bottle shaped struts, minimum reinforcement (code) HOME

16. ACI Code Provisions • Strength of struts • Strength of nodal zones • Strength of ties • Shear reinforcement requirements (Deep beams) HOME

17. ACI Code Provisions Strength of struts • Nominal compressive strength of a strut Where fce is the effective compressive strength of concrete in a strut or nodal zone Acs is the cross sectional area at one end of the strut = strut thickness x strut width • fce=0.85 βs f΄c HOME

18. ACI Code Provisions Strength of struts HOME

19. ACI Code Provisions Strength of struts • When compression steel is provided, strength is increased to • depends on the strain in concrete at peak stress • ACI A.3.5 • Transverse reinforcement for bottle shaped struts • Code says ‘it shall be permitted to assume the compressive force in the strut spreads at a slope of 2 longitudinal to 1 transverse to the axis of the strut’(cl.A.3.3) HOME

20. ACI Code Provisions Strength of struts • For ≤ 6000 psi, A.3.3 transverse reinforcement - axis of the strut being crossed by layers of reinforcement satisfying HOME

21. ACI Code Provisions Strength of struts • Rectangular or bottle-shaped strut? • Horizontal struts as rectangular, inclined as bottle-shaped HOME &page number

22. ACI Code Provisions Strength of Nodal zones • Nominal compressive strength of a nodal zone • ,effective strength of concrete in nodal zone • , is the smaller of (a) and (b) HOME

23. ACI Code Provisions Strength of Nodal zones • Unless confining reinforcement is provided in the nodal zone ,maximum (A.5.2) • , compressive strength of concrete in nodal zone • βn , factor for degree of disruption –incompatibility between strains in struts and ties HOME

24. ACI Code Provisions Strength of Ties (cl. A.4) • Nominal strength of a tie HOME

25. ACI Code Provisions Strength of Ties (cl. A.4) • Effective width of a tie, wt • Distribution of tie reinforcement • If placed in single layer, wt = diameter of the largest bars in the tie + 2*the cover to surface of bars • Or width of anchor plates HOME

26. ACI Code Provisions Strength of Ties (cl. A.4) • The axis of reinforcement in a tie shall coincide with the axis of the tie in STM • Anchor the reinforcement as required by mechanical devices, post-tensioning anchorage devices, standard hooks etc. • Ties must be anchored before they leave the extended nodal zone at a point defined by the centroid of the bars in the tie and the extensions of either the strut or the bearing area. HOME

27. Extended nodal zone showing the distribution of force

28. ACI Code Provisions ACI Shear Requirements for Deep Beams • Deep beams Beamswith clear span less than or equal to 4 times the total member depth or with concentrated loads placed within twice the member depth of the support • Design either by Non-linear analysis or by Strut and Tie method • Nominal shear ≤ (11.7.3) • Minimum reinforcement perpendicular to the span • Minimum reinforcement parallel to the span HOME

29. ACI Code Provisions • s and s2 may not exceed d/5 or 12 inches • For STM, bw is thickness of element b

30. Applications • Deep beams • Beam-column joints • Corbel

31. Deep beams • One of the principal application • Alternative solution –nonlinear analysis • Question: A transfer girder is to carry two 24in. Square columns, each with factored loads of 1200kips located at third points of its 36 ft span, as in the fig, The beam has a thickness of 2ft and a total height of 12ft. Design the beam for the given loads, ignoring the self weight. Use fc’=5000psi & fy=60000psi

32. Deep beams • Solution: • Span/depth =3.0 deep beam • Use strut and tie model • Step 1 : Define D-region • Entire structure as D-region • Thickness of struts and ties = thickness of beam = 2ft=24in • Assume effective depth =0.9h=0.9x12=10.8ft • Maximum shear capacity of the beam , = 0.75x10x√5000x24x(10.8x12/1000) =1650kips > Vu=1200kips • Thus design may continue Dividing by 1000, to convert to kips

33. Deep beams • Step 2 : Force Resultants on D-region boundaries • Reactions at supports = 1200kips (equilibrated by the column loads on the upper face of beam) • Assume centre to centre distance between horizontal strut and tie = 0.8h = 9.6ft • Angle between trial diagonal struts and horizontal tie =38.66 0 • Analyze the truss to find the forces in struts and tie

34. Deep beams • Step 3 : Truss model • Based on the geometry and loading, a single truss as shown, is sufficient to carry the column loads • The truss has a trapezoidal shape • Nodes that are not true pins and instability within the plane of truss. Not a concern in Strut and Tie models. Hence this is an acceptable solution • The truss geometry is established by the assumed intersection of the struts and ties used to determine θ

35. Deep beams • Step 4 : Selecting dimensions for struts and nodal zone • Two approaches 1) constant level of stress 2) minimum strut width • CCC node βn =1.0 • = 0.75 x 0.85 x 1.0 x 1.0 x 5000/1000 = 3.19 ksi • >2.08 ksi, demand from the column, smaller sizes possible • Width of the strut ac

36. Deep beams • Step 4 : Selecting dimensions for struts and nodal zone • wab=?, wtie=? • new c/c distance between horz strut and tie • Angle • Revised forces • iterate

37. Deep beams • Step 5 : Capacity of struts • Horizontal rectangular strut, inclined bottle shaped • Strut ac • Adequate • Strut ab • The capacity of strut abestablished at node b • The capacity of strut ac established at node b

38. Deep beams • Step 5 : Capacity of struts • Capacity at the end of the struts and at the nodes exceeds the factored loads • Hence the struts are adequate

39. Deep beams • Step 6 : Design of ties and anchorage • Selection of area of steel • Design of the anchorage • Validation that tie fits within the available tie width • Area of steel, • Provide 22 No.s No.11 bars, • Placing the bars in two layers of 5 bars each and three layers of 4 bars each, total tie width matching tie dimensions Note 2.5 in. clear cover, 4.5in. clear spacing b/w layers

40. Deep beams • Step 6 : Design of ties and anchorage • Anchorage length Ld, chapter 12 of ACI 318-11 • For no. 11 bars, • Length of nodal zone and extended nodal zone = 24 + 0.5 x 30.7 x cot 380 = 43.6 in. < Ld • Provide 900 hooks / mechanical anchorage • 1.5 in cover on both sides, side face reinforcement No.5 bars transverse & horizontal, 2db spacing between No.11 bars • required total thickness • Fit within the 24 in. beam thickness

45. Deep beams • Step 7 : Design details and minimum reinforcement requirement • Shear reinforcement requirement in deep beams- ACI

46. Deep beams • Step 7 : Design details and minimum reinforcement requirement • Av = 0.0025 x 24 x 12 =0.72 in2/ft • Providing No. 5 (0.625in,0.31in2) bars, s = 12 x 0.31x2faces /0.72 = 10.33 in. At spacing 10 in, Av provided = 12 x 0.31 x 2 /10 = 0.74 in2/ft

47. Deep beams • Step 7 : Design details and minimum reinforcement requirement • Avh = 0.0015 x 24 x 12 =0.43 in2/ft • Providing No. 4 (0.5 in,0.20in2) bars, s = 12 x 0.20x2faces /0.43 = 11.16 in. At spacing 10 in, Av provided = 12 x 0.20 x 2 /10 = 0.48 in2/ft

48. Deep beams • Step 7 : Design details and minimum reinforcement requirement • Read & understand RA 3.3 ACI 318-11 • Two No. 5 bars Av =0.62 in2 ; Two No.4 bars Avh = 0.4 in2 • This ensures sufficient reinforcement is present to control longitudinal splitting

49. Deep beams • Step 7 : Design details and minimum reinforcement requirement • Staggered hooks used for anchorage • Horizontal U-shaped No.4 bars @ 4 in (3db) across the end of the beam to confine No. 11 hooks

50. Column brackets or Corbel