strut & tie models (s-t-m)

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STRUT & TIE MODELS (S-T-M)

Module 2

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Topics

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

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• 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

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Strut and Tie design Methodology• Steps in design1) Define and Isolate D-regions #2) Compute the resultant forces on each D-region

boundary #3) Select a truss model to transfer the forces across a D-

region #4) Select dimensions for nodal zones #5) Verify the capacity of node and strut; for struts at

mid-length and nodal interface #6) Design the ties and tie anchorage #7) Prepare design details and minimum reinforcement

requirements #HOME

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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

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Strut and Tie design Methodology• Steps in design1) D-regions (ACI A)

Region extending on both sides of a discontinuity by a distance ‘h’

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Strut and Tie design Methodology• Steps in design2) 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

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Strut and Tie design Methodology• Steps in design3) 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

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Alternative truss models for a deep beam

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Strut and Tie design Methodology

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Alternative truss models for a deep beam

• Single tension tie• direct load path

• Complex layout•Upper tension tie• Lower tension tie

Greater number of transfer points & ties

More flexible truss

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Strut and Tie design Methodology• Steps in design4) 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

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Strut and Tie design Methodology• Steps in design4) 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

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Strut and Tie design Methodology

• Steps in design5) 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

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Strut and Tie design Methodology

• Steps in design6) Design of Ties and Anchorageo At service loads, stress in reinforcement well

below yield stress (crack control)o Geometry of tie – reinforcement fits within tie

dimensions, full anchoringo Anchorage – nodal and extended nodal zones +

available regions on far side

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Strut and Tie design Methodology• Length available

for anchorage of ties la

• Extended nodal zone

• Extend beyond or hooks for full development

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Strut and Tie design Methodology• Steps in design7) Design details and minimum reinforcement

requirementso Complete design demands the verification

Tie reinforcement can be placed in the section Nodal zones confined by compressive forces or ties Minimum reinforcement requirements

o Tie details – development length, mechanical anchorage

o Shear reinforcement – permissible shear force(code), controlled longitudinal cracking of bottle shaped struts, minimum reinforcement (code)

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ACI Code Provisions• Strength of struts

• Strength of nodal zones

• Strength of ties

• Shear reinforcement requirements (Deep

beams)

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ACI Code ProvisionsStrength 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

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ACI Code ProvisionsStrength of struts

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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)

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'fs

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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

•

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&page number

ACI Code Provisions

Strength of struts• Rectangular or bottle-shaped strut?• Horizontal struts as rectangular, inclined as

bottle-shaped

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ACI Code ProvisionsStrength of Nodal zones• Nominal compressive strength of a nodal zone

• ,effective strength of concrete in nodal zone

• , is the smaller of (a) and (b)

•

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cef

nzA

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ACI Code ProvisionsStrength 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

•

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ACI Code Provisions

Strength of Ties (cl. A.4)• Nominal strength of a tie

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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–

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ACI Code ProvisionsStrength 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.

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Extended nodal zone showing the distribution of force

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ACI Code Provisions

ACI Shear Requirements for Deep Beams– Deep beams Beams with 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

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• s and s2 may not exceed d/5 or 12 inches

• For STM, bw is thickness of element b

ACI Code Provisions

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Applications

• Deep beams• Beam-column joints• Corbel

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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

• • • •

• •

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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

dbfV wcn '10 Dividing by 1000, to convert to kips

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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

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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

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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

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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

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Deep beams• Step 5 : Capacity of struts– Horizontal rectangular strut, inclined bottle shaped– Strut ac

– Adequate– Strut ab

– The capacity of strut ab established at node b

– The capacity of strut ac established at node b

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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

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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 dimensionsNote 2.5 in. clear cover, 4.5in. clear spacing b/w layers

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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

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Deep beams

• Step 7 : Design details and minimum reinforcement requirement

• Shear reinforcement requirement in deep beams- ACI

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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

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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

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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

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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

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Column brackets or Corbel

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Column brackets & Corbels• Brackets – in precast construction - to support

precast beams at columns• When brackets are projected from a wall, rather

than from a column, they are properly called corbels• Both terms may be used interchangeably• Design - Vertical reaction Vu at the end of

supported beam• Horizontal force Nuc if adequate measures are not

taken to avoid horizontal forces by shrinkage, creep, temperature change

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Column brackets & Corbels• Bearing plates or angles on the top surface of

the bracket• Elastomeric bearing pads – frictional forces –

volumetric change• Account for horizontal forces• Strut and Tie model• The steel required by STM , main bars

anchorage

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Corbel

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Corbel

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Corbel

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Beam-Column Joints• Inadequate attention to the detailing of

reinforcement• Mainly at the connection of main structural

elements• The basic requirement at joint – all of the

forces existing at the ends of the members must be transmitted through the joint to the supporting members

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References:

• Design of concrete structures, by A H Nilson