Strut Tie Model (STM)

5/27/2018 Strut Tie Model (STM)

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THE STRUT-AND-TIE MODELOF CONCRETE STRUCTURES

By

Dr. C. C. Fu, Ph.D., P.E,

The BEST Center

University of Maryland

Presented to

The Maryland State Highway Administration

August 21, 2001


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Introduction

The Strut-and-Tie is a unified approach thatconsiders all load effects (M, N, V, T)simultaneously

The Strut-and-Tie model approach evolves as oneof the most useful design methods for shear criticalstructures and for other disturbed regions inconcrete structures

The model provides a rational approach byrepresenting a complex structural member with an

appropriate simplified truss modelsThere is no single, unique STM for most designsituations encountered. There are, however, sometechniques and rules, which help the designer,

develop an appropriate model


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History and Specifications

The subject was presented by Schlaich et al(1987) and also contained in the texts byCollins and Mitchell (1991) and MacGregor(1992)

One form of the STM has been introduced inthe newAASHTO LRFD Specifications (1994),which is its first appearance in a designspecification in the US

It will be included inACI 318-02 Appendix A


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Bernoulli Hypothesis

Bernoulli hypothesis states that: " Planesection remain plane after bending"

Bernoulli's hypothesis facilitates the flexuraldesign of reinforced concrete structures byallowing a linear strain distribution for allloading stages, including ultimate flexural

capacityN.A.


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St. Venants Principle

St. Venant's Principle states that: " Thelocalized effects caused by any load

acting on the body will dissipate orsmooth out within regions that aresufficiently away from the location of theload"


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B- & D-

Regionsfor

Various

Types ofMembers


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Design of B & D Regions

The design of B (Bernoulli or Beam) region iswell understood and the entire flexural

behavior can be predicted by simplecalculation

Even for the most recurrent cases of D(Disturbed or Discontinuity) regions (such asdeep beams or corbels), engineers' ability to

predict capacity is either poor (empirical) orrequires substantial computation effort (finiteelement analysis) to reach an accurateestimation of capacity


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STM

forSimpleSpanBeam


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Feasible Inclined Angle

Swiss Code: 0.5 Cot 2.0 (=26 to 64)

European Code: 3/5 Cot 5/3 (=31 to 59)

Collins & Mitchellsmin = 10 + 110(Vu/[fcbwjd]) deg

max = 90 - min deg

ACI 2002: min =25; (25 recom 65 here)

If small is assumed in the truss model, thecompression strength of the inclined strut isdecreased.


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STM of a Deep Beam

ACI Section 10.7.1 For Deep Beam:L/d < 5/2 for continuous span; < 5/4 for simple span

ACI Section 11.8: L/d


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DeepBeam

Stressand Its

STMModel


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Transition

from Deep Beam to Beam


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STM Modelfor aTwo-span

ContinuousBeam


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Basic Concepts

Strut-and-Tie Model: A conceptual frameworkwhere the stress distribution in a structure isidealized as a system of

ConcreteConnectionNode

ReinforcementTensionMember

Tie orStirrup

ConcreteCompression

Member

Strut


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Examples of STM Models


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Strut Angle of STM Model

A STM developed with struts parallel to theorientation of initial cracking will behave very well

A truss formulated in this manner also will make themost efficient use of the concrete because theultimate mechanism does not require reorientation ofthe struts


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Lower Bound Theorem

of PlasticityA stress field that satisfies equilibriumand does not violate yield criteria at any

point provides a lower-bound estimateof capacity of elastic-perfectly plasticmaterials

For this to be true, crushing of concrete

(struts and nodes) does not occur priorto yielding of reinforcement (ties orstirrups)


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Limitation of The Truss Analogy

The theoretical basis of the truss analogy isthe lower bound theorem of plasticity

However, concrete has a limited capacity to

sustain plastic deformation and is not anelastic-perfectly plastic material

AASHTO LRFD Specifications adopted thecompression theory to limit the compressivestress for struts with the consideration of thecondition of the compressed concrete atultimate


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Prerequisites

Equilibrium must be maintained

Tension in concrete is neglected

Forces in struts and ties are uni-axial

External forces apply at nodes

Prestressing is treated as a load

Detailing for adequate anchorage


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Problems

in STM Applications

1.How to construct a Strut-and-Tie

model?2.If a truss can be formulated, is it

adequate or is there a better one?

3.If there are two or more trusses for thesame structure, which one is better?


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Struts

A. Compression struts fulfill two functions inthe STM:1. They serve as the compression chord of

the truss mechanism which resistsmoment

2. They serve as the diagonal struts whichtransfer shear to the supports

B. Diagonal struts are generally orientedparallel to the expected axis of cracking


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Types of Struts

There are three types of struts that will bediscussed:

1. The simplest type is the prism which has aconstant width

2. The second form is the bottle in which the

strut expands or contracts along its length3. The final type is the fan where an array of

struts with varying inclination meet at orradiate from a single node


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Three Types of Struts


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Compression Struts


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Ties

Tensions ties include stirrups, longitudinal(tension chord) reinforcement, and any

special detail reinforcementA critical consideration in the detailing of theSTM is the provision of adequate anchoragefor the reinforcement

If adequate development is not provided, abrittle anchorage failure would be likely at aload below the anticipated ultimate capacity


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Nodes

Nodes are the connections of the STM,i.e., the locations at which struts and

ties convergeAnother way of describing a node is thelocation at which forces are redirected

within a STM


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Type ofSingular

Nodes(Schlaich

et al1987)


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Idealized Forces

at Nodal Zones


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SingularandSmeared

Nodes


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STM Model Design Concept

The successful use of the STM requires anunderstanding of basic member behavior andinformed engineering judgment

In reality, there is almost an art to theappropriate use of this technique

The STM is definitely a design tool forthinking engineers, not a cookbook analysisprocedure

The process of developing an STM for amember is basically an iterative, graphicalprocedure


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STMModelDesignFlow

Chart


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Methods for

Formulating STM Model

Elastic Analysis based on Stress

TrajectoriesLoad PathApproach

Standard Model


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Elastic

Analysisfor the

STMModel A


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Elastic Analysis

for the STM Models B & C


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Elastic Analysis Approach

Procedures1. Isolate D-regions2. Complete the internal stresses on the

boundaries of the element3. Subdivide the boundary and compute

the force resultants on each sub-length

4. Draw a truss to transmit the forces fromboundary to boundary of the D-region

5. Check the stresses in the individualmembers in the truss


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STM

Model CExampleusing

ElasticAnalysis


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STM Model C Example

Reinforcement


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LoadPath

Approach(Schlaich

et al.1987)


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Example ofDeterminingSTM Model

Geometry


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Factors Affecting Size of

Compression Strut

Location and distribution ofreinforcement (tie) and its anchorage

Size and location of bearing


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Nodal ZonesThese dimensions are determined for eachelement using(1) the geometry of the member and the STM,

(2) the size of bearings,(3) the size of loaded areas,

(4) the location and distribution of reinforcement, and

(5) the size of tendon anchorages, if any

Struts and ties should be dimensioned so thatthe stresses within nodes are hydrostatic, i.e.,the stress on each face of the node should bethe same


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Hydrostatic Nodal Zones


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Cracking of Compression Strut

bef=a+/6

T=C(1-a/bef)/4


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STM Models A & B for

Anchorage Zones


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STM Models C & D for

Anchorage Zones


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Examplesof Good

and PoorSTM

Models

Good Model is more closely approaches to the elastic stress trajectories Poor model requires large deformation before the tie can yield; violate the

rule that concrete has a limited capacity to sustain plastic deformation


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Nonlinear finite element comparison of

three possible models of a short cantilever

(d) behaves almost elasticallyuntil anticipated failure load

(c) requires the largest

amount of plasticdeformation; thus it is morelikely to collapse beforereaching the failure load level


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STM Model for a Ledged End


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Beam-Column Opening Joints


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Efficiency of Opening Joints


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T-Joints


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Concentrated Load on a

Bearing Wall


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STMModels

(a)Tensile Flange

w/Opening(b)

CompressionFlange

w/Opening


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STM Models(c) Web supported by Diaphragm

(d) Pier and Diaphragm w/Single Support


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STM Models(e) Other Model for Diaphragm

(f) Pier and Diaphragm w/Two Supports


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STMModels

(g) Piers ona Pile Cap


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Examples of STM Models &Reinforcement (Schlaich et al 1987)


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LimitingStressesfor Truss

Elements


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Limiting Compressive Stress in StrutAASHTO LRFD 5.6.3.3.3

'

1

'

85.01708.0

cc

cu ff

f +

=

where:

e1 = (es + 0.002) cot2 as

fcu = the limiting compressive stress

as = the smallest angle between the compressivestrut and adjoining tension ties (DEG)

es = the tensile strain in the concrete in thedirection of the tension tie (IN/IN)


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Simplified Values for Limiting CompressiveStress in Strut, f

cu

(Schlaich et al. 1987)

For an undisturbed and uniaxial state of compressive stress:

fcu = 1.0 (0.85 f c?) = 0.85 f c

?

If tensile strains in the cross direction or transverse tensile

reinforcement may cause cracking parallel to the strut withnormal crack width:

fcu = 0.8 (0.85 f c?) = 0.68 f c

?

As above for skew cracking or skew reinforcement:fcu = 0.6 (0.85 f c

?) = 0.51 f c?

For skew cracks with extraordinary crack width such cracksmust be expected if modeling of the struts departssignificantly from the theory of elasticitys flow of internal

forces:fcu = 0.4 (0.85 f c

?) = 0.34 f c?


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Strength of Compressive StrutAASHTO LRFD 5.6.3.3.3

Pr = F Pn (LRFD 5.6.3.2-1)

Pn = fcuAcs (LRFD 5.6.3.3.1-1)

where:

F = 0.70 for compression in strut-and-tie models

(LRFD 5.5.4.2.1)

Acs = effective cross-sectional area of strut

(LRFD 5.6.3.3.2)


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ACI 2002 STM Model

un FF

Design of struts, ties, and nodal zones shall be based on:

The nominal compressive strength of a strut without

longitudinal reinforcement:

ccuns AfF =

The effective compressive strength of the concrete

in a strut is:'

85.0 cscu ff =


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ACI 2002 STM Model

The nominal strength of a tie shall be taken as:

psepsystnt ffAfAF ++=

The nominal compression strength of a nodal zone shall be:

ncunn

AfF =

The strength of a longitudinally reinforced strut is:''

ssccuns fAAfF +=


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Findings of STM Model

The STM formulation that requires the leastvolume of steel will be the solution that bestmodels the behavior of a concrete member

This approach holds great promise for DOTsand design offices which could develop orobtain standard STMs for certain commonlyencountered situations

Standard reinforcement details based on anSTM could be developed for common situations

The STM then could be reviewed and revised ifany parameters change


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Hammerhead Pier Example


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Hammerhead Pier STM Model


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Spreadsheet Calculation of STMModel Examples

Abutment on Pile Model Example

Walled Pier Model Example

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Strut Tie Model (STM)