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8.26

Procedures for Load Path Approach

Find reactions

Subdivide loads and internal forces

- Replace stresses with resultants

- Replace asymmetrical stresses with

couple and resultant Provide struts and ties to provide load

path

Locate ties using practical dimensions

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8.27STM from Tests - Dapped Beam

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8.28Dapped Beam

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8.29Types of Nodes

(Schlaich et al. 1987)

C - CompressionT - Tension

TTT

CTT

CCT

CCC

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8.30Assumptions

Ties yield before struts crush (for ductility)

Reinforcement adequately anchored

Forces in struts and ties are uniaxial Tension in concrete is neglected

External forces applied at nodes

Prestressing is a load

Equilibrium must be maintained

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8.31Strut-and-Tie Model Design Procedure

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8.32Examples of Good and Poor

Strut-and-Tie Models

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8.33Factors Affecting Size of Strut

Width of the strut is affected by:

Location and distribution of reinforcement (tie)

and its anchorage Size and location of bearing

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8.35

a/d

V/bdfc

Source: Prestressed Concrete Structures by Collins & Mitchell

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8.36LRFD 5.2 - Definitions

Strut-and-Tie Model - A model used

principally in regions of concentrated

forces and geometric discontinuities todetermine concrete proportions and

reinforcement quantities and patterns

based on assumed compression struts in

the concrete, tensile ties in the

reinforcement, and the geometry of nodesat their points of intersection

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8.375.6.3.1 D-Regions

Strut-and-tie models may be used todetermine internal force effects near supports

and the points of application of concentrated

loads at strength and extreme event limit

states.

The strut-and-tie model should beconsidered for the design of deep footings and

pile caps or other situations in which the

distance between the centers of applied loadand the supporting reactions is less than about

twice the member thickness.

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8.385.8.1.1 D-Regions

Components in which the distance fromthe point of zero shear to the face of the

support is less than 2d, or components for

which a load causing more than of the

shear at a support is closer than 2d from the

face of the support, may be considered to bedeep components for which the provisions

of Article 5.6.3 and the detailing

requirements of Article 5.13.2.3 apply.

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8.39

Strength Limit State for STM

Pr = Pn (5.6.3.2-1)where:

Pr= Factored resistance

Pn = Nominal resistance of strut or tie

= Resistance factor for tension or compression (5.5.4.2)

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8.40

LRFD 5.6.3.3Unreinforced strut:

Pn=

fcuAcs (5.6.3.3.1-1)Reinforced strut:

Pn=

fcuAcs+ fyAss (5.6.3.3.4-1)

where:

= 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)

Ass= area of reinforcement in the strut

Strength of Struts

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8.41STM for Deep BeamLRFD Fig. C5.6.3.2-1

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8.42

LRFD 5.6.3.3.2

Determined by considering available concrete areaand anchorage conditions.

When anchored by reinforcement, strut may extendfrom the anchored bar.

C-T-T Nodea) Strut Anchored by Reinforcement

Effective Cross-Sectional Area of Strut, Acs

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8.43Effective Cross-Sectional Area of Strut, AcsLRFD 5.6.3.3.2

C-C-T Node

b) Strut Anchored by Bearing and Reinforcement

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8.44Effective Cross-Sectional Area of Strut, AcsLRFD 5.6.3.3.2

C-C-C Node

c) Strut Anchored by Bearing and Strut

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8.45

Limiting Compressive Stress in Strut

LRFD 5.6.3.3.3

where:

(IN/IN)tietensiontheofdirectiontheinconcretetheinstraintensilethe

(DEG)tiestension

adjoiningandstrutecompressiv

thebetweenanglesmallestthe

stressecompressivlimitingthefcot0.002

f85.1700.8

ff

s

s

cu

s2

ss1

c

1

ccu

0

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8.46Strength of TieLRFD 5.6.3.4.1

Pn = Astfy+ Aps ( fpe + fy)

where

Ast= Total area of longitudinal mild steel reinforcement

on the tieAps = Area of prestressing steel

fy = Yield strength of mild steel longitudinal

reinforcementfpe = Stress in prestressing steel due to prestress afterlosses

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8.47Development of Ties

If x < d fs = fy(x/d)

Critical

Section

=x

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8.48Development of Ties (ACI 318)

Limiting Stresses for STM Elements

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8.49

Element Limiting Stress

1 - CCC Node 0.85 f c 0.70 2 - CCT Node 0.75f c 0.70

3 - CTT or TTT Node 0.65f c 0.70

4 - Strut f cu 0.70

5 - Tie f yor (fpe+ fy) 0.90 or 1.00

Limiting Stresses for STM ElementsLRFD 5.6.3.3 - 5.6.3.5

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8.50

Crack Control Reinforcement

LRFD 5.6.3.6

Provide orthogonal grid of reinforcement

near each face of D-Region

Maximum Bar Spacing = 12 in.

Ratio As/ Ag 0.003 in each of theorthogonal directions

Crack control reinforcement, located

within tie, considered as part of tie

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8.51Summary

1. Visualize flow of stresses

2. Sketch an idealized strut-and-tie model

3. Select area of ties

4. Check nodal zone stresses

5. Check strength of struts6. Provide adequate anchorage for ties

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8.52

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8.53

Strut-and-Tie Model

8 54

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8.54

Strut-and-Tie Model

8 55

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8.55Design Examples

1. Two Column Bent Cap

2. Spread Footing

3. Pile Cap4. Dapped-End Beam

5. Hammerhead Pier