Strut and Tie

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Strut-and-Tie Model

• Background

• AASHTO LRFD Provisions • Design Example

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8.2

Background

z

STM is a Truss Analogy

z

Truss Analogy Used in Standard and

LRFD Specifications

V

_n

= V

_c

+ V

_s

V

_s

= [A

_s

f

_y

/s]d(cot

θ

)

- AASHTO Standard

V

_s

Æ

45º Truss

- AASHTO LRFD

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8.4

STM in Codes

z

CSA 23.3-84

z

OHBDC Third Edition, 1991

z

AASHTO LRFD - First Edition, 1994

z

CHBDC - 2000

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Quiz

z

A Three-Span Concrete Beam Is Built

Monolithically, with Continuous

Reinforcement Placed Only in the

Bottom of the Beam

z

How Will this Beam Perform Under

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8.6

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Under Service Loads

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-8.8

Under Service Loads

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-Observations

z

Reinforcement Becomes Active After

Concrete Cracks

z

Redistribution of Internal Stresses

Occurs After Concrete Cracks

z

After Cracking, Concrete Structures

Behave the Way they Are Reinforced

z

For Best Serviceability, the

Reinforcement Must Follow the Flow

of Elastic Tensile Stresses

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8.10

Strut-and-Tie Model (STM)

z

Valuable tool for the analysis and

design of concrete members,

especially for regions where the

plane sections assumption of beam

theory does not apply

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8.12

STM for D-Regions

Tee Beam Dapped Beam

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Past Practice

z

D-Regions Designed Based On:

» Experience

» Empirical Rules

» Rules of Thumb

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8.14

Basic Description of the

Strut-and-Tie Model

z A design tool for “disturbed” regions

where the flow of stresses is non-uniform and the usual rules of analysis do not

apply

z A rational approach to visualize the flow of

forces at the strength limit state based on the variable-angle truss analogy

z A unified approach that considers all load

effects simultaneously

z A highly flexible and conceptual method

that recognizes that several possible solutions may exist for any problem

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STM Basic Principle

z

Concrete is Strong in Compression

Æ

Compression Struts

z

Steel is Strong in Tension

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8.16

P

2 φ >

P

2 P

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Nodal

Zones

P

2 P

P

2 C

C

T

C C

Strut

Fill

Tie

Fill

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8.18

T

C

T

C

C C

P

2 φ

A f

_{s y}

>

T

P

2 φ

>

A

f

C

c cu

φ

A f

_{s y}

>

T

φ

>

A

f

C

c cu

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Basic Concepts

Visualize a truss-like system to transfer load to the supports where:

• Compressive forces are resisted by

concrete “struts”

• Tensile forces are resisted by steel

“ties”

• Struts and ties meet at “nodes”

For best serviceability, the model should follow the elastic flow of forces

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8.20

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8.22

Methods for Formulating

Strut-and-Tie Models

z Stress trajectories from elastic analysis

z Load path approach

z Experimentally

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8.24

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8.26

Procedures for Load Path Approach

z Find reactions

z Subdivide loads and internal forces

- Replace stresses with resultants

- Replace asymmetrical stresses with

couple and resultant

z Provide struts and ties to provide load

path

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8.28

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Types of Nodes

(Schlaich et al. 1987) C - Compression T - Tension TTT CTT CCT CCC

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8.30

Assumptions

z Ties yield before struts crush (for ductility) z Reinforcement adequately anchored

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

z External forces applied at nodes z Prestressing is a load

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8.32

Examples of Good and Poor

Strut-and-Tie Models

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Factors Affecting Size of Strut

Width of the strut is affected by:

• Location and distribution of reinforcement (tie)

and its anchorage

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8.34

Strut-and-Tie vs.

Traditional Analysis/Design

Traditional section analysis/design

z Linear strain over member depth z Uniform shear stress distribution z Not valid for D-regions

Strut-and-tie

z Regions with nonlinear strain distribution

» Deep beams, pile caps

» Brackets, beam ledges, P/T anchors » Shear span/member height < 2

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a/d V/bdf_c’

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8.36

LRFD 5.2 - Definitions

Strut-and-Tie Model - A model used

principally in regions of concentrated

forces and geometric discontinuities to

determine concrete proportions and

reinforcement quantities and patterns

based on assumed compression struts in

the concrete, tensile ties in the

reinforcement, and the geometry of nodes

at their points of intersection

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5.6.3.1 D-Regions

Strut-and-tie models may be used to

determine 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 be

considered for the design of deep footings and pile caps or other situations in which the

distance between the centers of applied load

and the supporting reactions is less than about twice the member thickness.

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8.38

5.8.1.1 D-Regions

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

support is less than 2d, or components for which a load causing more than ½ of the

shear at a support is closer than 2d from the face of the support, may be considered to be deep components for which the provisions of Article 5.6.3 and the detailing

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Strength Limit State for STM

P_r= ϕ P_n (5.6.3.2-1)

where:

P_r = Factored resistance

P_n = Nominal resistance of strut or tie

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8.40 LRFD 5.6.3.3 Unreinforced strut: P_n= f_cu A_cs (5.6.3.3.1-1) Reinforced strut: P_n = f_cu A_cs + f_y A_ss (5.6.3.3.4-1) where:

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

(LRFD 5.5.4.2.1)

A_cs= effective cross-sectional area of strut

(LRFD 5.6.3.3.2)

A_ss= area of reinforcement in the strut

Strength of Struts

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STM for Deep Beam

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8.42

LRFD 5.6.3.3.2

Determined by considering available concrete area and anchorage conditions.

When anchored by reinforcement, strut may extend from the anchored bar.

C-T-T Node

a) Strut Anchored by Reinforcement

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Effective Cross-Sectional Area of Strut, A

_cs

LRFD 5.6.3.3.2

C-C-T Node

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8.44

Effective Cross-Sectional Area of Strut, A

_cs

LRFD 5.6.3.3.2

C-C-C Node

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Limiting Compressive Stress in Strut

LRFD 5.6.3.3.3 where:

(

)

(IN/IN) tie tension the of direction the in concrete the in strain tensile the (DEG) ties tension adjoining and strut e compressiv the between angle smallest the stress e compressiv limiting the f cot 0.002 f 85 . 170 0.8 f f s s cu s 2 s s 1 c 1 c cu = = = + + = ′ ≤ + ′ = ε α α ε ε ε ε 0

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8.46

Strength of Tie

LRFD 5.6.3.4.1 P_n = A_st f_y + A_ps ( f_pe + f_y )

where

A_st = Total area of longitudinal mild steel reinforcement on the tie

A_ps = Area of prestressing steel

f_y = Yield strength of mild steel longitudinal reinforcement

f_pe = Stress in prestressing steel due to prestress after losses

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Development of Ties

If x <

_l

_d

Æ

f

_s

= f

_y

(x/

_l

_d

)

Critical Section

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8.48

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Element Limiting Stress ϕ 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_y or (f_pe + f_y) 0.90 or 1.00 LRFD 5.6.3.3 - 5.6.3.5

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8.50

Crack Control Reinforcement

LRFD 5.6.3.6

z Provide orthogonal grid of reinforcement

near each face of D-Region

z Maximum Bar Spacing = 12 in.

z Ratio A_s/ A_g ≥ 0.003 in each of the

orthogonal directions

z Crack control reinforcement, located

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Summary

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 struts

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8.54

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