Thiết kế bê tông cốt thép nâng cao (Đại học quốc gia Singapore)

CE5510 Advanced Structural
Concrete Design
- STRUT-AND-TIE METHODS -

Assoc Prof Tan Kiang Hwee
Department of Civil Engineering
National University of Singapore
2/16/2004


In this lecture

DEPARTMENT OF CIVIL ENGINEERING

We will explore
!the

concept of strut-and-tie models
!their applications to new construction
(and strengthening works)

2/16/2004

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

At the end of the lecture
You should be able to
!identify

cases where strut-and-tie
models are applicable or appropriate
!formulate strut-and-tie models in
structural concrete members
!design the reinforcement according to
the strut-and-tie models
2/16/2004

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

Contents
!B-

and D-regions

!Concept

of Strut-and-Tie Models

• Geometric Layout
• Design of Struts
• Nodes and Nodal Zones
• Design of Ties
• Detailing

2/16/2004

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

! Design

•
•
•
•
•
•

Examples for New Construction

High Wall
Corbel
Dapped-Beam
Transfer Girder
Deep Beam with Opening
(Stepped (Non-Prismatic) Beams)

! (Examples

for Strengthening Works)

• Dapped Beams
• Beam with Openings or Recesses

2/16/2004

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

Main (B-) & Local (D-) regions
D-region
B-region

2/16/2004

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

Main (B-) regions

!regions

of relatively uniform stresses
!Bernoulli hypothesis of linear strain
distribution applies
!internal forces or stresses are derived
from statics
!“Standard” methods of Codes apply

2/16/2004

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

Local (D-) regions

!significantly

non-linear strain

distribution
!near concentrated loads, corners,
bends, openings and other
discontinuities
!internal flow of forces well described
by strut-and-tie models
!conventionally design by thumb-rule
2/16/2004

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

Concept of Strut-and-Tie Models
!

Components

! concrete

compression

struts
! steel tension ties
! nodes (nodal zone) where
struts and ties meet
!

concrete

Dual purpose
! describe

essential aspects
of structural behaviour
! provide tools for structural
dimensioning

2/16/2004

steel

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

Geometric Layout of strut-and-tie models

Load path
?

Boundary
forces/stresses

follows the flow of internal forces in
the structure
2/16/2004

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

!

Major requirements
! S-T

model must be in equilibrium with applied
loads (statically admissible field)
! Strength of struts, ties and nodal zones must
equal or exceed forces in these members (safe)
! Sufficient to consider only axes of struts and ties in
the early design stage; need to consider widths in
general
! Struts must not overlap each other
! Ties may cross struts or other ties

! Angle between a strut and a tie joined at a node
should not be less than 25 degrees.
2/16/2004

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

Basic steps
!

!
!
!

Compute internal stresses
on boundaries, subdivide
boundary and compute
force resultants on each
sub-length; or
Compute action effects on
boundaries
Draw truss to transmit
forces
Check stresses in
individual truss member

2/16/2004

P

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

Some rules
for
estabilshing
strut-and tie
model

Elastic
stress
trajectories
2/16/2004

 Tan K H, NUS


Minimum steel content

DEPARTMENT OF CIVIL ENGINEERING

ΣFiliεmi=minimum

2/16/2004

"

×

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

Agreement with Crack Pattern

2/16/2004

×

 Tan K H, NUS


Superposition of models

DEPARTMENT OF CIVIL ENGINEERING

"
2/16/2004

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

Truss 2 can form only if truss

1 does not fail prematurely

2/16/2004

×

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

Exercise 1
!

Explore the application of strut-and-tie
model in the design of anchorage zone
of a post-tensioned beam
compression

or
tension

Principal compressive
2/16/2004Stress trajectories

Stress contours
 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

Exercise 2
!

A T-beam is post-tensioned with a cable anchored at
the centroid of the section at its end. Given that the
area of the flange is one-third of the overall crosssection, explain by sketching in the following figures,
how you would obtain the required reinforcement to
resist bursting tension in the web due to the
prestressing force.

x-section
2/16/2004

strut-&-tie model

reinforcement
 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

Elements of strut-and-tie model
!

Compression struts
! line

along centre-line of strut
! strut with width

!

Tension ties
! band

of steel reinforcement
! anchorage (hooks, development length)
!

Nodes
! bounded

by compressive forces (CCC)
! anchoring one tension tie (CCT)
! anchoring more than one tie (CTT, TTT)
2/16/2004

 Tan K H, NUS


Forces in struts and ties

DEPARTMENT OF CIVIL ENGINEERING

In general,
φFn ≥ Fu
φ : strength reduction factor
Fn : nominal strength of the member
Fu : force in the member due to factored
loads

2/16/2004

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

Struts
! Types

2/16/2004

of struts

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

!

Design of struts
Fns = fcuAc
fcu : effective compressive strength
fcu = ν fc’
ACI Code: φ fcu = φ ν fc’ = φSTM α1 βs fc’
(to ensure same load capacity as FIP Recommendations, consistency between AC1 1999 and 2002
Codes, & consistency between B-and D- regions)

2/16/2004

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

! Factors

affecting fcu

!Load

duration effects (α1 = 0.85)
!Cracking of struts
• Bottle-shaped struts
• Cracked struts
• Transverse tensile strains
!Confinement

from surrounding
concrete (e.g. pile caps)

2/16/2004

 Tan K H, NUS


DEPARTMENT OF CIVIL ENGINEERING

Prismatic strut

2/16/2004

 Tan K H, NUS