design of deep beams and joints
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Naveed Anwar
Buddhi S. Sharma
ACECOMS, AIT
Concrete Deep Beams,
Brackets and Joints
O-SCAAD-6July 12, 2002, AIT, Bangkok
Definition of
Deep Members
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Strain Profile – The Starting Point
• Section Capacity is represented by Stress
Resultants
• Stress Resultants are based on stress
Distribution
• Stress Distribution is based on Strain
Distribution
• Strain Distribution for a particular
deformation is not known for reinforced
concrete sections
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
The Axial-Flexural Stress Resultants
...),(1
....,1
...),(1
....,1
...),(1
...,1
121
3
121
2
121
1
i
n
i
ii
x y
y
i
n
i
ii
x y
x
x y
n
i
iiz
xyxAxdydxyxM
yyxAydydxyxM
yxAdydxyxN
The General Case: Linear or Non-linear Strain Distribution
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
The Axial-Flexural Stress Resultants
y
h
c
fc
Strain
Stresses fo
r
concrete and
R/F
Stresses fo
r
Steel
f1
f2
fn
fs NACL
Horizontal
Linear Strain
Distribution
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
The B and D regions
• If Strain is Assumed Linear then “B” Region– Plane sections remain Plane after Deformation
– “Bernoulli” assumptions apply
• If Strain is Non-linear: “D” Region: Disturbed Region– Zone where ordinary “flexural theory” does not apply
– Plane Sections do not remain plane after deformation
D B D
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Deep or Shallow
• Shallow Members:– Where most of the beam length is “B” Region
• Deep Members: – Where most of the beam length is “D” Region
• Thick Members:– Flexural Deformations are Predominant and shear
deformations can be ignored
• Thin Members:– Shear Deformations are Significant and can not be
ignored
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
What is a Deep Member ?
• Member in which most of the length is “D-
Region”
• Members that do not follow the ordinary
flexural-shear theories
• Members in which a significant amount of
the load is carried to supports by a
compression thrust joining the load and the
reaction
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Deep Members: Major Concerns
• Non linear Stress Distribution
• Possibility of Lateral Buckling
• Very Stiff Element
• Very Sensitive to Differential Settlement
• Reinforcement Development (Anchorage)
• High Stresses at Supports and Load Points
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Deep Members
• Deep Beams
• Shear Walls
• Pile Caps
• Brackets, Corbels
• Joints
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Design of Deep Members
• Empirical Methods– ACI Code Method
• The “Tie-Strut” Approach– Truss Analogy Method
– Truss Model Analysis
• Finite Element Analysis– Two Dimensional Analysis using Plane Strain
– Three Dimensional Analysis using Plates or Bricks
– Analysis modes
• Linear Analysis
• Non Linear Analysis
Basic Behavior of Deep Members
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
The Axial Stresses – True Deep Beams
Tension
Compression
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
The Axial Stresses – Semi Deep Beams
Tension
Compression
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Tension
Compression
The Axial Stresses – Mixed Beam
D B D
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Shear Stresses
Beam Model for Deep Members
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Modeling Using 1D Elements
Simple
Beam/Column
elements
Beam elements
with rigid ends
Beam elements
in “Truss
Model”
Membrane Model
for Deep Members
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Modeling Using 2D Elements
• Deep Beams are subjected to in-plane deformations so 2D elements that have transnational DOF need to be used
• A coarse mesh can be used to capture the overall stiffness and deformation of the beam
• A fine mesh should be used to capture in-plane bending or curvature
• General Shell Element or Membrane Elements can be used to model Deep Beams
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Modeling Using Membrane
Nodes: 4
DOFs: 2 (or 3) DOFs /Node Ux and Uy
2-Translation, 0 or 1 rotation
Dimension: 2 dimension element
Shape: Regular / Irregular
Properties: Modulus of Elasticity(E),
Poisson ratio(v),
Thickness( t )
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Modeling Using Shell Elements
Nodes: 4
DOFs: 5 or 6 DOFs /Node Ux and Uy
3 Translation, 2 or 3 rotation
Dimension: 2 dimension element
Shape: Regular / Irregular
Properties: Modulus of Elasticity(E),
Poisson ratio(v),
Thickness( t )
1
23
U1, R1
Node 3
U3, R3
U2, R2
U1, R1
Node 1
U3, R3 U2, R2
U1, R1
Node 4
U3, R3
U2, R2
U1, R1
Node 2
U3, R3
U2, R2
Shell
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Getting Results From Shell Model
f1, f2, …..fn are the nodal stresses at
section A-A , obtained from analysis
n
i
ii
i
n
i
i
n
i
i
iii
vAV
xFM
FP
fAF
1
1
1
f1
f2
f3
f4
f5
C
T
1x
x1
A
A
t
V
PM
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Connecting Beams to Slab
In general the mesh in the slab
should match with mesh in the
wall to establish connection
Some software automatically
establishes connectivity by using
constraints or “Zipper” elements
“Zipper”
Strut and Tie
Model for
Deep Members
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Tie-Strut Approach: Basic Concepts
• Basic Concept
– The Section is fully cracked
– Concrete takes not tension
– All Tension is taken by steel ties
– All Compression is taken by “struts” forming within
the concrete
– Strut and Tie provide a stable mechanism
– It is a “Lower Bound” solution
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Tie-Strut Approach: Basic Concepts
Conceptual TrussReal Truss
a) Simple Truss Model for V, Mx (Tie and Strut Mode)
L
d
LTies
Compressive
Struts
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Tie-Strut Approach in Use
• Truss analogy already in use– For shear design of “Shallow” and “Deep” beams
– For Torsion design of shallow beams
– For design of Pile caps
– For design of joints and “D” regions
– For Brackets and corbels
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
The Truss in Deep Members
Tension
Compression
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
The Truss in Deep Members
Tension
Compression
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
The Axial Stresses – Semi Deep Beams
Tension
Compression
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Tension
Compression
The Axial Stresses – Mixed Beam
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Truss Models and Forces
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Strut Tie Model Effect of Span:Depth Ratio
For L/D < 4
Load transferred by direct
Compression
For L/D > 4
Auxiliary Ties are required
for shear transfer
For L/D > 5
Beam tends to behave in
ordinary Flexure
L/d =2
L/a =1
L
d
a
L/d =1
L/a =0.5
L/d = 3
L/a = 1.5
L/d = 4
L/a = 2
L/d = 5
L/a = 2.5
L/d = 6
L/a = 3
L
d
a
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Strut Tie Model Effect of Strut Angle
Angle < 30 Deg.Too shallow, tension steelnot economical, strut toolong, anchorage difficult
Angle 35 - 45 DegGives the most
economicaland realistic design
Angle > 50 Deg.Too steep. Requires toomuch stirrups. Not good.
Angle = 18 Deg
Angle = 34 Deg
Angle = 45 Deg
Angle = 64 Deg
Not OK: Too Shallow
NOT OK: Too Steep and Expensive
OK: USed by ACI Code
OK: Most Ecconomical
Tension in Bottom Chord
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
The Basic Elements of Strut and Tie
• Basic Elements
– The Compression Struts in Concrete
– The Tension Ties provided by Rebars
– The Nodes connecting Struts and Ties
• Failure Mechanisms
– Tie could Yield
– Strut can Crush
– A Node could Fail
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Compression Struts
• Struts represent the compression stress field
with the prevailing compression in the
direction of the strut
• Idealized as prismatic members, or uniformly
tapered members
• May also be idealized as Bottled Shaped
members
• Transverse reinforcement is required for
prevention of failure after cracking occurs
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Types of Compression Struts
• Failure of Struts• By Longitudinal
Crushing
• Compression failure
of Struts
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Cracking of Compression Struts
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Tension Ties
• Represents one or several
layers of steel in the same
direction as the tensile force
• May fail due to
– Lack of End Anchorage
– Inadequate reinforcement quantity
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Nodal Zones
• The joints in the strut-and-tie model are know as nodal zones
• Forces meeting on a node must be in equilibrium
• Line of action of these forces must pass through a common point (concurrent forces)
• Nodal zones are classified as:– CCC
– CCT
– CTT
– TTT
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Hydrostatic Nodal Zones
Hydrostatic CCC Node Hydrostatic CCT Node
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Correct and Incorrect Truss
Correct Truss Incorrect Truss
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Using Truss Model
• Draw the beam and loads in proper scale
• Draw Primary Struts and Ties
– Struts angle between 35 to 50 degrees
– Each strut must be tied by “ties”
– The strut and ties model must be stable and determinate
• Assume dimensions of struts and ties
– Not critical for determinate trusses. Any reasonable sizes
may be used
• Make truss model in any software and analyze
• Design Truss Members
– Design rebars for tension members
– Check capacity of concrete compression members
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
How to Construct Truss Models
• For the purpose of analysis, assume the main truss layout
based on Beam depth and length
• Initial member sizes can be estimated as t x 2t for main axial
members and t x t for diagonal members
• Use frame elements to model the truss. It is not necessary to
use truss elements
• Generally single diagonal is sufficient for modeling but double
diagonal may be used for easier interpretation of results
• The floor beams and slabs can be connected directly to truss
elements
• Elastic analysis may be used to estimate truss layout
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
How to Construct Truss Models
C
t
H
t x 2t
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Simple Vs Modified Truss Model
L=2.5
a=1.6
d=1.4 h=1.6
T
P=10,000 kN
a) Simple "Strut & Tie" Model c) Modified Truss Model B
L=2.5
a=1.6
d=1.4
d=1.4 h=1.6
T
1
= tan-1 d/0.5L
= 48 deg
T = 0.5P/tan
T = 4502 kN
= tan-1 d/0.5(L-d1)
= 68.5 deg
T = 0.5P/tan
T = 1970 kN
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
A Space Truss Model for Pilecap
P1
P2
P4
P3
a2
a2
d
L2
L1
Main members
Secondary members
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Iterative Method for Truss Layout
• The truss layout can be found by using a
simple 2D truss analysis
• Draw trial truss using all possible strut tie
members
• Determine forces in the truss system
• Remove the members with small or no
forces and repeat
• Continue until the truss becomes unstable
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Getting Results from Truss Model
V
PM
)cos(
)sin(
)sin(
DV
xDCxTxM
DCTP
dct
C
T
D
Tension
Member
Compression
Member
xc
xt
xd
yst
f
TA
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Assuming Reinforcement
• Assume larger bars on the corners
• Assume more bars on predominant tension
direction/ location
• Assume uniform reinforcement on beam
sides
• Total Rebars ratio should preferably be more
than 0.8% and less than 3% for economical
design
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Interpretation of the Results
• Reinforcement should be provided along all directions where
truss members are in significant tension.
• This reinforcement should be provided along the direction of the
truss member
• The distribution of the reinforcement should be such that its
centroid is approximately in line with the assumed truss element.
• The compression forces in the struts should be checked for the
compressive stresses in the concrete, assuming the same area to
be effective, as that used in the construction of the model.
• The Bearing Stress should be checked at top of piles and at base
of columns
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Drawbacks of the Strut and Tie Approach
• Only guarantees stability and strength
• Gives no indication of performance at
service levels
• In appropriate assumed trusses layout may
cause excessive cracking
• Requires experience in judgment in truss
layout, member size assumption, result
interpretation and rebar distribution
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Designing as A Simple Flexural Member
• Approach– Design the Deep Member as “Big
Beam”
– Follow the normal axial-flexural
concept and provisions
• Input Needed– Mx , V
– Member Dimensions
• Problems– Does not consider the non-linear
strain distribution
– In efficient rebar distribution
– Does not consider Shear transfer
near ends
Deep BeamsSpecial Considerations
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Deep Members
• Behavior of Deep Beams
– What are Deep Beams?
– How do they behave?
• Design of Deep Beams
– The ACI Code Method
– The Tie and Strut Approach
– The Finite Element Analysis
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Deep Beams: ACI definition
• Beam is Deep for Flexure:
– Simple Span:
– Continuous Beam:
• Beam is Deep for Shear:
• Special Case
1.25/dln
5.2/dln
0.5/dln P
Deep
Beam Shallow Beam
ln
d
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Deep Beam or Veirendel Girder
Deep Beam
Deep Beam or
Veirendel Girder
Veirendel Girder
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
ACI Approach
• No Detailed Requirements Except “That Non
Linearity of Strain Distribution and Lateral
Buckling Must be Considered”.
• Flexure:
– No Special Requirements for design
– Specifies special limits on minimum steel
• Shear
– Special Provisions for single spans
– Special provisions for continuous beams
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Effect of Load Location
• Behavior of Deep Beams effected by the
application of load to the beam
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Shear Design: ACI Approach
• Ordinary Design Procedure
– When load is applied at the middle or at the bottom edge of the Beam, ordinary shear design provisions for shallow beams are used
• Special Design Procedure– When load is applied at the top, special design
provisions are used because load may form “arching” or “truss” mechanism
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Shear Design: ACI Approach
• Different for Simple and Continuous Beams
• Stirrups Required when
– For Single spans
– For Continuous spans.
• Critical Sections– Simple Span
– Continuous Beam: Face of Support
cVuV
cV5.0Vu
Load Conc.for a 0.15
for UDL l 0.15 n
d
d
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Shear Design: ACI Approach
Allowable shear in concrete
dbM
dVfV
dbfV
dbfVMax
w
u
uwcc
wcc
wcn
25009.1
2
8.
'
'
'
5.25.25.3
25009.1
2
52/103
2.
2/8.
'
'
'
'
u
u
w
u
uwcc
wcc
wcd
n
wcn
M
dVFwhere
dbM
dVfFV
dbfV
toisdlwhendbfd
lVMax
dlwhendbfVMax
Shallow Beams Deep Beams
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Modeling Openings in Beams
Plate-Shell Model Truss Model
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Nodal Zones within the Interaction of Members
Plastic Truss Model of a
Beam with horizontal
Web reinforcements
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Truss Model for Continuous Beam
Complete Model
Negative Moment Truss Positive Moment Truss
Brackets and Corbels
Special Considerations
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
What are Brackets and Corbels
• A short and deep member
connected to a large rigid
member
• Mostly subjected to a
single concentrated load
• Load is within ‘d’ distance
from the face of support
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Brackets or Corbels
• A short member that cantilevers out of a
column or wall to support a load
• Built monolithically with the support
• Span to depth ratio less than or equal to
unity
• Consists of incline compressive strut and a
tension tie
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Basic Stresses in Brackets
Tension Compression Shear
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Basic Stresses in Corbels
Tension Compression Shear
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Brackets using Strut and Tie Model
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Corbels using Strut and Tie Model
• Compute distance from column to Vn
• Compute minimum depth
• Compute forces on the corbel
• Lay out the strut and tie model
• Solve for reactions
• Solve for strut and tie forces
• Compute width of struts
• Reanalyze the strut and tie forces
• Select reinforcement
• Establish the anchorage of tie
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Structural Action of a Bracket
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Modes of Failure
• Yield of tension tie
• Failure of end anchorage of the
tension tie, either under the
load point or in the column
• Failure of the compression strut
by crushing or shear
• Local failure under bearing
plate
Failure due to poor detailing
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Design of Corbels ACI Method
• Depth of the outside edge of bearing area
should not be less than 0.5d
• Design for shear Vu, moment
and horizontal tensile
force of Nuc
Strength reduction Factor
850.
d)]-Nuc(h[Vua
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Design of Corbels ACI Method
uuc
ynuc
V.N
fAN
20
Area of Steel provided shall be
the greater of the two
nv
nf
A/A
AA
32
Provide Steel Area Avf to resist Vu
dbV
dbf.V
wn
wcn
800
20
Horizontal Axial Tension Force
should satisfy
Strut and tie are should not be less
than
ns AA. 50
y
cs f
f.bd/A 040
Ratio shall be
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Strut and Tie Method and the ACI Method
• Strut-and-Tie method requires more steel in
the tension tie
• Lesser confining reinforcement
• Strut-and-Tie method considers the effect of
the corbel on the forces of the column
• Strut-and-Tie method could also be used for
span to depth ratio greater than unity
Joints
Special Provisions
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Special Considerations in Joints
• Highly complex state of stress
• Often subjected to reversal of Loading
• Difficult to identify length and depth and
height parameters
• Main cause of failure for high seismic loads,
cyclic loads, fatigue, degradation etc
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Joints
• The design of Joints require a
knowledge of the forces to be
transferred through the joint and
the ‘likely’ ways in which the
transfer can occur
• Efficiency: Ratio of the failure moment of
the joint to the moment capacity of the
members entering the joint
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Basic Stresses in Joints – Gravity
Tension Compression Shear
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Basic Stresses in Joints – Lateral
Tension Compression Shear
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Strut and Tie Model
Tension Compression Strut and Tie Model
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Corner Joints
Opening Joints:
– Tend to be opened by the applied moment
• Corners of Frames
• L-shaped retaining walls
• Wing Wall and Abutments in bridges
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Corner Joints
• Closing Joints:
– Tend to be closed by the applied moment
• Elastic Stresses are exactly opposite as
those in the opening joints
• Increasing the radius of the bend increases
the efficiency of such joints
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Corner Joints
• T-Joints
• At the exterior column-beam connection
• At the base of retaining walls
• Where roof beams are continuous over
column
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Beam-Column Joints in Frames
• To transfer loads and moments at the end of
the beams to the columns
• Exterior Joint has the same forces as a T
joint
• Interior joints under gravity loads transmits
tension and compression at the end of the
beam and column directly through the joint
• Interior joints under lateral loads requires
diagonal tensile and compressive forces
within the joints
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Design of Joints-ACI
• Type 1 Joints: Joint for structures in non
seismic areas
• Type 2 Joints: Joint where large inelastic
deformations must be tolerated
• Further division into:
– Interior
– Exterior
– Corner
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
Design Stages for Type 1
• Providing confinement to the joint region by
means of beam framing into the side of the
joint, or a combination of confinement from
the column bars and ties in the joint region.
• Limiting the shear in the joint
• Limiting the bar size in the beam to a size
that can be developed in the joint
Summary
ACECOMS, AIT
Design of Deep Beams, Brackets and Joints
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