1.054/1.541 Mechanics and Design of Concrete Structures
Prof. Oral Buyukozturk
Spring 2004
Massachusetts Institute of Technology
Outline 8
1.054/1.541 Mechanics and Design of Concrete Structures (3-0-9)
Outline 8
Biaxial Bending
‰
R/C columns under biaxial bending
o The problem of columns under biaxial bending is nonlinear and the
number of unknowns is large. For any defined column the problem
may be expressed as:
( P , e , e ) = f ( c,θ , ε )
x
y
c
where f = a nonlinear function of the variables that can be derived
from the equilibrium equations and geometry of any given
column section and the stress-strain curves of the
materials,
P = axial load,
ex , e y = eccentricities measured parallel to x and y axes,
respectively,
θ = inclination of neutral axis,
c = depth of neutral axis measured from extreme compressive
fiber, and
ε c = failure strain of concrete in compression.
o A number of approximate methods which are based on simplifying
assumptions have been developed. However, for certain situations, the
simplifying assumptions may lead to inaccurate results, and the use
of presently available design charts is often limited.
o The use of computers to solve this problem with improved accuracy
has been based upon iterative analysis of trial sections until a
satisfactory result is achieved.
1/6
1.054/1.541 Mechanics and Design of Concrete Structures
Prof. Oral Buyukozturk
Spring 2004
Outline 8
o Apart from these considerations, the design process should not be
limited to a purely numerical evaluation of the loads, stresses, and
strains involved; basic design issues, such as seismic requirements,
architectural preferences, availability of material types and sizes,
economy, and constructability must be taken into account.
‰
Basic concept of analysis
o The analysis is based on the strain compatibility and equilibrium
equations for the column section.
o For a given neutral axis position, the strains, stresses, and forces in
the steel can be found. The resultant force in the concrete depends
on the shape of the stress block.
o Consider the following configuration of column section and its strain
distribution,
Column configuration
ex
y
Strain distribution
K xb
x
tx
1
4
3
h
ty
y
x
2
K yh
b
θ
εcu=0.003
εs1
c
εs2
tan θ =
K xb
fs1
εs3
K yh
εs4
fs2 Cc a = k c
1
fs3
fs4
2/6
N.A.
Pu
ey
1.054/1.541 Mechanics and Design of Concrete Structures
Prof. Oral Buyukozturk
Spring 2004
Outline 8
Strains in steel bars:
⎛
ε s1 = 0.003 ⎜⎜1 −
⎝
⎛
ε s 2 = 0.003 ⎜⎜1 −
⎝
t ⎞
tx
− y ⎟
K x b K y h ⎟⎠
t ⎞
b − tx
− y ⎟
K x b K y h ⎟⎠
{
}
ε s3 =
0.003
⎡ h (1 − K y ) − t y ⎤ cos θ + t y sin θ
⎣
⎦
c
εs4 =
⎡
⎫
⎛ K y h ⎞⎤
0.003 ⎧ K y h
⎡b (1 − K x ) − t x ⎦⎤ + h − t y ⎬ cos ⎢ tan −1 ⎜
⎨
⎟⎥
⎣
c ⎩ K xb
⎭
⎝ K xb ⎠⎦
⎣
o Determination of the stress of steel bars
For ε si < ε y , f si = Es ε si .
For ε si ≥ ε y , f si = f y .
o Equilibrium conditions
Forces in steel bars:
S1 = f s1 As1 = ( ε s1 Es ) As1
S2 = f s 2 As 2 = ( ε s 2 Es ) As 2
S3 = f s 3 As 3 = ( ε s 3 Es ) As 3
S4 = f s 4 As 4 = ( ε s 4 Es ) As 4
Force equilibrium:
∑F = 0; C
c
+ S1 + S2 − S3 − S4 = 0
Conditions of moment equilibrium are expressed in x and y directions.
⎛h
⎞
⎛h
⎞
⎛h
⎞
M ux = Pu e y = Cc ⎜ − y ⎟ + ( S1 + S 2 ) ⎜ − t y ⎟ + ( S3 + S 4 ) ⎜ − t y ⎟
⎝2
⎠
⎝2
⎠
⎝2
⎠
⎛b
⎞
⎛b
⎞
⎛b
⎞
⎛b
⎞
⎛b
⎞
M uy = Pu ex = Cc ⎜ − x ⎟ + S1 ⎜ − t x ⎟ − S 2 ⎜ − t x ⎟ − S3 ⎜ − t x ⎟ + S 4 ⎜ − t x ⎟
⎝2
⎠
⎝2
⎠
⎝2
⎠
⎝2
⎠
⎝2
⎠
3/6
1.054/1.541 Mechanics and Design of Concrete Structures
Prof. Oral Buyukozturk
Spring 2004
Outline 8
Analysis and design requires trials and iterations to find the
inclination and depth of the neutral axis satisfying the equilibrium
equations.
‰
Failure interaction curve
o The relationship expressed in
( P , e , e ) = f ( c,θ , ε )
x
y
c
describes the three dimensional failure surface (failure interaction
curve) if the concrete strain is taken to be ε u (usually 0.003). Any
combination of neutral axis depth c and inclination θ will give a
unique triplet of Pu, eux, and euy corresponding to a point on this
failure surface.
o Evaluation of column adequacy using a numerical scheme:
In order to declare the adequacy of the column section to resist a
given combination of P, ex, and ey only one point on the failure surface
need to be computed. Such point satisfies the following condition
Pu = P and
euy
eux
=
ey
ex
y
load point ray
eux
ex
b/2
given load point
euy
h/2
ey
α
x
0
ƒ
The procedure is summarized as follows:
4/6
1.054/1.541 Mechanics and Design of Concrete Structures
Prof. Oral Buyukozturk
Spring 2004
Outline 8
(a) Find the neutral axis inclination satisfying
euy
eux
=
ey
ex
.
(b) Set the neutral axis depth c equal to the neutral axis depth
computed from balanced failure condition for the section.
(c) Compute the value of Pu and update c using a modified secant
numerical method until Pu = P .
(d) Compute eux and euy and compare with ex and ey to decide
whether the section is adequate or not.
o Approximate method for the determination of the neutral axis
inclination θ
For rectangular column sections shown:
y
Compression region
Pu
α
h
θ
x
N.A.
b
Approximately,
θ = 90 − y + c −
where y =
c=
z
= inclination of N.A. to y axis
2
c
c2
+
− x 2 + cx
2
4
h
127 127b
(if = 1, θ = α )
=
h
b
−1 h − b
b
5/6
1.054/1.541 Mechanics and Design of Concrete Structures
Prof. Oral Buyukozturk
Spring 2004
Outline 8
z
2
x =α +c−
2
⎛h ⎞
⎛h ⎞
z = −13 ⎜ − 1⎟ + 39.4 ⎜ − 1⎟ + 63.6
⎝b ⎠
⎝b ⎠
‰
Approximate design methods
o Reciprocal load method:
1
1
1
1
=
+
−
Pu Pux Puy P0
where Pu = ultimate load under biaxial loading,
Pux = ultimate load when only ex is present,
Puy = ultimate load when only ey is present, and
P0 = ultimate load when ex = e y = 0 .
o Load contour method: ( M ux − M uy interaction curve in 2D)
For various loads of constant Pu ,
n
⎛ M ux ⎞ ⎛ M uy ⎞
⎟⎟ = 1
⎜
⎟ + ⎜⎜
⎝ M ux 0 ⎠ ⎝ M uy 0 ⎠
m
where M ux = Pu e y , M uy = Pu ex , and M ux 0 , M uy 0 are uniaxial flexural
strengths about x and y axes for the constant load level considered.
Æ Experiments suggest that m = n = α depends on column geometry.
Typically 1.15 < α < 1.55 for most rectangular columns with uniform
reinforcement.
Æ There is no single value that can be assigned to the exponent to
represent the true shape of the load contour in all cases.
6/6