BEAM COLUMNS -...
Transcript of BEAM COLUMNS -...
Prof. Dr. Zahid Ahmad Siddiqi
BEAM COLUMNS
Beam columns are structural members that are
subjected to a combination of bending and axial
stresses.
The structural behaviour resembles
simultaneously to that of a beam and a column.
Majority of the steel building frames have
columns that carry sizable bending moments in
addition to the usual compressive loads.
Prof. Dr. Zahid Ahmad Siddiqi
The sources of this bending moment are shown in
Figure 5.1 and explained below:
M=P×e
e
P
eP
e
P
a) Out-Of-Plumb b) Initial Crookedness c) Eccentric Load
Figure 5.1. Sources of Eccentricity in Columns.
Prof. Dr. Zahid Ahmad Siddiqi
It is almost impossible to erect the columns
perfectly vertical and centre loads exactly on
columns.
Columns may be initially crooked or have other
flaws with the result that lateral bending is
produced.
In some cases, crane beams parallel to columns-
line and other perpendicular beams rest on brackets
projecting out from columns. This produces high
values of bending moments.
Prof. Dr. Zahid Ahmad Siddiqi
Wind and other lateral loads act within the
column height and produce bending.
The bending moments from the beams are
transferred to columns if the connections are
rigid.
CONTROLLING DESIGN FACTOR:
SECOND ORDER EFFECTS
The elastic analysis carried out to calculate
deflections and member forces for the given
loads is called 1st order and analysis.
Prof. Dr. Zahid Ahmad Siddiqi
The high axial load present in the column
combined with this elastic deflection produces
extra bending moment in the column, as is clear
from Figure 5.2.
The analysis of structure including this extra
moment is called 2nd order analysis.
Similarly, other higher order analysis may also be
performed.
In practice, usually 2nd order analysis is
sufficiently accurate with the high order results of
much lesser numerical value.
Prof. Dr. Zahid Ahmad Siddiqi
δMaximum lateral
deflection due to
bending moment
(M)
PM
M
P
Extra moment = P×δ,
which produces more
deflections
Deflected shape
or elastic curve
due to applied
bending moment
(M)
Figure 5.2. Eccentricity Due to First Order Deflections.
Prof. Dr. Zahid Ahmad Siddiqi
The phenomenon in which the moments are
automatically increased in a column beyond the
usual analysis for loads is called moment
magnification or 2nd order effects.
The moment magnification depends on many
factors but, in some cases, it may be higher
enough to double the 1st order moments or even
more.
In majority of practical cases, this magnification
is appreciable and must always be considered for
a safe design.
Prof. Dr. Zahid Ahmad Siddiqi
1st order deflection produced within a member
(δ) usually has a smaller 2nd order effect called P-δ effect, whereas magnification due to sides-way
(∆) is much larger denoted by P-∆ effect (refer to
Figure 5.3).
P-Delta effect is defined as the secondary effect
of column axial loads and lateral deflections on
the moments in members.
The calculations for actual 2nd order analysis are
usually lengthy and can only be performed on
computers.
Prof. Dr. Zahid Ahmad Siddiqi
For manual calculations, empirical methods are
used to approximately cater for these effects in
design.
2nd order effects are more pronounced when loads
closer to buckling loads are applied and hence the
empirical moment magnification formula contains
a ratio of applied load to elastic buckling load.
The factored applied load should, in all cases, be
lesser than 75% of the elastic critical buckling load
but is usually kept much lesser than this limiting
value.
Prof. Dr. Zahid Ahmad Siddiqi
INTERACTION EQUATION AND
INTERACTION DIAGRAM
P ∆
P
Extra Moment
M = P×∆
M
Figure 5.3.
A Deflected Beam-Column.
The combined stress at any
point in a member subjected to
bending and direct stress, as in
Figure 5.3, is obtained by the
formula:
f = ± ±A
P
x
x
I
yM
y
y
I
xM
Prof. Dr. Zahid Ahmad Siddiqi
For a safe design, the maximum compressive
stress (f) must not exceed the allowable material
stress (Fall
) as follows:
f = ± ± ≤ FallA
P
x
x
I
yM
y
y
I
xM
+ + ≤ 1allAF
P
allx
x
FS
M
ally
y
FS
M
+ + ≤ 1maxP
P
max,x
x
M
M
max,y
y
M
M
This equation is called interaction equation
showing interaction of axial force and bending
moment in an easy way.
Prof. Dr. Zahid Ahmad Siddiqi
If this equation is plotted against the various terms
selected on different axis, we get an interaction
curve or an interaction surface depending on
whether there are two or three terms in the
equation, respectively.
1.0
1.0
0,0
Figure 5.4. A Typical Interaction Curve.
Prof. Dr. Zahid Ahmad Siddiqi
Pr = required axial compressive strength
(Pu in LRFD)
Pc = available axial compressive
strength
= φcPn, φc = 0.90 (LRFD)
= Pn / Ωc, Ωc = 1.67 (ASD)
Mr= required flexural strength (Mu in
LRFD)
Mc = available flexural strength
= φbMn, φb = 0.90 (LRFD)
= Mn / Ωb, Ωb = 1.67 (ASD)
Prof. Dr. Zahid Ahmad Siddiqi
AISC INTERACTION EQUATIONS
The following interaction equations are
applicable for doubly and singly symmetric
members:
If ≥ 0.2, axial load is considerable, and
following equation is to be satisfied:c
r
P
P
≤ 1.0
++
cy
ry
cx
rx
c
r
M
M
M
M
P
P
9
8
Prof. Dr. Zahid Ahmad Siddiqi
If < 0.2, axial load is lesser, beam
action is dominant, and the applicable
equation is:
c
r
P
P
≤ 1.0
++
cy
ry
cx
rx
c
r
M
M
M
M
P
P
2
MOMENT ADJUSTMENT FACTOR
(Cmx
or Cmy
)
Moment adjustment factor (Cm) is based on
the rotational restraint at the member ends
and on the moment gradient in the members.
It is only defined for no-sway cases.
Prof. Dr. Zahid Ahmad Siddiqi
1. For restrained compression members in
frames braced against joint translation (no
sidesway) and not subjected to transverse loading
between their supports in the plane of bending:
Cm
= 0.6 – 0.4 2
1
M
M
where M1
is the smaller end moment and M2
is
the larger end moment.
is positive when member is bent in
reverse curvature and it it is negative when
member is bent in single curvature (Figure 5.5b).
21 / MM
Prof. Dr. Zahid Ahmad Siddiqi
P
PM2
M1
a) Reverse Curvature
P
P
M2
M1
b) Single Curvature
Figure 5.5. Columns Bent in Reverse and Single Curvatures.
When transverse load is applied between the
supports but or sway is prevented,
for members with restrained ends Cm
= 0.85
for members with unrestrained ends Cm
= 1.0
Prof. Dr. Zahid Ahmad Siddiqi
K-VALUES FOR FRAME BEAM-COLUMNS
K-values for frame columns with partially fixed
ends should be evaluated using alignment charts
given in Reference-1.
However, if details of adjoining members are not
given, following approximate estimate may be
used:
K = 1.2 – 1.5 if sidesway is permitted with
partially fixed ends
K = 1 if sidesway is prevented but end
conditions are not mentioned
Prof. Dr. Zahid Ahmad Siddiqi
MOMENT MAGNIFICATION FACTORS
Moment magnification factors (B1
and B2) are
used to empirically estimate the magnification
produced in the column moments due to 2nd order
effects.
These are separately calculated for sway or lateral
translation case (lt-case) and for no-sway or no
translation case (nt-case).
Accordingly, the frame is to be separately
analysed for loads producing sway and not
producing sway.
Prof. Dr. Zahid Ahmad Siddiqi
Mlt = moment due to lateral loads producing
appreciable lateral translation.
B2
= moment magnification factor to take
care of Pu∆ effects for sway and
deflections due to lateral loads.
Mnt
= the moment resulting from gravity
loads, not producing appreciable lateral
translation.
B1
= moment magnification factor to take
care of Puδ effects for no translation
loads.
Prof. Dr. Zahid Ahmad Siddiqi
Mr
= required magnified flexural strength
for second order effects
= B1 M
nt+ B
2Mlt
Pr
= required magnified axial strength
= Pnt
+ B2
Plt
No-Sway Magnification
B1
= ≥ 1.011 er
m
PP
C
α−
Prof. Dr. Zahid Ahmad Siddiqi
where
α = 1.0 (LRFD) and 1.60 (ASD)
Pe1 = Euler buckling strength for
braced frame
= π2 EI / (K1 L)2
K1= effective length factor in the
plane of bending for no lateral
translation, equal to 1.0 or a
smaller value by detailed analysis
Prof. Dr. Zahid Ahmad Siddiqi
Sway Magnification
The sway magnification factor, B2, can be
determined from one of the following formulas:
B2
=
2
1
1
e
nt
P
P
∑∑− α
where,
α = 1.0 (LRFD) and 1.60 (ASD)
ΣPnt = total vertical load supported by
the story, kN, including gravity loads
Prof. Dr. Zahid Ahmad Siddiqi
ΣPe2 = elastic critical buckling
resistance for the story
determined by sidesway
buckling analysis
= Σπ2 EI / (K2 L)2
where I and K2 is calculated in the plane of
bending for the unbraced conditions
Prof. Dr. Zahid Ahmad Siddiqi
SELECTION OF TRIAL BEAM-
COLUMN SECTION
The only way by which interaction of axial
compression and bending moment can be
considered, is to satisfy the interaction equation.
However, in order to satisfy these equations, a
trial section is needed.
For this trial section, maximum axial compressive
strength and bending strengths may be
determined.
Prof. Dr. Zahid Ahmad Siddiqi
The difficulty in selection of a trial section for a
beam column is that whether it is selected based
on area of cross-section or the section modulus.
No direct method is available to calculate the
required values of the area and the section
modulus in such cases.
For selection of trial section, the beam-column
is temporarily changed into a pure column by
approximately converting the effect of bending
moments into an equivalent axial load.
Prof. Dr. Zahid Ahmad Siddiqi
Peq
= equivalent or effective axial load
= Pr
+ Mrx
mx
+ Mry
my
mx (for first trial) = 8.5 − 0.7K1xLx
my (for first trial) = 17 − 1.4K1yLy
mx = 10 − 14(d / 1000)2 − 0.7K1xLx
my = 20 − 28(d / 1000)2 − 1.4K1yLy
Prof. Dr. Zahid Ahmad Siddiqi
The above equation is evaluated for Peq and a
column section is selected from the
concentrically loaded column tables for that
load.
The equation for Peq is solved again using a
revised value of m.
Another section is selected and checks are then
applied for this trial section.
Prof. Dr. Zahid Ahmad Siddiqi
WEB LOCAL STABILITY
For stiffened webs in combined flexural and axial
compression:
If ≤ 0.125 λp
= yb
u
P
P
φ
−
yb
u
y P
P
F
E
φ75.2
176.3
For A36 steel, λp
=
−
yb
u
P
P
φ75.2
17.106
If > 0.125 λp
= yb
u
P
P
φ yyb
u
y F
E
P
P
F
E49.133.212.1 ≥
− φ
3.4233.28.31 ≥
−
yb
u
P
P
φFor A36 steel, λp
=
where λ = h / tw
and Py
= Fy
Ag