BEAM COLUMNS - weccivilians.weebly.comweccivilians.weebly.com/uploads/2/4/6/2/24623713/lec11.2_beam_col… · BEAM COLUMNS Beam columns are ... Majority of the steel building frames

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.


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.


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.


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.


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.


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.


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.


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.


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.


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


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.


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.


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)


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


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.


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


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


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


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.


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.


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−


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


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


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


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.


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.


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


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.


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

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BEAM COLUMNS - weccivilians.weebly.comweccivilians.weebly.com/uploads/2/4/6/2/24623713/lec11.2_beam_col… · BEAM COLUMNS Beam columns are ... Majority of the steel building frames