information and pool etabs manuals english e tn cfd bs 8110-97-006

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©COMPUTERS AND STRUCTURES, INC., BERKELEY, CALIFORNIA AUGUST 2002

CONCRETE FRAME DESIGN BS 8110-97

Technical Note

Column Design

This Technical Note describes how the program checks column capacity or de-

signs reinforced concrete columns when the BS 8110-97 code is selected.

Overview

The program can be used to check column capacity or to design columns. If

you define the geometry of the reinforcing bar configuration of each concrete

column section, the program will check the column capacity. Alternatively, the

program can calculate the amount of reinforcing required to design the col-

umn. The design procedure for the reinforced concrete columns of the struc-

ture involves the following steps:

Generate axial force/biaxial moment interaction surfaces for all of the dif-

ferent concrete section types of the model. A typical biaxial interaction

surface is shown in Figure 1. When the steel is undefined, the program

generates the interaction surfaces for the range of reinforcement from 0.2

to 10 percent.

Calculate the capacity ratio or the required reinforcing area for the fac-

tored axial force and biaxial (or uniaxial) bending moments obtained from

each load combination at each station of the column. The target capacity

ratio when calculating the required reinforcing area is taken as the Utiliza-

tion Factor Limit, which is equal to 0.95 by default. The Utilization Factor

Limit can be redefined in the Preferences.

Design the column shear reinforcement.

The following four subsections describe in detail the algorithms associated

with this process.

Generation of Biaxial Interaction Surfaces

The column capacity interaction volume is numerically described by a series

of discrete points that are generated on the three-dimensional interaction


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Concrete Frame Design BS 8110-97 Column Design

Generation of Biaxial Interaction Surfaces of 13

failure surface. In addition to axial compression and biaxial bending, the for-

mulation allows for axial tension and biaxial bending considerations (BS

3.8.4.1, 3.4.4.1). A typical interaction diagram is shown in Figure 1.

Figure 1: A Typical Column Interaction Surface

The coordinates of these points are determined by rotating a plane of linear

strain in three dimensions on the section of the column (BS 3.4.4.1). See Fig-

ure 1. The linear strain diagram limits the maximum concrete strain, εc , at the

extremity of the section to 0.0035 (BS 3.4.4.1).

33

22

22

33

2233




Calculate Column Capacity Ratio of 13

The formulation is based consistently upon the basic principles of ultimate

strength design and allows for any doubly symmetric rectangular, square, or

circular column section (BS 3.8.4).

The stress in the steel is given by the product of the steel strain and the steel

modulus of elasticity, ε s E s, and is limited to the design strength of the steel,

f y /γ s. The area associated with each reinforcing bar is placed at the actual lo-

cation of the center of the bar and the algorithm does not assume any simpli-

fications with respect to distributing the area of steel over the cross section of

the column (such as an equivalent steel tube or cylinder). See Figure 2.

The concrete compression stress block is assumed to be rectangular, with a

stress value of 0.67f cu /γ c . See Figure 2. The interaction algorithm provides

corrections to account for the concrete area that is displaced by the rein-

forcement in the compression zone.

Calculate Column Capacity Ratio

The column capacity ratio is calculated for each loading combination at each

output station of each column. The following steps are involved in calculating

the capacity ratio of a particular column for a particular loading combination

at a particular location:

Determine the factored moments and forces from the analysis load cases

and the specified load combination factors to give N , V 2, V 3, M 33, and M 22.

Determine the additional moments resulting from slender column effect.

Compute moments caused by minimum eccentricity.

Determine total design moments by adding the corresponding additional

moments to the factored moments obtained from the analysis. Determine

whether the point, defined by the resulting axial load and biaxial moment

set, lies within the interaction volume.

The following three subsections describe in detail the algorithms associated

with this process.





Figure 2: Idealized Stress and Strain Distribution in a Column Section

Determine Factored Moments and Forces

Each load combination is defined with a set of load factors corresponding to

the load cases. The factored loads for a particular load combination are ob-

tained by applying the corresponding load factors to the load cases, giving N ,V 2, V 3, M f33, and M f22.

Determine Additional Moments

If the column is in compression, the factored moments are magnified by

adding extra moments to account for the local stability effects (BS 3.8.3). If

the column axial force is tensile for a load combination, the additional mo-

ments are not considered for that load combination. Both the moments about

the major and minor axes are magnified. For simplicity, the following is de-

scribed for moments about either of the two axes. The program calculates the

magnified moments for a particular load combination at a particular point as:

M mag = ( ){ }ξ f M M ,M max add f f ∗+ , (BS 3.8.3.2)

where,

c





M f is the factored moment for a particular load combination at a

particular point. It is obtained by applying the corresponding load

factors to the load cases.

M add is the additional moment about a particular load combination at a

particular point. For both "Braced" and "Unbraced" columns, the

additional moment is obtained as follows:

≤

>=

tension.incolumn

andn,compressioincolumn

,0N if ,0

,0N if ,NaM

uadd (BS 3.8.3.1)

au is the deflection at the ultimate limit state. It is obtained as

au = β eKh. (BS 3.8.3.1)

β e =

2e

' b

l

000 ,2

1

. (BS 3.8.3.1)

l e is the effective length in the plane under consideration. It is

obtained from

l e = β l 0 , (BS 3.8.1.6.1)

where β is the effective length factor, and l 0 the unsupported

length corresponding to instability in the major or minor directionof the element, l 33 or l 22 in Figure 3. In calculating the value of

the effective length, the β factor is taken as 1. However, the pro-

gram allows the user to override this default value. Even if addi-

tional moments are considered, global P-delta analysis should be

completed for all frames, especially unbraced frames. The default

value of β is conservative for braced frames and for unbraced

frames for which P-delta analysis is performed. It may not be

conservative for unbraced frames if P-delta analysis is not per-

formed. In that case, a value greater than 1 for β is appropriate.

b' is the dimension of the column in the plane of bending consid-

ered.

h is also the dimension of the column in the plane of bending

considered.





Figure 3: Axes of Bending and Unsupported Length

K is the correction factor to the deflection to take care of the influence

of the axial force; K is conservatively taken as 1.

( )ξ f is a distribution function. This is used to modify the moment at any point

of a column by a certain fraction of M add , as M add is not added uniformly at all

points of the column. This function is consistent with BS Figure 3.20 for

braced frames and BS Figure 3.21 for unbraced columns. The function is

given for brced frames as follows (BS 3.8.3.2, BS Figure 3.20):





( )

( )

( )

( )( ) ( )

( )

−−

−

+

−−+

+−+

−

=

otherwise. ,

andhinged,isendJif ,62

hinged,isendIif ,62

hinged,areendsJandIbothif ,

2

116

14

62

11

4

621

14

f

ξ ξ

ξ ξ

ξ ξ

ξ ξ

ξ

The function is given for unbraced frames as follows:

( )

( )

( )( ) ( )

( ) ( )( )( )( )

>−−+

>−−+

+−

−

=

anyathingeno and, if ,

anyathingeno and,if ,

hinged,isendJif ,1-1

hinged,isendI if ,

hinged,areendsJandI both if ,

J I 32

I J

I J 32

J I J I

M M 231M M 1

M M 23M M 1M M

212

14

f

ξ ξ

ξ ξ

ξ ξ

ξ ξ

ξ ξ

ξ

In the above expressions,

ξ is the non-dimensional parameter to represent the location of thepoint being considered, ξ = x/L.

x is the distance of the point from the I end of the column.

L is the total length of the column.

M I is the absolute value of the end moment at the I end about the re-

spective axis of bending.

M J is the absolute value of the end moment at the J end about the re-spective axis of bending.

In addition to magnifying the factored column moments for major and minor

axes bending, the minimum eccentricity requirements are also satisfied. The

design moment is taken as:





M = max (M mag, N emin), (BS 3.8.24, BS 3.8.3.2)

where,

M is the design moment.

M mag is the magnified moment, which is obtained from the factored

moment and the additional moment by the procedure described

previously.

emin is the minimum eccentricity, which is taken as 0.05 times the

overall dimension of the column in the plane of bending consid-

ered, but not more than 20 mm (BS 3.8.2.4):

emin =20

h≤ 20 mm. (BS 3.8.2.4)

The minimum eccentricity is considered about one axis at a time (BS 3.8.2.4).

The sign of the moment resulting from the minimum eccentricity is taken to

be the same as that of the analysis moment.

It is assumed that the user performs a global P-delta analysis for both braced

and unbraced frames. For P-delta analysis, it is recommended that the load

combination used to obtain the axial forces in the columns be equivalent to

1.2 DL + 1.2 LL (White and Hajjar 1991).

Determine Capacity Ratio

A capacity ratio is calculated as a measure of the stress condition of the col-

umn. The capacity ratio is basically a factor that gives an indication of the

stress condition of the column with respect to the capacity of the column.

Before entering the interaction diagram to check the column capacity, the de-

sign forces N , M 33 and M 22 are obtained according to the previous subsections.

The point (N , M 33, M 22) is then placed in the interaction space shown as point

L in Figure 4. If the point lies within the interaction volume, the column ca-pacity is adequate; however, if the point lies outside the interaction volume,

the column is overstressed.

This capacity ratio is achieved by plotting the point L and determining the lo-

cation of point C. The point C is defined as the point where the line OL (if ex-

tended outwards) will intersect the failure surface. This point is determined by





three-dimensional linear interpolation between the points that define the fail-

ure surface. See Figure 4. The capacity ratio, CR, is given by the ratioOC

OL.

If OL = OC (or CR=1), the point lies on the interaction surface and the

column is stressed to capacity.

If OL < OC (or CR<1), the point lies within the interaction volume and the

column capacity is adequate.

If OL > OC (or CR>1), the point lies outside the interaction volume and

the column is overstressed.

Figure 4: Geometric Representation of Column Capacity Ratio

33

33

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Required Reinforcing Area of 13

The maximum of all the values of CR calculated from each load combination is

reported for each check station of the column, along with the controlling N ,

M 33, and M 22 set and associated load combination number.

Required Reinforcing AreaIf the reinforcing area is not defined, the program computes the reinforce-

ment that will give a column capacity ratio of the Utilization Factor Limit,

which is equal to 0.95 by default. This factor can be redefined in the Prefer-

ences.

In designing the column rebar area, the program generates a series of inter-

action surfaces for eight different ratios of reinforcing steel area to column

gross area. The column area is held constant and the rebar area is modified

to obtain these different ratios; however, the relative size (area) of each re-bar compared to the other bars is always kept constant.

The smallest and the largest of the eight reinforcing ratios used are taken as

0.2 percent and 10 percent. The eight reinforcing ratios used are the maxi-

mum and the minimum ratios, plus six more ratios. The spacing between the

reinforcing ratios is calculated as an increasing arithmetic series in which the

space between the first two ratios is equal to one-third of the space between

the last two ratios.

After the eight reinforcing ratios have been determined, the program develops

interaction surfaces for all eight of the ratios using the process described ear-

lier in this Technical Note in the section entitled "Generation of Biaxial Inter-

action Surfaces."

Next, for a given design load combination, the program generates a de-

mand/capacity ratio associated with the each of the eight interaction surfaces.

The program then uses linear interpolation between the interaction surfaces

to determine the reinforcing ratio that gives a demand/capacity ratio of the

Utilization Factor Limit, which is equal to 0.95 by default. The Utilization fac-tor can be redefined in the Preferences. This process is repeated for all design

load combinations and the largest required reinforcing ratio is reported.

If the required reinforcement is found to be less than the minimum allowed in

the code (0.4 percent), the program assigns the design reinforcement to be

0.4 percent (BS 3.12.5.3, BS Table 3.25).




Design Column Shear Reinforcement of 13

If the required reinforcement is found to be more than 6 percent for both

"braced" and "unbraced" frames (BS 3.12.6.2), the program declares a failure

condition.

Design Column Shear ReinforcementThe shear reinforcement is designed for each load combination in the major

and minor directions of the column. The following steps are involved in de-

signing the shear reinforcement for a particular column for a particular load

combination resulting from shear forces in a particular direction (BS 3.8.4.6):

Calculate the design shear stress and maximum allowed shear stress from

v =cv A

V , and (BS 3.4.5.2)

v max = min {0.8RLW cuf , 5 MPa}, where (BS 3.4.5.2, BS 3.4.5.12)

Acv = b d .

If v exceeds either 0.8RLW cuf or 5 N/mm2, the section area should be in-

creased (BS 3.4.5.2, BS 3.4.5.12). In that case, the program reports an

overstress.

RLW is a strength reduction factor that applies to light-weight concrete. It

is equal to 1 for normal weight concrete. The factor is specified in the con-

crete material properties.

Note

The program reports an overstress message when the shear stress exceed 0.8R LW cuf

or 5 MPa (BS 3.4.5.2, BS 3.4.5.12).

Calculate the design concrete shear stress from (BS 3.8.4.6)

'c v = v c + 0.6

M

Vd

A

N

c

≤ v c

c c v A

N 1 + (BS 3.4.5.12)

where,




Design Column Shear Reinforcement of 13

v c = RLW

4 / 13 / 1

s

m

21

d

400

bd

A100k k 79.0

γ

, (BS 3.4.5.4, Table 3.8)

RLW is a shear strength factor that applies to light-weight concrete. It is

equal to 1 for normal weight concrete. This factor is specified in the con-crete material properties.

k 1 is the enhancement factor for support compression and is taken con-

servatively as 1, (BS 3.4.5.8)

k 2 =

3 / 1

cu

25

f

, (BS 3.4.5.4, Table 3.8)

γ m = 1.25, (BS 2.4.4.1)

0.15 ≤ bd

A100 s ≤ 3, (BS 3.4.5.4, Table 3.8)

d

400 ≥ 1, (BS 3.4.5.4, Table 3.8)

M

Vd ≤ 1, (BS 3.4.5.4, Table 3.8)

f cu ≤ 40 N/mm2

, (BS 3.4.5.4, Table 3.8)

As is the area of tensile steel and it is taken as half of the total reinforc-

ing steel area, and

d is the distance from the extreme compression fiber to the centroid of

the tension steel of the outer layer.

If v ≤ 'c v + 0.4, provide minimum links defined by

yv v

sv

f 95 .0b4.0

s A ≥ , (BS 3.4.5.3)

else if 'c v + 0.4 < v < v max , provide links given by




yv

' c

v

sv

f 95 .0

b )v v (

s

A −≥ , (BS 3.4.5.3)

else if v > v max

a failure condition is declared. (BS 3.4.5.2, 3.4.3.12)

f yv cannot be taken as greater than 460 MPa (BS 3.4.5.1) in the calcula-

tion. If f yv is defined as greater than 460 MPa, the program designs shear

reinforcing assuming that f yv is equal to 460 MPa.

References

White. D. W., and J.F., Hajjar. 1991. Application of Second-Order Elastic

Analysis in LRFD: Research in Practice. Engineering Journal. American

Institute of Steel Construction, Inc. Vol. 28, No. 4.

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