©c s a 2002 c f d bs 8110-97 technical note column design...

Overview of 13

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

CONCRETE FRAME DESIGN BS 8110-97

Technical NoteColumn 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.

OverviewThe program can be used to check column capacity or to design columns. Ifyou define the geometry of the reinforcing bar configuration of each concretecolumn section, the program will check the column capacity. Alternatively, theprogram 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 interactionsurface is shown in Figure 1. When the steel is undefined, the programgenerates the interaction surfaces for the range of reinforcement from 0.2to 10 percent.

Calculate the capacity ratio or the required reinforcing area for the fac-tored axial force and biaxial (or uniaxial) bending moments obtained fromeach load combination at each station of the column. The target capacityratio when calculating the required reinforcing area is taken as the Utiliza-tion Factor Limit, which is equal to 0.95 by default. The Utilization FactorLimit can be redefined in the Preferences.

Design the column shear reinforcement.

The following four subsections describe in detail the algorithms associatedwith this process.

Generation of Biaxial Interaction SurfacesThe column capacity interaction volume is numerically described by a seriesof discrete points that are generated on the three-dimensional interaction

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 (BS3.8.4.1, 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 linearstrain 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 theextremity 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 ultimatestrength design and allows for any doubly symmetric rectangular, square, orcircular column section (BS 3.8.4).

The stress in the steel is given by the product of the steel strain and the steelmodulus of elasticity, εs Es, and is limited to the design strength of the steel,fy/γs. 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 ofthe column (such as an equivalent steel tube or cylinder). See Figure 2.

The concrete compression stress block is assumed to be rectangular, with astress value of 0.67fcu/γc. See Figure 2. The interaction algorithm providescorrections to account for the concrete area that is displaced by the rein-forcement in the compression zone.

Calculate Column Capacity RatioThe column capacity ratio is calculated for each loading combination at eachoutput station of each column. The following steps are involved in calculatingthe capacity ratio of a particular column for a particular loading combinationat a particular location:

Determine the factored moments and forces from the analysis load casesand the specified load combination factors to give N, V2, V3, M33, and M22.

Determine the additional moments resulting from slender column effect.Compute moments caused by minimum eccentricity.

Determine total design moments by adding the corresponding additionalmoments to the factored moments obtained from the analysis. Determinewhether the point, defined by the resulting axial load and biaxial momentset, lies within the interaction volume.

The following three subsections describe in detail the algorithms associatedwith this process.



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

Determine Factored Moments and ForcesEach load combination is defined with a set of load factors corresponding tothe 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,V2, V3, Mf33, and Mf22.

Determine Additional MomentsIf the column is in compression, the factored moments are magnified byadding extra moments to account for the local stability effects (BS 3.8.3). Ifthe column axial force is tensile for a load combination, the additional mo-ments are not considered for that load combination. Both the moments aboutthe major and minor axes are magnified. For simplicity, the following is de-scribed for moments about either of the two axes. The program calculates themagnified moments for a particular load combination at a particular point as:

Mmag = ( ){ }ξfMM ,Mmax addff ∗+ , (BS 3.8.3.2)

where,

c



Mf is the factored moment for a particular load combination at aparticular point. It is obtained by applying the corresponding loadfactors to the load cases.

Madd is the additional moment about a particular load combination at aparticular point. For both "Braced" and "Unbraced" columns, theadditional moment is obtained as follows:

≤>

=tension. in column

and n,compressio in column

,0Nif,0

,0Nif,NaM u

add (BS 3.8.3.1)

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

au = βeKh. (BS 3.8.3.1)

βe = 2

e

'bl

000,21

. (BS 3.8.3.1)

le is the effective length in the plane under consideration. It isobtained from

le = βl0, (BS 3.8.1.6.1)

where β is the effective length factor, and l0 the unsupportedlength corresponding to instability in the major or minor directionof the element, l33 or l22 in Figure 3. In calculating the value ofthe 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 becompleted for all frames, especially unbraced frames. The defaultvalue of β is conservative for braced frames and for unbracedframes for which P-delta analysis is performed. It may not beconservative 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 bendingconsidered.


Calculate Colum

Figure 3: Ax

K is tof th

( )ξf is a dis

of a columnpoints of thbraced framgiven for brc

n Capacity Ratio of 13

es of Bending and Unsupported Length

he correction factor to the deflection to take care of the influencee axial force; K is conservatively taken as 1.

tribution function. This is used to modify the moment at any point

by a certain fraction of Madd, as Madd is not added uniformly at alle column. This function is consistent with BS Figure 3.20 fores and BS Figure 3.21 for unbraced columns. The function ised frames as follows (BS 3.8.3.2, BS Figure 3.20):



( )

( )

( )

( )( ) ( )

( )

−−

−+−−+

+−+

−

=

otherwise. ,

and hinged, is end J if , 62

hinged, is end I if , 62

hinged, are ends J and I both if ,

21

16

14

6211

462

1

14

f

ξξ

ξξ

ξξ

ξξ

ξ

The function is given for unbraced frames as follows:

( )

( )( )

( ) ( )( ) ( )( )

( )( )

>−−+

>−−+

+−

−

=

end. any at hinge no and , if ,

end, any at hinge no and , if ,

hinged, is end J if , 1 -1

hinged, is end I if ,

hinged, are ends J and I both if ,

JI32

IJ

IJ32

JIJI

MM231MM1

MM23MM1MM

212

14

f

ξξ

ξξ

ξξξξ

ξξ

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.

MI is the absolute value of the end moment at the I end about the re-spective axis of bending.

MJ 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 minoraxes bending, the minimum eccentricity requirements are also satisfied. Thedesign moment is taken as:



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

where,

M is the design moment.

Mmag is the magnified moment, which is obtained from the factoredmoment and the additional moment by the procedure describedpreviously.

emin is the minimum eccentricity, which is taken as 0.05 times theoverall 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 tobe the same as that of the analysis moment.

It is assumed that the user performs a global P-delta analysis for both bracedand unbraced frames. For P-delta analysis, it is recommended that the loadcombination used to obtain the axial forces in the columns be equivalent to1.2 DL + 1.2 LL (White and Hajjar 1991).

Determine Capacity RatioA 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 thestress 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, M33 and M22 are obtained according to the previous subsections.The point (N, M33, M22) is then placed in the interaction space shown as pointL 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 ratio OCOL

.

If OL = OC (or CR=1), the point lies on the interaction surface and thecolumn is stressed to capacity.

If OL < OC (or CR<1), the point lies within the interaction volume and thecolumn capacity is adequate.

If OL > OC (or CR>1), the point lies outside the interaction volume andthe column is overstressed.

Figure 4: Geometric Representation of Column Capacity Ratio

33

33

22

22


Required Reinforcing Area of 13

The maximum of all the values of CR calculated from each load combination isreported for each check station of the column, along with the controlling N,M33, and M22 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 columngross area. The column area is held constant and the rebar area is modifiedto 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 as0.2 percent and 10 percent. The eight reinforcing ratios used are the maxi-mum and the minimum ratios, plus six more ratios. The spacing between thereinforcing ratios is calculated as an increasing arithmetic series in which thespace between the first two ratios is equal to one-third of the space betweenthe last two ratios.

After the eight reinforcing ratios have been determined, the program developsinteraction 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 surfacesto determine the reinforcing ratio that gives a demand/capacity ratio of theUtilization 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 designload combinations and the largest required reinforcing ratio is reported.

If the required reinforcement is found to be less than the minimum allowed inthe code (0.4 percent), the program assigns the design reinforcement to be0.4 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 failurecondition.

Design Column Shear ReinforcementThe shear reinforcement is designed for each load combination in the majorand minor directions of the column. The following steps are involved in de-signing the shear reinforcement for a particular column for a particular loadcombination 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 = cvAV

, and (BS 3.4.5.2)

vmax = 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 anoverstress.

RLW is a strength reduction factor that applies to light-weight concrete. Itis 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 0.8RLW 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)

'cv = vc + 0.6

MVd

AN

c ≤ vc

ccvAN

1 + (BS 3.4.5.12)

where,


Design Column Shear Reinforcement of 13

vc = RLW

4/13/1s

m

21

d400

bdA100kk79.0

γ, (BS 3.4.5.4, Table 3.8)

RLW is a shear strength factor that applies to light-weight concrete. It isequal to 1 for normal weight concrete. This factor is specified in the con-crete material properties.

k1 is the enhancement factor for support compression and is taken con-servatively as 1, (BS 3.4.5.8)

k2 = 3/1

cu

25f

, (BS 3.4.5.4, Table 3.8)

γm = 1.25, (BS 2.4.4.1)

0.15 ≤ bd

A100 s ≤ 3, (BS 3.4.5.4, Table 3.8)

d400

≥ 1, (BS 3.4.5.4, Table 3.8)

MVd

≤ 1, (BS 3.4.5.4, Table 3.8)

fcu ≤ 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 ofthe tension steel of the outer layer.

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

yvv

sv

f95.0b4.0

sA ≥ , (BS 3.4.5.3)

else if 'cv + 0.4 < v < vmax, provide links given by


References of 13

yv

'c

v

sv

f95.0b )vv(

sA −≥ , (BS 3.4.5.3)

else if v > vmax

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

fyv cannot be taken as greater than 460 MPa (BS 3.4.5.1) in the calcula-tion. If fyv is defined as greater than 460 MPa, the program designs shearreinforcing assuming that fyv is equal to 460 MPa.

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

Analysis in LRFD: Research in Practice. Engineering Journal. AmericanInstitute of Steel Construction, Inc. Vol. 28, No. 4.

©c s a 2002 c f d bs 8110-97 technical note column design...

Documents