Gouwens, Biaxial Bending Simplified

SP 50-10 BIAXIAL BENDING SIMPLIFIED

By

Albert J. Gouwens

Synopsis:

A comprehensive but simple design procedure is presented for the design of concrete columns subject to biaxial bending. The procedure is oriented towards hand computation but is suitable for incorpor.ation into computer programs.

The methods in the existing literature are summarized and evaluated. A new design procedure and an example problem showing its use are given.

A comparison of the proposed method was made with a column analysis based on a rigorous stress-strain approach. 67 columns were investigated at 10 to 16 different axial loads for each column. Square and rectangular columns with various sizes, fc, fy, percentages of steel, and bar arrangements were used. The comparison shows that the procedure gives moments which are 5% on the conservative side.

Keywords: axial loads; bending; biaxial loads; columns (supports); loads (forces); moment distribution; moments; reinforced concrete; reinforcing steels; stress-strain relationships; structural analysis; structural design.

233

reinforced concrete columns

------~· --------------------------------------------ACI Member Albert J. Gouwens, Manager - Computer Engineering Services, The Engineers Collaborative, supervises computer applications to engineering projects. He is a member of ACI-ASCE Committee 441, Reinforced Concrete Columns, 'and ACI Committee 340, Strength Design Handbook.

PURPOSE

The purpose of this article is to present a simple, but comprehensive and accurate design procedure for the design of concrete columns subjected to biaxial bending. The procedure is oriented towards hand computation, although it can also be incorporated into computer programs to increase the speed of computation.

PREVIOUS WORK

Two computer programs are readily available to calculate biaxial bending capacity of columns. Nieves• program (13) is based on an exact formulation. It is efficient for analysis but careless use can cause the program to be time consuming for design. The GouwensKripanarayanan program (11) uses an approximate procedure similar to the simplified version presented in this article which, due to its simplicity, enables the program to perform rapid design.

A number of papers have been written which give procedures based on fundamental strength of materials techniques for calculating the capacity of concrete columns subject to biaxial bending.

( H. Craemer (7) presented an iteration method of calculating the , capacity of members subjected to skew bending. The effects of ) axial load and of compression force are also presented • . ,

Tung Au (4) gives a similar presentation of members subject to skew , bending and presents design charts to simplify the solution of the

many equations and conditions involved.

\ K. H. Chu and A. Pabarcius (6) give a quite rigorous and completely ) general presentation of a procedure to calculate biaxial bending

1 capacity. It is even applicable to cross-sections of arbitrary · shape. Their procedure is undoubtably the most accurate type of an l approach that can be used. Such a method, however, is time consuming

even by computer, and impractical to use by hand.

, Fleming and Werner (9) give a design procedure which is based on the existence of a set of design curves for each concrete section. The design curves are basically equivalent to a three dimensional interaction surface for each column. This is an accurate but a voluminous approach. A complete set of design curves is not available in the literature.

biaxial bending simplified

We.ber (21) gives interaction curves for load vs. moment for square columns with evenly distributed reinforcement which are bent about their 45° axis. This is also a voluminous approach if interaction curves are provided for various bar arrangements and rectan9ular cross-sections. ACI "Ultimate Strength Design Handbook" (2) reproduces the curves of Weber's article.

Furlong (10) investigated many columns and plotted their capacity as contours of moment about two axes to give insights into the behavior of biaxially loaded columns. He described the variations of the moment capacities from the circular curve for different axial loads and for different percentages of reinforcement (from 1% to 4%). Also of particular interest is his observation that for small eccentricities about one of the axes, that the moment capacity about the other axis remains virtually constant.

Most attempts, including this article, at simplification of the problem of biaxial bending are based on the fact that the biaxial bending capacity can be found as a function of the uniaxial capacities of the cross section. The simplest but overly conservative approach is to assume a single straight line between the moment capacity about the two axes.

Pannell (15,-17) presents a method of calculating the biaxial capacity based on the uniaxial capacities. It requires the use of charts presented in the article and the use of a trigonometric interpolation formula.

Boris Bresler (5) suggests two methods, one the equation:

1 Pi (1)

which is exact for elastic materials but is merely an approximation for reinforced concrete columns. The equation has been used frequently and gives reasonable results for loads above balance condition load, but it is of questionable accuracy for loads less than balance.

The other equation proposed by Bresler is:

( Mx \ n +(~ \ n = 1. 0 Mxo} \Myo}

Equation (2) is discussed later.

(2)

235

A. Aas Jacobsen (1) presented a method of finding the capacity for bending about the diagonal in terms of bending about the main planes. He also separated the column's behavior into a compression failure

236 reinforced concrete columns

region and a tension failure region. These are key concepts necessary to develop an accurate solution to the problem.

The procedure given in this article is not intended to replace the many good procedures which have been presented in the existing literature, but rather, it is intended to give a simplified but accurate, conservative and rapid method of calculation.

The procedure given requires only curves such as those readily available for uniaxial bending (2, 8, 18, 19, 20).

REVIEW OF THE PROBLEM

The capacity of a column subject to biaxial bending can be represented by a 3-dimensional surface. Fig. 1 shows a typical surface.

Several sections can be cut on this surface which are familiar.

The curve P vs. Mx (9 = 90°) is shown in Fig. lb. The curve P vs. My (9 = 0°) is shown in Fig. lc. These load vs. moment interaction curves are readily available. The curve P vs. Md (9 = 450) is shown in Fig. ld. It represents the load as a function of the moment capacity for bending about the diagonal of the cross-section. The curve Mx vs. My (P = constant) is shown in Fig. le.

The focus of attention in this article is devoted to the study of the shape of the Mx vs. My interaction curves at various loads P.

For square columns bent about the axis at 45° from the other two, the moment capacity shown in Fig. 2, Mx = Bb Mxo and My = Bb Myo results in an easily understood concept to describe the shape of the biaxial moment capacity interaction.

Bb is mainly a function of axial load, and the reinforcement index pfY./fc, and to a lesser extent on the concrete cover, fy, f~, b/h, ana the distribution pattern of reinforcing.

If Bb = 0.707 (Fig. 2a) the interaction between the moment capacities is approximately circular. If Bb = 0.50 the interaction between the moment capacities is a straight line (Fig. 2b). As Bb approaches 1.0 (Fig. 2c), the moments about the two axes are nearly independent. The normal range of Bb is 0.51 to 0.80.

Rectangular columns bent about some axis approximately parallel to the dia~onal also results in moment capacities Mx = Bb Mxo and My = Bb Myo (Fig. 3a). The shape of the interaction curve for rectangular columns is an ellipse at Bb = 0.707. The interaction curve for rectangular columns can be transformed into approximately symmetrical shape by dividing the moment about each axis by the value of the maximum moment which occurs at uniaxial bending. The shape is only approximately symmetrical due to the fact that if constant cover is used on all sides of a column cross section, the ratio of the cover


to the overall column dimensions is different about the two axes. The same dissymmetry would occur in a square column if unequal cover were used on adjacent faces of the column.

Any approximate solution of the biaxial bending problem consists of two parts: 1) The determination of the value of Bb; and 2) the determination of an equation to represent the shape of the interaction curve Mx vs. My.

The curve for Mx vs. My has been approximated by several different forms of equations in previous articles. Even for square sections, there is no known exact equation for Mx vs. My.

Equation (2) given by Bresler can be expressed in terms of Bb as:

(3)

The above equation is plotted in Fig. 4a for several values of Bb.

Equation 3 is cumbersome to evaluate but gives results closest of any interaction expression to the actual interaction curve based

237

on fundamental strength of materials techniques. In reference (14), a comparison was made between equation 3 and fundamental analytical formulations for columns with various bar arrangements and for. ratios of h/b up to 3.0. Fig. 4b shows the type of deviation that was encountered. The error in over-estimation of column strength appears to be a maximum of 3%.

Pannell (15-17) gives a trigonometric interpolation formula which can be expressed in terms of Bb as:

_ (cos2 29 + {2'Bb sin2 29) My - Myo sec 9

(4a)

where

9 = tan-1 ( Myo x Mx ) Mxo ~ (4b)

For design, the value of My obtained by equation 4a must be greater than the value used to calculate Q in equation 4b.

Equations 4 result in a set of peculiar looking curves (Fig. 5) which are not verified by any of the curve shapes obtained by exact procedures (9, 10, 14). It appears that although the curves closely represent the moment capacity at Bb = 0.7, the curves give results in error by as much as 13% on the unconservative side for lower as well as higher values of Bb.


Meek (12) showed that the relationship between moment about two axes can be r-epresented by two straight lines. The scatter of his test results indicate that the two straight lines may actually be just as accurate a representation of column capacity as the curve calculated by exact procedures. Any error introduced by this assumption is on the conservative side. The straight lines when expressed in terms of Bb are given by two equations shown in Fig. 6.

The maximum difference between Meek's proposal and Bresler's proposal (equation 2) is approximately 7.9% at 9 = 26° and Bb = .80.

A second step is to determine the Bb value to be used for calculating the capacity at 9 = 45°.

Parme, Nieves, and Gouwens (14) presented curves of column bending capacity for diagonal bending for various bar arrangements, percentages and yield strengths. A set of curves which represented a lower bounds of many columns for Bb vs. load (P) are given.

The typical variation of Bb is given in Fig. 7.

Curves are given in reference (14) for various quantities and patterns of reinforcement.

Pannell (15, 16, 17) gives charts for N which is for Mx vs. My from a circular interaction curve. Bb by the equation:

Bb = (1-N) I~

a deviation factor It is related to

(5)

The above two references each require charts for each bar arrangement and reinforcement strength.

The procedure which follows results in an analytical representation of curves for Bb vs. load. The equations can be incorporated into a computer program or can easily be evaluated by hand.

METHOD OF SOLUTION

Curves have been derived which are analytically formulated as a function of the load (P) and the reinforcement index, pfylfc· They are shown to be quite accurate for all values of g, b/h, fy, fc and reinforcement arrangements.

A study was made using the PCA computer program by J. M. Nieves (10). It was noted that the minimum value of Bb defined as B25 occurred at or near a load of .25 fc bh for all columns despite a large range of fc, b/h, g, and pty/fc.

At loads greater than .25 fc bh, the value of Bb increases. At loads less than .25 fc bh, the value of Bb also increases.


For loads either greater than or less than .25 fc bh, the increase in Bb for an increment of load is greater for columns with a small percentage of reinforcement than it is for columns with a large percentage of reinforcement.

The increase in Bb for loads greater than .25 fc bh is nearly linear with the exception of loads in the range between minimum eccentricity and P0 as shown by the steep section at the right side of Fig. 7. In this region the exact value of Bb approaches 1.0 in most cases. This need not concern the designer, however, since the maximum load for design is at minimum eccentricity. No attempt was made to incorporate the phenomen of the rapid increase in Bb for such high loads since this method is intended only for design.

The increase in Bb for loads less than .25 f~ bh is not an easily represented funct1on. It has different shapes for different f~, and pfy/f~ values. The equation proposed in this region is on the conservat1ve side as is the equation for the upper load region but is more conservative.

The equation for P ~ .25 Cc is given by:

P/Cc-.25 Bb = B25 + .2 .85 + Cs/Cc

(6)

The equation for P < .25 Cc is given as:

Bb B25 + (.25- P/Cc) 2 (.85- Cs/2Cc) (7)

where Cc f~ bh (8)

Cs As fy (9)

and B25 is given by either equation (10) or (11) below. s25 is the value of Bb at P = .25 Cc.

A study was also made to determine an equation for the value of s25 . It, too, was found to have two distinct regions. One for Cs/C greater than .5 and one for Cs/Cc less than .5. The study 1nc~uded concrete strengths from 3,000 ps1 to 6,000 psi, various amounts of cover, b/h ratios up to 3:1 and various distributions of reinforce-

.,ment among column faces. As noted by Pannell (17) and by Parme, ~ieves and Gouwens (14), the capacity of 4 bar columns is distinctly 'ifferent from columns with more bars. In this study it was found •at the Bb values for 4-bar columns were consistently about 0.02 ~s than those for all other columns. For 4-bar columns, subtract 02 from equations 10 and 11.

239


In the region where Cs/Cc ~ .5:

825 = .485 + .03 CciCs

In the region where Cs/Cc < .5:

825 = .545 + .35 (.5- Cs/Cc) 2

( 1 0)

( 11)

The above equations, except for 4-bar columns, lead to a set of curves such as those shown in Fig. 8.

ACCURACY

The ratio (R) of the exact 8b to the approximate 8b for the 67 columns studied was 1.053 with a standard deviation (SD) of 0.053.

To evaluate the accuracy of the value of 8b calculated by the previous section, the exact 8b value was calculated using Nieves' (13) computer program.

Each of the 67 columns was studied for loads ranging from P = 0 to the P value corresponding to a minimum eccentricity moment. The load increments were 100 kips for the 20 x 20 and the 12 x 36 columns and 25 kips for the 10 x 10 columns. Depending on the amount of reinforcement, the above load increments result in 10 to 16 different load investigations for each column.

A summary is given in Table 1 for the various columns studied.

The accuracy of the procedure is given in terms of R and SD which are defined as above. The accuracy is given for loads less than P25 = .25 fc bh and for loads greater than P25• as well as for the entire range of P from 0.0 to the maximum load at minimum eccentricity. The column headed "No." gives the number of columns studied. The reinforcement arrangements are given by pattern designations A thru F which are schematically represented below the table.

It is interesting to note that the procedure is most accurate for the commonly used concrete strength of fc = 5,000 psi and for the nearly standard steel strength fy = 60 K~I. The procedure becomes more conservative for lower concrete strengths and for lower steel strengths.


EVALUATION OF RESULTS

Bars were placed at specific points and varied from symmetrical placement to all the bars in one face. The various extremes in the distribution of bars to various faces of the column indicate that the Bb value can be used for any distribution of reinforcement.

The 12" x 36" rectangular columns studied show that the procedure for calculating Bb is just as accurate for rectangular columns as it is for square columns.

The effect of the amount of cover was studied by using a 10" x 10" column with the same cover as the 20" x 20" column. Cover for both columns is 2". The distance between bars for the 10" columns is about 5". It is not recommended that the above procedure be used for columns with such large amounts of cover that the bars are spaced closer than 1/2 the column dimension, or for columns less than 8".

DESIGN PROCEDURE

The step by step design procedure is given below:

Given P, Mx, My, f~, fy

1) Choose band h of column. Calculate Cc·

2) Choose a bar arrangement and calculate Mx01Myo = Bm

3) Calculate p = As/bh and Cs.

4) Calculate B25 as a function of Cs/Cc·

5) Calculate Bb as a function of B25 , P, Cc, and Cs·

6) Determine which portion of the bilinear interaction curve as shown in Fig. 6 is to be used:

A. If My1My0 ~ Mx/Mxo the straight line and equation above the 450 division should be used.

B. If My/My0 < Mx/Mxo the straight line and equation below the ~50 division snould be used.

The equations given in figure 6 are restated below in a form more useful for design purposes.

241


6A) Find required My0 = My+ Mx I 8m (l-8b) TBbJ

68) or find required Mxo = ~x + My 8m l-8b TBbJ

DESIGN EXAMPLE

Design a column for p = 200k

Mx = 75 ft. -k

1) Choose 20 x 20 My = 350 ft. k

f' = c 6000 psi cc = fc bh = 6 x 20 x 20 = 24ook

.25 Cc = 600 >P fy = 60,000 psi

2) Try 12 #10 placed evenly on all 4 sides. cover = 2"

3) p = 3.81, Cs = 12 X 1.27 X 60 = 912 kips.

4) Cs/Cc = 912/2400 = .380 < .5

825 = .545 + .35 (.5-.38) 2 = .550

5) Bb=.550+(600-200 (.85-~) .568 2400 2

6) !1L = 350 = .85 Mx = 75 .182 Myo 412 Myo ill

75 (1.-.568) + 350 = 1 m .568 412

.182 (.76) + .85 = 0.97 < 1

Therefore design is adequate.


CONCLUSIONS

A purely analytical method has been presented for the analysis or iterative design of concrete columns subject to biaxial bending. Equations have been given which relate the capacity for bending about the diagonal to bending about the two major axes of the column.

The equations are compared to the exact values and are found to be in good agreement. The bilinear moment - moment interaction diagram is recommended to make a simple, but conservative and complete method of design.

Further analytical studies and comparisons with test results might indicate that the proposed equations could easily be included in the building code.

243


REFERENCES

1. Aas-Jakobsen, A., "Biaxial Eccentricities in Ultimate Load Design," ACI Journal, Proceedings V. 61, March 1964, pp. 293-315.

2. ACI Committee 340. "Ultimate Strength Design Handbook, Volume 2 Columns," Special Publication No. 17A, American Concrete Institute, Detroit, 226 pp.

3. Andersen, Paul, "Square Sections of Reinforced Concrete Under Thrust and Nonsymmetrical Bending," Engineering Experimental Station, Bulletin No. 14, Vol. XLII, No. 41, August 12, 1939, University of Minnesota, 42 pp.

co- 4'

~- 6.

(~}-· 7.

8.

Au, Tung, "Ultimate Strength Design of Rectangular Concrete Members Subject to Unsymmetrical Bending," ACI Journal, Proceedings V. 54, Feb. 1958, pp. 657-674.

Bresler, Boris, "Design Criteria for Reinforced Concrete Columns Under Axial Load and Biaxial Bending," ACI Journal, Proceedings V. 57, Nov. 1960, pp. 481-490.

Chu, Kuang-Han, and Pabarcius, Algis, "Biaxially Loaded Reinforced Concrete Columns," Proceedings, ASCE, V. 84, ST8, Dec. 1958, pp. 1865-1 to 1865-27.

Craemer, Hermann, "Skew Bending in Concrete Computed by Plasticity," ACI Journal, Proceedings V. 48, Feb. 1952, pp. 516-519.

"CRSI Handbook," Concrete Reinforcing Steel Institute, Chicago, 1972, Chapter 3, 176 pp.

9. Fleming, John F., and Werner, Stuart D., "Design of Columns Subjected to Biaxial Bending," ACI Journal, Proceedings V. 62, March 1965, pp. 327-342.

10. Furlong, Richard M., "Ultimate Strength of Square Columns Under Biaxially Eccentric Loads," ACI Journal, Proceedings V. 57, March 1961, pp. 1129-1140.

11. Gouwens, Albert, and Kripanarayanan, K. M., "Load Accumulation and Concrete Column Stack Design," Portland Cement Association, Chicago, 1973, 152 pp.

12. Meek, John L., "Ultimate Strength of Columns with Biaxially Eccentric Loads," ACI Journal, Proceedings V. 60, No.8, Aug. 1963, pp. 1 053-1 064.

13. Nieves, Jose M., "IBM 1130 Computer Program for the Ultimate Strength Design of Reinforced Concrete Columns," Portland Cement Association, Chicago, 1967, 58 pp.

f.

' (>\ biaxial bending simplified

~ Parme, Alfred L., Nieves, Jose M., and Gouwens, Albert J., "Capacity of Reinforced Rectangular Columns Subject to Biaxial Bending," ACI Journal, Proceedings V. 63, Sept. 1966, pp. 911-923.

245

~15. Pannell, Frederick N., "The Design of Biaxially Loaded Columns by Ultimate Load Methods," Magazine of Concrete Research:

17.

18.

19.

20.

Vol. 12, No. 35: July 1960, pp. 99-108.

Pannell, Frederick N., "Design of Biaxially Loaded Columns by Ultimate Load Method. - I," Concrete and Constructional Engineering, Oct., Nov., and Dec. 1960.

Pannell, Frederick N., "Failure Surfaces for Members in Compression and Biaxial Bending," ACI Journal, Proceedings V. 60, Jan. 1963, pp. 129-140.

"Ultimate Load Tables for Tied'Columns," Concrete Information, Portland Cement Association, Chicago, 1961, 29 pp.

"Ultimate Strength Design of Reinforced Concrete Columns," Portland Cement Association, Chicago, 1969, 49 pp.

"Biaxial and Uniaxial Capacity of Rectangular Columns," Advanced Engineering Bulletin 20, Portland Cement Association, Chicago, 1967, 29 pp.

21. Weber, Donald C., "Ultimate Strength Design Charts for Columns , \ with Biaxial Bending," ACI Journal, Proceedings V. 63, , :) Nov. 1966, pp. 1205-1230.

246

N

p

Bm

reinforced concrete columns

APPENDIX A

(NOTATION)

Area of reinforcement in a column.

Width of a rectangular column.

f(; bh

Asfy

Concrete design strength in psi.

Reinforcement yield strength in psi.

Depth of a rectangular column.

The bending moment capacity of a column about its diagonal.

The bending moment capacity of a column about its x-axis.

The bending moment capacity of a column about its x-axis without bending about the y-axis.

The bending moment capacity of a column about its y-axis.

The bending moment capacity of a column about its y-axis without bending about the x-axis.

Deviation factor from a circular interaction curve.

The axial load on a column.

The axial load capacity with only x-axis bending.

The axial load capacity with only y-axis bending.

The axial load capacity without any bending.

Agfbh

A factor relating Md to Mxo and Myo'

MxoiMyo

B25 The minimum value of Bb which occurs at P = .25 Cc.

~ Capacity reduction factor.


NOTATION

R Ratio of exact Bb to the approximate Bb given by Equation 6 or 7.

SO Standard deviation of the R values calculated.

9x• 9y = Ratio of distance between bars on the outside face to the column dimension in either the x or y direction.

247

TABLE 1 SUMMARY OF THE DESIGN PROCEDURE ACCURACY

Column Size f' f c y

20 X 20 5 60

20 X 20 5 60

20 X 20 3 60

20 X 20 6 40

10 X 10 5 60

12 X 36 4 60

SUMMARY FOR ALL COLUMNS

D • A

• • • • • B

• • • • •

Reinf. p% No. l> < l>?t;

Pattern R SD 1.00

A B c 9.00 12 1.057 .074 1.5

E, F 4.5 8 1.043 .049 1.0

A B c 9.00 12 1.079 .061 1.0

A, B c 9.00 12 1.109 .083 2.00

A, B 7.04 7 1.007 .032 .93

A B c D 8.33 16 1.086 .057

67

• •••• • • • • • • ~

D D.: • • • •

• ••••

• •••• -c D E F

REINFORCEMENT PATTERNS

p > p?._

R SD R

1.033 .014 1.043

1.017 .012 1.029

1.059 .010 1.066

1.046 .030 1.077

1.014 .020 1.011

1.057 .026 1.068

1.053

AllP SD

.050

.036

.039

.070 I

I

.026 I

.043

.053

~ co

... ~. :::::1 cr ... n Cl) c.. n 0 ::I n ~ r+ Cl)

n 2.. !: 3 ::I

"'

biaxial bending simplified 249

APPENDIX B

The accuracy of the approximate value of Bb is given in Tables Al to A6 for the columns studied.

TABLE Al - COLUMN SIZE 20 x 20, fy = 60, fc = 5

Reinforce- p < .25 fc bt p > .25 f' bt p% ment Rverage RVerage

Pattern Ratio Std. Ratio Std. (See exact B Dev. exact B Dev.

Table 1) Bb Bb

1.0 1.125 .138 1.007 .005

1.27 .1.113 • 121 1. 013 .004

1.56 A 1.103 .104 1 .021 .005

2.25 1.088 .079 1.032 .005

2.5 1.049 .049 1.022 .004

3.17 1.043 .041 1.028 .005

3.90 D 1.033 .036 1. 027 .005

5.63 1.032 .027 1.049 .006

4.0 1.022 .022 1.034 .006

5.08 c 1.024 .020 1.042 .007

6.24 1.026 .018 1.049 .007

9.00 1.029 .017 1.052 .010

SUMMARY 1.057 .074 1.033 0.014


TABLE A2 - COLUMN SIZE 10 x 10, fy = 60, f~ = 5

Reinforce- p < . 25 f; bt p > .25 f' bt ment AVerage -Average

Pattern Ratio Std. Ratio Std. p% (See exact B Dev. exact B Dev.

Table 1) Bb Bb

2.0 1. 027 . 028 1. 003 • 016

3.10 • 015 .024 1. 015 • 019

4.39 A . 998 . 029 1.009 • 021

5.99 1. 001 .042 1. 016 .026

3.20 1.009 .023 1 .015 .017

4.96 B .996 . 031 1. 015 .019

7.04 1. 001 .046 1.020 .022

SUMMARY 1.007 .032 1.014 .020



Reinforce- p < .25 f~ bt p > .25 f 1 bt ment Average Average

p% Pattern Ratio Std. Ratio Std. (See exact B Dev. exact B Dev.

Tab1e1) Bb Bb

.93 1.136 .1 01 1. 031 .007

1.18 1.133 .094 1.038 .006

1.44 B 1.132 .089 1.049 .009

2.08 1.124 .065 1.065 .011 --

2.31 1.101 .044 1.064 .007

2.94 c 1.100 .037 1. 071 .007

3.61 1.098 .029 1.077 .006

5. 21 1.116 .025 1 .1 00 .006

3.70 1.055 .009 1.052 .003

4.70 D 1.066 .007 1.064 .005

5.78 1.075 .009 1. 070 .007

8.33 1. 091 .026 1.076 • 012

2.31 1.046 .025 1.021 .008

2.94 A 1.036 . 015 1 .021 .011

3.61 1.031 .010 1 .019 .013

5.21 1.040 .008 1.030 .021

SUMMARY 1.086 .057 1.057 0.026



Reinforce- p < .25 f~ bt p > .25 f' bt ment ·Average Average


Table 1) Bb Bb

1.0 ' 1.137 .128 1. 051 .014

1.27 1.126 .106 1.054 .007

1. 56 A 1.117 .089 1.056 .005

2.25 1.097 .064 1.055 .005

2.50 1. 057 .038 1.042 .004

3.17 B 1.062 .033 1.052 .005

3.90 1.062 .027 1.056 .006

5.63 1.063 .026 1.056 .006

4.00 1. 055 • 013 1.067 .007

5.08 1.056 .012 1.068 .008

.6.24 c 1.057 • 013 1.068 .008

9.00 1. 061 .026 1. 061 .008

SUMMARY 1.079 .061 1.059 0.010



Reinforce- p < . 25 f~ bt p > .25 f' bt ment Average Average


Table 1) Bb Bb

1.0 1.132 • 132 .998 .005

1.27 1.134 .140 1.006 .005

1.56 A 1.133 • 142 1. 021 .017

2.25 1.128 .140 1.038 .017

2.50 1.089 .069 1.009 .002

3.17 B 1.090 .063 1.027 .007

3.90 1.093 .054 1.041 .005

5.63 1.098 .037 1.072 .014

4.00 1.089 .018 1.040 .002

5.08 c 1.098 • 012 1.058 .003

6.24 1.105 .008 1.073 .008

9.00 1.121 .011 1.093 .004

SUMMARY 1.109 .083 1.046 0.030



Reinforce- p < .25 f: bt p > .25 f~ bt ment Average AVerage


Table 1) Bb Bb

1.50 1.054 .071 .998 .008 Similar

1.90 to B 1.047 .058 1.005 .005 but with

2.34 only 6 1.043 .050 1. 014 .010 bars

3.38 1. 033 .041 1.020 .010

2.00 1.050 .057 1 • 011 .001 Similar

2.54 to B 1.047 .052 1.021 .004 but with

3.12 only 8 1.041 .043 1.026 .005 bars

4.50 1.025 .031 1. 027 .007

SUMMARY 1.043 .049 1.017 0.012


p

Fig. la--P-Mx-My interaction surface


p

Fig. lb--P-Mx interaction curve

p

Fig. ld--P-Md interaction curve

Fig. lc--P-M interaction curve y

p

Fig. le--M -M interaction curve X y


Mv

Mvo- /3b=:: I. f3bMYo-


--~,~-Mx

J3bMxo Mxo

Fig. 3a--Mx-My for a rectangular column

"--------'---~\ Mx/Mxo ,8~ 1.0

Fig. 3b--Mx/Mx0-Myo for a rectangular column


Mv -M .s

YO

.o.o

·~o

.25 .50 .75 Mx/Mxo

Fig. 4a--Plot of Eq. (3)

1,0

1,0

Fig. 4b--Comparison of Eq. (3) to actual interaction diagram

259


Mv -M .so

YO

·~o .25 .50 .75 1.0 M/Mxo

Fig. 5--Plot of Eq. (4)

~ (~+ .Mr_ =1.0 Mxo J3b J Mvo

.Mx.+ .Mr (1-.Sb\ = 1.0 Mxo Mvo A)

Mx/Mxo

Fig. 6--Straight line interaction approximation

f3b


I. Qt-----r------r------r-----1

us~----~-~----~----~-0. P0

AXIAL LOAD

Fig. ?--Variation of Bb with axial load

. 7 ~-+--+---+--t--::-

.2 .4 .6

P/Cc

.a

Fig. a--Simplified Bb as given by Eq. (6) through (10)

261

1.0

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Gouwens, Biaxial Bending Simplified