Uniaxial and Biaxial Bending of Reinforced Brickwork Columns

S R Davies

E A El t.r:üi'.v

AllSTRACT

In a previous paper lhe authors desc ri't- C'd 0 _'1 i ~erati vc 301uti 'J 11 fo r the analysie of rectangular re inforced bri ckwork cLlhllnJ :S ~;ubjec ted to bi axial bending. In this paper the method is devel op~d f'Jr~h f;r ar.d in teraction diagrame included which enabl e the designer to c:o"1s ider ~,j;'2 f'f reet of different eccentriclties of loading about both axes. ri~lC curves are sui table f o r solvi ng for both uniaxial and bia.xi a l bending .

A brief de scripti on of laboratory test~ on colu!l!ns i9 also included and a compariaon made of experimental and theoretic.::.l \"alucs .

Department of Civil Enginet' ring 3nd lluilding: Sc-i .:,]UI.:,

Un lversity of tdinbur gh, Klng's Bul1dlngs .

England

843

INTRODUCTION , Coluou1s of reinforced concrete loaded eccentrically about one axis have

been studied extensively and some recommendati ons have also been put forward for uniaxial bending af reinforced masonry. In the case of biaxial bending of reinforced concrete columns it ia usual to consider the two axes separately and to use some combination of twa separate uni axial cases .

Very li ttle work appears to have been done for the case of biaxial bending in co lumns of reinforced brickwork and a procedure for the design cf such columns ia outlined in this paper. Only one type of rectangular section ia considered and although t he number of interaction diagrama i8 limited to three cases it will be apparent that the method can be ext ended to other sections and additional load cases .

Biaxial bending can arise directly from an applied load which is eccentric to both axes ar indirectly from forces introduced by wind ar earthquake and it ia important that the columns are adequately designed to allow for these moments.

The problem is non-linear in that the bending produces a deformation which introduces secondary bending and the procedure used for obtaining a solution is based on an iterative method similar to that used by Farah and Huggins (1) for columns of r einfor ced concrete. The columns are first subdivided into sections and a triple iteration procedure used 1) across the section 2) between adjacent cross-sections 3) over the column l ength. ( 2) .

BASIS OF THE METHOD

Since the eccentricities at any sect ion will consis t of t he algebraic sum of the applied eccentricity and a component of the deflection it follows that a knowledge of the variat ion of the eccentricity implies a knowledge of the deflected formo

(a) Initially the following assumptions are made with respect to the central !Section.

1. The eccentricities 2. The corner strains

The forces and moments at the central section can now be calculated independently on the basis of the two assumptions and any differences corrected for, by changing the corner strains incrementally. 'llhe calculat ion of the force s and moments from the assumed comer strains is based on an i ntegration of the stresses over the sec t ion.

(b) Once the t wo set s of for ces and moments are wi thin acceptable limits the analysis proceeds to the next section by first ca1.culating the value of the eccentricity at the second section as a function of the previous value and the curvature. This requires a second iteration since the curvature is itself a f'unct ion of the hfO eccentrici ties.

The comer strain at the second section are now assumed and the first iteration repea t ed at section two until t he t wo sets of forces and. moments are in agreement.

(c) Finally the process reaches the top of the column and the calculated eccentricity compared with the known applled eccentricity. If the difference ia not within acceptable l~mits the whole process ~s repeated by increment ing the assumed values of the eccentrlclties at the central section.

844

This would be very tedious for hand calculation and a computer program has been developed in Fortran to carry out the iterative processes. A flow chart for this program is shown in Fig. 1. In this chart the second iteration between adjaeent cross- seetions has been omit t ed sinee it was found that, for the seetione eonsidered, the inclusion of this iteration did not greatly affeet the re eul t s.

MATERIAL PROPERTIES AND GEOMETRICAL DATA

Stress-strain relationships must be defined for the three materials used in the construction of the eolumns and for thi s present work the relationships used are as shown in Figs . 2 and 3.

Although the computer programme is wri tten in non-dimensional form it is based on columns of rectangular section similar to that shown in Fig . 4. The modification required to analyse other sectional forms would present no diff i eulties.

TREORETICAL RESULTS

Interact ion diagrams are shown in Figs . section for which dl = 0 .9T and d2 = 0 . 85B . of the axial l oad P of 0 . 2, 0 . 3 and 0.4.

5, 6 and 7 for a typical reetangular The diagrams are based on values

where P = N/f BT m

For these interaetion diagrams it has been as sumed that the grout and the brickwork have identical stress-strain charac teristi cs.

A set of these charts could be produced for other load cases and other rectangular sections with different values of d1 and d2 . ~~e charts sho~m have been derived using factors of safety of 2.5 for brickwork and 1.15 for steel so that ultimate values af the mornents are indicated .

The charts can be used in two ways. 1 . Knowing the applied axial load and the moments about both axes, it is first

necessary to select the appropriate chart for the axial load (or interpolate between two charts). Then by locating the interaction point using the known values of moments, the required amount of steel can be ascertained .

2. Knowing the ratio of M x

Example 1:

axial to M y

load and the area of s teel used , suitable values of the can be determined.

Consider a rectangular column with B = 1 OOmm , T ~ 200mm , d1 = O.9T and d2 = 0 . 85B . Assuming that the axial load (N) is 300 kN and the IDoments are Mx = 7.2kNm and My = 2.4kNm . The required area of steel can be deteImined 2 as shown below. The design stresses are taken as fm = 15N/mm2 and Fy = J..j.60N/mm .

Solution P = N/fmBT

M)fmBT2

300 x 100 15 x 100 x 200 = O.JO

7 .2 x 1000 x 1000 15 x 100 x 200 x 200

0.12

845

Example 2:

Mjfm TB2

From chart (Fig . 6)

A If BT s m

i. e . A s

2 .4 x 1000 x 1000 15 x 100 x 100 x 200

1~ x 10- 4

11 x 15 x 100 x 200 1000

0 .08

Using the same section and design stresses as given in Example 1 and assuming that the steel area ~s known to be 390mm2, then safe combínation of M and M can be determined fram the charts as shown below,

x y

Calculate N 300 0 · 3 f m BT 15 x 100 x 200

Therefore use ~

390 100 x 200 x 15

The broken line lying behleen A IB'T f values of 15 x 10- 4 and 10 x 10- 4 s rn

shown in Fig. 6 represents sa':e combinations of Mx and My'

EXPERIMENTAL INVESTIGATION

A test rig was constructeà for application ofaxial loads and biaxial bending to columns constructed of half s~ale brickwork . Details of the test rig are shown in Fig. 8 .

Initial tests indicated that there ',;ere inherent weaknesses in the rig construction which required modification. 1hese changes were concerned mainly wi th the support condi tioos at ei ther eod and the final arrangement the axial load was applied through steel balls placed at both end of the column.

The results obtained from three subsequent tests are shown below.

Test Ult. axial load Measured Values (kNrn) Theoretical Values (kNm) No. kN Mu Mu Mu Nu

x Y x y

235 . 8 10 .95 4.07 10 . 08 4 . 48

2 235 . 8 9 · 12 4 .98 8 . 19 4.71

3 294.8 7.08 3,01 8.65 4.27

The deflection profiles detennined using the computer programme, f~r constant axial load P and different combinations of Mx and My are shown in Figs . 9 and 10. The measured values obtained from the tests are also shown on the aame diagrama.

846

CONCLUSIONS

The limited number of practical tests completed to date shows that the theor etical resul ts are conservative . Additio llUl testing ia required before firm conclusion can be drawn but the results indicate that the theoretical approach gives results which are in elose agreement wi th the measure values.

REFERENCES

1. FARAH , A. and HIGGINS, M."' . ' Analysis af Reinforced concrete columns subjected to Longitudinal Load and biaxial bending '. ACI Journal, July 1969. pp. 569 -575 .

2. DAVIES , S.R. and EL1~IFY , E. A. 'Biaxial Bending of Reinforced Masonry Columns' . 5th International Masonry Conference, Washington, 1979.

ACKNOWLEDGEMENTS

The authors wish to thank the Londen Brick Company for donating bricks for the continuation of this project and the British Council for financiaI support given to the second author .

847

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