11th INTERNA TIONAL BRlCKlBLOCK MASONRY CONFERENCE

TONGJI UNIVERSITY, SHANOHAI, CHINA, 14 - 16 OCTOBER 1997

THE CAPACITIES OF SHORT COLUMNS OF GEOMETRIC SECTION CARRYING ECCENTRIC AXIAL LOAD

L Yang1 A J Be1l2 ME phippS3

1. ABSTRACT

An efficient computational method is presented for determining the ultimate capacities of short masonry columns of any geometric cross-section subjected to eccentric axial load. The method may use triangular, rectangular or parabolic stress blocks and in the case of the latter can use stress blocks with falling branches. Cross-section shapes are divided into triangular elements and direct formulations for the integrais of stress over these elements are presented. These formulations are used in an iterative process to locate the neutral axis of a section under eccentric axial load. With the neutral axis established, the capacity of the section is determined. Using the method proposed, capacity reduction factors have been obtained for short columns of geometric cross­section for ranges (lf eccentricities ofaxialload.

2. INTRODUCTION

Masonry columns of geometric section subjected to eccentric axial load are used increasingly for structural applications. The section shapes of these columns are not necessarily rectangular or symmetric and can be open, hollow or solid. Examples are I L T and X and derivatives thereof Current design codes, e.g. BS56281 can be used to determine capacities of columns of rectangular section but are not applicable to other shapes.

Masonry, Co1wnns, Geometric sections, Capacities, Nurnerical methods

lFormer Research Assistant, Department ofCivil & Structural Engineering, UMIST, PO Box 88, Manchester, England 2Senior Lecturer in Structural Engineering, Department of Civil & Structural Engineering, UMIST, PO Box 88, Manchester, England 3Professor of Structural Masonry, Department of Civil & Structural Engineering, UMIST, PO Box 88, Manchester, England

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Based on a rectangular stress block and assuming zero tensile strength of masonry, PhippS2 proposed a method for dealing with short and slender geometric colurnns subjected to uniaxial eccentric loading. Sawko and Rahman3 have developed a numerical procedure for predicting the load carrying capacity of slender rectangular columns eccentrically loaded in one direction using rectangular and parabolic stress blocks. Phipps and Lau4 presented an iterative method to determine the strength of short and slender geometric columns again with rectangular and parabolic stress blocks. Subse~uent1y, the numerical procedures used in the method were refined by Bell and Phipps .

In the present paper a more efficient numerical method is described which can be used for the analysis of short columns of geometric section subjected to eccentric loads. It can also be used as the basis of a method for dealing with slender columns of geometric section. The column cross section is divided into triangular elements to which a co­ordinate transformation is applied to enable the exact integration of stress over the cross-section for rectangular, linear and parabolic stress distributions. A trial neutral axis is moved around the section using a Newton RaphsQn approach until the force associated with the stress distribution is coincident with the eccentric applied load.

3. THEORETICAL METHOD

3.1 Stress strain relationships

The analysis assumes that masonry has zero tensile strength and that the distribution of strain across the section is linear. Three types of stress block may be used.

Rectangular stress block - stress is assumed to be distributed uniformly across the compressive area ofthe section as suggested in BS562S1

.

Triangular stress block - stress varies linearly across the compressive area of the section, i.e.

[1]

where I and B are stress and strain and Im and Bm are the maximum stress and the maximum strain in the masonry respectively.

Parabolic stress block - stress varies parabolically across the compressive area of the section according to the relationship obtained by Powell and Hodgkinson6

. This relationship, which has been found to be valid for a wide range of types of masonry is expressed as:

[2]

3.2 Forces and moments resulting from stress distributions

Consider a geometric column with principal axes (X, Y) as shown in Figure 1. The cross-section of the column is divided into n triangular elements with a suitable nodal numbering system

502

6,..-----+---~

5 .....:::::.......,,~-I-...,.....--'

x

FigureI Section divided into triangular elements

For the general case of li geometric colurnn subjected to an axialload with eccentricities ex and ey as shown in Figure 2, the neutra! axis is inclined. Assuming the neutra! axis to be as indicated, it cuts the elements comprising the section at a, b and c forming a new triangular e!ement, 9cb, and two quadrilateral e!ements, 3bclO and 49ba. The latter need to be subdivided as shown to retain a pattern of triangular e!ements. The three possib!e stress blocks, rectangular, triangular or parabolic, which may be assumed to act over the compression side ofthe neutra! axis are also shown in Figure 2.

COMPRESSION LOAD

6r--'T-----:::::=-t>-C

5 1"""''--.......... --.,..-----'

N

2 .---+-~--:=i-;_

TENSION

Figure 2 Stress blocks

Taking the parabo!ic stress block as being the most general, the stress at any point (x, y) may be written as:

[3]

where 8u is the strain at the point farthest from the neutra! axis; r is the distance of this point from the neutra! axis; u is the distance of point (x, y) from the neutra! axis which in turn is given by:

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IA*x+B*y +C*1 U = '-----,=====_-'-

JA *2 +B*2 [4]

A *, B* and C* are constants in the equation of the neutral axis and C), C2, C3, C4, C5

and C6 are constants.

Integrating the compressive stress distribution across the section and taking moments of it about the X and Y axes gives:

p = :tIL fdxdy i=l '

Mx = :t ff (C}x2 Y + C2y3 + C3X)'2 + c4X)' + Csl + CtY}txdy ; = } A,

My = :t ff (C}x 3 + C2X)'2 + C3X2 Y + C4X

2 + Csxy + c6x }ixdy ;= } A,

[5]

[6]

[7]

To simplify the evaluation of these integrais over each of the triangular elements, the coordinate system (X, Y) is transformed into a natural co-ordinate space (p, s) as shown in Figure 3 using the following transformation:

[8] i =l ;=1

where Xi and Yi are the co-ordinates of no de i of the element and the shape functions are defined by h)=J-p-s, h2=p .

y s 3

(0,1)

O x O (1,0) p

Figure 3 Co-ordinate transformation

Substituting equations [8] into equations [5] to [7], the integrations may be carríed out more easi1y with respect to the natural co-ordinate space. After manipulatíon7

, the values ofthe integrais in equations [5] to [7] may be re-expressed ínterms ofthe real co-ordinates ofthe triangular element as:

[9]

[10]

504

fI IJI ydxcry = -(YI + Y2 + Y3) [11]

A, 6

Jt X2~ = ~{2(XI - xS + 2(xI - xS + XI (-2xI + 6x2 + 6x3) + 2X2X3} [12]

f L, y2~ = ~~ {2(YI - yS + 2(YI - yS + YI (-2YI + 6Y2 + 6Y3) + 2Y2Y3} [13]

fI x2y~ = 1:1{X;(6YI +2Y2 +2Y3)2 + X; (2YI +6Y2 +2Y3)2 + .4j 120

X; (2YI +2Y2 +6Y3)+Xl x2(4YI +4Y2 +2Y3)+

x l x3(4YI +2Y2 +4Y3) +X2X3(2YI +4Y2 +2Y3)}

fI xy2~ = 1:12

(y;(6x I +2x2 +2x3r + y;(2xI +6x2 +2xS + A, 1201

y;(2xI +2x2 +6x3)+YIY2(4xI +4x2 +2x3)+

YIy3(4xI +2x2 +4x3)+Y2Y3(2xI +4x2 +2x3)}

ff/dxcry = ~ {(x; +x; + x; XXI +X2 +X3)+XIX2X3}

ft ldxdy = ~ {(YI2 + Y; + y; XYI + Y2 + Y3) + YIY2Y3 }

where J is the Jacobean ofthe transformation:

âc 0'

I.!J = ~ ~ = Y3(X2 - XI) + Y2(XI - x3) + YI(X3 - x2)

éS éS

[14]

[15]

[16]

[17]

[18]

Equations [5] to [7] may then be used to determine the force associated with the assumed neutral axis and its eccentricities relative to the princip"J axes of the section. If these eccentricities are the same as those of the applied load, the neutral axis is the correct one and the 10ad given by equation [5] is the capacity ofthe section. In general an initial assumed neutral axis will not be correct and it will be necessary to modify or move it in steps to obtain its correct position.

3.3 Location ofneutral axis

The equation of the line of the neutral axis may be written in one of the following forms:

x=b y=kx+b

for ey=O for sections symmetric about the x axis for all other cases

[19]

To locate the neutra! axis for any particular eccentric axial load it is necessary to determine the constants b and k in the above equations such that the centroid of the compression area ofthe section coincides with the point ofapplication ofthe load. This condition may be expressed in the forms oftwo functions:

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F(k,b) = P(k,b) - Mik,b) / ex = O

G(k,b) = P(k,b) - Mx(k,b) / ey = O [20]

The Newton Raphson method is used to solve these equations. Truncating a Taylor series expansion of the equations about values ki and bi, leads to incrementai corrections to ki and bi given by:

[21]

and hence to improved values of k and b:

[22]

Successive improvements to k and b are made until the neutral axis is such that the difference between the position of the centroid of the compression area of the section and the point ofapplication ofthe load is less than a prescribed small value.

An initial neutral axis from which the iterative scheme can proceed is that for the elastic section given by:

e e 1 a=-x x +-2... y +- = O

Iyy Ixx A [23]

where A is the area of the section. This initial neutral axis can be further improved simply by moving it parallel to itself until the distance between the position of the centroid ofthe compression area ofthe section and the point of application ofthe load is a minimum. Using this as the start point for the Newton Raphson iteration leads to rapid convergence to the correct neutral axis position and hence capacity ofthe column.

4. NUMERICAL EXAMPLES

The method described has been applied to short eccentrically loaded columns with the rectangular, I and T sections shown in Figure 4.

Figure 4 Column sections

Capacity reduction factors, i.e. ratios of eccentric load capacity to axial load capacity, have been determined for ranges of eccentricities assuming a parabolic stress block and these are presented in Tables 1, 2 and 3. In general less than ten iterations were required to obtain the necessary convergence.

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ex/B eyro 0.0 0.1 0.2 0.3 0.4

0.0 1.000 0.715 0.534 0.355 0.177

0.1 0.715 0.578 0.430 0.287 0.143

0.2 0.534 0.430 0.317 0.211 0.106

0.3 0.355 0.287 0.211 0.141 0.070

0.4 0.178 0.087 0.063 0.038 0.014

Table 1 Capacity reduction factors for rectangular section

e./B eyro 0.0 0.1 0.2 0.3 0.4

0.0 1.000 0.696 0.591 0.465 0.347

0.1 0.756 0.591 0.415 0.268 0.136

0.2 0.606 0.465 0.328 0.217 0.109

0.3 0.484 0.347 0.252 0.169 0.062

0.4 0.137 0.124 0.095 0.062 0.027

Table 2 Capacity reduction factors for I section

e./B eyro 0.0 0.1 0.2 0.3 0.4

-0.6 0.065 0.012

-0.5 0.152 0.056 0.010

-0.4 0.237 0.143 0.028

-0.3 0.318 0.259 0.099 0.005

-0.2 0.430 0.396 0.219 0.043

-0.1 0.603 0.575 0.326 0.114 0.008

0.0 1.000 0.630 0.382 0.185 0.041

0.1 0.788 0.536 0.347 0.228 0.089

0.2 0.467 0.378 0.282 0.188 0.094

0.3 0.088 0.073 0.053 0.036 0.018

Table 3 Capacity reduction factors for T section

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5. CONCLUSIONS

A new and effieient numerieal method has been presented to determine the load eapacities of short, eeeentrieally loaded masonry eolumns of geometrie seetion. The method ean make use of reetangular, linear or parabolie stress bloeks and eould be extended to other forms of non-linear stress relationships. The method is simple to apply and ean be generalised for use in the determination of the eapaeities of slender eolumns.

REFERENCES

l. BS5628 Code ofPractice for use ofmasonry: Part 1, Struetural use ofunreinforeed masonry, British Standards Institution, London, 1978.

2. Phipps, M. E., The design of slender masonry walls and eolumns of geometrie cross section to earry verticalload. Structural Engineer, 65A, (12), 1987.

3. Sawko, F. & Rahman, M. A., Numerical solution of eecentrieally loaded struts in non-tension material. Numerieal Methods in Fracture Mechanics, Pineridge Press, 1980.

4. Phipps, M. E. & Lau, C. K. 1., The load carrying capacity of masonry walls and columns of geometric cross section. British Masonry Society Proceedings, No.4, 1990.

5. Bell, A. 1. & Phipps, M. E., The strength of masonry columns of geometric cross section. Proceedings of the 9th International BrickIBlock Masonry Conference, Berlin, 1991.

6. Powell, R. & Hodgkinson, H. R, Determination of stress relationships of brickwork. Proceedings of the 4th International BrickIBlock Masonry Conference, Brugge, 1976.

7. Yang, L., Behaviour of masonry columns of geometric section subjected to eccentric axialload. PhD Thesis, UMIST, Manchester, 1995.

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