SIDE SWAY OF MULTISTORY AND MULTIBAY FRAMES UNDER LATERAL LOAD

TALAT H. ABDEL-LATEEF, SEDKY A. TOHAMY AND OMAR M. ABDEL-MOEZ Department of Civil Engineering, Minia University, Egypt

MAGDY I. SALAMA Department of Civil Engineering, Kafrelsheikh University, Egypt, magdy1000@hotmail.com

ABSTRACT: In this paper, the side sway of multistory and multibay frames under the action of lateral loads is

theoretically investigated. The lateral sway of the frames under study depends on many factors that depend on the

dimensions of the frame, the properties of the cross sectional area and the material of each frame element. The

analysis was performed by using the virtual work method. The straining actions which required applying the virtual

work method were obtained by using portal frame method assuming that the inflection points occurred at mid-height

of columns and mid-span of girders. The lateral sway of many types of frames is given in simple equations as a

function in all the factors which probably affect this sway for many types of lateral loads. Comparisons between the

results of the present analysis and those from finite elements analysis are given in this thesis.

Keywords: Lateral loads; multistory frames; multibay frames; side sway.

1. INTRODUCTION

From the structural engineer's point of view, high rise building structures (Schueller, 1997). may be defined as one

in which lateral forces equivalent to dynamic loads (Chopra, 2003) as wind or earthquake play an important or

dominant role in the structural design. In general, different structural systems have evolved for residential and office

buildings, which reflect their differing functional requirements. One of the most fundamental components of high

rise buildings is the rigid frame, which achieve its lateral stiffness from the rigidity of the joints between columns

and beams. Excessive lateral sway that a building's structural system may be able to withstand still to be reduced to

the acceptable limits for human use. To attain lateral deflection of multistory and multibay frames under lateral

loads, it is required to solve statically indeterminate structure from higher degree of indeterminacy. When using

finite element analysis (Felton, 1996; Bathe, 2003), the effect of both flexural and axial stiffness of the columns and

girders is not clear. Semi-exact and simple equations were derived in this paper to obtain the lateral sway as a

function in all factors affecting its value by using energy approach.

1

2. ASSUMPTIONS

Consider a multistory frame as shown in Fig. 1. submitted to lateral static load with the following assumptions:

1) The material of the frame is made is linearly elastic.

2) The frame is fixed at the base.

3) The frame is perfectly straight in its plane before loading.

Aci

Aci+1

Aci-1

Acn

Ac1

ci

Ici+1

Ici-1

Icn

Ic1

I

Agn

Ign

Agi

Igi

Agi+1

Igi+1

Agi-1

Igi-1

Ag1

Ig1

Aci

Aci+1

Aci-1

Acn

Ac1

ci

Ici+1

Ici-1

Icn

Ic1

I

b

hh

hh

floor no. i

no. i+1

no. i-1

no. n

h

floor no. 1

H=

n.h

b

Fig. 1. Frame model under study

In the beginning of the study, we pointed the main factors that may affect the analysis of the multistory frames

under lateral loads. These factors depend on the dimensions of the frame, the properties of the cross sectional area

and the material of each frame element, which are:

1) Bay width (b).

2) Height of each floor (h).

3) No of floors (n).

4) No of bays (n1).

5) Cross sectional area of the columns (Aci).

6) Moment of inertia of the columns (Ici).

7) Cross sectional area of the girders (Agi).

8) Moment of inertia of the girders (Igi).

9) Shear shape factor (k`).

10) Modulus of elasticity of the material, of which the frame is made, (E).

11) Shear modulus (G).

12) Lateral load (P-Load system) which may be concentrated loads at each floor level.

2

3. METHOD OF ANALYSIS

By applying the energy method (Virtual work method) to the multistory and multibay frames under lateral loads,

the maximum lateral sway "s" at the higher point of the frame can be determined by the following equation

(1)

Where N0, Q0 and M0 are the straining actions produced by the P-load system, N1, Q1 and M1 are the straining actions

produced by the unit load acting at the upper point of the frame.

The third term on the right hand side of (1) which represents the contribution of the shearing forces to the total

deflection (lateral sway) is very small compared to the first and second terms.

(2)

The first term on the right hand side of (1) which represents the contribution of the normal forces to the total

deflection (lateral sway) can be divided into two terms which represent the effect of the normal forces in columns

and girders as follows

(3)

Also, the second term on the right hand side of (2) which represents the contribution of the normal forces of the

girders is very small. Thus, the contribution of the normal forces is mainly from columns.

(4)

The second term on the right hand side of (1) which represents the contribution of the bending moments to the

total deflection (lateral sway) is the main term of the deflection and also can be divided into two terms which

represent the effect of the bending moments in columns and girders as follows

(5)

Substituting (2), (4) and (5) in (1), we can obtain

(6)

Equation (6) can be written in this form

(7)

Where s1, s2 and s3 represent the contribution of the normal forces in the columns (the effect of the cross sectional

area of the columns), the contribution of the bending moments in the columns (the effect of the moment of inertia of

the columns) and the contribution of the bending moments in the girders (the effect of The moment of inertia of the

girders) respectively.

3

To apply the analysis of multistory and multibay frames under lateral static loads by using the virtual work method

(Shames, 1985; Williams, 1978; Coates, 1990; Smith, 1980), the straining actions due to the lateral static load (P-

load system) and due to unit load at the upper point of the frame under study are required. These straining actions

were obtained by using portal frame method (Scarlat, 1996) assuming the inflection points form exactly at mid-span

of columns and mid-span of beams.

4. THEORETICAL EQUATIONS

4.1 Frame with One Bay

Referring to Figs. 2(a) and 2(b) which represent the normal forces and the bending moments of the frame under

study with one bay due to P-load system and due to unit load at the upper point of the frame respectively, substituting

in (6), we can obtain

(a)

N , M 1 1(b)

h

P1

P

i=1

i=2

i=3

M

M/2

M (h)0*

M (2h)0*

M (3h)0*

M (2.5h)0*

P2

3

b

hh/

2

M ((i-0.5)h)0*

b

x

M ((i-0.5)h)0*

b

M/4

M/2

M/2hM/2h

N , M 0 0

M/2

h

1 ton

i=1

i=2

i=3

h/2

3 h

b

hh/

2

(i-0.5)h b

x

h/4

1/2

h/2

h/2

(i-0.5)h b

1/2 2.5 h

2 h

h

Overall B. M.

Overall M . D.1

Fig. 2. Straining actions of multistory frame with one bay

4

(8a,b.c)

Where,

M0*(x) = external bending moment function (the frame as a cantilever)

x = the height from the higher point of the frame

Applying the same method to determine the lateral sway at any floor level no. m from the higher floor of the

frame, we can obtain

(9)

Where,

(10a,b,c)

4.2 Frame with Two Bays

Referring to Figs. 3(a) and 3(b) which represent the normal forces and the bending moments of the frame under

study with Two bays due to P-load system and due to unit load at the upper point of the frame respectively,

substituting in (6), we can obtain

(11a,b,c)Where,

Ic1i, Ic2i are the moment of inertia of the exterior and interior columns respectively.

5

Fig. 3. Straining actions of multistory frame with two baysAlso, the lateral sway at any floor level no. m from the higher floor of the frame can be obtained referring to (9)

(12a,b,c)

4.3 Frame with Arbitrary Number of Bays n1

By using the same steps which used in the determination of the maximum lateral sway of multistory frame with

one bay and two bays, we obtained the maximum lateral sway for the multistory frame with multibay as follows

N , M

0 0(a)

N , M

1 1(b)h

P1

P

i=1

i=2

i=3

M

DM/2

M (h)0*

M (2h)0*

M (3h)0*

M (2.5h)0*

P2

3

b

hh

/2

M ((i-0.5)h)0*

2 b

x

M ((i-0.5)h)0*

2 b

M/8

M/4

M/4h

M/4h

M/4

h

1 ton

i=1

i=2

i=3

h/2

3 h

hh

/2

(i-0.5)h2

b

x

h/8

1/4

h/4

h/2

(i-0.5)h2

b

1/2

2.5 h

2 h

h

M/2h

b

bb

1/4

Overall B. M.

Overall M . D.1

6

(13a,b,c)Then,

(14)

(15)

The lateral sway at any floor level no. m from the higher floor of the multistory frame with multibay as follows

(16)

(17)

Thus, the maximum deviation for vertical line between adjacent floors No. m and m+1 can be obtained from the

previous equations. This maximum deviation "s" can be written as in the following:

(18)

7

(19a,b,c)

Then,

(20)

(21)

5. LOAD SYSTEMS

5.1 General Load System

By assuming the lateral load as a different concentrated load at each floor level and assuming no relation between

these concentrated loads as a general case, the external bending moment (cantilever bending moment) can be

expressed as function of the distance from the higher point of the frame. So that, this function will be the summation

of the effect of each concentrated load as follows

(22a,b,c)

Substituting (22 a,b,c) in (13a,b,c), we can obtain

8

(23a,b,c)

Then,

(24)

(25)

Substituting (22 a,b,c) in (16), we can obtain

(26)

9

(27)

5.2 Practical Load System

By assuming the lateral load as a concentrated load P at each floor level and P/2 at the upper floor level which

the practical case for the wind load, the external bending moment (cantilever bending moment) can be expressed as a

function of the distance from the higher point of the frame as follows

(28)

Then,

(29a,b)

Substituting (28) and (29 a,b) in (13a,b,c), we can obtain

(30a,b,c)

Then,

(31)

10

(32)

If the cross sectional area of the columns, the moment of inertia of the columns, and the moment of inertia of the

girders are the same in all the floors and equal to Ac, Ic and Ig respectively, the equation (32) will be in the following

explicit form

(33)

(34)

(35)

Substituting (28) and (29 a,b) in (16), we can obtain

(36)

6. COMPARISON OF RESULTSSeveral steel frames were analyzed in order to discuss the derived equations and to evaluate the ability of present

work to accurately compute lateral displacement comparing with the more exact analysis using finite element method

[10].

Three frames, shown in Fig. 4, represent realistic designs and were chosen to include varies number of bays and

floors and varies bay width.

11

To compute the results from the investigated equations, a simple spread sheet program (EXCEL) was developed

which clarify the effect of each input data and the simplicity to change any (see appendix). Modulus of elasticity for

all frame elements was taken equal to 2100 t/cm2.

Fig. 5. shows distinctly the accuracy of the results for the lateral sway computed from the predicted equations

compared with that computed by finite element program. Also, comparison of lateral roof displacement was shown in

Table 1.

4.00 m 4.00 m

(b)

15 @

3.0

0 m

B.F.

I. 24

0B.

F.I.

260

B.F.

I. 28

0B.

F.I.

220

B.F.

I. 30

0

7.00 m

10

@ 3

.00

m

B.F.

I. 24

0B.

F.I.

260

B.F.

I. 28

0

(a)

5.00 m 5.00 m5.00 m

(c)

20 @

3.0

0 m

B.F.

I. 24

0B.

F.I.

260

B.F.

I. 28

0B.

F.I.

360

B.F.

I. 34

0B.

F.I.

320

B.F.

I. 30

0

Fig. 4. (a) 10-story, one bay frame, (b) 15-story, two-bay frame, (c) 20-story, three-bay frame

12

0.0

3.0

6.0

9.0

12.0

15.0

18.0

21.0

24.0

27.0

30.0

0 10 20 30 40 50 60 70

heig

ht (

m)

Centerline Displacement (mm)

S1

S2

S3

P. W.

F. E. Method

(a)

0.0

3.0

6.0

9.0

12.0

15.0

18.0

21.0

24.0

27.0

30.0

33.0

36.0

39.0

42.0

45.0

0 10 20 30 40 50 60 70 80 90 100

110

120

130

140

150

heig

ht (

m)

Centerline Displacement (mm)

S1

S2

S3

P. W.

F. E. Method

(b)

0.0

6.0

12.0

18.0

24.0

30.0

36.0

42.0

48.0

54.0

60.0

0 10 20 30 40 50 60 70 80 90 100

110

120

heig

ht (

m)

Centerline Displacement (mm)

S1

S2

S3

P. W.

F. E. Method

(c)

Fig. 5. Comparison of lateral centerline displacements resulting from present work and finite element analysis for: (a) 10-story, one bay frame, (b) 15-story, two-bay frame, (c) 20-story, three-bay frame

Model F. E. M. P. W. % Diff.

(a) One-bay frame,10-story, b=7.0 m

(b) Two-bay frame,15-story, b=4.0 m

(c) Three-bay frame,20-story, b=5.0 m

Table 1. Comparison of lateral roof displacement with F. E. M.

142.60 142.30 -0.21%

107.99 105.65 -2.17%

58.73 58.76 0.05%

7. CONCLUSION

In this research, a theoretical analysis is developed to determine the side sways for multistory and multibay

frames under lateral loads. An approximate method is presented for the predication of the lateral deflection for any

number of floors and for any number of bays by means of energy.

From the previous results the following conclusions may be drawn.

(1) The proposed method of analysis offers a number of relatively simple expressions from which the lateral sway

can be easily calculated for multistory and multibay frames under lateral loads that saving time and attractive for

13

use in preliminary design and studying structures subjected to dynamic loads.

(2) The present analysis describes the relation between the side sway and all the factors that affect in it (i.e. the

lateral sway function in all factors that affect in it). Thus, the effect of each factor can be investigated from these

relations as follows:

- The maximum sway depends mainly in the contribution of the bending moment of girders, the bending moment in

columns and the normal forces in columns which represent the contribution of the moment of inertia of girders, the

moment of inertia of columns and the cross sectional area of the columns.

- The contribution of the normal forces in girders approximately equals to zero which represent the contribution of

the cross sectional area of the girders. Also, the contribution of the shear forces in columns and the shear forces in

girders are very small compared with the total value of the lateral deflection.

(3) The great advantage of the present analysis is the determination of the lateral sway by using hand calculations

only (without the use of computer computations) or by using simple spread sheet program [EXCEL Program]

which gave us the ability and the simplicity to change any of the input data (cross sectional area and moment of

inertia of columns and moment of inertia of girders at each floor, bay width, height of the floor and the lateral

load value) and obtain the lateral sway immediately in no time that save time and gave us the effect of our

changing data in the lateral sway value.

Finally, the present analysis provides sets of equations and simple programs, which can be of great help for design

purposes especially for preliminary design and to verify the accuracy of the results obtained from finite element

computer programs. The comparison of the results with the finite element method confirms the accuracy of such

equations.

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APPENDIX

The spread sheet of model shown in Fig. 4 (a)

A c I c I g

cm 2 cm 4 cm 4

1 102.2 10892.85 33740 0.000 0.246 0.370

2 102.2 10892.85 33740 0.009 0.738 1.111

3 102.2 10892.85 33740 0.040 1.230 1.852

4 113.5 14350.86 33740 0.099 1.307 2.593

5 113.5 14350.86 33740 0.211 1.680 3.334

6 113.5 14350.86 33740 0.385 2.053 4.075

7 126.42 18596.6 33740 0.570 1.872 4.816

8 126.42 18596.6 33740 0.876 2.161 5.557

9 126.42 18596.6 33740 1.275 2.449 6.298

10 126.42 18596.6 33740 1.780 2.737 7.039

5.243 16.471 37.048

i

Sway = 58.76 mm

S1 S2 S3

REFERENCES

[1] Schueller, W. (1997). High-Rise Building Structures, John Wiley & Sons, Inc., New York.[2] Chopra, A. K. (2003). Dynamics of Structures Theory and Applications to Earthquake Engineering, Prentice-Hall, Inc., New Delhi.[3] Felton, L. P., and Nelson, R. B. (1996), Matrix Structural Analysis, John Wiley & Sons, Inc., New York.[4] Bathe, K. J. (2003). Finite Element Procedures, Prentice-Hall, Inc., Newjersy, U. S. A., seventh printing.[5] Shames, I. H., and Dym, C. L. (1985). Energy and Finite Element Methods in Structural Mechanics, Hemisphere Publishing Corporation, United States of America,[6] Williams, N. and Lucas, W. M. (1978). Structural Analysis for Engineers, McGraw-Hill Book Company, Inc.[7] Coates, R.C., Coutie, M.G., and Kong, F. K. (1990). Structural Analysis, Tomas Nelson and sons Ltd, Hong Kong.[8] Smith, M. J., and Bell, B. J. (1980). Theory of Structures, George Godwin, London.[9] Scarlat, A. S. (1996). Approximate Methods in Structural Seismic Design , E. & FN Spons, an imprint of chapman & Hall, London. U K.[10] Sap2000, advanced 14.0.0, (2009), Computer Software for Static and Dynamics Finite Element Analysis of Structures, Computer & Structures

Inc., Berkely, California, U.S.A.

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