variation of deflection of steel high rise structure due to p- delta effect considering global...

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 12, December 2013)

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Variation of Deflection of Steel High-Rise Structure Due to

P- Delta Effect considering Global Slenderness Ratio

Yousuf Dinar1, Nazim Uddin Rahi

2, Pronob Das

2

1 Graduate Student, Department of Civil Engineering, University of Asia Pacific, Bangladesh

2 Student, Department of Civil Engineering, University of Asia Pacific, Bangladesh

2Structural Designer, JPZ Consulting Limited, Bangladesh

[email protected]

[email protected]

[email protected]

Abstract—This paper evaluates deflection of the steel high rise

structure due to the P-Delta effect considering the global

slenderness of the whole structure. For easy and quick design

only Linear Static analysis is performed and secondary

loading effect is neglected in several underdeveloped and

developing countries of South Asia. Using STAADPro v8i, 40

different model is simulated to observe the severity of the P-

Delta phenomenon against standard Linear Static method. 4

different storey were combined with 5 varying span in both

direction for varying the slenderness of the structure. During

analysis lateral load imposed with UBC94 to perform the

seismic events in two directions in the seismic moderate risk

zone of Bangladesh using Bangladesh National Building Code

(BNBC) corresponding coefficients however wind load is

omitted to observe the seismic event effect in Steel high-rise

structure solely assuming outcome decision would be same if

the simulation would done for wind load also. This analysis

reveals how crucial side of the structure generates different

deflections with changing slenderness. Test results were

evaluated by storey deflection (in mm) and percentage of

variation of deflection was performed by comparing P-Delta

outputs with Linear Static Method outputs.

Keywords— P-Delta analysis, Linear Static Method,

Slenderness, Steel high-rise, deflection

I. INTRODUCTION

Generally Structural designers are prone to use linear static

analysis, which is also known as first order analysis, to

compute design forces, moments and displacements

resulting from loads acting on a structure. First order

analysis is performed by assuming small deflection

behavior where the resulting forces, moments and

displacements take no account of the additional effect due

to the deformation of the structure under vertical load prior

to imposing lateral loads. P-Delta is a non-linear (second

order) effect that occurs in every structure where elements

are subject to axial loads. It is a genuine “effect” that is

associated with the magnitude of the applied axial load (P)

and a displacement (delta). If a P-Delta affected member is

subjected to lateral load then it will be prone to deflect

more which could be computed by P-Delta analysis not the

linear static analysis. The magnitude of the P-delta effect is

related to the magnitude of axial load, stiffness/slenderness

of the structure as a whole and slenderness of individual

elements. Here during analysis for easy visualization only

slenderness of the whole structure is judged keeping other

two factors constant. Again excessive vertical loads buckle

the compressive member and make them unsuitable as load

bearer before coming lateral loads. When lateral loads

appear it do not find the initial undeflected shape but

deflected shaped member left by vertical loads.

Global slenderness ratio is the ratio of the height of the

building and radius of gyration of the building. Again it is

possible to simply divide the height of the building by the

width of the building for a quick estimation of

the slenderness ratio what way is adopted for this study. If

the building is too slender, it will be prone to deflect much,

where the middle portion gives way even as the top and

bottom remain solid like each and every slender member.

On the other hand, a very thick building which is opposite

of slender, may be so heavy that it causes structural

problems itself. The self-weight of thick building can be a

significant issue in deflection of tall buildings.

In summary it could be noted that linear static analysis

determines algebraic combination of forces, moments and

deflections due to vertical and lateral loads on the other

mailto:[email protected]



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hand a primary load case (vertical loads) is revised just

before combining with effects of lateral loads during the P-

Delta analysis based on the deflections which generates a

severe changes in high rise structures and eventually

difference increases with slenderness. As P-Delta effect is

not always severe and several complexities is involved so it

is not always preferred for analysis unless the P-Delta

effect becomes a issue. That is one of the reasons that P-

Delta is not known and performed in underdeveloped and

developing countries. The ideas of this paper lays here that

how crucial side of the structure generate deflections

changes with changing slenderness from the point of view

of plan span in two direction and overall structure height.

Finally percentage of variation is presented against

slenderness ratio to show how displacement changes with

changing slenderness. Force unit is KN while

displacements are measured in mm.

II. METHODOLOGY

P-Delta is an effect considered while designing high-rise

structure but sometimes it is avoided because of complexity

involve in this. Without considering the real slenderness,

bay and height parameter during decision, is really a

serious fact. The ideas of this study evolve here and that is

to show that slenderness changes with two different

parameters: bay and height, again displacement varies

unexpectedly with increasing slenderness. It may make a

guideline for designer to allow P-Delta during design after

peoper justification while knowing the effects properly. By

controlling slenderness, the magnitude of the P-delta effect

is often “managed” such that it can be considered

negligible and then “ignored” in design; for instance, at the

structure level by the use of more or heavier bracing, at the

element level by increasing member size. Slenderness

effects are extremely important in designing compression

members. It was decided that the best way to evaluate the

P-Delta effects in high-rise structure is simulating different

cases by both P-Delta analysis and basic analysis which is

chosen later Linear Static analysis. To vary the slenderness

two is chosen; one is to change the storey height and

another is varying the bay in both directions. Both were

adopted for vary the slenderness.

While changing the height a problem was faced that is what

will be the interval. It is really time consuming and

unmanageable to conduct research such a hugh amount of

cases and to simplify the analysis, four most used in sub

continental steel structure is taken for all cases: 7, 14, 20

and 30, according to A.S. Moghadam and A. Aziminejad.

One of the benefits for using these storeys is no need of

specialized structural system which makes the model

simple and easy to evaluate the slenderness effects. For

storey 20 and 30 double bracing was used to reduce

excessive displacement for both Linear Static and P-Delta

analysis. The next problem is how to change the

slenderness by bay increment. It may also face the same

fate if it varies in unit pace so increment of bay is done by

adding additional bay in each direction. By such step 5

different bay cases were developed to study.

To meet the objective displacement in which is how crucial

side of the structure generate different deflections with

changing slenderness, displacements in several point have

to be taken. Displacement in top is normally maximum

which helps to identify percentage of variation with

slenderness, later storey displacements helps to observe the

changing trend. It is not continent to take the displacement

of different points while changing the bay for study

purpose so same point in each case is taken for data

collection makes the study successful in nature.

III. DESCRIPTION OF P-DELTA ANALYSIS

There are two options by which the slenderness effect can

be accommodated. One of the options is to perform an

exact analysis which will take into account the influence of

axial loads and variable moment of inertia on member

stiffness and fixed end moments, the effect of deflections

on moment and forces and the effect of the duration of

loads which is known as P-Delta analysis. Again structures

subjected to lateral loads often experience secondary forces

due to the movement of the point of application of vertical

loads. This secondary effect, commonly known as the P-

Delta effect, plays an important role in the analysis of the

structure shown in Figure 1. by generating additional

deflection due to calculating 2nd

order loading effect in two

separate steps on the other side, Linear Static generates 1st

order loading effects only in one step.

Figure 1: (a) Linear Static analysis is performed in one step

(b) P- Delta analysis is performed in two steps



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In STAADPro, a unique procedure has been adopted to

incorporate the P-Delta effect into the analysis. The

procedure consists of the following steps:

1. First, the primary deflections are calculated based

on the provided external loading.

2. Primary deflections are then combined with the

originally applied loading to create the secondary

loadings. The load vector is then revised to

include the secondary effects.

lateral loading must be present concurrently with the

vertical loading for consideration of the P-Delta effect. The

Repeat Load facility has been created with this requirement

in mind. This facility allows the user to combine previously

defined primary load cases to create a new primary load

case.

3. A new stiffness analysis is carried out based on

the revised load vector to generate new

deflections.

4. Element/Member forces and support reactions are

calculated based on the new deflections.

P-Delta effects are calculated for frame members only not

for finite elements or solid elements.

IV. DESCRIPTION OF MODELS

Four three dimensional building models of Figure 2 are

used as the basic models in this study. The buildings have

7, 14, 20 and 30 stories. The lateral load resisting system of

7 and 14 story buildings is consists of steel moment

resisting frames, while the 20 and 30 story buildings have a

dual moment resisting and braced frame system to reduce

excessive deflection into acceptance limit. The plan of four

different storey of buildings is varied into five different bay

group: 25 by 20 meter, 30 by 25, 35 by 30, 40 by 35 and 45

by 40 as shown in Figure 3.

Bay length of buildings in each direction is 5 and their

story height is 3 meters. The floors are assumed to be rigid

in their plane. The lateral load seismic is considered in both

directions of the structure using UBC94 by providing

seismic coefficient of seismic zone 2, moderate risk rated

arena of Bangladesh to perform both Linear Static and P-

Delta analysis separately. Accidental load is taken into

account for both two major analyses to ensure load

eccentricities are considered in analysis. The bracing of 20

and 30 story buildings are in four bays in each direction

that are the first and last bay of the perimeter frames. The

column and bracing sizes are W14X90 for all position and

the slab thickness is 152.4 mm reinforced concrete. All

beams are of same size W27X34 of A342 Grade. The

concrete strength is assumed to be 24 MPa with yield

strength 414 MPa where Modulus of Elasticity (Young’s

Modulus) is 248200 MPa. The model is assumed to be

situated in Dhaka city so according to Bangladesh National

Building Code (BNBC) seismic zone 2 is taken. Therefore,

each column is subjected to both in compression and

tension during the shaking in alternative sequence. Higher

bending moment governs to the columns due to

compression than the tension.

Figure 2: Three-dimensional frame models of the four

different storeys

Figure 3: Five different model spans: (a) 5X4, (b) 6X5,

(c) 7X6, (d) 8X7 and (e) 9 meter X8 meter



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The reason is the P-Delta which is added to the lateral

effect in the case of compression but is deducted in the case

of tension. So, if the P-delta effects are observed in the

compression side of the structure then maximum results

will be found.

V. CASE STUDY

To investigate the effect of P-Delta with slenderness five

different bay groups in four different standard storey

geometrical possibilities were examined: 7, 14, 20 and 30

storey.

During study, total 20 different case, or geometrical

possibilities, were simulated through both Linear Static and

P-Delta analysis shown in Table I.

The load deformation responses of the numerical model

specimens were followed through to failure by means of

the deflection in each storey of each case of a particular

column. A particular frame, in each and every case with

two different analysis procedure, in crucial side of the

structure is observed and value taken from it to meet the

objectives of the study.

TABLE I

VARIATION OF GLOBAL SLENDERNESS RATIO FOR DIFFERENT CASES

Storey 7 Storey 14 Storey 20 Storey 30

5 m X 4 m 1.05 3.5 5 7.5

7S5X4 14S5X4 20S5X4 30S5X4

6 m X 5 m 0.84 2.80 4 6

7S6X5 14S6X5 20S6X5 30S6X5

7 m X 6 m 0.70 2.33 3.33 5

7S7X6 14S7X6 20S7X6 30S7X6

8 m X 7 m 0.60 2 2.86 4.28

7S8X7 14S8X7 20S8X7 30S8X7

9 m X 8 m 0.525 1.75 2.5 3.75

7S9X8 14S9X8 20S9X8 30S9X8

VI. RESULTS AND DISCUSSIONS

P-Delta and Linear Static analysis of 20 cases, in total 40

models reveals that P-Delta effects significantly influence

the displacement and get higher value than the Linear Static

analysis. The variation particularly identified when the

slenderness ratio is comparatively increasing by increasing

the storey and reducing the bay in both direction. Variation

is observed in several sections: Variation of horizontal

displacement in top, variation of storey displacement of P-

Delta analysis and percentage of variation against

slenderness ratio to systematically scrutinize the

displacement characteristics due to P-Delta effects with

respect to slenderness

A. Variation of horizontal displacement in top:

Maximum displacement due to lateral loads occurs in the

top storey of the structure and to identify this particular

column was selected which is present for each different

case. Increasing displacement for P-Delta analysis against

Linear Static is clearly observed from two different

analyses conducted in this study: P-Delta and Linear Static,

which is found to be increasing as the slenderness is

increasing due to height increment in different cases Figure

4 and Figure 5.

Figure 4: The comparison horizontal displacement in top

considered four storey cases with their varying span using

P-Delta Analysis

However this increasing trend is found to be decreasing in

storey 20 and storey 30 under P-Delta analysis where the

slenderness ratio dropped from 5 to 2.5 in storey 20 and 6

to 3.75 in storey 30 due to increment in bay number which

decreases slenderness. Latter scenarios however not



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occurred same for the Storey 7 and Storey 14 group and

seem much following the general trend like Linear Static

analysis.

Figure 5: The comparison horizontal displacement in top

considered four storey cases with their varying span using

Linear Static Analysis

B. Variation of storey displacement of P-Delta analysis:

(i) Variation of Storey displacement for Storey 7 for

different bay cases:

Storey displacements for storey 7 group are found to be

increasing in nature as slenderness increases due to height

increment and bay increment Figure 6. It is not found the

same as it has been expected that displacement might be

decreased as slenderness reducing due to increment of bay.

6X5, 7X6, 8X7 bay groups are tens to give similar

displacement in top due to less deference in slenderness

ratio.

Figure 6: The comparison of story horizontal displacement

for different bay cases: 5X4, 6X5, 7X6, 8X7 and 9X8

considering P-Delta effect for Storey 7

May be for storey 7 slenderness effects due to bay might be

started working. A significant fluctuation is also seen in the

trend for 7S9X8 in between 1st floor to 4

th floor

(ii) Variation of Storey displacement for Storey 14 for


Storey displacements for storey 14 group are found to be

increasing in nature same as the storey group 7 but bay 9X8

shows significant increase in displacement. In storey

groups 14, bay groups: 6X5, 7X6, 8X7 shows almost same

values and it might be caused by the similarity of

slenderness. It is not found the same as it has been expected

that displacement might be decreased as slenderness is

reducing due to increment of bay. May be for storey 14

slenderness effects due to bay might are started working. A

significant fluctuation is also seen in the trend for 7S9X8 in

between 1st floor to 4

th floor Figure 7 same to Storey groups

7. It will be the same trend for Linear Static analysis.




(iii) Variation of Storey displacement for Storey 20 for


From storey groups 20 and onwards, trend of increasing

displacement against increment of bay is found to be

opposite Figure 8. Relatively lower bay cases are showing

larger displacement and vice-versa. It might be caused by

slenderness ratio range which is found in a range of 2.5 to 5

for all cases in this storey group. It is also found that the

deflections are particularly tend to vary after storey 15 and

onwards. The outcome for this storey group is opposite of



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Linear Static analysis and necessity of P-Delta is

established.




(iv) Variation of Storey displacement for Storey 30 for


For the storey group 30 the inverse trend of displacement

against increment of bay is established properly and after

gradual increment till half of total storey displacement

ranges widely till remaining 15 storey Figure 9. These

outcomes establish importance of P-delta against for high

rise slender structure where it governs. It seems

displacement effects for slender ratio 3.75 to 7.5 are a quite

unitary in nature.




C. Variation of displacement percentage against

Slenderness ratio:

All 40 model and 20 case, are studied to describe how

crucial side of the structure generate different deflections

with changing slenderness and obviously to present the

priority of P-Delta analysis over Linear Static analysis

percentage of variation must be seen keeping Linear Static

analysis outcomes as base Figure 10. It seems with

increasing slenderness all storey group variation between

Linear Static and P-Delta will be maximized and vice-versa

where in storey group 7 the slenderness 0.525 to 1.05

influence the data to vary 9 to 20%, storey group 14 the

slenderness 1.75 to 3.5 influence the data to vary 10 to 54%

and for tripling slenderness outcomes varies almost triple

too. The double braced storey cases: 20 and 30 reveal these

high valued slender group need serious attention as

showing with slenderness double increment displacement

data varies almost triple which is a matter to be considered.

For storey 20 if slender varies from 2.5 to 5, variation

reaches 25 to 60% while for slenderness increment 3.75 to

7.5 for storey 30 causes displacement variation 33 to 90%;

almost triple.

Figure 10: Variation of displacement percentage against

Slenderness ratio

VII. CONCLUSION AND FUTURE WORK

This paper presented the variation of displacement with

slenderness considering P-Delta analysis keeping Linear

Static analysis outcomes as base. Variation of displacement

for each case under two analysis procedure identified that

differences begin develop upward as the bay increases

making slenderness decrease and it continues in

slenderness ratio 0.525 to 1.05 for storey 7. On the other



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side, For storey 14 where slenderness ratio varies 1.75 to

3.5 keeping same trend of increasing but in faster pace is a

mentionable fact, makes it a moderate level zone where

performing P-Delta is beneficial. Although double steel

bracing were used in periphery direction for reducing

excessive horizontal displacement of Storey 20 and 30,

generate higher displacement and under study it shows that

after slenderness ratio 2.5 a different trend generates in

storey 20 cases. Storey 20 which ranges from 2.5 to 5,

reveals that with increasing slenderness by reducing

number of bay causes significant displacement increment(

double) The trend found here established in storey group 30

where the displacement varies in quite a large scale (triple)

by increasing slenderness from 3.75 to 7.5 from decrement

of number of bay. Table II showing below shows a scenario

of the total outcomes in short. Although the maximum and

minimum slenderness ratio in every storey group is same

that is two but the variation of results is not following same

trend to each other. It changes dramatically from minimum

to maximum in each storey group without maintain specific

trend like their corresponding slenderness ratio do whereas

the dramatic changes multiplied for each storey group

increment. So, due to wide displacement variations with

increasing slenderness P-Delta Analysis is required for

structures higher that 7 storey.

In coming days, with displacement effects of forces and

moments could be viewed with respect to slenderness,

different structural system could be simulate to evaluate the

P-Delta response against different structural point of view.

Like the Linear Static analysis other dynamic analysis

could be simulate to suggest the designers the most suitable

analysis for high-rise structure with storey limit.

TABLE II RESEARCH SCENARIO OF THE OUTCOMES

7

14

20

30

Slenderness

Range

Max 1.05 3.5 5 7.5

Min 0.525 1.75 2.5 3.75

Results

varies

Max 20 54 60 90

Min 9 10 25 33

Bay

Increment

Trend Up Up Down Down

Results

Variation

Low Low High High

Bracing No No Yes Yes

Necessity of P-Delta

Analysis

Low High High High

A. Links and Bookmarks

For more inquiry about P-Delta analysis, its effects in high

rise structure, how to develop the analysis procedure, how

to proceed and basic characteristics, following links will be

beneficial and informative for researchers, designers and

students

1.www.bentley.com/enUS/Training

2. www.en.wikipedia.org/wiki/P-Delta_Effect

3.www.cscworld.com/getattachment/...Analysis

4. www.communities.bentley.com/products/structural

References

[1] A. Rutenberg, “Simplified P-Delta Analysis for Asymmetric Structures”, Struct. Div. ASCE, P1993-2013 (1987).

[2] Goto, Y. and Chen, W.F. “Second order Analysis for frame design”,

Journal of Structural Engineering, ASCE, 113, 7 (1987).

[3] Rutenberg, A. “A Direct P-Delta Analysis Using Standard Plane Frame Computer Programs”, Computer and Structures, 14, 1-2

(1987).

[4] A.S. Moghadam and A. Aziminejad , “Interaction of Torsion and P-Delta effects in Tall Buildings”, In Proceedings of the 13th World

Conference on Earthquake Engineering

[5] Nixon, D. and Beaulieu D. “Simplified Second Order Frame Analysis”, Canadian Journal of Civil Engineering, 2, 4, (1975).

[6] Wilson, E.L., Eeri, M. and Habibullah, A. “Static and Dynamic

Analysis of Multi Story Building Including P-Delta Effects”,

Earthquake spectra, 3, 2 (1987).

[7] BNBC (2006) Bangladesh National Building Code, Housing and Building Research Institute, Mirpur, Dhaka, Bangladesh.

[8] Bently System, StaadPro V8i, Pennsylvania, USA.

[9] Chen, W.F. and Lui, E.M. “Stability Design of Steel Frame”, CRC Press, Boca Raton, FL (1991).

[10] Wynhoven, J. H. and Adams, P. F., “Behavior of Structures Under

Loads Causing Torsion”, J. Structural Div. ASCE 98, No. ST7, 1361-1376, July 1972.

http://www.en.wikipedia.org/wiki/P-Delta_Effect

http://www.communities.bentley.com/products/structural

http://en.wikipedia.org/wiki/Exton,_Pennsylvania

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