Variation of deflection of steel high rise structure due to p- delta effect considering global slenderness ratio

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)

Variation of Deflection of Steel High-Rise Structure Due to
P- Delta Effect considering Global Slenderness Ratio
Yousuf Dinar1, Nazim Uddin Rahi2, Pronob Das2
1

Graduate Student, Department of Civil Engineering, University of Asia Pacific, Bangladesh
2
Student, Department of Civil Engineering, University of Asia Pacific, Bangladesh
2
Structural Designer, JPZ Consulting Limited, Bangladesh
1

yousuf_dinar@yahoo.com
2
rahicox@gmail.com
2
pronob20@yahoo.com

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.

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 PDelta 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
Slenderness, Steel high-rise, deflection

Static

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.

Method,

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

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

1


hand a primary load case (vertical loads) is revised just
before combining with effects of lateral loads during the PDelta 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 PDelta 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.

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.

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.

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 PDelta 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 1 st
order loading effects only in one step.

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.

Figure 1: (a) Linear Static analysis is performed in one step
(b) P- Delta analysis is performed in two steps

2


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.

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.

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.
Figure 2: Three-dimensional frame models of the four
different storeys

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 PDelta 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

Figure 3: Five different model spans: (a) 5X4, (b) 6X5,
(c) 7X6, (d) 8X7 and (e) 9 meter X8 meter

3


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.

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 PDelta analysis and percentage of variation against
slenderness ratio to systematically scrutinize the
displacement characteristics due to P-Delta effects with
respect to slenderness

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.

A. Variation of horizontal displacement in top:
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.

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.

TABLE I
VARIATION OF GLOBAL SLENDERNESS RATIO FOR DIFFERENT CASES

Storey 7

9mX8m

3.5

5

7.5

14S5X4

20S5X4

30S5X4

0.84

2.80

4

6

14S6X5

20S6X5

30S6X5

0.70

2.33

3.33

5

14S7X6

20S7X6

30S7X6

0.60

2

2.86

4.28

7S8X7

8mX7m

1.05

7S7X6

7mX6m

Storey 30

7S6X5

6mX5m

Storey 20

7S5X4

5mX4m

Storey 14

14S8X7

20S8X7

30S8X7

0.525

1.75

2.5

3.75

7S9X8

14S9X8

20S9X8

30S9X8

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

4


occurred same for the Storey 7 and Storey 14 group and
seem much following the general trend like Linear Static
analysis.

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 4th floor
(ii) Variation of Storey displacement for Storey 14 for
different bay cases:
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 4th floor Figure 7 same to Storey groups
7. It will be the same trend for 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
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 7: The comparison of story horizontal displacement
for different bay cases: 5X4, 6X5, 7X6, 8X7 and 9X8
considering P-Delta effect for Storey 14
(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


5


Linear Static analysis and necessity of P-Delta is
established.

C. Variation of
Slenderness ratio:

displacement

percentage

against

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.

(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.

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


6


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.

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 PDelta 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).

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.

[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.

TABLE II
RESEARCH SCENARIO OF THE OUTCOMES

Slenderness
Max
Range
Min
Results
Max
varies
Min
Bay
Trend
Increment Results
Variation
Bracing
Necessity of P-Delta
Analysis

7
1.05
0.525
20
9
Up
Low

14
3.5
1.75
54
10
Up
Low

20
5
2.5
60
25
Down
High

30
7.5
3.75
90
33
Down
High

No
Low

No
High

Yes
High

Yes
High

7

Variation of deflection of steel high rise structure due to p- delta effect considering global slenderness ratio

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