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- 1. 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
- 2. 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) 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
- 3. 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) 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
- 4. 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) 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
- 5. 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) 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 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 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 different bay cases: 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 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 5
- 6. 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) 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. Figure 8: The comparison of story horizontal displacement for different bay cases: 5X4, 6X5, 7X6, 8X7 and 9X8 considering P-Delta effect for Storey 20 (iv) Variation of Storey displacement for Storey 30 for different bay cases: 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 Figure 9: The comparison of story horizontal displacement for different bay cases: 5X4, 6X5, 7X6, 8X7 and 9X8 considering P-Delta effect for Storey 30 6
- 7. 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) 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

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