A Study on the Effect of Second Order Geometric Nonlinearity for Asymmetrical Building

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A Study on the Effect of Second Order Geometric Nonlinearity for Asymmetrical Building

Shaik Mahummad Suhail

PG Student, Dept. of Civil Engineering Annamacharya Institute of Technology and Sciences Kadapa, India

Smt. G. Bhagyamma

Assistant Professor, Dept. of Civil Engineering Annamacharya Institute of Technology and Sciences Kadapa, India

Abstract In this new millennium land acquisition is more of a relentless task in every place. With the booming population across the world, the next best option is to opt for vertical expansion which is much more feasible and practical to accommodate more households. When these high rise structure are preferred, the safety and stability requirements of the structure are very much crucial so a special care and attention is given to each and every single aspect.

For convenience of computation, the preferred analysis is linear static analysis as the elastic behaviour is simple to predict structural behaviour, with certain properties. Since it cuts down the complex calculations, the principle of superposition is valid, different loads and/or combination of loads can be applied. Hence, we can easily find the responses for the given loads. But this is not always the case as the structural elements are flexible and due course our analysis becomes inconsiderate towards the important interaction between the vertical loads and horizontal displacements which nevertheless is an important stability parameter. While considering this interaction one has to consider nonlinearity and nonlinear analysis is complex and the accuracy is tough to achieve. In this study a structure is chosen such that, it is not symmetrical.

Conventionally there are few such structure which are symmetrical in the real world and majorly the structures are such that the orientation, plan are skewed because of this observation, for the current study a structure is chosen in such a way that it is asymmetrical and to relate to it with the prevailing situation. The structure consists of frame and shearwall system. The structure is firstly analysed and designed without the second order geometrical nonlinear effects (P-Delta effects). In the second analysis the same model is analysed and designed with the application of geometric nonlinear effects and the response of the structure has increased considerably in the story displacements, drifts, time period and the moments this is because of the relative decrease in the stiffness of the structure. With this increase in forces and displacements the design requirements have been enlarged so as to account for such second order geometric effects.

Keywords Geometric Nonlinearity, Shear Walls, Storey stiffness, lateral displacements, Asymmetric Structure, Etabs

  1. INTRODUCTION

    Generally, the analysis of buildings is done by using linear elastic methods, which is first order structural analysis. In the first order analysis, the evaluation of internal forces and displacement are dependent on material and section properties here the buckling and yielding are not considered. In the first order elastic analysis, the superposition of forces is possible and the forces are proportional to the applied loads. These form of linear elastic behaviour cant be assured in the real world and it cannot account for the second order geometric and/or material nonlinear effects are applied on deflections

    from the analysis. This kind of geometric nonlinearity can be analysed by performing through iterative processes which is merely practicable by using analysis software.

    a. Linear Analysis:

    Previously and conventionally we use linear static analysis, the linear static analysis of a structure involves the solution of the system of linear equations are represented by:

    K u = r (1)

    Where, K is the stiffness matrix,

    r is the vector of applied loads, and

    u is the vector of resulting displacements

  2. P-DELTA (SECOND ORDER GEOMETRIC EFFECT) The P-Delta effect is the second order effect on shears and

    moments of frame members and on the shear walls due to the action of the vertical loads, interacting with the lateral displacement of buildings, resulting from the seismic and wind forces. The structures behave flexible against applied seismic and/or wind lateral loads as the columns are subjected to compressive loads.

    There are two types of P-Delta effects,

    1. P- or P- 'big delta' effect.

    2. P- effect or P- 'small delta' effect.

      The P- effect or P- 'small delta' effect is concerned with load displacements of structural elements in between end nodes. The P- or P- 'big delta' effect is associated with the global load-displacement of the structure.

      In a first-order analysis only gravity loadings are considered. The structures are analyzed for each loading to obtain results and superimposed. However, this method does not provide accurate results. In low-rise and medium-rise buildings P- Delta analysis is not essential as the displacements are small. In taller and slender building structures having greater flexibility. The P-Delta effects are more significant.

      Fig. 1. P-Delta

      If so, then it requires larger sectional size of structural components to sustain the augmented moments and shears. In

      extreme cases such as very flexible, tall building structures subjected to large gravity loading, the P-Delta effects are severe enough to cause catastrophic collapse. In the design of multi-storey building structures, assessment of P-Delta effects is more important to predict whether P-Delta effects are significant, if so, P-Delta effects are accounted for analysis and design of structural elements. The P-Delta effect is critical in case of nonlinear analysis of multi-storey buildings. If the P-Delta effect causes sufficiently large lateral displacements in structures then P-Delta analysis is essential. The P-Delta effects are not considered in analysis and design of building components of high-rise building structures if the stability coefficient as obtained by the below equation is equal to or less than 0.10

      (2)

      Where,

      P – Total vertical design load (kN), – Design storey drift,

      V – Seismic shear force (kN), h – Height of storey (mm),

      Cd – deflection amplification factor.

  3. SHEAR WALLS

    Shear walls which form a part of the lateral load resisting system, are vertical members cantilevering vertically from the foundation, designed to resist lateral forces in its own plane, and are subjected to bending moment, shear and axial load. Unlike a beam, a wall is comparatively thin and deep, and is subjected to substantial axial forces. The wall should be designed as an axially loaded beam, capable of forming reversible plastic hinges (usually at the base, as shown in Fig.2 with sufficient rotation capability.

    The code recommends that the thickness of any part of the wall should preferably be not less than 150 mm. Walls that are thin are susceptible to instability (buckling) at regions of high compressive strain.

    A minimum of 4 nos 12 mm bars arranged in at least two layers should be provided near each end face of the wall. The concentrated vertical flexural reinforcement near the ends of the wall must be tied together by transverse ties, as in a column, to provide confinement of the concrete, and to ensure yielding without buckling of the compression bars when a plastic hinge is formed.

    Where the extreme fibre compressive stress in the wall exceeds 0.2 fck, boundary elements should be provided along the vertical boundaries of walls. These ar portions along the wall edges that are strengthened by longitudinal and transverse reinforcement, and may have the same or larger thickness as that of the wall web.

    Fig. 2. Lateral loads on shear walls

  4. BACKGROUND AND MOTIVATION

    There have been a lot of case studies regarding P-delta effects. Those structures have been investigated with different seismic zones, with and without shearwall and of different stories for p-delta effects but the models chosen in those studies were symmetrical and that is not always the case in real life, the structure are not of symmetry in plan and elevation. So as to examine the effects of second order geometric nonlinearity here the structure is chosen such that it is not of symmetrical geometry the L/D ratio is 2.9. The structure is longer in one direction than the other. Which also make the structure more slender. Proceeding with this perception in mind. The structure of 25 stories has been analysed and designed. The consideration of this kind of effects are vital in the analysis and design of the high rise structures

  5. SCOPE OF THE PROJECT

    The buildings are prone to deform laterally from original position with an eccentricity during an earthquake. When building structures are subjected to seismic, wind loading causing the structure to deform, the resulting eccentricity of the total gravity load due to inclined axes of structure causes the extra moments at the base. In taller and slender building structures having greater flexibility, the P-Delta effects are more significant. If so, then it requires larger sectional size of structural components to sustain the augmented moments and shears. In extreme cases such as very flexible, tall building structures subjected to large gravity loading, the P-Delta effects are severe enough to cause catastrophic collapse. In the design of multi-storey building structures, assessment of P-Delta effects is more important to predict whether P-Delta effects are significant, if so, P-Delta effects are accounted for analysis and design of structural elements. The P-Delta effect is critical in case of nonlinear analysis of multi-storey buildings. If the P-Delta effect causes sufficiently large lateral displacements in structures then P-Delta analysis is curial.

  6. MODEL DISCRIPTION

    Here a G+24 model structure is considered, as it can be seen from the architectural plan the position of the shear walls have been changed this is to control the rotation of the structure which has come down to 2% as this location of the

    shear walls have been adopted for both the model with and without Second order effects.

    TABLE I. Model Configuration

    Structure data

    Length of the Structure, L =

    33.3 m

    Breadth of the Structure, b =

    11.49 m

    No. of Storeys =

    G+24+Terrace+OHT

    Height of each storey =

    3m

    Height of the Structure, h =

    81 m

    Type of Structure =

    Frame and shearwall

    Primary beam =

    230x450mm

    Secondary beam =

    230x300mm

    shear wall =

    300mm

    slab thickness =

    150mm

    Natural frequency, fa =

    0.282 Cycles/s

    Diaphragm used

    Semi-rigid

    Grade of concrete used

    For beams

    M30

    For shear walls

    M40

    Grade of steel

    Fe 500

    Modules of elasticity

    5000fck

    TABLE II. Loading

    SEISMIC LOAD DATA:

    Acc. to IS 1893

    Location

    Mumbai

    Zone factor

    0.16

    Importance factor

    1.2

    Site type

    II

    R factor (SMRF)

    5

    Damping

    5%

    WIND DATA:

    Acc. to IS 875

    Basic wind speed, Vb =

    44 m/s

    Probability factor, K1 =

    1

    Category =

    3

    Topography Factor, K3 =

    1

    Importance factor, K4 =

    1

    GUST FACTOR =

    3.29

    GRAVITY LOADING: (kN/m2)

    Dead

    Live

    For general slab

    1.5

    2

    For sunken slab

    3

    2

    For staircase slab

    3

    3

    For lobby slab

    2

    3

    For terrace slab

    2

    3

    A. Architectural plan

    Fig. 4. Etabs model of typical plan

    Fig. 5. 3D model of G+24 model

  7. RESULTS AND DISCUSSIONS

    The structure is made into two models i.e., without the second order geometric effects and with the second order geometric effects and the results clearly reveal that the effects of second order geometric nonlinear effects are influence the response of the structure and results of all the different types of analysis such as dynamic earthquake and wind analysis, creep analysis, buckling, auto construction sequence analysis and P-Delta analysis for reinforced concrete structures are obtained and mentioned here.

    1. Modal analysis:

      Modal analysis is linear analysis at all times. It is be based on the stiffness of the entire unstressed structure, it also determines the system's undamped free vibration mode shapes and frequencies. These natural modes provide a fantastic insight into the structures behaviour.

      TABLE III. Model Comparison

      Without 2nd order effects

      With 2nd order effects

      Time Period

      3.481

      3.631

      Translation in X Dir

      70%

      70%

      Translation in Y Dir

      67%

      68%

      Rotation

      2.50%

      1.70%

      Mass Participation

      95%

      95%

      The structure satisfies the model criteria of the IS 1893 cl.7.7.5.2. that the first mode should be in translation, here it is in, here it is in Y direction, second mode is in X direction and the third mode is in translation. All the modes are within the limits. The mass participation ratio is also above 95%. While the behavior of the structure doesnt show any drastic changes when the second order geometric nonlinear effects are applied but we can observe that the time period of the structure has increased.

    2. Storey displacements:

      When the structure is subjected to the lateral forces like earthquake and winds in the form of lateral loads the displacement exerted by the structure is in the form of lateral displacements and this displacement are measured in the form of storey displacements which is different for each storey depending on the intensity of lateral load, terrain category, topography, seismic zone and other factors this lateral displacements vary. The comparison of the model for the lateral displacement are shown for each and very floor individually along X-direction and in Y-direction independently.

      Here, the maximum storey in the Y direction is 49mm and the limit of lateral storey displacement for the earthquake loading according to IS 16700:2017 clause 5.4.1 should be less than h/250 i.e. 78000/250 which is 315mm and it is under the limit.

      TABLE IV (a). Storey displacement

      Percentage increase in displacement after the application of second order effects

      Story

      For Earthquake loading

      X direction

      Y direction

      26

      3.543

      17.996

      25

      3.698

      23.032

      24

      3.721

      23.211

      23

      3.750

      23.394

      22

      3.782

      23.584

      21

      3.816

      23.773

      20

      3.847

      23.953

      19

      3.875

      24.119

      18

      3.896

      24.263

      17

      3.905

      24.377

      16

      3.906

      24.455

      15

      3.898

      24.489

      14

      3.880

      24.475

      13

      3.854

      24.403

      12

      3.827

      24.269

      11

      3.797

      24.062

      10

      3.761

      23.781

      9

      3.722

      23.413

      8

      3.670

      22.961

      7

      3.616

      22.417

      6

      3.557

      21.777

      5

      3.486

      21.048

      4

      3.388

      20.214

      3

      3.256

      19.286

      2

      3.104

      18.260

      1

      2.838

      17.087

      Average

      3.669

      22.619

      Fig. 6. Storey displacements for earthquake loading

      The maximum storey in the Y direction is 115.19mm and the limit of lateral storey displacement for the wind loading according to IS 16700:2017 clause 5.4.1 should be less than h/500 i.e. 78000/500 which is 156mm and it is under the limit.

      Percentage increase in displacement after the application of second order effects

      Story

      For Wind loading

      X direction

      Y direction

      26

      3.206

      14.991

      25

      4.573

      17.214

      24

      4.565

      17.290

      23

      4.568

      17.373

      22

      4.577

      17.466

      21

      4.586

      17.570

      20

      4.595

      17.682

      19

      4.603

      17.805

      18

      4.608

      17.931

      17

      4.605

      18.057

      16

      4.597

      18.182

      15

      4.583

      18.298

      14

      4.561

      18.401

      13

      4.534

      18.485

      12

      4.501

      18.545

      11

      4.465

      18.569

      10

      4.426

      18.554

      9

      4.385

      18.489

      8

      4.340

      18.367

      7

      4.282

      18.173

      6

      4.212

      17.899

      5

      4.133

      17.528

      4

      4.022

      17.049

      3

      3.885

      16.446

      2

      3.707

      15.690

      1

      3.450

      14.760

      Average

      4.33

      17.57

      Percentage increase in displacement after the application of second order effects

      Story

      For Wind loading

      X direction

      Y direction

      26

      3.206

      14.991

      25

      4.573

      17.214

      24

      4.565

      17.290

      23

      4.568

      17.373

      22

      4.577

      17.466

      21

      4.586

      17.570

      20

      4.595

      17.682

      19

      4.603

      17.805

      18

      4.608

      17.931

      17

      4.605

      18.057

      16

      4.597

      18.182

      15

      4.583

      18.298

      14

      4.561

      18.401

      13

      4.534

      18.485

      12

      4.501

      18.545

      11

      4.465

      18.569

      10

      4.426

      18.554

      9

      4.385

      18.489

      8

      4.340

      18.367

      7

      4.282

      18.173

      6

      4.212

      17.899

      5

      4.133

      17.528

      4

      4.022

      17.049

      3

      3.885

      16.446

      2

      3.707

      15.690

      1

      3.450

      14.760

      Average

      4.33

      17.57

      TABLE IV (b). Storey displacement

    3. Storey drifts:

      The storey drift is the lateral displacements of the structure relative to the storey/floor above and below the storey considered. The comparison the model for the lateral drifts are shown for each floor along X and Y-direction individually, now the percentage increase in the drifts is shown in graphs.

      TABLE V (a). Storey drift

      Percentage increase in drift after the application of second order geometric effects

      For Earthquake loading

      Story

      X direction

      Y direction

      26

      2.259

      8.004

      25

      2.524

      5.833

      24

      2.547

      5.765

      23

      2.545

      5.807

      22

      2.685

      5.765

      21

      2.829

      5.794

      20

      2.989

      6.414

      19

      3.173

      9.956

      18

      3.397

      13.531

      17

      3.579

      17.168

      16

      3.759

      19.259

      15

      3.837

      20.460

      14

      3.971

      21.594

      13

      3.974

      22.613

      12

      4.098

      23.526

      11

      4.046

      24.300

      10

      4.079

      24.901

      9

      4.012

      25.254

      8

      3.918

      25.337

      7

      3.994

      25.098

      6

      3.865

      24.602

      5

      3.909

      23.721

      4

      3.679

      22.520

      3

      3.596

      21.182

      2

      3.442

      19.531

      1

      4.082

      17.653

      G

      0.800

      15.877

      Average

      3.392

      17.091

      Percentage increase in drift after the application of second order geometric effects

      For Earthquake loading

      Story

      X direction

      Y direction

      26

      2.259

      8.004

      25

      2.524

      5.833

      24

      2.547

      5.765

      23

      2.545

      5.807

      22

      2.685

      5.765

      21

      2.829

      5.794

      20

      2.989

      6.414

      19

      3.173

      9.956

      18

      3.397

      13.531

      17

      3.579

      17.168

      16

      3.759

      19.259

      15

      3.837

      20.460

      14

      3.971

      21.594

      13

      3.974

      22.613

      12

      4.098

      23.526

      11

      4.046

      24.300

      10

      4.079

      24.901

      9

      4.012

      25.254

      8

      3.918

      25.337

      7

      3.994

      25.098

      6

      3.865

      24.602

      5

      3.909

      23.721

      4

      3.679

      22.520

      3

      3.596

      21.182

      2

      3.442

      19.531

      1

      4.082

      17.653

      G

      0.800

      15.877

      Average

      3.392

      17.091

      Fig. 7. Storey displacements for earthquake loading

      Fig. 8. Storey drift for earthquake loading

      TABLE V (b). Storey drift

      Percentage increase in drift after the application of second order geometric effects

      For Earthquake loading

      Story

      X direction

      Y direction

      26

      26

      2.778

      25

      25

      4.639

      24

      24

      4.563

      23

      23

      4.430

      22

      22

      4.426

      21

      21

      4.403

      20

      20

      4.455

      19

      19

      4.525

      18

      18

      4.639

      17

      17

      4.754

      16

      16

      4.727

      15

      15

      4.849

      14

      14

      4.853

      13

      13

      4.814

      12

      12

      4.875

      11

      11

      4.847

      10

      10

      4.741

      9

      9

      4.708

      8

      8

      4.714

      7

      7

      4.699

      6

      6

      4.589

      5

      5

      4.567

      4

      4

      4.423

      3

      3

      4.326

      2

      2

      4.050

      1

      1

      4.550

      G

      G

      1.316

      Average

      3.392

      17.091

      Fig. 9. Storey drift for wind loading

      The lateral drifts for the dynamic load cases as per IS 16700:2017 cl 5.4.1 are restricted to the limit of 1/250 i.e.

      0.004, here for the dynamic seismic load case and for the wind load case the maximum story drift ratio is under this limit, without and even with the second order geometric effects it is about 0.00095 and 0.00098 respectively. The percentage increase from the above graphs

    4. Storey stiffness:

      The soft storey concept is related to a discontinuity in the stiffness of building. According to IS 1893: 2002 a soft storey is one in which the sum of the lateral translational stiffness is lower than 70% of that in the storey above or less than 80% of the average lateral translational stiffness of three stories above.

      The stiffness for every storey has been calculated for the dynamic seismic loading in both the X-direction and Y- direction and it has been compared.

      TABLE V (b). Storey drift

      Percentage decrease in stiffness after the application of second order geometric effects

      For Earthquake loading

      Story

      X direction

      Y direction

      26

      2.904

      17.408

      25

      3.294

      20.747

      24

      3.395

      21.255

      23

      3.559

      21.701

      22

      3.829

      22.375

      21

      4.156

      23.086

      20

      4.504

      23.602

      19

      4.840

      24.132

      18

      5.145

      24.574

      17

      5.396

      24.975

      16

      5.730

      25.415

      15

      5.720

      25.898

      14

      5.814

      26.418

      13

      5.868

      26.924

      12

      5.914

      27.461

      11

      6.174

      27.914

      10

      6.206

      28.323

      9

      5.997

      28.691

      8

      5.954

      28.960

      7

      5.917

      29.099

      6

      5.799

      29.076

      5

      5.667

      28.797

      4

      5.505

      28.180

      3

      5.416

      27.199

      2

      4.893

      25.803

      1

      7.139

      24.525

      G

      0.940

      21.908

      The stiffness is more in X direction than that of Y direction of about 55.75% when the second order geometric effects are not incorporated and about 65% when the second order geometric effects are incorporated. The decrease in the storey stiffness has been observed in both the X and Y direction the stiffness in y direction is more given the geometry of the building.

      Fig. 10. Storey stiffness of each floor

    5. Overturning Moments:

      When lateral loads act on a structure, for adequate stability of the structure as a whole should be safeguarded at the foundation level, these are accounted as overturning moments which try to rotate the structure. These overturning moments are individually shown both in X and Y direction with and without the consideration of second order geometric effects.

      TABLE VI. Overturning moments

      Without 2nd order effects

      With 2nd order effects

      Percentage Increase(%)

      Load Case

      MX

      MY

      MX

      MY

      X

      Y

      EQ X

      9164.8913

      108545.3

      9361.706

      110375.3

      2.1

      1.6

      EQ Y

      58207.732

      11332.76

      62245.8

      11786.49

      6.9

      4.0

      The overturning moments for the dynamic seismic loading can be observed that they are increasing when the second order geometric effects are taken into consideration, the percentage increase in the other load cases like dead, live and wind are relatively less.

  8. CONCLUSIONS

From the post-processing results and discussions as mentioned above, for each parameter the variation for the two models is clearly visible such as

    • The model behaviour of the structure is same but the time period increases with the incorporation of second order effects, which in turn says that the stiffness has decreased.

    • In case of the lateral displacements their magnitude is clearly increasing and its prudent with the increased displacements the delta is going to increase which is going to increase the moments at the base hence it warrants the heavier design.

  • The lateral drifts are increasing and with the increase of lateral displacements, it has a significant effect on the aesthetic on non-structural members like partition/curtain walls that needs to be checked and also on the architectural members like glazing/brick walls and this result of increased drifts may reduce the serviceability of the building with higher lateral force. In order to maintain the aesthetic of non-structural elements. We need to mitigate this second order geometric effects by accounting for the P-delta and incorporating it in the design.

  • There is the redistribution of stiffness is observed clearly from the virtual work diagram when the second order geometric nonlinear effects are taken into consideration then the shearwall need to work more than the nominal and depending on the available results we can provide the steel accordingly and can be made effective.

  • There were no changes in the deflection, base shear of the structure, when the second order geometric nonlinear effects were incorporated.

  • The overturning moments are coming out to be slightly increased when the second order geometric nonlinear effects are taken into consideration this is because of the increased displacements.

  • From the results and response of the building, the shear walls have been designed and the percentage increase in the required area of steel is more when second order geometric nonlinear effects will be taken into account. It is to be provided so as it can resists the heightened responses which are clearly visible in the results of the building.

  • The second order nonlinear effects are higher for the high rise structures. Hence it is crucial that these kind of nonlinear effects should be taken into the analysis and design of the structures so that these effects and responses to be accounted for in turn making the design effective and efficient.

REFERENCES

  1. IS 1893 (part 1) (2002) Indian Standard Criteria for Practice for Earthquake Resistant Design of Structures General Provisions and Buildings (Fifth Revision).

  2. IS 875:1987(part1,2,3),dead, live & wind loads, Code of practice for design loads for building structure

  3. IS 456:2000, Indian standard Plain and reinforced concrete Code of Practice, Bureau of Indian standard, 2000, New Delhi.

  4. A. N. Pattar P-delta effect on multi-storey buildings i- managers Journal on Structural Engineering, Vol. 6 l No. 3 l September – November 2017.

  5. Rajat Sharma P-Delta Effects on Tall RC Buildings with and Without Shear Wall International Research Journal of Engineering and Technology (IRJET) Vol: 04 Issue: 11 | Nov – 2017.

  6. T. Avinash Investigation of the effects of p-delta on tubular tall buildings International Journal of Civil Engineering and Technology (IJCIET) Vol: 8, Issue 2, February 2017.

  7. Payal N.Shah Nonlinear dynamic analysis on stepped RCC building considering P-delta effect Journal of Engineering Research and Application.

  8. Bhavani Shankar Study on effects of p-delta analysis on RC structures International Research Journal of Engineering and Technology (IRJET) Vol: 04 Issue: 08, Aug -2017.

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