ch_3 analysis procedures

7/29/2019 Ch_3 Analysis Procedures

1/22

21

Chapter 3

ANALYSIS PROCEDURES

3.1 General

The job of a structural engineer is to ensure that the buildings modelled are built to

withstand the gravity forces, wind forces earthquake forces etc. The modelled building

frames shall be analysed for the response. The response may be member internal forces

and displacements. This necessitates the use of appropriate structural analysis procedures.Two analysis procedures are briefly described in this chapter: Firstly, code-specified [58]

equivalent lateral load procedure (static analysis) and secondly, the time history analysis

(dynamic analysis), a performance based analysis technique, The first is a procedure that

is analysed to mimic real loads caused by earthquakes, while the later is meant to test the

building performance against an actual earthquake excitation. Six, nine, twelve and

fifteenstorey steel building plane frames were analysed applying both analysis

procedures. The goal is to determine which method of analysis will produce the best

results with the minimal analysis specifications.

Typically, seismic loads are resisted by axial member forces in the

bracings, gravity forces (dead loads and imposed loads) and partly seismic forces are

resisted by beam shear forces and bending moments and column moments and

compressive forces. To determine member forces and displacements SAP2000 Version 11

is used. In this analysis beam-column connections are assumed to be fully restrained and


2/22

22

geometrical nonlinearity in the frames is ignored by considering bi-linear material

nonlinearity as shown in Fig. 3.1.

Fig.3.1: The simplest hysteretic model with no stiffness shown after yield occursSource:http://www.ruaumoko.co.nz/Gif/Loops/Loop02.GIF.(Force-displacement curve, Fy = yield force,k0=initial

stiffness,r=stiffness reduction factor)

The design of BRBF is not governed by any building code but recommended

provisions are available. Structural Engineers Association of California (SEAOC) group

in association with various research agencies developed the recommendations.

Researchers and manufacturers have developed several types of BRBs and they are

commercially available in the United States.

The design dead and live loads of 9.87kN/m and 13.77kN/m were used

for analysis and design of above referred frames respectively. The damping ratio for

dynamic analysis was assumed to be 5% of the critical damping. The importance factor of

1.0 and zone factor 0.36 was used to obtain design base shear. The beams and columns

were designed as per IS: 800-2007 and seismic provisions [5,58] . The BRBs were designed

as per FEMA-450, in which the response modification factor 5 for MRFs and 8 for

BRBFs respectively. The yield stress of the structural steel was taken as 250MPa.


3/22

23

The building model frames considered are shown in Fig. 3.2.1 -3.2.8 and the details are

shown in Table 3.1.

Fig. 3.2.1: Six storey steel moment resisting frame model with nodal numbers and element numbers

(Generated from SAP 2000 V.11.analysis model)


4/22

24

Fig. 3.2.2: Six storey steel buckling restrained braced frame model with nodal numbers and element numbers



5/22

25

Fig. 3.2.3: Nine storey steel moment resisting frame model with nodal numbers and element numbers



6/22

26

Fig. 3.2.4: Nine storey steel buckling restrained braced frame model with nodal numbersand element numbers (Generated from SAP 2000 V.11.analysis model)


7/22

27

Fig. 3.2.5: Twelve storey steel moment resisting frame model with nodal

numbers and element numbers (Generated from SAP 2000 V.11.analysis model)


8/22

28

Fig. 3.2.6: Twelve storey steel buckling restrained braced frame model with nodal

numbers and element numbers (Generated from SAP 2000 V.11.analysis model)


9/22

29

Fig.3.2.7. Fifteen storey steel moment resisting frame model with nodal

numbers and element numbers (Generated from SAP 2000 V.11.analysis

model)


10/22

30

Fig. 3.2.8: Twelve storey steel buckling restrained braced frame model with

nodal numbers and element numbers (Generated from SAP 2000 V.11.analysis

model)


11/22

31

Table 3.1: Steel Model Building Frame Details.

ModelBuildingFrame

No. ofStoreys

StoreyHeight(m)

No. of BaysBay

Width(m)Total FrameHeight(m)

MRF6 6 3.000 4 6.000 18.000MRF9 9 3.000 4 6.000 27.000MRF12 12 3.000 4 6.000 36.000MRF15 15 3.000 4 6.000 45.000BRBF6 6 3.000 4 6.000 18.000BRBF9 9 3.000 4 6.000 27.000

BRBF12 12 3.000 4 6.000 36.000BRBF15 15 3.000 4 6.000 45.000

Note: MRF6=six-storey moment resisting frame; 4th digit indicates number of storeys.BRBF6=six-storey buckling restrainedbraced frame; 5th digit indicates number of storeys.

3.2 Equivalent Lateral Force Procedure

The Equivalent Lateral Force (ELF) Procedure involves applying static forces on

the structure and analysing how it reacts to these forces. Also, the forces are usually

applied at the joints of the members, which make the crosssectional members act like

twoforce members. The most important force in this procedure is the base shear, or the

sum of all the lateral forces affecting the structure. The strength or capacity of the

members must be able to withstand the base shear. To find out if the appropriate members

are selected or not, the story drift check has to be performed. Other factors, such as the

seismic response coefficient, response modification factor and importannc factor must be

taken into account when following this procedure. The Indian seismic code carefully

enumerates the equations and conditions that must be satisfied when following the ELF

procedure.


12/22

32

3.2.1 Seismic Response Coefficient

The formula for seismic response coefficient (Ah) given in the code is used to

determine the base shear V. This formula forms the basis of the design spectrum of the

ELF procedure. According to IS: 1893(Part-1)-2002, the base shear can be obtained by

multiplying the seismic response coefficient by the structures seismic weight. This

includes the dead load and other loads, such as live and snow loads.

3.2.2 Response Modification Factor

When a structure is proposed to analyse and design, it is expected that thebuilding has to sustain permanent damage. Even the best designed buildings are

susceptible to inelastic deformation. Keeping this point in mind, the goal of the engineer

is to design a building that will not collapse for higher earthquakes. The response

modification factor(R) accounts the ability of the structure to absorb energy without

collapse. The more ductile the structure is, the higher its modification factor. Response

reduction factor depends on the perceived seismic damage performance of the structure.

It is characterised by ductile or brittle deformations. However, the ratio (I/R) shall not be

greater than 1.0.

3.2.3 Importance Factor

It is factor assigned to each structure according to its Occupancy Category.

Importance factor (I) depending upon the functional use of the structures, characterised

by hazardous consequences of its failure, post-earthquake functional needs, historical

value, or economic importance.


13/22

33

3.2.4 Analysis Procedure

ELF procedure comprises steps 1 to 7 as described below:

1 Evaluate the approximate period Ta of the fundamental vibration mode using an

expression for steel frame building

Ta = 0.085 h0.75 (3.1)

where

h = Height of building, in m. This excludes the basement storeys, where basement

walls are connected with the ground floor deck or fitted between the buildingcolumns. But, it includes the basement storeys, when they are not so connected.

2 The design horizontal seismic coefficient (Ah) for a structure shall be determined

by the following expression:

Ah=(Z/2)( I/R)(Sa/g) (3.2)

Z=Zone factor for the Maximum Considered Earthquake (MCE) and service life

of structure in a zone. The factor 2 in the denominator of Z is used so as to reduce

the Maximum Considered Earthquake (MCE) zone factor to the factor for Design

Basis Earthquake (DBE).

Sa/g= Average response acceleration coefficient.

3 The seismic resultant design base shear( Vb ) shall be computed by the following

expression;

Vb=Ah W (3.3)

W=Seismic weight of building.


14/22

34

4 Vb shall be distributed over the height of the structure into a number of storey

forces.

5 Internal forces and displacements of the structure under the force Vb, by using a

static analysis shall be established

6 These seismic action effects shall be combined to other action effects (gravity

loading in the seismic situation)

7 Carried out all seismic checks required for the structural elements.

3.2.5 Analysis Implementation

The ELF Procedure was the last step in the analysis process before proportioning

of frame members. This includes calculating the story forces for each individual level,

assigning it in SAP2000 and running the simulation to get our deflection by elasticity test

result. Once this result has been obtained, we are able to test if the model building

frames have to satisfy the requirement for max allowable story drift. An equivalent static

analysis is conducted to determine the member forces in the frame members and the

horizontal drifts in the storeys. It is assumed that all connections are rigid (moment)

connections in the frame.

The forces in the members of MRFs are computed by using ELF procedure and

presented in Appendix-A and the member forces of BRBFs are computed by using ELF

procedure and presented in Appendix-B.


15/22

35

3.3 Non-Linear Time History Analysis

The Nonlinear Time History (NTH) analysis differs from the ELF procedure intwo respects. First, time history analysis with elasto-plastic braces is used rather than

response spectra analysis. Second, an optimization procedure is used to determine brace

areas rather than approximating the fundamental natural period with a formula that is

independent of brace area. The nonlinear time history analysis is described in this

chapter.

3.3.1 Structural Model

The structural model consists of a horizontal displacement degree-of-freedom

(Ui), at each storey i where i = 1 is the top storey and i = n is the bottom storey as shown

in Fig; 3.3.. The equations of motion are given in Equation 3.4.

M+C+R=F (3.4)

where M is the mass matrix, C is the damping matrix, Ris the resistance vector,

F is the force vector, U is the displacement vector, is the velocity vector, and is the

acceleration vector.

The resistance vector is a nonlinear function of the displacement vector, however;

a linear approximation of the resistance vector in the vicinity of the current displacement

vectorU* is shown in Equations 3.5 and 3.6.

RR* +K*U (3.5)

U =U-U* (3.6)


16/22

36

3.3.2 Stress and Strain

It is assumed that the behavior of BRB is the same in tension and compression,

and that the behavior is elastic-perfectly-plastic (elasto-plastic) as shown in Fig. 3.4.The

current strain in the braces of story i is given in Equation 3.7.

Fig. 3.3: Structural model of buckling restrained braced frame


17/22

37

*i ={L /(L2+H2)} (ui*- ui+1

*) for i


18/22

38

ki* =0 if *i=y

ki*=EAiL2/(L2+H2)3/2 otherwise (3.10)

If there are 5 storeys in the frame, then the current stiffness matrix is given in

Equation 2.11.

(3.11)

3.3.4 Resistance Vector

The current horizontal resistance force provided by the braces in storey i is given

in Equation 3.12.

ri* =LAi

*/(L2+H2)1/2 (3.12)

If there are five storeys, the current resistance vector is given in Equation 3.13.

(3.13)

For a building subject to horizontal ground acceleration (g), the effective

horizontal force is the same at every storey. If there are five stories, the force vector is

given in Equation 3.14.


19/22

39

(3.14)

where 1 is a vector of one's.

3.3.5 Mass Matrix

It is assumed that the mass is constant with time and is the same for all storeys.

Let the mass at each story be m, then the mass matrix is given in Equation 3.15.

(3.15)

where I is the identity matrix

3.3.6 Damping Matrix

It is assumed that the damping matrix is constant with time and is mass-

proportional. Let the damping ratio be , and let the natural circular frequency of the

fundamental elastic mode be . If there are five stories, then the damping matrix is given

in Equation 3.16.

(3.16)


20/22

40

The damping ratio is specified by the user, and for the fundamental elastic

mode is determined by inverse vector iteration which begins with a starting vector of

one's, v0 = 1, and iterates as shown in Equation 3.17.

vk+1=K-1Mvk=Mk

-1vk (3.17)

where the vectorvk+1 becomes the mode shape vector for the fundamental mode and Kis

the elastic stiffness matrix (no plastic yielding).The natural circular frequency is

calculated at each iteration from the Rayleigh quotient as shown in Equation 3.18 and it

stops when the change in is negligible.

=(vkTvk/vk

TMvk)1/2 =(vk

TKvk/mvkTvk)

1/2 (3.18)

3.4Nonlinear Response History Analysis

Numerical integration procedure with direct use of Taylors series is used to

integrate Equation (3.4) which is a second-order nonlinear differential equation. Average

acceleration has been proven to be a stable integration technique equivalent to the

trapezoidal rule. The ground acceleration (g) changes with time. It is normally specified

as a set of discrete values at equally spaced time steps ranging from 1 to n steps. This

set of discrete values is called an accelerogram. Let gj be the ground acceleration at the

jth time step, and let t be the length of time between time steps.

Average acceleration assumes that the acceleration throughout each time step is

constant. Thus, the acceleration at all times between time step j and time step j+1 is given

in Equation 3.19.

= (j+ j+1) (3.19)


21/22

41

Acceleration is then integrated to get velocity given by Equation 3.20.

= (j+ j+1)+ j (3.20)

where the constant of integration is determined such that at t = 0, the velocity is equal to

the velocity at time step j. Now velocity (Equation 3.20) integrates to get displacement

given in Equation 3.21.

(3.21)

where the constant of integration is determined such that at t = 0, the displacement is

equal to the displacement at time step j. Evaluating the previous two Equations 3.20 and

3.21 at t = t is shown in Equations 3.22 and 3.23.

(3.22)

(3.23)

From the above two Equations

(3.24)

(3.25)

These are the basic equations used by the Newmarks method to update velocities

and accelerations from time step j to time step j+1. Writing Equations 3.4, 3.5, and 3.6 at

time step j+1 are shown in Equations 3.25, 3.26, and 3.27.

(3.26)

(3.27)

(3.28)


22/22

42

Substituting Equations 3.24, 3.25, 4.27, and 4.28 into Equation 4.26 gives Equation 3.29.

(3.29)

Rearranging Equation 4.25 gives Equation 3.29,

(3.30)

where:

(3.31)

(3.32)

Substituting Equations 3.14, 3.15, and 3.16 into Equations 3.31and 3.32gives Equations

3.33and 3.34.

(3.33)

(3.34)

The forces in the members of MRFs are computed by using NTH analysis and

presented in Appendix-C and the member forces of BRBFs are computed by using ELF

procedure and presented in Appendix-D.

ch_3 analysis procedures

Documents