Appendix a 5 Storey

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APPENDIX A

APPENDIX PUSHOVER ANALYSIS OF 5-STORY MODEL FRAME



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A-1 Introduction

This appendix presents the procedure for obtaining the pushover curve of the 5-story

frame model used in this study based on the approximate method of Ramirez et al., 2001.

In addition, results from pushover analysis of the frame using the program SAP2000

(Version 7.5) are graphically presented for comparison.

A-2 Approximate Construction of Pushover Curve

Consider the 5-story frame (half structure) subjected to an arbitrary distribution of lateral

load as shown in Figure A-1. The work done by the forces applied to the structure must

be equal to the energy dissipated at plastic hinge locations.

Figure A-1: Five-story frame and assumed collapse mechanism.



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In stating the latter condition, the base shear strength of the frame yV , defined as the base

shear at the collapse stage, can be derived as

( )( )

( )1

1

11 2

N

y pc pbi i N ii i

i

V n M n M

h

χ

λ =

=

⎡ ⎤= ⋅ + + ⋅ ⋅

⎢ ⎥⎣ ⎦⋅ ∑∑ (A-1)

where pc M = plastic moment at base of first-story column; pbi M = plastic moment of

beam at level i ; n = number of spans; N = number of stories; ih = height of level i

above the hinge at the base of the structure; iλ = a force distribution factor that depends

on the lateral force pattern (first mode, modal, uniform, etc.) utilized to push over the

structure. Parameteri

χ is given by

1

1 2i

i

χ α

=−

(A-2)

where i pb ba e L= = length factor for beam hinge location at level i; pbe being the

distance of hinge from the beam end, and b L the beam length.

Assuming a pattern of lateral load proportional to the first mode of the frame under

elastic conditions, iλ is calculated as

1

1

1

i ii N

m m

m

w

w

φ λ

φ =

=

∑ (A-3)

The yield displacement can then be obtained from the relation between the base shear

strength, yV , and the fundamental period of the building under elastic conditions, 1T , as

2

1 1214

y

y

V g D T

W π

⎛ ⎞⎛ ⎞= ⋅ Γ ⋅ ⋅⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠ (A-4)

where



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11

1

1

N

m m

m

W

w φ =

Γ =

∑ (A-5)

is the first modal participation factor, and

2

1

1

12

1

1

N

i i

i

N

i i

i

w

W

w

φ

φ

=

=

⎛ ⎞⎜ ⎟⎝ ⎠=∑

∑ (A-6)

the first modal weight.

Calculations

• The following data pertain to the 5-story frame at hand.

- Spans: 1n = ; 1.321 mb L =

- Stories: 5 N = ; 1 1.092 m H = ; 2 3 4 4 1.194 m H H H H = = = =

- Column properties: 0.121 mcd = ; 19.456 kN-m pc M =

- Beam properties: 0.098 mb

d = ; 7.898 kN-m pb

M =

- Location of plastic hinges with respect to member end:

At beams: 0.159 m2

c pb b

d e d = + = ; At columns: 0.121 m pc ce d = =

- Floor weights: 1 2 6.850 kNW W = = ; 3 4 6.939 kNW W = = ; 5 6.806 kNW =

- From eigenvalue analysis:

1 0.419 sT = ; { } [ ]1 1.000 0.850 0.640 0.380 0.120 T φ =

• Evaluation of term ( )1

N

pbi i

i

M χ =

⋅∑

Parameter ia and i χ are calculated according to ( )2i pb b c b ba e L d d L= = +



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and Equation A-2, respectively.

Floor i pbi M

(kN-m)iα i χ pbi i M χ

(kN-m)

5 7.898 0.120 1.316 10.397

4 7.898 0.120 1.316 10.397

3 7.898 0.120 1.316 10.397

2 7.898 0.120 1.316 10.397

1 7.898 0.120 1.316 10.397

pbi i M χ =∑ 51.984

• Evaluation of term ( ) ( )1

1 2 N

pc pbi i

i

n M n M χ =

⎡ ⎤+ + ⋅ ⋅⎢ ⎥

⎣ ⎦∑

( ) ( )1

1 2 142.881 kN N

pc pbi i

i

n M n M χ =

⎡ ⎤+ + ⋅ ⋅ =⎢ ⎥

⎣ ⎦∑

• Evaluation of term ( )1

N

i i

i

hλ =

⋅∑

Assuming a first-mode lateral load pattern, iλ is described by Equation A-3.

Floor i iW

(kN)1iφ ih 1i iW φ

(kN) iλ i ihλ

(m)

5 6.806 1.000 5.747 6.806 0.331 1.901

4 6.939 0.850 4.553 5.898 0.287 1.306

3 6.939 0.640 3.359 4.441 0.216 0.725

2 6.850 0.380 2.165 2.603 0.127 0.274

1 6.850 0.120 0.972 0.822 0.040 0.039

1i iW φ =∑ 20.570 i ihλ ⋅ =∑ 4.245

Substituting in Equation A-1 yields the base shear strength:



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( )1

142.881 33.66 kN4.245

yV = ⋅ =

The yield displacement is calculated by Equation A-4 upon noting that

1

26.867W =

and 1 1.306Γ =

:

( ) ( )2

2

9.81 33.661.306 0.419 0.071 m

4 26.867 y D

π

⎛ ⎞ ⎛ ⎞= ⋅ ⋅ ⋅ =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

A-3 Pushover Analysis in SAP2000

Pushover analysis of the frame was performed by using program SAP2000 (Version 7.5).

The analysis included gravity load and P-∆ effects. The hinges at beams and columns are

modeled to have elastoplastic behavior without considering the effect of axial load. The

assumed lateral load distribution is proportional to the first mode. The results of the

pushover analysis are graphically compared to the predictions of the approximate

analysis in Figure A-2.

Figure A-2: Comparison of idealized elastoplastic pushover curve based on plasticanalysis and pushover curve calculated using SAP2000 for five-story frame.

Appendix a 5 Storey

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