Issue 74

E. Sharaf et alii, Fracture and Structural Integrity, 74 (2025) 262-293; DOI: 10.3221/IGF-ESIS.74.17

T f T KT 

(9)

where T is the transformation matrix that takes the form:

1 0 0 0 0 1 0 0

     

     



trans T

(10)

Then the shear stiffness matrix of the shown frame is obtained as follows:

1 K f  

(11)

The structure mass is condensed to one mass at the first-floor level, then the structure mass matrix takes the form:

m

1 0 0 0

 

  

 

M

(12)

where m 1 is the equivalent mass of the structure. Then the first natural frequency of this frame is obtained by:

2

1 eq K M   

0

(13)

Then the equivalent first-floor stiffness is obtained as follows:

2 m   eq

(14)

K

eq

1 1

which takes the following form:

2

1 2 1 IcIc L IbIc Lh Ib IcLh IbIc Lh IbIbhh         1 2 1 2 Ib Ib h h 2 16 4 4

1 1 2 EIc Ic Ic L Ib Ic Lh Ib Ic Lh Ib Ic Lh 2 1 2 2 2 1 6 4 4 4



K

(

)

(15)

eq

3

h

1

1 2

1 2 1

2 1 2

2 2 1

1 2 1 2

where K eq is the lateral equivalent stiffness, E is the concrete modulus of elasticity, Ic 1 and Ic 2 are the moments of inertia of the first and second floors columns, respectively, L is the frame span, Ib 1 and Ib 2 are the moments of inertia of the first and second floors girders, respectively, and h 1 and h 2 are the heights of the first and second floors, respectively. By substituting in equation (15) with Ic 1 =Ic 2 and Ib 1 =Ib 2 to obtain a more simplified form as follows:

2 2 h Ic L Ib hh IbIcLh IbIcLh       2 1 1 1 1 2 1 1 1 EIc Ic L Ib hh IbIc Lh IbIc Lh 1 1 2 3 2 2 2 1 1 1 1 1 2 (6 ( 16 8 4 ( ( 4 2 4 )) 1 1 1 1 2

))



K

(16)

eq

By substituting into equation 16 with I b1 = α *Ic 1 .

2 2 h h L L h L h h h h L L h L h              1 1 2 1 3 ( (4 2 2 1 1 2 1 2 (16 8 4 ) 2 4 )) EIb 2

6



K

(17)

eq

where α is defined as the beam-to-column moment of inertia ratio. This method is applied on a broader scale to include multi-story frames, as shown in Fig. 4. The equivalent lateral stiffness is derived using the same equation, based on the sections and heights of the first and second floors. This indicates that the

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