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Calculation of Stiffness in Structural elements

1) AIM: Compute lateral stiffness of the one-story frame with an intermediate realistic stiffness of the beam. The system has 3 DOFs as shown. Assume L = 2h and Elb = Elc Solution : if $U_{1}$ $U_1$ = 1 $U_{1}$ $U_1$ = 24 $\frac{E I_{c}}{h^{3}}$ $(EI_c) /h^3$ $'U_2'$ = 6 $(EI_C)/h^2$ $U_3$ = 6 $(EI_C)/h^2$ …

Sanjana Reddy Pandillapalli
updated on 16 Jul 2021

AIM:

Compute lateral stiffness of the one-story frame with an intermediate realistic stiffness of the beam. The system has 3 DOFs as shown. Assume L = 2h and El_b= El_c

Solution :

if $U_1$ = 1

$U_1$ = 24 $(EI_c) /h^3$

$'U_2'$ = 6 $(EI_C)/h^2$

$U_3$ = 6 $(EI_C)/h^2$

1 st equation :

If $U_2$ = 1

$U_1 = (6 EI_C)/h^2$

$U_2 =(4EI_b )/(2h) + (4EI_C)/ h$

$U_3 = (2EI_b)/(2h)$

2nd equation :

If $U_3 = 1$

$U_1 = (6 EI_C)/h^2$

$U_3 =(4EI_b )/(2h) + (4EI_C)/ h$

$U_2= (2EI_b)/(2h)$

This is solved using a 3 X 3 matrix

*According to stiffness matrix method :

$((U_2 ), (U_3 ))$ = - $[[6h^2 , h^2 ],[h^2,6h^2]]^-1$ $((6h),(6h))$ $U_1$

$-(1/(36h^4-h^4)) [[-6h^2,h^2],[h^2,-6h^2]] 6h ((1),(1))U_1$

$-(6h)/(35h^4) [[6h^2,-h^2],[-h^2,6h^2]]((1),(1))U_1$

$-(6h)/(35h^4)((5h^2),(5h^2))U_1$

= $-(6h)/(35h^4) xx 5h^2 ((1),(1))U_1$ $=((U_2),(U_3))$

$((U_2),(U_3))$ = $-6/(7h) ((1),(1))$ $U_1$ `

substituting the above equation in the equation given below

$f_s=(24U_1 +6hU_2 + 6hU_3) (EI_C)/h^3$

= $f_s = (24 (EI_C)/h^3 -6/(7h) (EI_C)/h^3 [6h ,6h] ((1),(1)))U_1$ `

= $((24EI_C)/h^3 -6/(7h) (EI_C)/(h^3) . 12h)U_1$

= $((EI_c)/h^3 (24-6/(7h)).12h) U_1$

= $((EI_C)/h^3 (24)(-6/(7h)).12h)U_1$

= $(EI_C)/h^3 (24-72/7)U_1$

= $F_S = (EI_C)/h^3 96/7 U_1$

AIM:

For the following structures:

Determine the number of degrees-of-freedom for dynamic analysis
Establish the equation of motion
Calculate their natural frequencies

*The degree of freedom is 1

*Equation of motion

= mu + fd+fs= p(t)

= mu + ku = P(t)

= $((12EI)/ h^3 ) U + Mu = P(t)$

* Natural frequency

= $F_n = 1/T_n$

= $T_n = (2 pi )/w_n$

= $F_n = w_n /(2pi)$ = $sqrt ( k/M)$ $. 1/(2pi)$

= $w_n = sqrt (k/m$

= $w_n =sqrt (((12EI)/ (h^3 m))$

*The degree of freedom is 1

*as the beam is rigid ( i.e flexural rigidity M.I = infinity

* K= $sum_(column) (12EI )/h^3$

= $(36EI)/h^3$

*Equation of motion

= mu + fd+fs= p(t)

= mu + ku = P(t)

=mu + $(36EI)/h^3$ u = p(t)

* Natural frequency

= $F_n = 1/T_n$

= $T_n = (2 pi )/w_n$

= $F_n = w_n /(2pi)$ = $sqrt ( k/M)$ $. 1/(2pi)$

= $w_n=sqrt(k/m$

= $w_n=sqrt((36EI)/(mh^3))$

* Natural frequency

= $F_n = 1/T_n$

= $T_n = (2 pi )/w_n$

= $F_n = w_n /(2pi)$ = $sqrt ( k/M)$ $. 1/(2pi)$

= $w_n = sqrt(k/m$

$therefore w_n =sqrt(g/u$

$T_n = 2pi sqrt(u/g$

$f_s/K_e = f_s/k_s + f_s/k_b$

$K_e = (K_s K_b )/(K_s+K_b$

$K_b = (3EL)/L^3$

$K_e = (K(3EI)/L^3 )/(K + (3EI)/L^3$

$K_e = (3EIK )/((3EI) +L^3K)$

*as the beam is rigid i.e EI = infinity

K= $sum_(column) (12EI )/h^3$

k ( column ) = $(24EI)/h^3$

$K_e = K + K ( column )$

$K_e = K+ (24 EI)/h^3$

$K_e = (Kh^3 + 24 EI )/h^3$

$Ke = K_a + K _b$

$K_e =(EA)/(sqrt (h^2+L^2)$ $+ (48EI)/L^3$

Read more Projects by Sanjana Reddy Pandillapalli (47)

Project 2

Objective:

AIM: A 100 m WBM road needs to be constructed inside the Skill Lync Education center premises to allow access to the center for students and faculties.Create the following Project (under same EPS/OBS as project-1) and Resource :Project Name - Education Centre RoadAssume - Calendar as 6 x 24 and start date on 22nd Feb 2021…

22 Jun 2022 07:15 AM IST

Project 1

Objective:

AIM: Skill-Lync wants to construct a school as a part of its corporate social objective. You are the project manager. Develop a realistic schedule to construct the main school building (G+2). It will have a parking facility with internal roads and the main building complete with architectural, furniture, and electrical…

29 Jan 2022 07:00 PM IST

Project 2

Objective:

AIM : To create a dynamo script of multi-storeyed complex structure Procedure : *Open a new file *family generic model *In reference plan select model line *Open dynamo, *select elements models >(Select from the Revit ) *curve element *Curve by sweep >connect the element curve and arc Result : *We have successfully…

29 Dec 2021 07:45 PM IST

Project 1

Objective:

Aim: To Model an apartment with 4 units on each floor (2- 3 BHK) of 7 stories with a stilt parking facility. Design and model the structural elements as required. 1. Print BOQ and estimates from the model for different trades2. Print layouts for floor plans trade wise and one including all trades3. Print 3D views from…

21 Dec 2021 06:15 PM IST

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