Structural dynamics

ANSWER 1

STATIC ANALYSIS :

• LOAD IS NOT VARIABLE WITH TIME AND NEGLECTED DUE TO SLOW APPLYING OF LOAD 
• MOTION OF THE STRUCTURE CAN BE DISEGARDED 
• THERE IS NO NECESSITY TO COUNT INTERIA AND DAMPING FORCES IN CASE OF STATIC ANALYSIS , BUT THE ELASTIC FORCE SHOULD BE ACCOUNTED.
• THERE IS NO INTERIA EFFECT ON STATIC ANALYSIS.

DYNAMIC ANALSIS: 

• LOAD IS VARIABLE WITH TIME DUE TO FAST APPLYING OF LOAD AND VIBRATION .
• MOTION CAN NOT BE  DISREGARDED ( CAN'T BE CONSIDERED AS STATIC .) 
• INTERIA AND DAMPING FORCES SHOULD BE ACCOUNTED FOR THE CALCULATION OF DYNAMIC ANALYSIS .
• INTERIA EFFECT IS CONSIDERED IN DYANMIC ANALYSIS .

 ANSWER 2: EXPLAIN THE FOLLOWING WITH RELEVANT FORCE DISPLACMENT GRAPH:

a) ELASTIC BEHAVIOR.

b)INELASTIC BEHAVIOR .

c) PLASTIC BEHAVIOR .

d) NON-LINEAR INELASTIC BEHAVIOR .

 

BELOW IS THE FORCE -DISPLACEMENT GRAPH EXTRACTED  FROM DYNAMIC OF STRUCTURES THEORY AND APPLICATION TO EARTHQUAKE ENGINEERING A.K CHOPRA WITH THE FOLLOWING EXPALNATION OF THE LISTED BEHAVIOR : 

 

 

1) ELASTIC BEHAVIOR : ELASTIC BEHAVIOR IS THE PROPERTY OF THE MATERIAL UPTO WHICH THE MATERIAL CAN REATAIN IT'S SHAPE .FROM THE ABOVE MENTION THE GRAPH IT IS CLEARLY MENTIONED THAT  UPTO A CERTAIN POINT FROM "O" THE GRAPH IS LINEAR AND SHOWN ITS ELASTIC PROPERTIES .

2) INELASTIC BEHAVIOR : INELASTIC BEHAVIOR IS THE PROPERTY OF THE MATERIAL WHEN THE MATERIAL CAN'T RETAIN IT'S PREVIOUS SHAPE AFTER A CERTAIN DEFORMATION .IN THE ABOVE MENTIONED GRAPH INELASTIC BEHAVIOR HAS BEEN SHOWN IN THE UPPER SHAPE OF THE CURVE UPTO POINT "a" .

3) PLASTIC BEHAVIOR : PLASTIC BEAHVIOR OCCURS WHEN THE MATERIAL COMPLETELY LOST IT'S DUCTILITY. IN THE ABOVE MENTIONED GRAPH PLASTIC BEHAVIOR OCCUR DURING UNLOADING AFTER POINT "a".

4) NON-LINEAR ELASTIC BEHAVIOR : 

NON-LINEAR INELASTIC BEHAVIOR OCCURS AFTER THE LINE FROM "O" TENDS TO BENDS TOWARDS POINT "a" .

ANSWER 3:  EXPLAIN MASS, STIFFNESS  AND DAMPING  COMPONENT IN THE EQUATION OF MOTION .

STIFFNESS ,DAMPING ,AND MASS COMPONRNT :

THE SDOF SYSTEM CAN BE VIEWED AS A COMBINATION OF THREE PURE , INDEPENDENT SYSTEM : 

1) THE STIFFNESS COMPONENT : THE FRAME IS WITHOUT  DAMPING OR MASS . THE EXTERNAL FORCE "fs"  ON THE STIFFNESS COMPONENT IS RELATED TO DISPLACEMENT "u"  BY   "fs=Ku  , if the system linearly elastic.

 2) THE DAMPING COMPONENT : THE FRAME WITH ITS DAMPING PROPERTIES BUT NO STIFFNESS OR MASS . THE EXTERNAL FORCE ON THE DAMPING  COMPONENT IS RELATED TO VELOCITY BY "fd= CV ".

 3) THE MASS COMPONENT : THE ROOF MASS WITHOUT THE STIFFNESS OR DAMPING OF THE FRAME. THE EXTERNAL FORCE "fi"  ON THE MASS COMPONENT IS RELATED TO THE ACCELERATION BY fi =m ( m^double dot.) 

EQUATION OF MOTION : EXTERNAL FORCES :

IN FIG . SHOW IDEALIZED ONE STORY FRAME INTRODUCS EARLIER SUBJECTED TO EXTERNALY APPLIED FORCE p(t)   IN  THE DIRCETION OF DOF "u" . THIS NOTATION INDICATES THAT THE FORCE "p" varies WITH TIME "t".   THE RESULTING DISPLACMENT OF THE MASS ALSO VARIES WITH TIME (t) , IT IS DENOTED BY u(t) . 

WE DRIVE THE DIFFERENTIAL EQUATION GOVERNING THE DISPLACEMENT u(t)  BY TWO METHOD USING 

1. NEWTON'S SECOND LAW OF MOTION .

2. DYNAMIC EQUILIBRIUM . 

 USING THE NEWTON'S SECOND LAW OF MOTION : 

EXTERNAL FORCE= p(t) 

ELASTIC OR (INELASTIC) FORCE (fs) 

DAMPING RESTING FORCE f(D) 

DISPLACEMENT FORCE U(t) 

VELOCITY U(t) 

1. ACCELERATION U"(t) 

THE RESULTING FORCE ALONG THE X-AXIS 

P-fs -f(D) =mu"  OR 

mu"+ f(D) + fs = p(t) 

put this equation in fs =k(u) and f(D) = CU" 

NOW RESULT : mu"+cu"+fs =p(t)  OR

                         mu" +cu+ku= p(t)

ANSWER 4) PROVIDE RELATIONSHIP BETWEEN NATURAL PERIOD (Tn) & NATURAL FREQUENCY (f) , AND PROVIDE THEIR DEFINITIONS.

1. NATURAL PERIOD OF VIBRATION (Tn) : 

THE TIME REQUIRED FOR THE UNDAMPED SYSTEM TO COMPLETE ONE CYCLE OF FREE VUBRATION (UNIT IN SECOND ) .

IT IS RELATED TO THE NATURAL (Wn), IN UNITS OF RADIANS PER SECONDS .

Tn = 2`(pi/Wn).

2.NATURAL CYCLIC FREQUENCY (f) .

 

fn =(1/Tn) 

THEREFORE ; THE RELATION BETWEEN NATURAL PERIOD OF VIBRATION OF THE SYSTEM (Tn) AND NATURAL FREQUENCY (fn)  IS fn = (1/Tn ) .`

ANSWER 5. EXPLAIN THE DETAILS ABOUT RESPONSE SPECTRUM & ITS GRAPH .

RESPONSE SPECTRUM : 

• It’s the central concept in earthquake engineering, provides a convenient means to summarize the peak response of all possible linear SDF systems to a particular component of ground motion.
• It provides a practical approach to apply the knowledge of structural dynamics to the design structures and development of lateral force requirements in building codes.
• A plot of the peak value of a response quantity as a function of the natural vibration period Tn of the system, or a related parameter such as circular frequency ωn or cyclic frequency fn, is called the response spectrum for that quantity. 
• Each such plot is for Single Degree of Freedom systems having a fixed damping ratio Շ, and several such plots for different values of Շ are included to cover the range of damping values encountered in actual structures. 
• A variety of response spectra can be defines depending on the response quantity is plotted. Consider the following peak responses.

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• The deformation response spectrum is a plot of u_o against T_n for fixed Շ (damping ration). A similar plot for ů_o is the relative velocity response spectrum, and for ű^t_o is the acceleration response spectrum.
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