Deflection of Curved Members

Department of Civil Engineering, ISM Dhanbad Structural Analysis-I Practical Manual Deflection of Curved Members Aim of the experiment : To determine the elastic displacement of the curved members experimentally

and verification of the same by theoretical results. Apparatus required : Various curved members, viz. quadrant of a circle, semicircle with straight arm,

quadrant of a circle with straight arm and circle, dial gauge, weights, scale and etc. Castiglia lianos nos first theorem theorem is used used to find find the elasti elasticc displa displacem cement entss of curved curved member members. s. Theory : Castig Theorem states that the partial derivative of the total strain energy of a linearly elastic structure expressed in terms of displacements with respect to any displacement ! j at coordinate " is equal to the force P force P j at coordinate ". The theorem may be expressed symbolically as ∂U ∂∆ j

= Pj

#n all cases the horizontal ∆$,horizontal and vertical deflection ∆$,vertical due to vertical load % are to be determined. These deflections are obtained by using Castiglianos first theorem where strain energy due to bending bending only is ta&en into account. The results results obtained for the four curved members members is given as below ' a( )uadr )uadrant ant of a cir circl clee *ixed at + and free at $ having radius -( and sub"ected sub"ected to a concentrate concentrated d load % at free end.

$ W -

+

*igure .a

Prepared by Puja rajhans

Department of Civil Engineering, ISM Dhanbad Structural Analysis-I Practical Manual

WR

π

1

0 EI ∆$, vertical /

Vertical displacement of load / WR

1

2 EI

Horizontal displacement of load =

∆$, horizontal /

b( 3emicircle with straight arm *rom + to $ is a semi circle of radius -, $ to C is a straight length of y W $

C y

-

+

Vertical displacement of loaded point C *igure .b W

7 EI

6 2 y 1 + 1 R , 2π y 2 + 5 y R + π R 2 (4

C, vertical /

∆

WR

2

EI

,π y + 2 R (

displacements of loaded point C / ∆ C, horizontal / 8orizontal c( )uadrant with a straight leg *rom + to $ is a quadrant of a circle of radius - and from $ to C, straight length of y. Prepared by Puja rajhans


+ W -

$ y C *igure .c WR 1

π

0 EI

+

WR 2 y EI

Vertical displacement of load point + / ∆+, vertical / WR

2 EI Horizontal displacement of load point + /

, R + y ( 2

∆+, horizontal /

d( Circle of radius -

W $

-

+ *igure .d



=

WR

1

0π EI

,π 2 − 5(

∆$, vertical

Vertical displacement of loaded point $ / Procedure:

. 9lace a load on the hanger to activate the member and treat this as the initial position for measuring deflections. 2. *ix the dial gauges for measuring horizontal and vertical deflections. 1. 9lace the additional loads at the steps mentioned in the table below for each case and tabulate the values of dial gauge reading against the applied loads. 0. 9lot the graph load Vs deflection for each case to show that the structure remains within the elastic limit. . ;easure the value of - and straight length in each case. *ind width and depth of steel section and calculate the value of # as bd 1<2.

Observations and table :

%idth of section mm( b

/

=epth of section mm( d

/ bd

1

.2

>east moment of inertia

/

? @
/ 2 x A7.

a(

)uadrant of a circle

3l.

+dditional

@o.

load &g(

=ial gauge reading mm(

=eflection mm(

8orizontal

Vertical

8orizonta

Vertical

direction

direction

l direction

direction



b(

)uadrant with 3traight leg

3l.

+dditional

@o.

load &g(

c(

d(


=eflection mm(

8orizontal

Vertical

8orizonta

Vertical

direction

direction

l direction

direction

3emi'circle with straight leg

3l.

+dditional

@o.

load &g(


=eflection mm(

8orizontal

Vertical

8orizonta

Vertical

direction

direction

l direction

direction

Circle =ial gauge reading mm(

=eflection mm( Prepared by Puja rajhans

Department of Civil Engineering, ISM Dhanbad Structural Analysis-I Practical Manual Results :


Deflection of Curved Members

Recommend Documents