Energy Methods Tut 2012 SOLUTIONS

ME 2114 Solutions to Tutorial 4

Energy Methods

Q 1) Q  1)

      2

                        2 4        ⇒     ⇒    ,

Q 2) Q  2)

Resolve the forces to get the axial forces

                2     1     2         1

Q 3)

                         1        2 6 2 6           For Rod AB

,

For Beam BC

Q 4)

Neglecting the Strain Energy due to shears and axial forces, Strain energy of the bar is given by

By Castigliano’s Theorem

For AB :

For BC :

       

       2    2                 ;    1 ;  2

          11        



           cos        .    .    

Q 5)

Curved beam loaded perpendicular to its plane ; Semi‐circular beam OAB, Angle ACO=90’ ; Radius CO=R

     . .       . .  11                       0             .          1∅ 0  ;

Moment M at F :

Twisting Moment T at F :

NB: For any point D on portion AB With

, no contribution to

Q 6)

3

;

  1 

Resolve the forces to get the axial forces

                          21 2 2     1 5 2   2 2 6  102  3  8 6     21 4  23 18   16   23    0 ⇒    

     

Compatibility (or BC) Q 7)

     ,   ∈ 0,      2      1  1  ∆              3            ∆                                           ⇒     ∆      ,

Compatibility (or BC)

(a)=(b)

  /2                         0  Q 8) Let

,

,

,

4

   sin,        sin    1     sin1  0    sin   0.298   .   .   .       sin           Q 9)

         By Castigliano’s Theorem

            0             0             1      For AB :

;

For BC :

;

;

;

  1    1 

              1  0     2.69   2.07  5   

 6          1     1  0     2.07   2.57  2        0.422 ↑           &      0.145

Solving (1) & (2) simultaneously =>

5