Module 5 : Force Method - Introduction and applications
Lecture
3 : Analysis of Statically Indeterminate Structures by Energy Method
5.4
Analysis of Statically Indeterminate Structures by Energy Method
Let a statically indeterminate structure has degree of indeterminacy as n . On the selected basic determinate structure apply the unknown forces , ..... and . Using the Eq. (4.16) the displacement in the direction of is expressed by
( j = 1, 2, .. ……n) (5.1)
The equations (5.1) will provide the n linear simultaneous equations with n unknowns , ..... and . Since the is known, therefore, the solution of simultaneous equations will provide the desired
( j =1, 2,…., n ).
For structures with members subjected to the axial forces only (i.e. pin-jointed structures), the equation (5.1) is re-written as
(5.2)
where P is the force in the member due to applied loading and unknown ( j =1, 2,…., n ); and L and AE are length and axial rigidity of the member, respectively.
For structures with members subjected to the bending moments (i.e. beams and rigid-jointed frames), the equation (5.1) is re-written as
(5.3)
where M is the bending moment due to applied loading and unknown ( j =1, 2,…., n ) at a small element of length dx ; and EI is the flexural rigidity.
, 
..... and 
. Using the Eq. (4.16) the displacement 
in the direction of 
is expressed by 
( j = 1, 2, .. ……n) (5.1) 
, 
..... and 
. Since the 
is known, therefore, the solution of simultaneous equations will provide the desired
( j =1, 2,…., n ). 
(5.2)
( j =1, 2,…., n ); and L and AE are length and axial rigidity of the member, respectively. 
(5.3)
( j =1, 2,…., n ) at a small element of length dx ; and EI is the flexural rigidity. 
