1. STABILITY ANALYSIS OF
RIGID FRAMES
Omkar M. Salunkhe.
M.Tech-Structures-I
(132040010)
VJTI,Mumbai

2. 1. The buckling of rigid jointed frame implies the
buckling of its compression members.
2. Rigid frame is separated in to isolated frame
members.
3. In analysis of rigid frame , the results for the
beam-column derived can be directly applied to
determine the critical load for isolated frame
member.

3. Rigid frame subjected to end moment and axial thrust

4. Isolated beam-column
subjected to axial thrust
& moment at one end
only.

5. The rotation
0
0
at the joint B is given by
=e/L( Lcosec L-1)+M0/PL( Lcot L-1)----{1}
For beam element BC
0=M0 L1/2EI1
M0=2EI1 0/L1 ----{2}

6. Eliminate M0from {1} and {2}
----(3)
0
At the buckling of the frame rotation
becomes very large i.e. It tends to infinity. This
occurs when the denominator vanishes i.e.
(1/PL)( 2EI1/L1)(1- Lcot L)+1=0

7. ( Lcot L-1)=PLL1/2EI1
=1/2(P/EI)[(I/I1)(L1/L)]L2
=1/2( L2)[(I/I1)(L1/L)]
Lcot L=1+1/2( L2)[(I/I1)(L1/L)]----{4}

8. For typical case where I=I1
L=L1
Then equation {4} becomes
cot L=1/( L)+( L)/2----{5}
By trial and modification , the lowest root of
transcendental equation is given by
L=3.59
Therefore,
Pcr=(3.59)2EI/L2
Pcr= 2EI/(0.875L)2----{6}

9. Closed rigid frame subjected to end moment and axial
thrust

10. Isolated beam-column
subjected to axial thrust
& moment at both end.

11. The rotation at the end of columns are given by
A =[MAL/3EI]
1(
)+[MBL/6EL]
2(
)
B =[MAL/6EI]
1(
)+[MBL/3EL]
2(
)---{1}
Where
1(
)=
2(
)=
2 = L=
, Pe = 2EI/L2

12. In this case due to symmetry
MA=-MB= M0
B=-
A= 0
Above equation becomes,
0=
M0 L/6EI [2
= M0 L/6EI
1(
)+
2(
)]
----{2}

13. For compatability , the rotation 0 of the column
must be same as that of horizantal member is
given by
0=-M0
L1/2EI1----{3}
Equate {2} and {3}, we get
=
----{4}

14. For typical case I=I1
L=L1
then equation {4} becomes
=
or tan
=-
The lowest root of this transcendental
equation is given by
=( L)/2=2.02916
Therefore,
Pcr=(4.0583)2EI/L2
Pcr= 2EI/(0.774L)2----{5}

15. THANK YOU.....