Stiffness Matrix

Structural Analysis 2

1. Subject:- S tructural A nalysis-II Subject code:- 2150608 Guided by:- Prof. Vikunj Tilva Prof. Pritesh Rathod Name Enrollment No. Deshmukh Bhavik Hasmukhlal 151103106002 Gain Yogesh Budhabhai 151103106004 Kotila Jayveer Vanrajbhai 151103106008 Mistry Aditya Pramodbhai 151103106009 Patel Nirmal Dalpatbhai 151103106012

2. Topic: Analysis of Beam Stiffness Method Outline Introduction Procedure Properties Types of supports

3. Introduction The systematic development of slope deflection method in this matrix is called as a stiffness method. This method is a powerful tool for analysing indeterminate structures. Stiffness method of analysis of structure also called as displacement method. In the method of displacement are used as the basic unknowns.

4. Procedure Determine degree of kinematic indeterminacy of structure. Select unknown displacement Restrain all the joints to set fully restrained structure under given condition. For analysis of the restrain structure to get ADL. Generate stiffness matrix of given structure apply unit positive displacement for members and add all the displacement for members meeting at a joint.

5. 6 . Superposition equation . {AD} = {ADL} + [S] {D} {AD} = Joint action or forces given in structure {ADL}= Force analysis of restrained structure under given loading [S] = Stiffness method {D} = Unknown 7. Determine the final moments. 8. Calculation for SF and draw SF, BM diagram.

6. Properties of stiffness matrix Stiffness matrix is a square matrix of order n*n, where n is equal to KI. Stiffness matrix is symmetrical matrix. Hence, sij = sji . Sii =represents action due to unit positive displacement and while other displacement are 0. Sii is the principle diagonal element. Stiffness matrix does not exist for unstable structure. Stiffness matrix is non-singular matrix [s] is not equal to 0 for stable structure. Sii is the action at joint due to unit value of displacement at J joint

7. Analysis given beam by stiffness method. Take EI is constant. A B 1.5 m 1.5 m 4 m C 6 kn /m 100 kn

8. Step 1: Degree of kinematic indeterminancy Dki = 2 Step 2: Select unknown displacement D₁= θ B D₂= θ C

9. Step 3: Restrain the Structure to get kinematical determinate structure Step 4: Free analysis of restrained structure to get ADL Mab = - Wl^2 / 12 = - 80 kn.m Mba = + 8 kn.m Mbc = -W*l / 8 = - 37.5 kn.m Mcb = +37.5 kn.m 6 kn /m 100 kn 4 m 3 m ADL1 = 80-37.5 = 42.5 ADL2 = 37.5

10. Step 5 : Stiffness matrix 1. Apply unit rotation at joint B S₁₁= (4EI/L)+(4EI/L) = (4EI/4)+(4EL/3) = 2.33EI S ₂₁= 2EI/L = 2EI/3 = 0.67EI B S11 S21 4 m 3 m ←

11. Apply unit rotation at C S12= 2EI/L = 2EI/3 = 0.67EI S22= 4EI/L = 4EI/3 = 1.33EI C S12 S22 4 m ←

12. Superposition equation : { AD } = { ADL } + [ S ] { D } 42.5 37.5 2.33 0.67 0.67 1.33 θ B θ C 0= 42.5 + 2.33EI θ B+ 0.67EI θ C 0= 37.5 + 0.67EI θ B+ 1.33EI θ C Θ B= -22.22/EI Θ c= -11.85/EI = + EI

13. Final End moment : Mab = Mfab + 2EI / L (2 Ѳ a + Ѳ b – 3∆/L ) = - 8 + 2EI / 4 (-11.85/EI ) = -85.92 kn.m Mba = 68.16 kn.m Mbc = 68.15 kn.m Mcb = 0 kn.m Span moments : AB= wl^2/8 = 60*4^2/8 = 120 kn.m BC= wl /4 = 100*3/4 = 75 kn.m

14. A B C 120 85.92 68.16 75 + - - +

15. Thank You