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Lecture 23.pptx
- 1. Structural Analysis II Lecture No-23 By Iqbal Hafeez Khan Assistant Professor Department of Civil Engineering SISTec GN 1 Unit-4 Matrix method of Structural Analysis
- 2. Matrix method • Matrix method are based on the concept of replacing the actual continuous indeterminate structure by a mathematical model made up from structural element having non elastic and inertial properties that can be expressed in matrix form. • The method is carried out using either a stiffness matrix or a flexibility matrix . 2
- 3. Flexibility matrix • The force method is also known as flexibility matrix method • In this method the unknown to be selected is redundant reaction for internal force. 3 W L A B C 𝑅𝐵
- 4. Stiffness matrix • The displacement method is also known as stiffness matrix method • In this case the structure is analysed by taking unknown displacement as redundant. 4 W L A B C
- 5. Action and Displacement corresponds • An action or a force is commonly a single force or a moment . • The action can be external or internal. • The displacement can be translation or rotation. • The displacement and action are said to be corresponding when they are of analogous type and are located on the same point on the structure 5
- 6. 6 P A ∆ 𝜃 M • ∆ is corresponding to P but not solely causes by P • 𝜃 corresponds to M but not solely causes by M
- 7. Static indeterminacy • Static indeterminacy means an excess of unknown actions whether external or internal forces as compared to the number of static equation of equilibrium available. • In the flexibility method static indeterminacy must be considered 7 L A B 𝑅𝐵
- 8. Kinematic indeterminacy • Kinematic indeterminacy of a structure means unknown joint displacement in a structure. • Kinematic indeterminacy is considered in the stiffness matrix method • Kinematic indeterminacy = 2 = 1 rotation + 1 translation 8 L A B
- 9. Relationship between flexibility & stiffness 9 Consider a spring shown in fig if F is the displacement produced by unit load then the displacement caused by force A is D A D 1 F 𝐴 𝐷 = 1 𝐹 FA = D 𝐴 = 𝐹−1 𝐷 (i) A D 1 F
- 10. 10 Consider a spring as shown in fig if S is the stiffness of spring (load per unit deflection) A D S 1 𝐴 𝐷 = 𝑆 1 A = SD (ii) 𝐷 = 𝑆−1 𝐴 Eq (i) = Eq (ii) 𝐹−1𝐷 = 𝑆𝐷 𝐹−1 = 𝑆 So Flexibility is inverse of stiffness A D S 1
- 11. 11
- 12. 12 Deflection Formulae 𝛿𝐵 = 𝑃𝐿3 3𝐸𝐼 𝜃𝐵 = 𝑃𝐿2 2𝐸𝐼 𝛿𝐶 = 𝑃𝑥2 6𝐸𝐼 (3𝐿 − 𝑥) P A B C L 𝒙 L A B w kN/m 𝛿𝐵 = 𝑤𝐿4 8𝐸𝐼 𝜃𝐵 = 𝑤𝐿3 6𝐸𝐼
- 13. Sign Convention Downward deflection = Negative Upward deflection = Positive Clockwise rotation= Positive Anticlockwise rotation= Negative 13 𝛿𝐵 = 𝑀𝐿2 2𝐸𝐼 𝜃𝐵 = 𝑀𝐿 𝐸𝐼 A B L M
- 14. Find out the deflection at point B and C. 14 12 t 8 t 𝟒 𝐦 𝟐 𝐦 5 𝐦 A E C D B 3 𝐦
- 15. Deflection at B deflection at B due to 12 kN load = −𝑃𝑙3 3𝐸𝐼 + −𝑃 𝑙2 2𝐸𝐼 x 2 = −12 x 43 3𝐸𝐼 + −12 x 42 2𝐸𝐼 x 2 = −448 𝐸𝐼 deflection at B due to 8 kN load = −𝑃𝑥2 6𝐸𝐼 (3𝑙 − 𝑥) = −8 x 62 6𝐸𝐼 (3 x 9 − 6) = −1008 𝐸𝐼 15 12 t 8 t 𝟒 𝐦 𝟐 𝐦 5 𝐦 A E C D B 3 𝐦
- 16. Total deflection at B = −1456 𝐸𝐼 16 12 t 8 t 𝟒 𝐦 𝟐 𝐦 5 𝐦 A E C D B 3 𝐦
- 17. Deflection at C deflection at C due to 12 kN load = −𝑃𝑙3 3𝐸𝐼 + −𝑃 𝑙2 2𝐸𝐼 x 10 = −12 x 43 3𝐸𝐼 + −12 x 42 2𝐸𝐼 x 10 = −1216 𝐸𝐼 deflection at B due to 8 kN load = −𝑃𝑙3 3𝐸𝐼 + −𝑃 𝑙2 2𝐸𝐼 x 5 = −8 x 93 3𝐸𝐼 + −8 x 92 2𝐸𝐼 x 5 = −3564 𝐸𝐼 17 12 t 8 t 𝟒 𝐦 𝟐 𝐦 5 𝐦 A E C D B 3 𝐦
- 18. Total deflection at B = −4780 𝐸𝐼 18 12 t 8 t 𝟒 𝐦 𝟐 𝐦 5 𝐦 A E C D B 3 𝐦
- 19. Steps in flexibility matrix method • Calculation of degree of indeterminacy • Selection of determinate structure • Analysis of determinate structure under load • Analysis of determinate structure for unit value of redundant • Determination of redundant by compatibility equation Q = − F −1[DQL] [Q]= Redundant matrix [F]= Flexibility matrix DQL =Displacement matrix due to given loading in released structure in coordinate direction 19
- 20. Thank You 20