1. Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials

2. Standard Cases Cantilever with point load at free end Cantilever with point load in span Cantilever with UDL Strength of Materials

3. 1. Cantilever with point load at free end Section at distance X Strength of Materials W L X W

4. Cantilever with point load at free end Taking Moment about Section X and equating with Differential Deflection Equation Integrate Strength of Materials 1

5. Cantilever with point load at free end Again integrate Apply BCs, When X=L, dy/dx=0. in eq 1, we get When X=L, y=0. in eq 2, we get Strength of Materials 2

6. Cantilever with point load at free end Put values of C1, C2 in eq 1 and eq 2 to get Slope and Deflection equation Slope Equation Deflection Equation Strength of Materials

7. Cantilever with point load at free end Slope at free end, put x=0 in slope equation, we get Deflection at free end, put x=0 in deflection equation Strength of Materials

8. 2. Cantilever with point load in span Section at distance X Strength of Materials W L X W a b a b

9. Cantilever with point load in span Taking Moment about Section X and equating with Differential Deflection Equation Integrate Strength of Materials 1

10. Cantilever with point load in span Again integrate Apply BCs, When X=L, dy/dx=0. in eq 1, we get Put x=L, y=0 in eq 2, we get Strength of Materials 2

11. Cantilever with point load in span Put values of C1, C2 in eq 1 and eq 2 to get Slope and Deflection equation Slope equation Deflection equation Strength of Materials

12. Cantilever with point load in span Strength of Materials Slope at free end, put x=0 in slope equation, we get Deflection at free end, put x=0 in deflection equation

13. 3. Cantilever with UDL Section at distance X Strength of Materials w L ab w X

14. Cantilever with UDL Taking Moment about Section X and equating with Differential Deflection Equation Integrate Strength of Materials 1

15. Cantilever with UDL Again integrate Apply BCs, When X=L, dy/dx=0. in eq 1, we get Put x=L, y=0 in eq 2, we get Strength of Materials 2

16. Cantilever with UDL Put values of C1, C2 in eq 1 and eq 2 to get Slope and Deflection equation Slope equation Deflection equation Strength of Materials

17. Cantilever with UDL Strength of Materials Slope at free end, put x=0 in slope equation, we get Deflection at free end, put x=0 in deflection equation

18. Numerical:1 Strength of Materials

19. Numerical:2 Strength of Materials

20. Workout Example 1 A cantilever of 3m length and uniform rectangular cross section 150 mm wide and 300 mm deep is loaded with 30 kN load at its free end. In addition to this it carries a udl of 20 kN per meter run over its entire length, calculate: 1.The maximum slope and deflection 2.The slope and deflection at 2m from fixed end. Strength of Materials

21. Steps to Solve -Draw diagram -Apply Macaulays Method to get Differential deflection equation -Integrate it to get Slope and deflection equations. -Apply BCs to get C1=225 and C2=472.5. -Max slope at x=0, Θ max = 3.33 x 10 - 3 rad -Max deflection at x=0, y max =7mm -Θ x=1 = 3.062x10 -3 rad and y x=1 =3.753mm Strength of Materials