standard cases for slope and deflection (cantilever) lecture no-4 j p supale mechanical engineering...
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![Page 1: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/1.jpg)
Strength of Materials
Standard Cases for Slope and Deflection (Cantilever)
Lecture No-4J P Supale
Mechanical Engineering DepartmentSKN SITS LONAVALA
![Page 2: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/2.jpg)
Strength of Materials
Standard Cases
• Cantilever with point load at free end • Cantilever with point load in span• Cantilever with UDL
![Page 3: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/3.jpg)
Strength of Materials
1. Cantilever with point load at free end
Section at distance X
W
L
X
W
![Page 4: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/4.jpg)
Strength of Materials
Cantilever with point load at free end
Taking Moment about Section X and equating with Differential Deflection Equation
Integrate
2
2
d yM EI Wx
dx
212
dy WEI x Cdx
1
![Page 5: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/5.jpg)
Strength of Materials
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
31 26
WEIy x C x C 2
2
1 2
WLC
3
2 3
WLC
![Page 6: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/6.jpg)
Strength of Materials
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
221
2 2
dy W WLx
dx EI
2 331
6 2 3
W WL WLy x x
EI
![Page 7: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/7.jpg)
Strength of Materials
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
2
max 2
dy WL
dx EI
3
max 3
WLy
EI
![Page 8: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/8.jpg)
Strength of Materials
2. Cantilever with point load in span
Section at distance X
W
L
X
W
a b
a b
![Page 9: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/9.jpg)
Strength of Materials
Cantilever with point load in span
Taking Moment about Section X and equating with Differential Deflection Equation
Integrate
2
2( )
d yM EI W x b
dx
21( )
2
dy WEI x b Cdx
1
![Page 10: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/10.jpg)
Strength of Materials
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
31 2( )
6
WEIy x b C x C 2
2
1 2
WaC
2
2 ( 3 )6
WaC a L
![Page 11: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/11.jpg)
Strength of Materials
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
221
( )2 2
dy W Wax b
dx EI
2 3 231
( )6 2 6 2
W Wa Wa Way x b x L
EI
![Page 12: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/12.jpg)
Strength of Materials
Cantilever with point load in span
Slope at free end, put x=0 in slope equation, we get
Deflection at free end, put x=0 in deflection equation
2
max 2
dy Wa
dx EI
3 2
max ( )3 2
Wa Way L a
EI EI
![Page 13: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/13.jpg)
Strength of Materials
3. Cantilever with UDL
Section at distance X
w
L
a b
w
X
![Page 14: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/14.jpg)
Strength of Materials
Cantilever with UDL
Taking Moment about Section X and equating with Differential Deflection Equation
Integrate
2 2
2
( )
2
d y W x bM EI
dx
31( )
6
dy WEI x b Cdx
1
![Page 15: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/15.jpg)
Strength of Materials
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
41 2( )
24
WEIy x b C x C 2
3
1 6
WaC
3
2 ( 4 )24
WaC a L
![Page 16: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/16.jpg)
Strength of Materials
Cantilever with UDL
Put values of C1, C2 in eq 1 and eq 2 to get Slope and Deflection equation
Slope equation
Deflection equation
331
( )6 6
dy W Wax b
dx EI
3 341
( ) ( 4 )24 6 24
W Wa Way x b x a L
EI
![Page 17: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/17.jpg)
Strength of Materials
Cantilever with UDL
Slope at free end, put x=0 in slope equation, we get
Deflection at free end, put x=0 in deflection equation
3
max 6
dy Wa
dx EI
4 3
max ( )8 6
Wa Way L a
EI EI
![Page 18: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/18.jpg)
Strength of Materials
Numerical:1
![Page 19: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/19.jpg)
Strength of Materials
Numerical:2
![Page 20: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/20.jpg)
Strength of Materials
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 deflection2. The slope and deflection at 2m from fixed end.
![Page 21: Standard Cases for Slope and Deflection (Cantilever) Lecture No-4 J P Supale Mechanical Engineering Department SKN SITS LONAVALA Strength of Materials](https://reader035.vdocuments.us/reader035/viewer/2022081503/5697bfe81a28abf838cb674f/html5/thumbnails/21.jpg)
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, Θmax = 3.33 x 10-3 rad- Max deflection at x=0, ymax=7mm- Θx=1 = 3.062x10-3 rad and yx=1=3.753mm
Strength of Materials
2 2
230 20
2
d y xMxx EI x
dx