Strength of Materials

Standard Cases for Slope and Deflection (Cantilever)

Lecture No-4J P Supale

Mechanical Engineering DepartmentSKN SITS LONAVALA

Strength of Materials

Standard Cases

• Cantilever with point load at free end • Cantilever with point load in span• Cantilever with UDL

Strength of Materials

1. Cantilever with point load at free end

Section at distance X

W

L

X

W

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

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

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

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

Strength of Materials

2. Cantilever with point load in span

Section at distance X

W

L

X

W

a b

a b

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

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

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

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

Strength of Materials

3. Cantilever with UDL

Section at distance X

w

L

a b

w

X

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

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

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

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

Strength of Materials

Numerical:1

Strength of Materials

Numerical:2

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.

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