Home Work #8

HW 8-Problem #1

The beam is subjected to a uniform load w. Determine the placement a of the supports so that the shear

stress in the beam is as small as possible. What is this stress?

HW 8-Problem #1

For 0<x<a

For a<x<(L-a)

For (L-a)<x<L

0 yF 0 vwx wxv

0 yF 02

wl

vwx wxwl

v 2

0 yF 0)2(2 wl

vwx wxwlv

4/

)2(

la

waal

w

2/

2/

2/

0

2

8

b

b

d bdydydQ

2

12

1bdI

bd

wl

bbd

dbllw

8

3

121

)8/()4/2/(

3

2

V

w

x

HW 8-Problem #2

The fence board weaves between the three smooth fixed posts. If the posts remain along the same line, determine the maximum bending stress in the board. The board has a width of 150mm and a thickness of 12mm. E=12Gpa. Assume the displacement of each end of the board relative to its center is 75mm.

HW 8-Problem #2

2/)('' pxxMEIy For 0<x<L/2

12'

4Cx

PEIy

213

12CxCx

PEIy

Y=0 at x=0; C2=0

x=L, dy/dx=0; C1=-PL2/16)

1612(

1)( 23 xL

Px

P

EIxy

At x=1.2m; y= -0.075 m39

23

)012.0)(15.0)(121

)(10*12(

)2.1()4.2(16

)2.1(12075.0

PP

P=67.5 N MPaI

My25.11max

HW 8-Problem #3

Determine the equation of the deflection of the beam and specify the slope at B and deflection at C. EI is constant. (please ignore x1, x2 and x3).

For 0<x<a

For a<x<2a

0 VF

HW 8-Problem #3

0 MFxM o25.1 wawaxFxMM o

0)( vaxwF )2( xawv

0)2

9)((

M

xawwFxM o

22 5.022 wxwaxwaM

For 0<x<a MEIy '' 122' 5.05.1 CwaxxwaEIy

21322

6

175.0 CxCwaxxwaEIy

For x=0 y=0; C2=0 For x=0 dy/dx=0; C1=0

322

6

175.0 waxxwaEIy

HW 8-Problem #3

For a<x<2a MEIy ''

3223' 2

6Cxwaawxx

wEIy

432234

3

1

24CxCxwaawxx

wEIy

For x=a3

22322' 26

5.05.1 Cxwaawxxw

waxxwaEIy

432234322

3

1

246

175.0 CxCxwaawxx

wwaxxwaEIy

6

3

3

waC

24

4

4

waC

At point B x=2aEI

wa

dx

dy

6

7 3

At point C x=a EI

way

12

7 4

The beam is supported by a pin at A, a spring having a stiffness k at B, and a roller at C. Use method of superposition to determine the force the spring exerts on the beam. EI is constant.

HW 8-Problem #4

HW 8-Problem #4