1 Department of Civil and Environment Engineering CGN 4980/CGN 6939 FE/Graduate Seminar Review Examples Fall 2005.

1

Department of Civil and Environment Engineering

CGN 4980/CGN 6939

FE/Graduate Seminar Review Examples

Fall 2005

2

Determine the force in each member of the truss and state if the members are in tension or compression.

Solution:

3

yFFyAF

xAF

xFF

4

5

FDC

FDE

6

FAB

yAF

xAF

7

FFE

yFF

xFF

FFB

8

FFE

yFF

xFF

FFB

9

FBE

FBA

FBC

10

FECFEF

FED

11

FCD

FCB

12

The rod has a weight W and rests against the floor and wal for which the coefficients of static friction are A and B, respectively. Determine the smallest value of for which the rod will not move.

Given:

Find:

Solution:

13

FA

NA

NBW

L sin

FA

FB

Impending Motion at All Points

14

FA

NA

NBW

L sin

FA

FB

0FN

:0M

NF

:0F

NF

:0F

BB

A

AB

y

BA

x

cos2

LWcosLsinL

0W

0

Equilibrium Eqs.

15

FA

NA

NBW

L sin

FA

FB

slipping must occur at A & B

A

BA

BA

AB

BAA

BBB

AAA

μ2

μμ1tan

μμ1

WμN

μμ1

WN

ngsubstituti and

NμF

NμF

16

The three bars have a weight of WA = 20 lb, WB = 40 lb and WC = 60 lb, respectively. If the coefficients of static friction at the surfaces are as shown, determine the smallest horizontal force P needed to move block A.

Given:

Find:

Solution:

17

NAD

FAD

T

WABCABC

18

NAD

FAD

WABAB

FBC

NCB=WCC+Tsin+Tsin

NCB

FCB

T

WCC

If blocks A & B move first

lb 81.82N and lb 46.36T

017

15TN.5

017

15TNμ

017

15TF

:0F

06017

8TN

:0F

CB

CB

CBCB

CB

x

CB

y

19

NAD

FAD

WABAB

FBC

NCB=WCC+Tsin+Tsin

NCB

FCB

T

WCC

If blocks A & B move first

lb 69.27P

.2N.5NP

0NμNμP

:0F

lb 141.82N

060NN

060NN

:0F

ADCB

ADADCBCB

x

AD

CBAD

CBAD

y

20

FAB

T

T

If blocks A move first

017

15T.3N

017

15TNμ

:0F

000117

8TN

:0F

AB

ABAB

x

AB

y

NAB

NAD

NAD

FAD

WAA

FAB

NAB

WCBCB

0NμNμ-P

:0F

020N-N

:0F

ADADABAB

x

ABAD

y

lb 63.52P

lb 139.05N

lb 119.05N

lb 40.48T

AD

AB

Therefore block A moves first

21

Given: rod,

Find: x

Solution:

Determine the distance x to the center of mass of the homogeneous rod bent into the shape shown. If the rod has a mass per unit length of 0.5 kg/m, determine the reactions at the fixed support O.

22

1.44mL

2

3

x4

91

27

8

1

0

x4

91L

dxdx

dy1dLL

gintegratin

x2

3

dx

dy

xy

xx'

dxdx

dy1dydxdL

1

0

1

0

2

2

1

2

13

222

dx

23

.546m1.44

.786

dL

dLx'

x

Centroid

.786

2

5

x4

91

1215

642

3

x4

91

27

8

1

0

dxx4

91xdLx'

L

L

1

0L

24mN 3.85M

0.44(.546)0.5(9.81)1M

0F

7.06NO

0.440.5(9.81)1O

0F

0O

0F

mEquilibriu

o

o

x

y

y

y

x

x

25

Locate the centroid of the quarter circle shown in the figure.

x

y

m 1

m 1

2x=y

26

27

28

29

30

Find the Centroid

31

12

3

32

123

33

Each of the three members of the frame has a mass per unit length of 6 kg/m Locate the position (x, y) of the center of gravity. Neglect the size of the pins at the joints and the thickness of the members. Also, locate the reactions at the pin A and roller E.

Solution:

34

1

2

3

35

Segment L(m) x'(m) y'(m) x'L(m2) x'L(m2)1 8 4 13 32 1042 7.211 2 10 14.42 72.113 13 0 6.5 0 84.5

28.211 46.42 260.61

mL

Lyy

mL

Lxx

24.9211.28

61.260'

65.1211.28

42.46'

1

2

3

36

0=A

0=F

1319.N=A

0=WE+A

0=F

342.=E

0=W(8)E

0=M

N 1660.5=1)(6)28.211(9.8=W

x

x

y

yy

y

y

y

A

∑

∑

∑1

2

3

37

Determine the moment of inertia Iy for the slender rod. The rod’s density and cross-sectional area A are constant. Express the results in terms of the rod’s total mass m.

38

dx

x