STATICS EXERCISE BOOK (For BSc students in Mechanical Enigineering) Compiled by Dr. Tamás Insperger Department of Appli
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STATICS EXERCISE BOOK (For BSc students in Mechanical Enigineering)
Compiled by Dr. Tamás Insperger Department of Applied Mechanincs Budapest University of Technology and Economics
Budapest 2009
Problem 1 Force F is applied at point B. Determine the moment of force F about point C! F = [ 3, 4, 0 ] [N] ;
rB = [8, 3, 0 ] [m] ;
1
rC = [1, 2, 0 ] [m]
Problem 2 Determine the moment of the load lifted by the trolley about point A for the lower and the upper position of the load.
2
Problem 3 The spatial force system consists of three forces. Determine the moment vector of the force system about point O! FA = 10 kN ,
FB = 15 kN ,
FC = 20 kN ,
a = 1 m,
b=1m,
z a
b
FC
c O
x
FB
FA
y
3
c=3m.
Problem 4 Reduce the force F to point B and to point A!
4
Problem 5 Calculate the moment of the given couple to points A and B and to the origin of the coordinate system! F = [ 2, 1, 0 ] [N] ;
rA = [4, 1, 0 ] [m] ;
5
rB = [3, 5, 0 ] [m]
Problem 6 (a) Determine the resultant of the planar force system consisting of two forces by construction! F1 = 200 N ,
F2 = 250 N .
F1
F2
(b) Determine the resultant of the planar force system consisting of two parallel forces by construction! F1 = 200 N ,
F2 = 100 N .
F1
F2
6
Problem 7 Determine the resultant of the planar force system consisting of three forces by construction! F1 = 200 N ,
F2 = 250 N ,
F3 = 150 N
F1
F3 F2
7
Problem 8 Determine the resultant of the given force system! F1 = 200 N ,
F2 = 800 N ,
F3 = 600 N .
F3
F1
F2 3m
8
Problem 9 Determine the resultant of the planar force system consisting of four forces by construction! F1 = 200 N ,
F2 = 250 N ,
F3 = 150 N,
F4 = 150 N.
F1
F3 F4
F2
9
Problem 10 The rigid beam is loaded four forces. Determine the resultant of the force system by calculation! F1 = 150 N , F2 = 600 N , F3 = 100 N , F4 = 250 N
F1
1,6 m
F2
1,2 m
F3
F4
2,0 m
10
Problem 11 Determine the resultant of the distributed force system! (Both the magnitude and the location!) L=2m,
p = 5 kN/m .
(a)
y
O
p
x
p
x
L
(b)
y
O
L
11
Problem 12 Determine the location of the centroid of the given plane figures! The sizes are given in mm.
(a) 20 20 100
20
70
(b)
60 100
20
30
(c) 20 100 20 20
70
12
Problem 13 Determine the location of the centroid of the bodies!
(a)
a
a a a
a a (b)
a
a a a a a
(c)
a
a a a
a a
13
a = 0,5 m .
Problem 14
The structure in the drawing is in equilibrium! Determine force B and the reaction forces! F1 = 1200 N
14
Problem 15
Determine the reaction forces for the given planar structure! F = 600 N ,
p = 300 N/m , M = 300 Nm,
(a)
A
F
B 2a
a
y x (b)
F
A
B a
2a y x (c)
B
p
A
a
2a y x (d)
M
A 2a
B a
y x
15
a=2m.
Problem 16
Determine the reaction forces!
16
Problem 17
Determine the reaction forces, if G1 = 10000 N and G2= 24000 N!
17
Problem 18
Determine the reaction forces for the structure!
F = 600 N C 1m A
B 1m
3m
18
Problem 19
Determine the reaction forces for the structure!
1,5 m D F = 1 kN C 1m A
B 1m
y
3m
x
19
Problem 20
Determine the reaction forces for the structure!
0,5 m
0,5 m
C 0,5 m
200 N
400 N 0,5 m
A
B
20
Problem 21
Determine the reaction forces for the structure! Make a separate drawing of the beam D-C and give the forces acting on it. a=1m
F = 1 kN 2a
2a
a
D
a B
G
C
3a
A
21
Problem 22
Calculate the reaction forces for the structure and the force in the rope D-E!
C F = 10 kN 1m
D
E
1m A
B 1m
y x
22
Problem 23
Determine the reaction forces! Make a separate drawing of the horizontal beam and give the forces acting on it! a = 0,5 m ,
b = 0,6 m ,
F = 18 kN ,
p = 30 kN/m .
C b
F p D
E b B
A a
a
2a
y x
23
Problem 24
Determine the reaction forces and the forces in the bars of the given truss! a = 1 m, F1 = 4 kN ,
C 3
a
F2
R (kN)
1 2 3 4 5 6 7
24
a
7
B
6
x
bar
F1
5
D
2
a
y
E
4
1
A
F2 = 6 kN .
a
a
Problem 25
Determine the reaction forces and the forces in the bars of the given truss! a = 1 m,
F = 10 kN .
a
B
a 2
6
a
7
5
3
1
F 4
A bar
+/-
R (kN)
1 2 3 4 5 6 7
25
Problem 26
Determine the reaction forces and the forces in the bars! a=1m,
F = 1 kN .
(a)
F a B
A a
a
(b)
F
a A
B a
a
26
Problem 27
Determine the reaction forces and the forces in the bars! a = 0,6 m ,
F = 750 N .
a
B
a 2
a
5
600
3
1
4
A bar
+/-
R (N)
1 2 3 4 5
27
F
Problem 28
Determine the stress resultant diagrams for the given beams! F = 4 kN ,
a = 0,5 m .
(a)
F
A
B a
a
V
Mh
(b)
F
A
B a
a
V
Mh
28
Problem 29
Determine the stress resultant diagrams for the given beams! p = 4 kN/m ,
a = 0,5 m .
(a)
p
A a
B
a
V
Mh
(b)
p
A a
B
a
V
Mh
29
Problem 30
Determine the stress resultant diagrams for the given beams! (a)
A
90 N
30 N B
a
a
a
a=0,3 m
V
Mh
(b)
A
1 kN
1 kN B
a
a
a
a=1 m
V
Mh
30
Problem 31
Determine the stress resultant diagrams for the given beams! (a)
3 kN
6 kN
A a
a
2a
3 kN
B 2a
a=150 mm
V
Mh
(b)
p = 10 kN/m a
A
4a
B
a=1 m a
V
Mh
31
Problem 32
Determine the stress resultant diagrams for the given beams! (a)
3 kN
A a
1 kN 1 kN a
a=2m
V
Mh
N
(b)
3kN
6kN
4kN/m a=2m
a
2a
V
Mh
N
32
Problem 33
Determine the stress resultant diagrams for the given structures! (a)
3 kN 0,5 m
A
N
1m Mh
V
V
Mh
(b)
B 0,5 m
A 1m
N
4 kN Mh
V
V
Mh
33
Problem 34
Determine the reaction forces and give the stress resultant diagrams! a=2m,
F1 = 4 kN , F2 = 10 kN ,
M = 20 kNm ,
A
p = 3 kN/m .
B p
a
M
a
F1
V
Mh
N
34
a
a
F2
Problem 35
Determine the limit values of force F such that the body is standing in equilibrium on the frictional slope! The coefficient of friction between the slope and the block is µ. µ = 0,2 ,
G = 100 N (weight).
G µ F 30o
35
Problem 36
The ladder is placed on the frictional ground (µ = 0,3) leaning against the smooth wall (with frictionless connection). How high can one climb up the latter without making the ladder to slip?
smooth sima G
5m h
frictional érdes 2m
36
Problem 37
Determine the limit values of force F such that the system remains in equilibrium! G1 = 30 kN ,
G2 = 15 kN ,
µ1 = 0,3 ,
µ2 = 0,2 .
A
45o µ2
2
B F
1 µ1
37
Problem 38
The mechanical system of vertical arrangement presented in the figure is in equilibrium. Determine the limit values of force F such that the system remains in equilibrium! G1 = 2 kN ,
G2 = 5 kN ,
µ1 = µ2 = 0,2 ,
µ3 = 0 .
µ3 45o 1 µ1 F 2 µ2
38
Solutions:
Problem 1:
⎛0⎞ ⎜ ⎟ M C = ⎜ 0 ⎟ Nm ⎜ 25 ⎟ ⎝ ⎠
Problem 2:
⎛ 0⎞ ⎛ 0 ⎞ ⎜ ⎟ ⎜ ⎟ M A1 = ⎜ 0 ⎟ kNm , M A 2 = ⎜ 0 ⎟ kNm ⎜ −2 ⎟ ⎜ −1, 2 ⎟ ⎝ ⎠ ⎝ ⎠
Problem 3:
⎛ 15 ⎞ ⎛ −20 ⎞ ⎜ ⎟ ⎜ ⎟ F = ⎜ 10 ⎟ kN , M O = ⎜ −60 ⎟ kNm ⎜ 30 ⎟ ⎜ 15 ⎟ ⎝ ⎠ ⎝ ⎠
Problem 4:
⎛ 0 ⎞ ⎛ 38,5 ⎞ ⎛ 0 ⎞ ⎛ 38,5 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ FrB = ⎜ 0 ⎟ N , M B = ⎜ 0 ⎟ Nm , FrA = ⎜ 0 ⎟ N , M A = ⎜ 55 ⎟ Nm ⎜ −110 ⎟ ⎜ 0 ⎟ ⎜ −110 ⎟ ⎜ 0 ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
Problem 5:
⎛0⎞ ⎛0⎞ ⎛0⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ M A = ⎜ 0 ⎟ Nm , M B = ⎜ 0 ⎟ Nm , M O = ⎜ 0 ⎟ Nm (the couple is a free vector) ⎜9⎟ ⎜9⎟ ⎜9⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
Problems 6-9: Construction Problem 10: Fe = 600N , xe = 3,133m Problem 11: (a) Fe = 10kN , xe = 1m (b) Fe = 5kN , xe = 4 / 3m Problem 12: (a) xS = 28,53mm , yS = 58, 24mm (b) xS = 16, 21mm , yS = 53,10mm (c) xS = 36, 25mm , yS = 50mm Problem 13: (a) xS = 0, 4643m , yS = 0, 4643m , zS = 0, 4643m (b) xS = 0,375m , yS = 0,375m , zS = 0,375m (c) xS = 0, 4167m , yS = 0,5m , zS = 0,5m Problem 14: Ax = 1200N , Ay = 500N , B = 500N
Problem 15: (a) Ax = 0N , Ay = −300N , B = 900N (b) Ax = 0N , Ay = 200N , B = 400N (c) Ax = 0N , Ay = 400N , B = 800N (d) Ax = 0N , Ay = 50N , B = −50N (Couple can only be balanced by couple!) 39
Problem 16: Ax = 770N , Ay = 0N , Bx = 770N , By = 350N Problem 17: Ax = −58kN , Ay = 34kN , Bx = 58kN Problem 18: Ax = 450N , Ay = 450N , Bx = −450N , By = 150N Problem 19: Ax = 1125N , Ay = 1125N , Bx = −1125N , By = 125N Problem 20: Ax = 150N , Ay = 250N , Bx = 250N , By = −50N Problem 21: Ax = 333,3N , Ay = 333,3N , Bx = −333,3N , By = 666, 7N Dx = 0N , Dy = −166, 7N , Gx = 333,3N , G y = 500N Problem 22: Ax = −10kN , Ay = −5kN , By = 5kN , D = 5kN Problem 23: Ax = 35kN , Ay = 42kN , Bx = −35kN , By = 42kN , C y = −36kN Dx = 35kN , Dy = 42kN , Ex = −35kN , E y = 6kN Problem 24: R1 = −2,83kN , R2 = 6kN , R3 = 2,83kN , R4 = −4kN , R5 = 5, 66kN R6 = 4kN , R7 = −5, 66kN Problem 25: R1 = −14,14kN , R2 = 10kN , R3 = 10kN , R4 = −10kN , R5 = −14,14kN R6 = 20kN , R7 = 10kN Problem 26: (a) Ax = 1kN , Ay = 0,5kN , By = −0,5kN , R1 = −0,5kN , R2 = −1kN , R3 = 1,118kN (b) Ax = 0kN , Ay = 0,5kN , By = 0,5kN , R1 = −0, 707kN , R2 = 0,5kN , R3 = −0, 707kN Problem 27: R1 = −918, 6N , R2 = 1024,5N , R3 = 918, 6N , R4 = −1299N , R5 = 0N Problem 28: (a) V
(b)
2
V
4 0
[kN]
[kN]
-2
Mh
Mh 2 0
[kNm]
[kNm]
-1
40
Problem 29: (a)
(b) V
V 2
0,5
[kN]
[kN]
-1,5
-2
Mh 2
Mh
[kNm]
[kNm]
0,25
-0,5
Problem 30: (a)
(b)
50
V
0,2813
1
V
[N]
-10
[kN]
0
-40
-1
Mh
Mh
[Nm]
[kNm]
-3 -15
-1
Problem 31: (a)
(b)
V 2,25
3
20
V
10 [kN]
[kN]
-10
-0,75
-20
-6,75
Mh
0,9
[kNm]
Mh 5
5
[kNm]
-0,1123 -0,3375
-15
Problem 32: (a) V
(b) V
3
[kN]
[kN]
Mh
11
-5
-6 12
Mh
6
[kNm] [kNm]
N
-3,125
N
2 1
0
[kN] -3
[kN]
41
Problem 33: (a) 3 kN
0 0,5 m
A
+3
N
[kN]
1m
V [kN]
+3
Mh
0
[kNm]
+1,5
0 V
Mh +1,5
Problem 33: (b) B
0 0,5 m
A 1m
+8
N
[kN]
4 kN
V [kN]
+8
Mh
0
[kNm] +4
V
Mh +4
-4
Problem 34: V
10 6
4
2
[kN] -2
Mh
6
3,33 4
-4
3,33
6
[kNm] -14 N
10
[kN]
42
Problem 35: Fmin = 33,83N , Fmax = 87,88N Problem 36: h = 3, 75m Problem 37: Fmax = 15, 25N , Fmin = −18,375N Problem 38: Fmax = 4, 4kN , Fmin = −0, 0667kN
43