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y2

F

2

Resultant and Equivalent of Force Systems

A force on a body produces translation while a couple produces rotation. The forces acting on a body may be in the same plane (coplanar) or in different planes (non-coplanar) or in space ; see plate 1 and plate 2. The coplanar forces may be parallel (like or unlike) or concurrent. If a number of forces are acting on a body, Fig. 1.1, to find their resultant R (in magnitude, direction `' with horizontal and line of action), find first their algebraic sum of horizontal and vertical components i.e., X and Y, then

F3

y3

R

x2

x3 Y

R X y

Rigid body a1 a3 F1 x1 y1 a A a2 a4 x

O

y4

F4 x4

Fig. 1.1

R=

( X ) 2 + ( Y ) 2 ,

tan =

Y X

...(1.1)

To find the line of action of R (at `a' w.r.t. a point A or a line), find the algebraic sum of the moments of all the forces about A, then MA = F1 . a1 = R . a ; a=

M A R

...(1.2)

The force systems are illustrated in A, B & C (graphical methods) and D in space, as follows:

3

4

20 N A B N = 15 N 40 N C D E

Engineering Mechanics

c a R 0 a 0 b 30 N E 0 c 0 15 N 0 e d e b d Force or vector diagram O

Space diagram (ii) Unlike II forces

Gusset plate

20 = Q L ­ irons

(angle-irons)

kN

Q

R = 46 kN 20° L ir s ons P=3 0 kN

=

R= 20° 46 kN P=3 0 kN (i) Parallelogram law of forces

20

kN

Resultant of two forces

R = 46 kN kN 20° 20 P=3 = 0 kN Q

(ii) Triangle of forces

0 B 200 N Aa 0 0 a¢ 100 N

kN

300 b N Block A

a C c 0 a¢

N 110 R= aa¢ =

b

R O d c

e

250 N 0 D d 200 N

0 60° e E 30°

Force polygon (will not close) R = aa¢

No. of forces on Block A

= 67

a R = oe e 4 a 25 kN 25 kN 25 kN 3 D C A 60° B 16 m 22 m 0 47° 0d 45° E x 0 = 43 m 25 kN c 0 0 a b

b

O pole

c

R = 67 kN e

Space diagram

d Vector or force diagram

See Example 1.3, Fig. 1.3 & 1.4, page 8 (single equivalent tug boat, R) A . Resultant of Coplanar Non Concurrent Forces

Resultant and Equivalent of Force Systems

N

5

40

Q = 300 N B

A g a C 180 ­ g 180 ­ a C R = 500 N a B C A = = Sin a Sin b Sin g B W RA RB

R

B

A b Block A

g

R

90° = 2 60° W = 400 N 00 N

30° 60° of forces Spherical ball of mass 40 kg rests in a smooth Dr groove W = RA = RB Sin 90° Sin 60° Sin 30°

Lami's Theorem:

Block A in equilibrium under three forces

A

(t)

A

kN

10 kN B

kN

tie

45°

(C)

18

150 N D E 180 N a E R a¢ b

A B

160 N

jib

27

2m

30°

10 kN

C

D of forces at A

Jib crane

C

30°

B

200 N R = 150 N = aa¢ c

d No. of concurrent forces R = Resultant = aa¢ E = Equilibrant = a¢a

PLATE 1 B. Equilibrium Under Coplanar Concurrent Forces

A

=3

P = 400 N

b

180 ­ b

30°

D

6

f 1 kN C 2.5

2.2

Engineering Mechanics

1 kN D 1.5 kN

4

RA a b c d e

II t

1 kN

B 2.8

RE

oc

los

ing

lin

e

RA

=1

.7 k

NA

1.7 0 H 0G 0 0 F 0 a b 4m 4m Cl os ing lin e

I 1.1

J

1.

4m 0 d

O pole a b c d

f, g, h, i

0 c 0e E 4m RE = 3.8 kN

Funicular polygon

Vector diagram or Force polygon to find reactions

j

(i) Truss ­ Space diagram (with Bow's notation)

e Superposition of force Maxwell or º polygons at each joint Force diagram

[See Ex. 3.1 & Fig. 3.1, page 30]

RB = 8 kN B

8

(c) e (C) = E (D)

3.5

h

1.1

J

3.5 H

3.5

G

I 1.7

f i E 3 kN a RA

RA = 7 kN

A

6.3

(ii) Cantilever Truss ­ Space diagram (with bow's notation)

[See Ex. 3.3 & Fig. 3.14, page 34] C. Equilibrium Under Coplanar Non Concurrent Forces

5.8

R

B

F 2 kN

j g

Maxwell diagram gives also reactions in cantilever trusses

D

z TA = ? T B = ? TC = ?

z 40 kN O (x, y) of R = ? R.C.C. mat 20 kN 30 kN 15 kN

4m C A G

2mx

y

2m 3m

2.5 m

2m

y

2m

B W = 50 kN

2m

m x 2.5 Column loads

(i) Concurrent forces

(ii) Parallel forces

Resultant and Equivalent of Force Systems

z R ­ C at O = ? z B Cables 3m E

m 1.5

7

50 kN (in x-y plane) TE, TD = ? RA = ? C 20 kN crate y x Forces for f = 0 : Coplanar concurrent at C Non Coplanar concurrent at B (iv) Non concurrent, Non parallel forces 30° x 1m 1m 20 1m 1m 20 kN kN-m 60 kN (in x-z plane) (II to y-axis)

3m 2m A

3m

1.5

m

f

D y

(iii) Space frame-Derrick

D. Spatial Force System [Forces in three-dimensions] PLATE 2

Also,

MA = (X)y ­ (Y)x

...(1.3)

This Eqn. can be solved for the ratio of y to x corresponding to any point `O' along the line of action of R as indicated by the dashed line, that is the force can be applied at any point on its line of action. This is called the principle of tansmissibility of force. From Eqs. (1.2) and (1.3), R . a = (X )y ­ (Y ) x This is called `Varignon's Theorem'. Example 1.1. Determine the resultant of the forces in Fig. 1.1. ...(1.4) i.e., the moment of the resultant is equal to the algebraic sum of the moments of its components.

Fig. 1.1

Solution: + X = 150 ­ 100 sin 45° = 79.29 N + Y = 50 ­ 8 ­ 100 cos 45° = ­ 28.71 or 28.71 N R= tan =

79.29 2 + 28.712 = 84.32 N

28.71 = 0.3621, 79.29

= 19.9°

8

Moment of the resultant (at a distance `a' from C) = Moment of the components (about C) R . a = MC + MC = 150 × 1 + 100 × 1 ­ 50 × 1 ­ 8 × 1 = 192 N.m a=

Engineering Mechanics

Fig. 1.1a

MC 192 = = 2.27 mm R 84.32

Example 1.2. The resultant (100 kN) of four forces and three of these are shown in Fig. 1.2. Determine the fourth force.

Fig. 1.2

Solution:

+ X = ­ 70 cos 60° ­ 120 cos 60° + 50 cos 50°

= ­ 62.88 = ­ 81.6 or 62.88 kN or 81.6 kN

F4 Y4 q X4

+ Y = 70 sin 60 ­ 120 sin 60° ­ 50 sin 50° RX = 100 cos 45° = 70.71 kN RY = 100 sin 45° = 70.71 kN X4 = 70.71 ­ (­ 62.88) = 133.59 kN Y4 = 70.71 ­ (­ 81.6) = 152.31 kN F4 = tan =

133.59 2 + 152.312 = 202.6 kN, Fig. 1.2a

Fig. 1.2a

152.31 = 1.14, = 48.746°. 133.59

Example 1.3. (a) State `Varignon's Theorem'. (b) Four tugboats exert 25 kN each (as shown in Fig. 1.3) to bring an ocean liner to the pier. Determine the point on the hull where a single, more powerful tugboat should push to produce the same effect as the original four boats.

Resultant and Equivalent of Force Systems

Boat - 1 4 3 60° 16 m 22 m 36 m 63 m 66 m 30 m 30 m Boat - 2 Boat - 3

9

Hull of ship 1

1 Boat - 4

Fig. 1.3

Solution: (a) The Varignon's (French Mathematician) Theorem states that "if a number of coplanar forces acting on a body, the algebraic sum of the moments of all the forces about any point is equal to the moment of their resultant about the same points, i.e. F1. a1 = R.a" or simply ``the moment of a force about any point equals the algebraic sum of the moments of its components about the same point, i.e. F . a = X . x + Y . y, where F =

x 2 + y 2 ".

(b) To find the resultant `R' due to the four forces acting on the hull of the ship, Fig. 1.4.

Y 25 kN F1 60° 12.5 O 16 m 25 kN F2 5 4 3 15 22 m 63 m 43 m 66 m a SY x R q SX 22 m 45° 30 m 30 m X SX = 45.18

21.65

20

25 kN F3 SY = 48.97

q = 47.3 R = 66 kN

Fig. 1.4(a)

17.68

25 kN 1 1 F4

Fig. 1.4

+ X = 12.5 + 15 + 17.68 = 45.18 kN

+ Y = ­ 21.65 ­ 20 ­ 25 + 17.68 = ­ 48.97 or 48.97 kN R=

45.18 2 + 48.97 2 = 66.63 kN, Fig. 1.4(a)

17.68

10

= tan­1

Engineering Mechanics

48.97 = 47.3°. 45.18

Let the line of action of R cut the X-axis at x. Taking moments about `O' (and using Varignon's theorem), M0 = X.y + Y.x, + M0 = 48.97 × x y=0

+ M0 = 12.5 × 16 + 15 × 22 + 20 × 63 + 25 × 159 ­ 17.68 × 22 ­ 17.68 × 129 = 3095.32 kN.m 48.97 x = 3095.32, x=

3095.32 = 63.2 m 48.97

From Fig. 1.4, the resultant R will intersect the hull at a distance of (x ­ a) from O : tan =

22 = 1.084, a

a = 20.3 m,

63.2 ­ 20.3 43 m

i.e., where a single, more powerful tug boat should push, exerting a force of 66.63 kN. See the graphical solution in Plate 1­A, page 5. Example 1.4. Find the resultant of the four forces acting as shown in Fig. 1.5.

50 100

70.7 kN 200 kN

45° 50 70.7 kN 45° 200 kN 30° a A A 1m 1m 30° 100 kN 80 kN 50 x 1m

30° 173.2 q 1.5 m 80 R 309

1.5 m

Fig. 1.6(a)

86.6 30° SY 100 kN q SX R

80 kN

Fig. 1.5

Fig. 1.6

Solution: See Fig. 1.6

+ X = 50 + 173.2 + 86.6 = 309.8 kN

+ Y = ­ 50 + 100 ­ 50 ­ 80 = ­ 80 or 80 kN

Resultant and Equivalent of Force Systems

R=

x 2 + y 2 = 309.8 2 + 80 2 = 320 kN, Fig. 1.6(a)

11

= tan­1

80 = 14.48° 309.8

To find the line of action of the resultant R, take moments about A (and applying Varignon's theorem), MA = R . a = X.y + Y.x, + y = 0, MA = 50 × 1.5 ­ 100 × 1 + 173.2 × 1.5 + 50 × 1 = 284.8 kN a=

M A 284.8 = = 0.89 m, R 320

x=

M A 284.8 = = 3.56 m 80 Y

Note: a = x sin = 3.56 sin 14.48° = 0.89 m Example 1.5. A rigid bar AB is subjected to a system of parallel forces as shown in Fig. 1.7. Reduce the given system of forces to an equivalent (a) single force, (b) force and moment at A, (c) force and moment at D, and (d) force and moment at B.

25 kN A 1m C 2m D 2m 15 kN

B

10 kN

20 kN

Fig. 1.7

Solution: (a) For an single force on AB: + Resultant R = 10 ­ 25 + 20 ­ 15 = ­ 10 or 10 kN To find its location : MA = R . a + MA = 25 × 1 ­ 20 × 3 + 15 × 5 = 40 kN.m a=

M A 40 = = 4 m, Fig. 1.8 R 10

Fig. 1.8 Single Force on AB To find force and couple at any point other than E, introduce equal and opposite forces R at the required point, which will not affect the equilibrium:

12

Engineering Mechanics

(b) A force R at E may be treated as another force at A and a couple as shown in Fig. 1.9(b)

Fig. 1.9. Force and Moment at A

(c) A force at E another force at D and a couple as shown in Fig. 1.10(b)

R 4m A 3m R R = 10 kN A 3m D 10 × 1 = 10 kN.m (b) B D 1m E (a) R = 10 kN

Fig. 1.10 Force and Moment at D

(d) A force at E another force at B and a couple as shown in Fig. 1.11(b)

R = 10 kN R 4m A (a) E 1m R 10 kN A (b) 10 × 1 = 10 kN.m = M B B

Fig. 1.11 Force and Moment at B

From Fig. 1.7, +

MB = 10 × 5 ­ 25 × 4 + 20 × 2 = ­ 10 i.e., 10 kN.m

Thus, the equivalent system at any point other than E is the same resultant force R and a moment which is equal to the algebraic sum of the moments of all the forces acting on the bar AB about that point.

Resultant and Equivalent of Force Systems

Note: A single force F (acting at A in Fig. 1.12) can be looked upon as a force of the same magnitude and direction shifted parallel to itself (by AB = a) accompanied by a couple of moment (M) = force × shift (= F . a) i.e., F at A = F at B + F . a

13

Contrariwise, a force, F and a couple of moment M (= F . a) acting at point (B) is equivalent to a single force (F) of the same magnitude and direction shifted parallel to its original line of action, the shift being =

M (i.e., = `a' in Fig. 1.12). F

Fig. 1.12

Example 1.6. Replace the system of a force and couple shown in Fig. 1.13 by a single force.

100 N 50 N

0.4 m 50 N x 100 N 100 N

Fig. 1.13

Fig. 1.14

Fig. 1.15

Solution: Since a force another force (shifted suitably) and a couple, Fig. 1.14, shift `x' to counteract the existing couple : 100 × x = 50 × 0.4 x = 0.2 m or 1.8 m from the support The equivalent system is as shown in Fig. 1.15. Example 1.7. Replace the system of a force and couple shown in Fig. 1.16 by a single force on the line AB.

120 N

r= 10 0m

A

96

A

O

50 mm

O C 60°

B

120 N

B

Fig. 1.16

Fig. 1.17

Fig. 1.18

F

m

60°

0

N

14

shift `x' to counteract the existing couple: 960 × x = 120 × 0.2 x = 0.025 m or 25 mm ; OC = The equivalent system is as shown in Fig. 1.18.

Engineering Mechanics

Solution: Since a force another force (shifted suitably) and a couple Fig. 1.17,

25 = 50 mm cos 60°

Example 1.8. An angle is subjected to a force-couple system as shown in Fig. 1.19. Reduce the system (a) to an equivalent system at A, (b) as a single resultant anywhere. Solution: (a) At A, X = 80 cos 45° = 56.56 N Y = 50 + 80 sin 45° ­ 150 = ­ 43.44 or 43.44 N R= at +

56.56 2 + 43.44 2 = 71.32 N

= tan­1

43.44 = 37.53° 56.56

M = ­ 300 ­ 80 sin 45° × 3 + 150 × 4 = 130.32 Nm

Hence, the equivalent system at A is a `force-couple system' of R and M, Fig. 1.20.

50 N

50 N

B

3m

300 Nm 80 N q R 45° 3m

300 Nm

80 N

2.3 m a q M R

A 3m

45° 1m 150 N

A

1m 150 N

Fig. 1.19

Fig. 1.20

(b) The force-couple system at A can be reduced to a single resultant R acting at B, Fig. 1.20, in the same direction at a distance, a=

130.32 Nm M = = 1.83 m R 71.32 N a 183 . = = 2.3 m cos cos 37.53°

AB =

Principle of Transmissibility. The point of application of a force may be transmitted along its Line of Action to another point, without changing its effect on any rigid body. This

Resultant and Equivalent of Force Systems

is known as the principle of transmissibility of force and it enables to regard a force acting on a rigid body as a Sliding Vector. In Fig. 1.21. the force P on the body can be applied at A or B, without changing the reactions at the bearing support at C or roller support at D.

P C Pinned RCH RCY RD B A D P

15

Roller

Fig. 1.21

PROBLEMS

1.1. Six forces are acting along the sides of a regular hexagon of side 100 mm as shown in Fig. P 1.1. Find the resultant and its distance from A. [185. 2N, 57.32° , 276.5 mm]

2 4 2

A 1 3m 2

2

Loads in kN 2 3m 1

3m

3m

B

Fig. P 1.1 intersect AB.

Fig. P 1.2 [15.46 kN, 75° ; 4.4 m from A]

1.2. For the loaded truss shown in Fig. P 1.2, find the resultant load and where its line of action will 1.3. Determine the resultant of the forces acting on the dam shown in Fig. P 1.3 and locate its intersection with the base AB. For safe design, the intersection should occur within the middle-third of the base. Is the design safe ?

2m MWL

[137.12 kN,

79.92° ; 2.15 m from B ; Yes]

Dam

P = 50 kN 2m A

c.g. TWL Q = 30 kN W = 120 kN 5m Base of dam 60° B 1m

Fig. P 1.3

16

1.4.

Engineering Mechanics

(a) Determine completely the resultant of the five forces shown in Fig. P 1.4. The squares are 100 mm × 100 mm. (b) Give an equivalent system at `O'.

Y

[0.8 N , Y = 67 m ; 0.8 N , 53.6 Nm

at O]

20 0

100

N

N

20

0

N

100

N

X O 282 N

Fig. P 1.4 1.5. A force of 500 N is acting at A in Fig. P 1.5 produces a moment of 1200 Nm about O. Find the y-intercept of the force. Give an equivalent system with the same force acting at `O'. [3 m ; 500 N and 1200 Nm at O]

Fig. P 1.5

Fig. P 1.6

1.6. A 50 N force is applied to a corner plate as shown in Fig. P 1.6. Determine an equivalent forcecouple system at A. [50 N, 3.08 Nm] 1.7. Find the single resultant of the forces acting on the pulley in Fig. P 1.7. Give its intercept with the axes. [1020 N ], (0.1, 0.02) m] 1.8. An electric light fixture weighing 15 N is hung from two strings from the roof and wall as shown in Fig. P 1.8. Find the tension in the strings. [7.8, 11 N]

Resultant and Equivalent of Force Systems

Roof 60° Wall 45°

17

Light 15 N

Fig. P 1.7

Fig. P 1.8

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