Momentosvvbvbnvbnvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvn

UCV - INGENIERÍA CIVL - ESTÁTICA
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1
Ingeniería Civil
Encontrar el momento de una fuerza con respecto
a un eje específico.
Definir el momento de un par.
Presentar métodos para determinar la resultante de
sistemas de fuerzas concurrentes.
Indicar como reducir una carga simple distribuida a
una fuerza resultante.
2
Ingeniería Civil
MMMMoooommmmeeeennnntttt ooooffff aaaa FFFFoooorrrrcccceeee aaaabbbboooouuuutttt aaaa SSSSppppeeeecccciiiiffffiiiieeeedddd AAAAxxxxiiiissss
For moment of a force about a point, the
moment and its axis is always perpendicular
to the plane.
A scalar or vector analysis is used to find the
component of the moment along a specified
axis that passes through the point.
3
Ingeniería Civil
Scalar Analysis
According to the right-hand rule, My is directed
along the positive y axis
For any axis, the moment is
Force will not contribute a moment
if force line of action is parallel or
passes through the axis.
4
Ingeniería Civil
Vector Analysis
For magnitude of MA,
MA = MOcosθ = MO
·ua
where ua = unit vector
In determinant form,
5
Ingeniería Civil
EEEExxxxaaaammmmpppplllleeee: Determine the moment produced by
the force F which tends to rotate the rod about
the AB axis.
6
DOCENTE: EDUAR RODRÍGUEZ 1

Ingeniería Civil
Solution:
Unit vector defines the direction of the AB axis
of the rod, where
For simplicity, choose rD
The force is
7
Ingeniería Civil
8
EEEEXXXXEEEERRRRCCCCIIIISSSSEEEE
Determine the moment of the force F about the
OA axis. Express the result as a Cartesian vector.
Ingeniería Civil
9
F4-14. determine the magnitude of the moment
of the force F=(300i-200j+150k)N about the OA
axis. Express the result as a Cartesian vector.
Ingeniería Civil
4-55. determine the
moment of the force
F about an axis
extending between
A and C. Express
the result as a
Cartesian vector.
((((11111111....55552222iiii++++8888....66664444jjjj))))llllbbbb....fffftttt 10
Ingeniería Civil
4-56. determine the moment produced by the
force F about segment AB of the pipe assembly.
Express the result as a Cartesian vector.
11
Ingeniería Civil
Couple
MMMMoooommmmeeeennnntttt ooooffff aaaa CCCCoooouuuupppplllleeee
◦ two parallel forces
◦ same magnitude but opposite direction
◦ separated by perpendicular distance d
Resultant force = 0
Tendency to rotate in specified direction
Couple moment = sum of moments of both couple
forces about any arbitrary point
12

Ingeniería Civil
SSSSccccaaaallllaaaarrrr FFFFoooorrrrmmmmuuuullllaaaattttiiiioooonnnn
Magnitude of couple moment
M = Fd
Direction and sense are determined by right hand
rule
M acts perpendicular to plane containing the forces
VVVVeeeeccccttttoooorrrr FFFFoooorrrrmmmmuuuullllaaaattttiiiioooonnnn
For couple moment,
M = r X F
If moments are taken about point A, moment of –F
is zero about this point
r is crossed with the force to which it is directed
13
Ingeniería Civil
Equivalent Couples
2 couples are equivalent if they produce the
same moment.
Forces of equal couples lie on the same plane
or plane parallel to one another.
14
Ingeniería Civil
Resultant Couple Moment
Couple moments are free vectors and may be
applied to any point P and added vectorially.
For resultant moment of two couples at point P,
MR = M1 + M2
For more than 2 moments,
MR = Σ(r X F)
15
Ingeniería Civil
EEEExxxxaaaammmmpppplllleeee:::: Determine the couple moment acting
on the pipe. Segment AB is directed 30°below
the x–y plane.
16
-111133330000jjjjNNNNmmmm
Ingeniería Civil
Take moment about point O,
M = rA X (-250k) + rB X (250k)
= (0.8j) X (-250k) + (0.66cos30ºi
+ 0.8j – 0.6sin30ºk) X (250k)
= {-130j}N.cm
Take moment about point A
M = rAB X (250k)
= (0.6cos30°iiii – 0.6sin30°k)
X (250k)
= {-130j}N.cm
17
Ingeniería Civil
18
A
44443333....9999NNNN....mmmm
B
D
EEEEjjjjeeeerrrrcccciiiicccciiiioooossss
C
Determine la
magnitud y la
dirección del
momento de par
que actúa sobre el
engranaje.

Ingeniería Civil
19
FP 4-19. Determine el momento de par resultante
que actúa sobre la viga.
777744440000NNNNmmmm
B
C
E D
Ingeniería Civil
F4-20 Determine el momento de par resultante que
actúa sobre la placa triangular.
20
D A
C
Ingeniería Civil
21
4-87 Determine la magnitud requerida de la fuerza
F, si el momento del par resultante sobre la viga
debe ser igual a cero.
Ingeniería Civil
F4-20 Determine el momento de par resultante que
actúa sobre el ensamble de tubos.
22
Ingeniería Civil
4.99. Determine la distancia d entre A yB de modo que el
momento de par resultante tenga una magnitud MR=20 N· m.
23
333344442222mmmmmmmm
Ingeniería Civil
4.101 Determine las magnitudes de los momentos de par M1,
M2 y M3 de modo que el momento de par resultante sea igual
a cero.
24

Ingeniería Civil
25
Simplification of SSSiiimmmpppllliiifffiiicccaaatttiiiooonnn ooofff aaaa ffffoooorrrrcccceeee aaaannnndddd ccccoooouuuupppplllleeee
ssssyyyysssstttteeeemmmm
Ingeniería Civil
SSSSiiiimmmmpppplllliiiiffffiiiiccccaaaattttiiiioooonnnn ooooffff aaaa ffffoooorrrrcccceeee aaaannnndddd ccccoooouuuupppplllleeee ssssyyyysssstttteeeemmmm
An equivalent system is when the external effects are the same
as those caused by the original force and couple moment
system
External effects of a system is the translating and rotating
motion of the body
Or refers to the reactive forces at the supports if the body is
held fixed
26
Ingeniería Civil
SSSSiiiimmmmpppplllliiiiffffiiiiccccaaaattttiiiioooonnnn ooooffff aaaa ffffoooorrrrcccceeee aaaannnndddd ccccoooouuuupppplllleeee ssssyyyysssstttteeeemmmm
Procedure for Analysis
1. Establish the coordinate axes with the
origin located at point O and the axes
having a selected orientation
2. Force Summation
3. Moment Summation
27
Ingeniería Civil
EEEExxxxaaaammmmpppplllleeee:::: A structural member is subjected to
a couple moment M and forces F1 and F2.
Replace this system with an equivalent
resultant force and couple moment acting at
its base, point O.
28
(-111166666666....66667777iiii-666655550000jjjj++++333300000000kkkk))))NNNNmmmm
Ingeniería Civil
Express the forces and couple moments as
Cartesian vectors.
29
Ingeniería Civil
Force Summation.
30

Ingeniería Civil
FFFFuuuurrrrtttthhhheeeerrrr SSSSiiiimmmmpppplllliiiiffffiiiiccccaaaattttiiiioooonnnn ooooffff aaaa FFFFoooorrrrcccceeee aaaannnndddd CCCCoooouuuupppplllleeee SSSSyyyysssstttteeeemmmm
Concurrent Force System
A concurrent force system is where lines of
action of all the forces intersect at a
common point O
31
Ingeniería Civil
Coplanar Force System
Lines of action of all the forces lie in the
same plane.
Resultant force of this system also lies in
this plane.
32
Ingeniería Civil
Parallel Force System
Consists of forces that are all parallel to the
z axis
Resultant force at point O must also be
parallel to this axis
33
Ingeniería Civil
Reduction to a Wrench
3-D force and couple moment system have an
equivalent resultant force acting at point O.
Resultant couple moment not perpendicular
to one another.
34
Ingeniería Civil
Example: The fix crane is subjected to three
coplanar forces. Replace this loading by an
equivalent resultant force and specify where
the resultant’s line of action intersects the
column AB and boom BC.
35
Ingeniería Civil
Force Summation
36

Ingeniería Civil
37
Solution
For magnitude of resultant force,
= + = + R Rx Ry
F F F
4.16
( )2 ( )2 (3.25)2 (2.60)2
=
kN
For direction of resultant force,

 = ÷

F q
= - -
Ry
F
tan 1 tan 1
38.7
2.60
3.25
=

÷

÷

 
Rx
Ingeniería Civil
Moment Summation
Summation of moments about point A,
38
Ingeniería Civil
Moment Summation
Principle of Transmissibility
39
Ingeniería Civil
-444477770000NNNNmmmm 40
Ingeniería Civil
41
Ingeniería Civil
-444477770000NNNNmmmm 42

Ingeniería Civil
43
Ingeniería Civil
44
Ingeniería Civil
(-
15i+225j)Nm
45
FR=-180k-80k+50k=(-210k)N
Mro=(1.25i)x(-180k)+(1.25i+0.5j)x(-80k)
+(2.0i+0.5j)x(50k)
=225j+100j-40i-100j+25i
=(-15i+225j)Nm
Ingeniería Civil
46
Ingeniería Civil
47
Ingeniería Civil
48

Ingeniería Civil
49
Ingeniería Civil
RRRReeeedddduuuuccccttttiiiioooonnnn ooooffff aaaa SSSSiiiimmmmpppplllleeee DDDDiiiissssttttrrrriiiibbbbuuuutttteeeedddd LLLLooooaaaaddddiiiinnnngggg
Large surface area of a body may be subjected to
distributed loadings
Loadings on the surface is defined as pressure
Pressure is measured in Pascal (Pa): 1 Pa = 1N/m2
Uniform Loading Along a Single Axis
Most common type of distributed
loading is uniform along a
single axis.
50
Ingeniería Civil
Magnitude of Resultant Force
Magnitude of dF is determined from differential area
dA under the loading curve.
For length L,
Magnitude of the resultant force is equal to the total
area A under the loading diagram.
51
Ingeniería Civil
Location of Resultant Force
MR = ΣMO
dF produces a moment of xdF = x w(x) dx
about O
For the entire plate,
Solving for
52
Ingeniería Civil
EEEExxxxaaaammmmpppplllleeee:::: Determine the magnitude and location
of the equivalent resultant force acting on the
shaft.
53
Ingeniería Civil
For the colored differential area element,
For resultant force
54

Ingeniería Civil
For location of line of action,
Checking,
55
Ingeniería Civil
56
F4.33337777 Determina la fuerza resultante y
determine el punto donde actúa medida desde A
44440000....5555kkkkNNNN yyyy 1111....22225555mmmm
Ingeniería Civil
FFFFPPPP 4.44441111 Determine the resultant force and specify
where it acts on the beam measured from A
57
Ingeniería Civil
FFFFPPPP 4.44440000 Determina la fuerza resultante y
58
1111555555550000llllbbbb yyyy 5555....00003333fffftttt
Ingeniería Civil
F4.44441111 Determina la fuerza resultante y
59
Ingeniería Civil
60
111166660000NNNN yyyy 3333....2222mmmm

Ingeniería Civil
61
2222....00006666mmmm
Ingeniería Civil
62
4-146. La distribución de la carga de suelo sobre el fondo de
una losa de edificio se muestra en la figura. Reemplace esta
carga por una fuerza resultante equivalente y especifique su
ubicación, medida desde el punto O.
Ingeniería Civil
63
4·152. El viento ha depositado arena sobre una plataforma
de manera que la intensidad de la carga puede ser
aproximada por la función w = (0.5) N/m. Simplifique esta
carga distribuida a una fuerza resultante equivalente y
especifique la magnitud y la ubicación de la fuerza, medida
desde A.
1111222255550000NNNN yyyy 8888mmmm
Ingeniería Civil
64
4·153 El concreto húmedo
ejerce una presión distribuida a
lo largo de la pared de la
cimbra. Determine la fuerza
resultante de esta distribución
y especifique la altura h en que
debe colocarse el puntal de
soporte de modo que esté
posicionado sobre la línea de
acción de la fuerza resultante.
El ancho de la pared es 5m.
Ingeniería Civil
65
4·156. Reemplace la carga por una fuerza equivalente
resultante y un momento de par equivalente actuando
en el punto B.
555577776666....66663333llllbbbb,,,, 2222888800000000llllbbbbppppiiiieeee
Ingeniería Civil
66
Información tomada del texto de:
HIBBELER, Rusell. Ingeniería Mecánica, Estática. 12 a ed

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