Applied mechanics

1. CHAPTER No. 3
BEAMS AND SUPPORT REACTIONS
CONTENT OF THE TOPIC:
- Definition of statically determinate beam
o Types of beam supports
o Types of Beams
o Type of Loading
- Procedure To Find The Support Reactions Of Statically Determinate Beam
- Compound beam
- Concept of virtual work
Definition of Beam:
A beam is horizontal or inclined member carrying transverse or inclined loads and supported at
ends or anywhere. It is a structural member for the frame or structures of steel or concrete which
has one dimension (length) considerably larger than the other two dimensions.
If support reactions can be determined by using the conditions of equilibrium only, then the beam
is known as statically determinate beam. If support reactions cannot be determined by using the
conditions of equilibrium only, then the beam is known as statically indeterminate beam.
Definition of span:
Centre to centre distance between the two end supports is called span.
Types of beam supports:
1) Simple support:
It is a theoretical case in which the ends of the beam are simply supported or rested over
the supports. The reactions are always vertical as shown in Fig.1 below
Fig.1 Simple Support
It opposes downward movement but allows rotation and horizontal displacement or
movement.
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2. 2) Pin or hinged Support:
In such case, the ends of the beam are hinged or pinned to the support as shown in Fig.2
below.
Fig.2 (A) Hinged Support Fig.2 (B) Hinged Support
The reaction may be either vertical or inclined depending upon the type of loading. If the
loads are vertical the reaction is vertical as shown in Fig. 2 (A) and when the applied
loads are inclined the reaction is inclined as shown in Fig. 2 (B).
The main advantage of hinged support is that the beam remains stable i.e. there is only
rotational motion round the hinge but no translational motion of the beam i.e. hinged
support opposes displacement of beam in any direction but allows rotation.
3) Roller Support:
In such cases, the end of the beam is supported on roller as shown in Fig. 3 below.
Fig. 3 Roller Support Fig. 4 Fixed Support
The reaction is always perpendicular to the surface on which rollers rest or act as shown
in Fig. 3. The main advantage of the roller support is that, the support can move easily in
the direction of expansion or contraction of the beam due to change in temperature in
different seasons.
4) Fixed Support:
It is also called as Built-in-supports. It is rigid type of support. The end of the beam is
rigidly fixed in the wall as shown in Fig. 4 below.
It produces reactions Ra in any direction and a moment Ma as shown in Fig. 4 above.
Types of Beams:
The types of beam are depends upon the types of supports over which it will rest.
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3. 1) Simply Supported Beam:
A beam supported or rested freely on the supports at its both ends is known as simply
supported beam. Such beam can support load in the direction normal to its axis. The
support reactions are always vertical (as shown in the Fig. 5 Ra and Rb).
Fig. 5 Simply Supported Beam
2) Cantilever Beam:
One end of the cantilever beam is rigidly fixed in the wall as shown in Fig. 6 below. Such
supports are known as fixed support (as explained in above). It is a type of rigid
support. It produces reactions Ra in any direction and a moment Ma as shown in Fig. 6.
Fig. 6 Cantilever Beam
3) Overhang Beam:
The beam is supported on hinged support and roller support. The beam has overhang
on one end i.e. to the right or left of the beam and on both sides as shown in Fig. 7
below.
Fig. 7 (a) Overhang Beam (overhang on both sides)
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4. Fig. 7 (b) Overhang Beam (to right) (c) Overhang Beam (to left)
4) Continuous Beam:
Such beams are supported at more than two points as shown in Fig.8 below. It is also
called as multi-span beam.
Fig. 8 Continuous Beam
Type of Loading:
1) Concentrated load or Point Load:
A Concentrated load or Point Load is one which is considered to act at a point, as
shown in Fig.9 below. Following Fig. 9 shows three concentrated forces F1, F2 and F3
acting on a simply supported beam.
Fig. 9 Point Loads acting on the beam
2) Distributed loads:
There are three types of distributed loads:
a) Uniformly distributed Load
b) Uniformly varying Load
c) Non-Uniformly distributed Load
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5. a) Uniformly distributed Load:
If a load which is spread over beam in such a manner that rate of loading ‘w’ is
uniform along the length (i.e. for each unit length the magnitude of load is uniform) as
shown in Fig.10 below.
Fig. 10 uniformly distributed Load acting on the beam
Note: If ‘w’ N/m is the Uniformly distributed Load on beam AB as shown in Fig. 11
above then the total load say (W=w x l) is acting at the midpoint say c as shown in
Fig. 11 below.
Fig. 11 Conversion of U.D.L. into Point Load
b) Uniformly varying Load
A Uniformly varying Load is one which is spread over a beam in such a manner that
rate of loading varies from point to point along the length of the beam as shown in
Fig. 12 below.
Fig.12 Uniformly varying Load acting on the Beam
For such loading it is zero at one end i.e. end A and increases uniformly at other end
i.e. end B.
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6. Note: The equivalent Concentrated or point load for this case is the area of the
triangle or average loading intensity multiplied by the length, which is acting at a
distance of (2/3) l form A or (1/3) l from B of the support as shown in Fig. 13 below.
Fig. 13 Conversion of U.V.L. into Point Load
c) Non-Uniformly distributed Load
If the load distributed on the beam is such that the load per unit length is not constant,
then it is called as Non-Uniformly distributed Load. Different types of loading are
shown in the Fig. 14 below
Fig. 14 (a)
Fig. 14 (b)
Procedure To Find The Support Reactions Of Statically Determinate Beam
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7. If support reactions can be determined by using the conditions of equilibrium only, then the
beam is known as statically determinate beam. If support reactions cannot be determined by
using the conditions of equilibrium only, then the beam is known as statically indeterminate
beam.
1) Such problems are treated as the problem to be a co-planar, non-con-current equilibrium
force system.
Following equilibrium conditions are used
ΣM = 0, ΣFy = 0, ΣFx = 0
2) When the beam is simply supported, the reactions at the supports are vertically upwards.
3) Taking summation of moments (either at A i.e. ΣMA or either at B i.e. ΣMB) of all given
forces about any support, assuming the reaction at the other support as vertically upwards.
Equating algebraic sum of these moments to zero i.e. ΣM = 0 and calculate unknown reaction
RA or RB and then using equations ΣFy = 0 find the reactions of the other support.
4) When one of the reaction as pinned or hinged and other support is on roller, the reaction at
the roller support is always perpendicular to the roller line. Find the reaction at the roller
support by taking moment of all the forces about hinge support and equate it to zero.
5) Then using equations ΣFy = 0 and ΣFx = 0 find the reactions at the hinge support.
6) To find the reaction at hinge support in magnitude and direction, use the equation
R = √ΣFy 2 + ΣFx2
And
ΣFy
ΣFx
θ = tan -1 (
)
Concept of virtual work:
1) Consider a force (P) is acting on a body which get displaces through a distance (s) due to
applied force.
Then,
Work done = Force X Displacement
W = F.s
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8. 2) But if the body is in equilibrium, under the action of a system of forces, then the work
done is zero.
3) If we assume that the body, which is in equilibrium, undergoes a small imaginary
displacement (virtual displacement) some work will be imagined to be done. Such
imaginary work is called as virtual work. This concept is useful to find out the unknown
forces in the structures.
Principle of virtual work:
“ If system of forces acting on a body (or a system of bodies) be in equilibrium and the system to
be imagined to undergo a small displacement consistent with the geometrical conditions, then the
algebraic sum of the virtual works done by all the system is zero”.
i.e. mathematically,
ΣW = 0
Types of virtual work:
1) Linear virtual work:
If a force (F) causes a displacement (virtual displacement) in its direction of line of
action, then its virtual work is given as,
WV = F x δ
Sign convention:
Upward forces are considered as positive, while downward forces are considered as
negative
QUESIONS
1. What are the different types of beam support? Explain the reactions exerted by each type
of support.(4 Mks) or
2. What are the different types of supports? Indicate with neat sketch of reactions offered by
them.
3. Explain the concept of virtual work? (3 Mks)
Chapter No. 3 Beam Page 8