THEORY OF STRUCTURES-I [B. ARCH.]

THEORY OF STRUCTURES-I [B. ARCH.]

1. Eng. Kishor K. Ade Lecture 2 Theroy of Structur-I
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THEORY OF STRUCTURES -I [B. ARCH.]

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UNIT-I : FORCES
Applied Mechanics:
Applied Mechanics is considered the backbone of all engineering
programs. It deals with the basic concepts of force, moment and its
effect on bodies at rest or in motion. It helps us understand how
different bodies behave under the application of different kinds of
loads or forces.
The Branch of Science which deals with the study of different laws
of Mechanics as applied to the Solution of Engineering Problems is
called Applied Mechanics.
Mechanics can broadly be classified into two branches
1. Statics i.e. study of bodies at rest.
2. Dynamics i.e. study of bodies in motion.
Rigid Bodies are those which do not change their size and shape under
the effects of forces acting over them.

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Deformable Bodies
Deformable Bodies are those that undergo deformation in size and
shape under the effects of forces acting over them. In the Real world no
solid body is PERFECTLY RIGID as everybody changes its size and
shape under the effects of forces but many a times the deformation is
negligible enough for the body to be considered Rigid. Good
Knowledge of Materials is essential in the study of Mechanics of
Deformable Bodies.
As can be seen the study of Mechanics begins with the study of
Forces.
Forces may be of many types like gravitational, magnetic, frictional
etc. but we will herein deal only with Static Forces which are
contact forces, but do not cause relative acceleration in the bodies
in question.

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Force
In Physics, a force is any external effort or agency that causes an object to
undergo a certain change, either concerning its movement, direction, or
geometrical construction.
As can be seen the study of Mechanics begins with the study of Forces. Forces
may be of many types like gravitational, magnetic, frictional etc. but we will
herein deal only with Static Forces which are contact forces, but do not cause
relative acceleration in the bodies in question.
Characteristics of a Force:These are the elements by which a
force is fully represented.
1.A force has magnitude measured in kN. Or N 1 kN. = 1000 N.
2.A force has direction and the direction is measured in angle. Angle is
measured with the Horizontal Zero Degrees in the Standard Co–ordinate
system.
3. A force has a point of application.
4.A force has a sense of pull or push.

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Effects of different forces on a Building:

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Effects of a Force
A force may cause a body to change its state of motion.
a.A body in motion can be brought to rest.
b.A body in uniform motion may be accelerated or retarded.
c.A body in a state of rest can be made to move.
 A force may change the shape of a body.
a.It may compress a body and thus cause internal stresses(tensile) to be
set up within the body.
b.It may cause tension in a body and thus cause internal
stresses(compressive) to be set up within the body.
c.It may cause both the above.
d.It may twist, bend, stretch or even distort the body.

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Principle of Transmissibility of Force
The principle states that when a force acts upon a body, it’s effect is the
same whatever point in its line of action is taken as the point of
application provided that the point is connected with the rest of the
body in the same invariable manner.



Effect of the force on the body is unchanged whether the point of
application is 1 or 2 or 3.

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Principle of Superposition of Forces
The effects of two forces acting simultaneously on a body are the
same as the effect of the two forces acting independently.
For a structure, the load effects caused by two or more loads are the
summation of the load effects caused by each load independently.

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Representation of a Force






Vector Representation: A force can be represented graphically by a
vector as shown above. Hence the force is Force AB of 60N @ 45˚
upward.
 Bow’s Notation: A force can be designated by two capital letters
written one on either side of the force as shown above. So force P2 is
Force CD and force P1 is Force AB

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Force Systems
When a number of forces are acting together we call them a system of forces.
A.Coplanar Force Systems: When a system of forces is such that all
forces lie in one plane then it is called a Coplanar Force System.
Collinear force system: Forces in this system lie along a single line.
Concurrent force system: Forces in this system intersect at a single point.
Non Concurrent Parallel force system: Forces parallel to each other.
Non Concurrent Non Parallel force system: Forces are coplanar bur not
parallel.
Non Coplanar Force Systems: When forces in a system do not lie in
one plane.
Concurrent force system: Forces in this system intersect at one point.
Non Concurrent force system: Forces in this system do not intersect at one
point.

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Composition of Forces:
A Resultant force is a single force which can replace two or more
forces and produce the same effect on the body as the forces. Many
forces can be composed into one single Resultant Force and this is
known as Composition of forces.




When a single force acts on a body which is free to move, the body
moves in the direction of the force and the distance travelled by the
body in unit time is directly proportional to the magnitude of the force.
So when many forces are acting on the body, the body moves in a
direction of the Resultant and the distance travelled is proportional to
the magnitude of the Resultant.

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Resolution of a Force:
As many forces can be composed into one single Resultant, so can a
single Force be replaced by two forces acting in directions which
will produce the same effect as the single force. This breaking of the
force into two forces is called the Resolution.
A Force can be resolved into
1.Two mutually perpendicular components.
2.Two non-perpendicular components.
When the force is resolved into two mutually perpendicular
components, generally the two components are horizontal and vertical.
The Horizontal component is denoted by F
H and Vertical component is
denoted by F
V.

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As seen in the above figure,
F
H= F Cosθ (Horizontal Component) &
F
V= F Sinθ (Vertical Component)

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Triangular law of forces:
The law states that, If three forces acting at a point are in
equilibrium, they can be represented in magnitude and direction by
the sides of a triangle taken in order.





When the triangle law is applied to three forces in equilibrium, the
resulting triangle will be a closed figure, i.e. all the vectors will be
head-to-tail. Such a vector diagram implies that the resultant force is
zero.

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Law of polygon of forces:
Law of polygon of forces states that, If a number of forces acting
simultaneously on a particle/ body (concurrent forces) can be
represented in magnitude and direction by the sides of a polygon
taken in order, their resultant may be represented in magnitude
and direction by the closing side of the polygon taken in opposite
order.





W.r.t. the above figure,F1, F2, F3 & F4are the forces and R =
Resultant force

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Conditions of equilibrium:
For concurrent forces system: ΣF
H= 0, ΣF
V= 0
For Non-concurrent forces system: ΣFH= 0, ΣF
V= 0, ΣM = 0
Couple:
A couple is a pair of forces, equal in magnitude, oppositely directed,
and displaced by perpendicular distance or moment. The simplest kind
of couple consists of two equal and opposite forces whose lines of
action do not coincide. This is called a simple couple. The forces have
a turning effect or moment called a torque about an axis which is
normal(perpendicular) to the plane of the forces. The SI unit for the
torque of the couple is KNm.
Example: Steering wheel, Bicycle handle or pedaling.

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Supports:
Definition: The structural element that bears the weight / load of the
spanning member and keeps it upright, providing resistance from
gravity.

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Types of supports:
1.Simple support– A support providing only vertical resistance to the
ends of a spanning members, while provides horizontal & rotational
translation. E.g.–A plank over a trench. & No. of reactions-1(Vertical
only)


2. Hinged / pinned support–A support providing vertical & horizontal
resistance to the ends of a spanning member, while provides rotational
translation only. E.g.–A horizontal member, singly nut-bolted / pinned
at the ends & No. of reactions-2 (Vertical& horizontal)

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CONT…

3. Roller support–A non-frictional support providing only vertical
resistance to the ends of a spanning member, while provides horizontal &
rotational translation. If spiked shoes are compared with roller skates,
spiked shoes provide horizontal resistance in the form of friction, while
roller skates don’t, because of wheels; hence you can roll on them. E.g.–
Flyover / bridge beams & No. of reactions-1(Vertical only)


4. Fixed support– A support providing vertical, horizontal & rotational
resistance to the end of a spanning member. E.g.–Beams in a RCC
structure. & No. of reactions-3 (Vertical, Horizontal & rotational)

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Statically determinate structures
Structures which can be analysed just by use of basic equilibrium
equations (ΣF
H= 0, ΣF
V= 0,ΣM = 0) are called as statically determinate
structures. E.g. Simply supported beam, Cantilever beam, Overhang
beam.
Statically indeterminate
Structures which cannot be analysed just by use of basic equilibrium
equations (ΣF
H= 0, ΣF
V= 0,ΣM = 0) and require special equations like
compatibility relations, are called as statically indeterminate structures.
E.g. Fixed beam, Propped cantilever beam, Continuous beam, Frames /
Trusses.
Degree of indeterminancy
The number of reactions excess over the number of equilibrium
equations available is called as Degree of indeterminancy. Thus, Degree
of indeteminancy = No. of reactions– no. of equilibrium equations

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Example of a fixed beam (Statically indeterminate,Degree of
indeterminancy =3)







No. of a reactions = 3 + 3 = 6
 No. of equilibrium equations = 3 (ΣF
H= 0, ΣF
V= 0,ΣM = 0)
Thus, Degree of indeteminancy = 6–3 =3

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CONT…
Example of a propped cantileverbeam (Statically
indeterminate, Degree of indeterminancy = 1)







No. of a reactions = 3 + 1 = 4
 No. of equilibrium equations = 3 (ΣF
H= 0, ΣF
V= 0,ΣM= 0)
Thus, Degree of indeteminancy = 4–3 =1

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Clear span & effective span:
The clear horizontal distance between supports is called as clear span of
the beam / slab. The centre to centre horizontal distance between
supports is called as effective span of the beam / slab. Thus Effective
span = ½ thickness of support 1 + clear span + ½ thickness of support 2




Dead load calculation:
Cross sectional area x density of material

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Live load calculation:
Live load is assumed as per occupancy of the building, which depends
on the footfall. In terms of live load, staircases and balconies are more
loaded as compared to the normal floors.


Load transfer:
W.r.t. the sketch below,
Let w
1 = self wt. of slab
w
2= self wt. of beam B1
w
3= Total self wt. of slab & Beam B1
w
4= Total self wt. on Beam B2

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The slab is connected to the beam B1 throughout the entire span (L2) of the
beam. Hence the slab transfers the load (w1) to the beam B1 in the form of
an UDL. This beam transfers the load (w2= slab load + Beam B1 self wt.)
to the peripheral beam B2 in the form of a point load equal to W1= w3x L2/
2. The peripheral beam B2 transfers the load to the column in the form of a
point load equal to (W1/2 + w4xL3/2)

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Problems for practice–
Unit II& III-Centre of
gravity–Moment of
Inertia
For the right given
sections
 a. Find out C.G. along
both axes.
b. Also calculate Moment
of Inertia along both axes
passing through its C.G.



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Problems for practice–UnitV–Shear force & Bending
Moment

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