Sem-3^rd

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BY – ER. ABHISEK MOHANTY

(Lect. In the civil engineering department)

AY-2021-2022

CHAPTER/MODULE –Basic Concepts

SUB-Structural Mechanics

• Force- The push or pull on an object with mass causes it to change its velocity. Force is an external agent capable of changing a body's state of rest or motion. It has a magnitude and a direction.

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• Force is an external agent capable of changing a body’s state of rest or motion. It has a magnitude and a direction. The direction towards which the force is applied is known as the direction of the force, and the application of force is the point where force is applied.

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• The Force can be measured using a spring balance. The SI unit of force is Newton(N).

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• Common symbols:

SI unit: Newton

In SI base units: kg·m/s2

Other units: dyne, poundal, pound-force, kip, kilo pond

Derivations from other quantities: F = m a

Dimension: LMT-2

What are the Effects of Force?

• In physics, motion is defined as the change in position with respect to time. In simpler words, motion refers to the movement of a body. Typically, motion can either be described as:

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Change in speed

Change in direction

The Force has different effects, and here are some of them.

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Force can make a body that is at rest to move.

It can stop a moving body or slow it down.

It can accelerate the speed of a moving body.

It can also change the direction of a moving body along with its shape and size.

Unit of Force

• In the centimeter gram second system of unit (CGS unit) force is expressed in dyne.
• In the standard international system of unit (SI unit) it is expressed in Newton (N).

Types of Force

Force is a physical cause that can change an object’s state of motion or dimensions. There are two types of forces based on their applications:

1. Contact Force
2. Non-Contact Force

Force-

Moment-The turning effect of the force on the body on which it is acting is measured by the moment of a force. The moment of a force depends on the magnitude of the force and the distance from the axis of rotation.

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The moment distribution method is a structural analysis method for statically indeterminate beams and frames developed by Hardy Cross. It was published in 1930 in an ASCE journal. The method only accounts for flexural effects and ignores axial and shear effects.

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Support connection-Imagine a beam extending from the wall. How much weight can the beam handle before it breaks away or falls ‘off’ the wall? It depends on the way it’s attached to the wall. We model these real world situations using forces and moments.For example, the grand canyon skywalk lets people walk out over the grand canyon. You want to be sure that the skywalk is so the people on it are safe

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We call the skywalk a cantilever beam and turn the real world beam into a 2d model with constrains. So we can use the same terminology, it is a fixed constraint, preventing horizontal movement, vertical movement, and rotation.

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Reaction forces and moments are how we model constraints on structures. They are external forces. There are 3 different kinds of constraints we will focus on in this course and they each have different reaction forces and moments:

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 Pinned (Frictionless)                      

• Two reaction forces acting perpendicularly in the x and y directions.
• Pinned constraint and then its free body diagram shown:

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• Moment rotating about fixed constraint (usually a wall), use right hand rule to find its direction
• This is also called a cantilever beam.
• Fixed constraint and then FBD shown

• Fixed-
• Two reaction forces acting perpendicularly in the x and y directions

• Single reaction force acting in the y direction
• No moment is created
• This can be the ground that the object rests on as well
• Free body diagram shown for roller

Roller (there are multiple kinds

• Notice that the Fixed restraint is the most restrictive and the roller is the least restrictive. You put a force to show how the restraint restricts motion. The roller only keeps the object from moving vertically, so there is only 1 force. The pinned restraint doesn’t allow horizontal or vertical movement, hence the two forces. The fixed beam restricts vertical translation, horizontal translation, and rotation, so there is a moment and two forces. Note that this applies only to 2d restraints.

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Roller Support

A roller support allows rotation about any axis and translation (horizontal movement) in any direction parallel to the surface on which it rests. It restrains the structure from movement in a vertical direction. The idealized representation of a roller and its reaction are also shown in Table 3.1.

Rocker Support

The characteristics of a rocker support are like those of the roller support. Its idealized form is depicted in Table 3.1.

Link

A link has two hinges, one at each end. It permits movement in all direction, except in a direction parallel to its longitudinal axis, which passes through the two hinges. In other words, the reaction force of a link is in the direction of the link, along its longitudinal axis.

Fixed Support

A fixed support offers a constraint against rotation in any direction, and it prevents movement in both horizontal and vertical directions.

The two conditions that are important for equilibrium:-

• The sum or resultant of all external forces acting on the body must be equal to zero.
• The sum or resultant of all external torques from external forces acting on the object must be zero.
• The two conditions given here must be simultaneously satisfied in equilibrium. In essence, for an object to be in equilibrium, it should not experience any acceleration (linear or angular). So both the net force and the net torque on the object must be zero.

Particles and Rigid Bodies

Particles.  A particle is a body whose size does not have any effect on the results of mechanical analyses on it and, therefore, its dimensions can be neglected.  The size of a particle is very small compared to the size of the system being analysed.

Rigid body.  A body is formed by a group of particles.  The size of a body affects the results of any mechanical analysis on it.  A body is said to be rigid when the relative positions of its particles are always fixed and do not change when the body is acted upon by any load (whether a force or a couple).  Most bodies encountered in engineering work can be considered rigid from the mechanical analysis point of view becase the deformations that take place within these bodies under the action of loads can be neglected when compared to other effects produced by the loads.  All bodies to be studied in this book are rigid, except for springs.  Springs undergo deformations that cannot be neglected when acted upon by forces or moments.  For the analyses ini this book, only the effects of the deformations of springs on a rigid body interacting with the springs are considered but the springs themselves will not be analysed as a body.

FREE BODY

Mechanical analysis of a structure, in general, is started by applying Newton’s law to the whole structure, or to part of the structure. To see what actually happens to any particular part of a structure, that part has to be isolated from the other parts of the body. The concept of isolating the part which is the target of analysis, is a very important concept in mechanical analysis. The part which is isolated is called a free body.

Free-body Diagram (FBD)

The isolation of a mechanical system is achieved by cutting and isolating the system from its surroundings.  The isolation enables us to see the interactions between the isolated part and the other parts.   The part which has been cut (imaginarily), forms a free body.  A diagram that portrays the free body, complete with the system of external forces acting on it due to its interaction with the parts which have been removed, is called the free-body diagram (FBD) of the isolated part.

The FBD of a body system shows all loads acting on the external boundary of the isolated body.  The loads include active forces applied on the free body by the parts removed from it.  Note that the internal forces acting on a structure becomes external forces when exposed by the imaginary “cutting” process carried out to form the free body.

 Procedure To Draw A FBD

A FBD is drawn by following the following steps in order:

1. Show the body which has been isolated from its surroundings by drawing its outline.

2. Show, on the drawing, all loads acting on the body. The loads consist of the active forces couples (which cause the tendency to move) and the reactive forces and couples (caused by any constraint and tend to prevent motion).

3. Indicate the magnitude and sense of each load.  Use letters to indicate unknown magnitudes.  For unknown sense, choose the sense arbitarily but every pair of interacting loads must obey Newton’s third law.

4. Show all dimensions which are necessary to calculate moments.

It must be stressed here that only a correctly drawn FBD will produce the correct solution.  The most important step taken to solve problems in mechanics is drawing the FBD.

 EQUILIBRIUM EQUATIONS FOR A RIGID BODY-

A rigid acted upon by any applied load will tend to translate dan rotate about a particular axis.  The tendency to translate is due to the action of the resultant force on the body and the tendency to rotate is due to the action of the resultant couple.  Equilibrium will occur on the body if the resultant force, as well as the resultant couple, are both zero.  Mathematically, equilibrium is determined by the conditions:

∑F = 0                                                                                 

∑M = 0

Where ∑F is the resultant force and ∑M is the resultant moment.

The first expression of Equation 3.1 says that, for equilibrium, the sum of all forces acting on the body is zero.  The second expression says that the sum of the moment about any axis must be zero. The two conditions manifested by the Equation are necessary and sufficient for equilibrium.  It is said to be necessary because fulfilling the first condition only will result in equilibrium in terms of rotation.  Similarly, if only the second condition is fulfilled, equilibrium occurs in terms of translation.  The two conditions are said to be sufficient because equilibrium takes place when both conditions are met without the need for any additional conditions.

• Force- It is an agent which produces or tends to produce motion in a body. It can change or retard the motion of a body. It can give rise to internal stresses in the body. Its unit in S.I is newtons (N) or kg-f in MKS. 1 kg-f = 9.81 N.

• Lami’s Theorem- This law states that if three coplanar forces acting at a point in equilibrium, then each force is proportional to the sine of angle between the other two forces.

Mathematically, ^P/_{sin α} = ^Q/ _{sin β} = ^R/ _{sin γ}

• Moment of a force- It is the product of force and perpendicular distance of a point               about which the moment is required and the line of action of force. It causes turning effect. M = P X l

• Couple- Two equal and opposite forces whose lines of actions are different form a couple. It is the product of force and the perpendicular distance between the lines of actions of the two forces. It produces a rotating motion on the body.

Moment of a couple = P x

• Centre of gravity- The point through which the whole mass of the body acts without taking the position of the body into consideration.
• Centre of gravity of a uniform rod is its middle point.
• C.G of a rectangle lies at the intersection of its diagonals.
• C.G of a triangle lies at the intersection of the three medians.
• C.G of a trapezium with parallel sides a and b lies at a distance of ^h/₃(^2a+3/_a+b).
• C.G of a semi circle lies at a distance of ^4r/3п from its base measured along the vertical radius.
• C.G of a hemisphere lies at a distance of ^3r/₈ from its base measured along the vertical radius.
• C.G of a right circular solid cone lies at a distance of ^h/₄ from its base measured along its vertical axis.

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• Moment of Inertia (I) - It is the moment of the moment or second moment of mass or area of body.

I = m k², where k is the radius of gyration.

o   The moment of inertia of a thin disc of mass m and radius r about an axis passing through its C.G and perpendicular to the plane of disc is given as, I = mr²/2

o   The moment of inertia of a thin disc of mass m and radius r about its diameter is given as, I = mr²/4

o   The moment of inertia of a thin rod of mass m and length l about an axis passing through its C.G and perpendicular to its length is given as, I = ml²/12

o   The moment of inertia of a thin rod of mass m and length l about a parallel axis through one end of the rod is given as, I = ml²/3

o   The moment of inertia of a rectangular section having width b and depth d is given as, I_XX = bd³/12

I_YY = db³/12

o   The moment of inertia of a hollow rectangular section having width b and depth d is given as, I_XX = BD³/12 - bd³/12

I_YY = DB³/12 - db³/12

o   The moment of inertia of a circular section of diameter D is given by, I_XX = I_YY = пD⁴/64

o   The moment of inertia of a triangular section of height h about an axis passing through its C.G and parallel to its base is given as,

I = bh³/36

o   The moment of inertia of a triangular section of height h about its base is given as,

I = bh³/12.