Quantities

In structures there are 2 group of quantities:
  • Scalar quantities and
  • Vector quantities
Scalar quantities are defined by a number e.g.
10 kilograms (10 kg)7 square metres (7 m2)
5 metres (5 m)3 cubic metres (3 m3)
25 seconds (25 s)2 litres (2 L)

Vector quantities need more description than magnitude alone. Forces are vector quantities. These quantities possess magnitude and direction.These are known as vector quantities and include such things as force, displacement, velocity and acceleration. Vector quantities possess both magnitude and direction and can be represented by a line drawn to scale in a certain direction with an arrowhead indicating its sense. This line is called a vector (see Figure 1.2).

Scalars are added algebraically (for example, the distances of 5 m + 7 m are simply added to give 12 metres); however, vectors must be added geometrically because their line of action and direction are just as important as their magnitude. When vectors are added, the solution is known as the resultant. This is the one vector which could replace all the others without changing the overall effect.
For instance, if a man walked 1000 metres east from A to B and then turned and walked 1000 metres south from B to C, he would arrive at the same place as if he had walked 1414 metres south-east (i.e. from A directly to C) as shown in opposite Figure.

If the vector diagram ABC is drawn to a suitable scale, the resultant vector AC will be to the same scale, and the magnitude of this resultant vector can easily be obtained by measuring the length of the vector (line) AC.

This method works equally well for other vector quantities. For example, a force has magnitude, line of action, direction, and point of application and all of these factors can be represented by a single line as shown in the following diagram

Forces

An interaction that causes an acceleration of a body is called a force, in general terms, anything that pushes or pulls. A force is needed to keep a body (car) moving at constant velocity. Similarly, a body is in its natural state when it is at rest. For it to move it must be propelled in some way by a push or pull. Otherwise, it would naturally stop moving. (Try to push over a brickwall by leaning against it. The wall does not move (hopefully) but you still apply a force to the wall and the wall reacts with the same force back, otherwise the system would not be in balance (equilibrium).)

A force causes the acceleration of a body. It tends to change the state of rest or uniform motion of the body upon which its acts.

In Structures we dealt with statics; all building components in a static equilibrium. The requirements for a body to be in equilibrium are as follws:

  1. The vector sum of all the external forces that act on a body must be zero
  2. The vector sum of all the external moments that act on a body must be zero

These vector equations, are each equivalent to three independent scalar equations, one for each direction of the coordinate axes:

  1. Sum of all vertical forces must be zero
  2. Sum of all horizontal forces must be zero
  3. Sum of all rotational (moments) forces must be zero
We will deal with these equation in more detail in the following sections.

Mass, force and gravity

Mass is a measure of the inertia of a body. In other words the resistance that the body offers to having its speed or position changed by the application of a force.
We know that a given force produces different magnitudes of acceleration. If you put a tennis ball and a bowling ball on the floor and give both the same kick, the tennis ball receives a much lager acceleration than the bowling ball.

The weight (W) of a body is a force that pull the body directly towards the center of the Earth. the force is primarily due to an attraction, called gravitational attraction (acceleration). (remember Newton's famous story about the apple falling from the tree.)

The gravitational acceleration on Earth is ~9.81 m/s². However, in our subject we round this figure up to 10 m/s&#178. This fits well the decimal system and makes conversion from kilogram (kg) to newton (N) easy. To convert from kg to N you need to multiply the kg by 10 to obtain newtons and to convert N to kg you divide the newtons by 10.

All this can be summerised in a simple vector equation, which is called Newton's second law of motion:

Force = mass x acceleration
F = m x a

The weight of a body on earth is the force that the earth exerts on the mass of a body, i.e. The force of attraction called the gravitational force.

The equation for the weight is W = m x g

Where g = acceleration due to gravity. The gravitational acceleration is 9.8 m/s². In our subject we round this up to 10 m/s&#178. This figure fits the decimal system and make the convertion from a mass unit to a weight unit much easier.

Forces

Forces are represented by a line drawn to a force-scale in a certain direction with an arrowhead indicating it sense as shown below.

A force is completely defined by its:

Magnitude
The unit for the magnitude or size of a force is the newton (N) and multiples or sub-multiples of the newton. e.g. kilonewton (kN), meganewton (MN) and giganewton (GN)

Line of Action and Direction
The line of action of the force (usually indicated with reference to the horizontal or vertical lines) and an arrowhead (indicating the way it acts along the line of action) must be clearly indicated.

Point of Application (Position)
The point of application or position (the location) is the point where a force is applied to the body. It could be described as four metres to the left of the support.



Concurrent coplanar force systems
Coplanar means in the same plane like the force shown above drawn on a sheet of flat paper. Forces that lie in the same plane are two-dimensional with a x-direction and a y-directions.
A concurrent coplanar force system is a system of two or more forces whose lines of action all intersect at a common point. However, all of the individual vectors might not actually be in contact with the common point. (as can be seen in this diagram)





Non-concurrent coplanar force system

Non-concurrent coplanar force system consists of a number of vectors that do not meet at a single point. These systems are basically a jumble of forces and take considerable care to resolve.






Parallel coplanar force system
A parallel coplanar force system consists of two or more forces whose lines of action are all parallel. This is commonly the situation when simple beams are analysed under gravity loads. The resolution of the forces can be done graphically but the preferred method is the analytical method.



Geometrical Coordinate
A coordinate system is defined by three things: an origin, the directions of the axes, and a distance scale (which is usually but not always the same for each axis). Coordinates refer to a set of numbers to place a fixed point in space. For example, a point in the Cartesian plane is uniquely determined by its x-axis coordinate and its y-axis coordinate. The Cartesian plane is used to resolve a force system or to manipulate forces.

Zero (0) is at the origin of the Cartesian plane and the coordinates are shown along the x-axis and y-axis.

The point in the first quadrant has the coordinates
X = +2; Y = +5,

in the second quadrant has the coordinates
X = -4; Y = +2,

in the third quadrant has the coordinates
X = -3; Y = -5

and in the fourth quadrant has the coordinates
X = +4, Y = -6.

The Cartesian plane is used for the graphical “resolution” of forces. As all forces are vectors their magnitude and direction is easily recognised if they are drawn to scale in the Cartesian plane as shown below.


In the opposite diagram are the Vectors (forces) shown pointing towards the crosses shown in the previous diagram. By counting the hrizontal and vertical components you can determine the resultant using Pythagoras theorem. First look at the horizontal components (x-axis) of the 4 forces
1st Quadrant = +2,
2nd Quadrant = -4,
3rd Quadrant = -3,
4th Quadrant = +4.
Add these components
+2 -4 -3 +4 = -1.

The same applies for the vertical components
(y-axis) adding these components
1st Quadrant = +5,
2nd Quadrant = +2,
3rd Quadrant = -5,
4th Quadrant = -6.
Add these components
+5 +2 -5 -6 = -4.



We can also add the forces graphically as shown in the opposite diagram. The vectors (forces) are arranged in a head-to-tail fashion. Howerver, the addition of more than two forces is a matter of Structures 2.


Parallelogram method of summing two (2) force vectors

The parallelogram of forces method is a graphical methods to find the resultant of a coplanar force system. Two or more concurrent forces can be replaced by a single resultant force that is statically equivalent to these forces.

The illustration above shows two force vectors. In order to resolve these forces graphically, one must first extend the lines of action of two concurrent forces until they intersect. (Forces can move along the line of action without changing the effect of the forces. If a tractor pulls an object with a rope, the length of the rope is of no importance in regard of the pulling force.) Then parallel lines are offset starting at the head of the force vectors. These lines intersect at point A. The line joining 0 to A is the Resultant.

The Triangle of Forces Method is another graphical method developed to find the resultant of a coplanar force system. It is an alternative method derived from the parallelogram method.Since the opposite sides of a parallelogram are equal, a force triangle may also be found instead of using the parallelogram method. This method is quite useful because it can be simultaneously applied to any number of concurrent forces. Force vectors are arranged tail-to-head and the line joining the tail to the head of the all force vectors is the resultant. This method is shown below.


Problem:

Find the resultant of the following two coplanar forces:
5.8 kN acting at 30° to the horizontal to the right and
3.5 kN acting at 20° to the vertical to the left

Solution:

(a) graphical

First select an appropriate force scale (e.g. 10 mm = 1 kN). Next step would be to draw the forces at the selected scale on graph paper. Remember to add them tail-to-head. Connect the tail end of the first drawn force with the head of the last drawn force an measure the distance. The distance in millimeter divided by 10 gives you the magnitude of the resultant in kN.

Answer: The resultant is 7.28 kN at an angle of 58° to the horizontal to the right

(b)analytical

In general we use the sin and cos function to calculate the components of the forces and add them together. Whether you have to add or subtract the components depents on the direction (see geometrical coordinates). We can then use Pythagors theorem to calculate the resultant.

Forces Fh = F x cos A Fh = F x sin A
5.80 x cos 30°
5.80 x 0.866
= + 5.023 kN
5.80 x sin 30°
5.80 x 0.500
= + 2.900 kN
3.50 x cos 70°
3.50 x 0.342
= - 1.197 kN
3.50 x sin 70°
3.50 x 0.940
= + 3.29 kN
Sum+ 3.826 + 6.190

Having calculated the x and y component we can now use Pythagoras theorem to calculate the resultant.

In the above table the reference angle must always refer to the horizontal [sinA=cos(90-A)]

To find the angle of the resultant in reference to the horizontal we use the tan-function.

    By using the invert funtion of your calculator you find the angle of 58.28° to the horizontal.

Simplified calculation of the resultant of two forses

To find the resultant of to forces we can use the following formula:

To find the angle in reference to the 5.8 kN force:

Resultant

consider the above forces F1 = 5.8 kN and
F2 = 3.5 kN and the angle between the forces is
180°- (90° + 20° - 30°) = 100°.
Substituting these figures in the above equation.



Angle

The calculation refers to the cos of the angle between the 5.8 kN force and the resultant R = 7.276 kN. The inverse of cos = 0.8809 is 28.25°. The angle of the 5.8 kN to the horizontal (30°) must be added and the is 58.25°, which worked out the same as the other figures.

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