Basic principles of statics

Structural system is concerned with the strength, stiffness and stability of structures such as buildings, dams, bridges and retaining walls.

Although a building is constructed from the foundation upwards, the designer usually starts designing from the top, the roof and works his way downwards.

Stages of Structural Design 2 distinct stages in structural design. 1st experience, intuition and knowledge, an

imaginative choice of preliminary design in terms of layout, materials and erection methods.

2nd is the estimation of the various forms of loading are made

3rd the chosen design is subjected to detailed analysis based on the principles of statics.

Statics = branch of mechanics & deals with forces on bodies, which are 'at rest' (static equilibrium). Another branch, dynamics, deals with moving bodies, such as parts of machines

Newton’s Laws of Motion

Newton's First Law of Motion

An object at rest tends to stay at rest and an object in motion tends to stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

Newton’s Second Law Consider another

example involving balanced forces - a person standing upon the ground. There are two forces acting upon the person. The force of gravity exerts a downward force. The floor of the floor exerts an upward force.

Static Equilibrium

Forces acting in one plane (i.e., coplanar) and in equilibrium must satisfy one of the following sets of conditions:

Fx=0 Fx=0 Fy=0 Ma=0

Fy=0 or Ma=0 or Ma=0 or Mb=0

Ma=0 Mb=0 Mb=0 Mc=0

where F refers to forces and M refers to moments of forces.

Static Determinacy

If a body is in equilibrium under the action of coplanar forces, the equations of statics above must apply.

In general then, 3 independent unknowns can be determined from the 3 equations.

But, if applied and reaction forces are parallel (i.e., in one direction only) only 2 separate equations obtain and then only two unknowns can be determined. Such systems of forces are said to be statically determinate.

Force

A force is any cause which tends to alter the state or rest of a body or its state of uniform motion in a straight line.

A force can be a quantitatively as the product of the mass of the body, which the force is acting on, and the acceleration of the force.

F = ma where F = applied force m= mass of the body ( kg) a = acceleration caused by the force (m/s2)

The Sl units for force are therefore kg m/s2 which is designated a Newton (N). The following multiples are often used:

1kN = 1,000N, 1MN = 1,000,000N

Gravitational ForceAll objects on earth tend to accelerate toward the center of the earth due to gravitational attraction, hence the force of gravitation acting on a body with the mass (m) is the product of the mass and the acceleration due to gravity (g), which has a magnitude of

9.81 m/s².F = mg = v g where:F = force (N) m= mass ( kg)

g = acceleration due to gravity (9.8m/s²) v = volume (m³) = density ( kg/m³)

Vector

Most forces have magnitude and direction and can be shown as a vector.

Its point of application must also be specified.

A vector is illustrated by a line, whose length is proportional to the magnitude to some scale and an arrow which shows the direction.

Vector Addition The sum of 2 or more vectors is called the resultant.

The resultant of 2 concurrent vectors is obtained by constructing a diagram of the two vectors.

The vectors to be added are arranged in tip-to-tail fashion. Where 3 or more vectors are to be added they can be arranged in the same manner and this is called a polygon. A line drawn to close the triangle or polygon (from start to finishing point) forms the resultant vector.

The subtraction of a vector is defined as the addition of the corresponding negative vector.

Illustration of Vector Addition

Vector Resolution In analysis and calculation it is often

convenient to consider the effects of a force in other directions than that of the force itself, especially along the Cartesian (xx-yy) axes. The force effects along these axes are called vector components and are obtained by reversing the vector addition method.

Fy is the component of F in the 'y' direction Fy = F sin

Fx is the component of F in the 'x' direction Fx = F cos

Sample of Vector Resolution

P

Q

SA

Concurrent Coplanar Forces Concurrent Forces have their line of action

meeting at one point Coplanar forces lie in the same plane Non-coplanar forces have to be related to a 3 dimensional space and require 2 items of

directional data together with the magnitude. 2 Coplanar nonparallel forces will always be

concurrent.

Elements of Coplanar Force Resolution

There are many ways in which forces can be manipulated.

It is often easier to work with a large, complicated system of forces by reducing it an ever decreasing number of smaller problems.

This is called the "resolution" of forces or force systems.

This is one way to simplify what may otherwise seem to be an impossible system of forces acting on a body.

Coplanar Force Systems Certain systems of forces are easier

to resolve than others. Coplanar force systems have all the

forces acting in in one plane. They may be concurrent, parallel, non-concurrent or non-parallel. All of these systems can be resolved by using graphic statics or algebra.

Concurrent Coplanar Force System

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. These are the most simple force systems to resolve with any

one of many graphical or algebraic options.

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 analyzed under gravity loads. These can be solved graphically, but are combined most easily using algebraic methods.

Non-concurrent and Non-parallel System

A non-concurrent and non-parallel system consists of a number of vectors that do not meet at a single point and none of them are parallel. These systems are essentially a jumble of forces that require considerable care to resolve.

Concurrent & Parallel Forces