ARE SS 3- Structural Fundamentals

For an unrestrained material, the general formula for calculating thermal stress is:

total deformation, equals the coefficient of linear expansion, Times the original length, times the change in temperature.

For rectangular sections, the moment of inertia about the central axis parallel to the base is:

Moment of inertia, equals, the base of the rectangular section, times the height of the rectangular section cubed, all divided by 12.

For concurrent forces, who's the lines of action pass through a CommonPoint, both the magnitude and direction must be taken into account.

The resultant of these forces can be found graphically or algebraically.

Third, connect the point of concurrency, to the opposite corner of the parallelogram with a line.

The resulting line is the resultant who's magnitude and direction, can be found by scaling the length of it, and measuring it's angle.

These external loads are called stresses.

The structural design of buildings is primarily concerned with selecting the size, configuration, and material of components to resist, with a reasonable margin of safety, external forces acting on them.

If the load is continually increased, the material will ultimately rupture.

The unit stress, just before this rapture occurs, is called the ultimate strength of the material.

The simplest combination of forces is represented by Cole linear forces. The magnitudes of collinear forces are added directly in the same direction of force.

Therefore, if two collinear forces, 1000 pound, and 1800 pounds, Are colinear, and in the same direction, the resulting force will be 2800 pounds.

Second, the sum of all horizontal forces acting on the body must equal zero.

Third, the sum of all the moments acting on a body, must equal zero.

The moment of inertia equations show that a beams depth has a greater bearing on the beams resistance to bending, then it's widths or total area.

This explains why a board, placed on edge between two supports, is much stronger then the same board placed on its side.

As a force is applied to a material, the deformation, or strain, is directly proportional to the stress, up to a certain point.

This is known as Hooks law, named after Robert Hook, the British mathematician in physicist who first discovered Hooks law.

Concurrent forces

Those whose lines of action meet at a common point.

Colinear forces

Those whose vectors lie along the same straight line.

Statical moment

To find the centroid of unsymmetrical areas, the statical moment must be used.

Torsion

Torsion is a type of sheer, in which a member is twisted.

Resultant forces

Two or more concurrent forces, combined into one force, so that the one force produces the same effect on the body, as the concurrent forces.

If the forces are collinear, the resultant is simply the sum of the forces

, with forces acting upward or to the right considered positive, and forces acting downward or to the left considered negative.

Other types of stresses consist of torsion, bending, and combined stresses.

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The portion of the structure under study is called a free body diagram, to which the principles of equilibrium can be applied.

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Would has a very low coefficient of expansion, while plastics and acrylics have very high coefficients of expansion.

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Once the area of the rod is noon, this can be divided into the total force, 10,000 pounds.

10,000 pounds of force, divided by the area of 1.23 inches, equals 8130 pounds per square inch.

Combined loads

A column resisting loads from above, and lateral wind loads, is subjected to both compression, and bending.

The line of action on a force, is a line concurrent with the force vector.

A force acting anywhere along the line of action can be considered equal, or unchanged, as long as the direction and magnitude do not change. This is the principle of transmissibility.

The principle of transmissibility.

A force acting anywhere along the line of action, can be considered equal, or unchanged, as long as the direction and magnitude do not change.

Force

A force is any action applied to an object.

Moment

A moment is the tendency of a force, to cause rotation about a point.

A special type of nonconcurrent forces that is commonly found in architectural applications, is a parallel force system.

A parallel for system may be acting on a beam.

A trust is a collection of sets of concurrent, coplanar forces.

A space frame is an example of a combination of sets of concurrent, non-coplanar forces.

At any stress up to the elastic limit, the material will return it to its original size, if the force is removed.

Above the elastic limit, there will be permanent deformation, even once the force is removed.

For example, at a scale of 1 inch equals 2000 pounds, A line 2 inch long represents a force of 4000 pounds.

And 8000 pound force would therefore be shown with a line 4 inches long, if 1 inch equals 2000 pounds.

Strain and deformation

As a force is applied to a material, the material changes size. For example, a tensile force, causes a ride to elongate and narrow. It compressive force, causes a material to shorten and widen.

Elastic limit.

At a certain point, the material will begin to change links at a faster ratio, then the applied force. This point is called the elastic limit.

Bending

Bending is a combination of tension and compression, like the type that occurs in Beams.

It is most common to use the neutral axis, the axis passing through the centroid, as the axis of reference.

But the moment of inertia about an axis through the base of a figure, is also useful when calculating for the moment of inertia for unsymmetrical sections.

Compression

Compression is stress in which the particles of the member are pushed together, and the members tend to shorten.

To find the resultant algebraically, sketch a forced triangle. Since the forces are in equilibrium, The triangle must close.

Four, the resultant will then be the third side of the triangle. Both magnitude and direction can be solved with trigonometry, using the law of cosines, and signs, or the Pythagorean theorem for a right triangle.

Coplanar forces

Coplanar forces are forces, his lines of action all fly within the same plane. Non-coplanar forces do not lie within the same plane.

A force has both the direction and magnitude, and as such is called a vector quantity.

Direction is shown by using a line with an arrowhead. And magnitude is indicated by establishing a convenient scale.

The most important part of the materials elastic limit, is that stress and strain are directly proportional, up to the elastic limit.

Engineering practices set by building codes, establish the working stresses to be used in calculations, at some point below the yield point.

Equilibrium

Equilibrium is said to exist when the resultant of any number of forces acting on a body, is zero.

The modulus of elasticity

Every material has a characteristic ratio of stress to strain, called the modulus of elasticity, which is a measure of a materials resistance to deformation, or it's stiffness.

Three fundamental principles of equilibrium, apply to all buildings:

First, the sum of all vertical forces acting on a body must equal zero.

For example, if 10,000 pounds is applied to a rod that is 1 1/4 inch in diameter, what is the stress in the rod?

Firstly, the area of the rod must be found. Half the diameter is .625 inches, which can be squared in multiplied by pi, to find the area. Area equals 1.23 inches.

Non-concurrent forces

Have lines of action, that do not pass through a CommonPoint.

The statical moment of a plane area, with respect to an axis, is the product of the area, times the perpendicular distance from the centroid of the area, to the axis.

If a complex, unsymmetrical area, is divided into two or more simple parts, the statical moment of the entire area is equal to the sum of the statical moments of the parts.

If a force causes a clockwise rotation, the moment is said to be positive.

If a force tends to cause a counterclockwise rotation, the moment is said to be negative.

Centroid

In all solid bodies, there is a point at which the mass of the body can be considered concentrated. This is the center of gravity, or the centroid.

Freebody diagram's

In analyzing structures, it is sometimes convenient to extract a portion of the structure, and represent the forces acting on it with force vectors.

If the material is restrained at both ends, a change in temperature causes an internal thermal stress. This can be calculated as follows:

Internal stress, lowercase f, equals the modulus of elasticity, times the coefficient of expansion, times the change in temperature.

Components of a force

Just as a resultant can be found for two or more forces, so can a single force be resolved into two components.

To find it graphically, draw the lines of force to any convenient scale, such as 1 inch equals 100 foot pounds, and in the direction they are acting.

Second, draw a line parallel to each force, starting with the head of the force vector, to form a parallelogram.

Sheer

Sheer is a stress, in which the particles of a member, slide past each other.

The yield point is the point, above the elastic limit, at which the material continues to deform, with very little increase in load.

Some materials, such as wood, have poorly defined elastic limits, and no yield points.

Statics

Statics is the branch of mechanics that deals with bodies in a state of equilibrium.

Strain

Strain is the deformation of a material, caused by external forces.

Strain is the ratio of the total change, in length of a material, to its original length.

Strained can be calculated with the following formula: unit stream, equals total deformation, strain, divided by the original length.

For these three conditions, stress is expressed as force per-unit area.

Stress, lowercase f, is found by dividing the total force, P, by the total area of which the force is applied to.

Two force members

Structural members subjected to collinear forces, such as tension or compression, are said to be two force members.

There are three basic types of stresses: tension, compression, and sheer

Tension and compression stresses are known as normal stresses. All stresses consist of these basic types, or some combination thereof.

Tension

Tension is stress in which the particles of the member, tend to pull apart under load.

Four example, a 10 pound object on the ground, is acted on by gravity to the magnitude of 10 pounds.

The ground, in turn, exerts an upward force of 10 pounds, and the object is in equilibrium.

In architecture, external forces are called loads, in result from the weights of such things as people, wind, snow, or building materials.

The internal structure of a building material must resist external loads, with internal forces of their own, that are equal in magnitude, and opposite in sign.

The modulus of elasticity and is the ratio of stress to strain for a given material.

The modulus of elasticity he is a measure of a materials resistance to deformation, or it's stiffness.

Calculating the modulus of elasticity

The modulus of elasticity, E, equals the unit stress, lowercase f, divided by the unit strain.

Moment of inertia

The moment of inertia is a measure of the bending stiffness of a structural members cross-sectional shape, similar to how the modulus of elasticity, is a measure of the stiffness of the material of a structural member.

In more exact terms, the moment of inertia about a certain axis of a section, is the sum of all the small areas of the section, multiplied by the square of the distance from the axis, to each of these areas.

The moment of inertia is commonly designated Capital I, and it's units are inches to the fourth power.

Therefore, a moment is the product of the force, times the perpendicular distance to the point about which it is acting.

Understanding moments is important, because a system in equilibrium has a sum of moments about any point being zero.

Thermal stress

When a material is subjected to a change in temperature, it expands if he did, or contracts if cooled.

With tension and compression, force acts perpendicular to the area of the material resisting the force.

With sheer, the force acts parallel to the area resisting the force.

Yield point

With some materials, there is also a point slightly above the elastic limit, at which the material continues to deform, with very little increase in the load.