Physical body

A physical body (= physical object) is an identifiable collection of matter, which may be more or less constrained to move together by translation or rotation, in 3-dimensional space.

1   Study

Classical mechanics is concerned with the set of physical laws describing the motion of physical bodies under the action of a system of forces.

Classical mechanics consists of solid mechanics_ and fluid mechanics_. Fluid mechanics consists of aerodynamics_ and hydrodynamics_.

2   Properties

All physical bodies, as matter, have mass and volume.

2.1   Stress

Stress is the average force per unit area that some particle of a body exerts on an adjacent particle, across an imaginary surface that separates them. For example, when a solid vertical bar is supporting a weight, each particle in the bar pushes on the particles immediately below it.

At the atomic level, tension is produced when atoms or molecules are pulled apart from each other and gain electromagnetic potential energy.

Examples of things designed to withstand tension include diving boards, spider silk, ropes_, and rubber bands_.

Further reading with great images: http://www-materials.eng.cam.ac.uk/mpsite/interactive_charts/strength-toughness/basic.html

2.1.1   Uniaxial normal stress

Normal stress (\sigma) is force (P) divided by the area perpendicular to the force (A_0).

\begin{equation*} \sigma = P / A_0 \end{equation*}

For example, if a prismatic bar has a circular cross section with diameter d = 50mm and an axial tensile load P = 10kN, then the normal stress is:

\begin{equation*} \sigma = P/A_0 = P/(\pi \times (d/2)^2) = 4(10\times10^3)/\pi(50\times10^{-3})^2 \times N/m^2 \end{equation*}

If the cross-sectional area of a member is doubled, the ability of that member to restrain the tension forces is also doubled. Unlike a tension member, the ability of a column to restrain compression forces is not simply a function of the cross-section area, but a combination of the materials strength, the column length, and the cross-sectional shape of the column. Shorter columns are stronger that longer columns, and symmetrical cross-sections are stronger than asymmetrical cross sections.

Unit are force per unit area (pascal):

\begin{equation*} N / m^2 = Pa \end{equation*}

One pascal is very small, so we usually work in mega-pascals (Pa x 10^6).

The axial force P must act through the centroid of the cross-section, otherwise the bar will bend and you need a more complicated analysis. Further, the stress must be uniformly distributed over the cross section, and the material should be homogeneous.

When a bar is stretched, stresses are tensile (taken to be positive). If the forces are reversed, stresses are compressive.

2.2   Deformation

Deformation is the transformation of a body from a reference configuration to a current configuration, where a configuration is a set containing the positions of all particles of the body.

A strain is a normalized measure of deformation representing the displacement between particles in the body relative to a reference length. Such a measure does not distinguish between rigid body motions (translations and rotations) and changes in shape (and size) of the body

The amount of stretch or compression along material line elements or fibers is the normal strain, and the amount of distortion associated with the sliding of plane layers over each other is the shear strain, within a deforming body.

2.2.1   Normal strain

Normal strain \epsilon is the change in length \delta divided by the original length L_0.

\begin{equation*} \epsilon = \delta / L_0 \end{equation*}

When the bar is elongated, strains are tensile. When the bar shortens, strains are compressive.

Any strain (deformation) of a solid material generates an internal elastic stress, analogous to the reaction force of a spring, that tends to restore the material to its original non-deformed state.

Modulus is how well a material resists deformation. Modulus is measured by calculating stress and dividing by elongation, and would be measured in units of stress divided by units of elongation. But since elongation is dimensionless, it has no units by which we can divide. So modulus is expressed in the same units as strength, such as N/cm2.

Toughness is a measure of the area underneath the stress-strain curve. It is a measure of the energy a sample can absorb before it breaks. (Since strength is proportional to the force needed to break the sample, and strain is measured in units of distance (the distance the sample is stretched), then strength times strain is proportional is force times distance, and as we remember from physics, force times distance is energy.)

Strength tells how much force is needed to break a sample, and toughness tells how much energy is needed to break a sample. A material that is strong is not necessarily tough. A material like this which is strong, but can't deform very much before it breaks is called brittle. Materials that are strong and tough will elongate before breaking; deformation allows a sample to dissipate energy. For example, spider silk.

2.2.1.1   Elongation

Elongation is a type of deformation in which tensile stress causes a sample to deform by stretch.

Percent elongation is the length of the sample after it stretched (L) divided by its original length (L_0). Two other important measure are ultimate elongation and elastic elongation. Ultimate elongation is the amount you can stretch the sample before it breaks. Elastic elongation is the percent elongation you can reach without permanently deforming your sample; how much can you stretch it, and still have the sample snap back to its original length once you release the stress on it.

2.2.2   Tension and compression

Gravity acts upon all beams. Since all materials are flexible to some degree, beams tends to sag of their own weight, even more as loads are applied. This means that the upper part of a beam between two supports is squeezed together and is compressed along the top surface, while the lower part is selected and is said to be in "tension". In a cantilever, the situation is exactly reversed and the forces are strongest just over the support.

Fibrous materials, such as wood, resists tensile strength well, as does wrought iron, steel, and Kevlar_. Beams of these materials can span significant distances. The tensile forces along the bottom of a beam are determined by the length of the span and the load placed on the beam. Eventually, given a span and a load sufficiently great, the tensile strength of the material will be exceeded by ?; the beam will crack at the bottom or deform along the top (or both) and will collapse.

Crystalline material, such as stone and solid concrete, has great compressive strength but less tensile strength than fibrous materials. Therefore, a wooden beam over a given span can carry a load that would crack a stone beam carrying the same load. (Of course, the stone beam starts off being far heavier by itself.) Anything that has to support weight from underneath must have good compressional strength.

Builders solve this by placing something that will take the tensile forces within the beam of concrete. The Romans placed iron rods in the formwork into which the liquid concrete is poured. The result is reinforced concrete. The steel is placed where the tensile forces accumulate -- on the bottoms of beams and at the top of cantilevers. The Greeks also faced this problem. (see text for example)

Buckling occurs when compression overcomes an object's ability to endure that force. Snapping is what happens when tension surpasses an object's ability to handle the lengthening force.

The best way to deal with these powerful forces is to either dissipate them or transfer them. With dissipation, the design allows the force to be spread out evenly over a greater area, so that no one spot bears the concentrated brunt of it. In transferring force, a design moves stress from an area of weakness to an area of strength. As we'll dig into on the upcoming pages, different bridges prefer to handle these stressors in different ways.

4   References

 [1] Roth 1993_.
 [2] Gordon. 1978. Structures or Why things don't fall down.

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A shear load is a force that tends to produce a sliding failure on a material along a plane that is parallel to the direction of the force. When a paper is cut with scissors, the paper fails in shear.

In a reinforced concrete beam, the main purpose of rebar_ to increase the shear strength.

A shear stress, denoted tau, (Greek: tau), is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts.

Wherever a member is subjected to a tension, compression or bending force (bending moment), the member is deformed by the force, irrespective of how strong the material is or how large the section. The amount of deformation does, however, depend on material strength and the size and shape of the section.

An Oregon beam would deflect 32 mm soon after the one tonne point load is applied at a mid-span. If this load is maintained, the deflection may gradually increase to three times the initial deflection after a period of 20 to 24 months. This increase in deflection, with time, without increase in load, is called “creep”. This characteristic is significant with timber, but can be ignored in other structural materials like steel.

If the same load is applied to a steel universal beam, the spontaneous deflection is approximately 1 mm. The long term deflection will also be 1 mm.

A bow pulled back is storing a lot of energy. With an arrow in place, that energy is transferred (mostly) to the arrow, and it happens much more slowly. Without an arrow, the bulk of the energy gets absorbed by the string and limbs, and it happens much more quickly, so it's more of a sudden shock. Sudden shocks can be more damaging.

Ex-competitive archer here. For similar reasons, for most all compound bows it's dangerous to use wooden arrows, because the wood cannot absorb the rapid acceleration delivered by the compound system without high risk of the wood arrow breaking (recurve and long bows don't accelerate so violently when released) and possibly driving a wooden shard into your arm. Aluminum, carbon fiber, fiberglass and other types of arrows allow for higher levels of either strength or flexure to absorb this more sudden and more violent acceleration.