Direct Stress and Strain

Intoduction to terms used in "Materials" and the concepts of Direct Stress and Strain

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Contents

Introduction
Load
Stress
Principle Of St. Venant
Strain
Hooke's Law. The Principle Of Superposition.
Young's Modulus Or (modulus Of Elasticity)
Page Comments

Introduction

The following page introduces some of the terms associated with the study of "The Strength of Materials" and in particular those associated with Direct Stress.

Load

The simplest type of load $\inline F$ is a direct pull or push, known technically as Tension and Compression.

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Acceleration is the rate of change of velocity as a function of time,it is vector. Acceleration is the second derivative of position with respect to time or, alternately, the first derivative of the velocity with respect to time.

Examples of these types of load are:

A rope hanging from hanging from a beam and carrying a load is in Tension. The forces on the rope are the wight of the load acting downwards and the pull of the beam at the other. Since there is no movement these forces must be equal and opposite.
The piers of a bridge. The weight of the bridge presses down on the pier and the ground pushes up. As above, since there is no movement, the forces are equal and opposite.

In pin-jointed structural framework some members will be in Compression and some in Tension depending upon the loads through the joints at the ends of the member.

If a member is in motion the loading may be cause partly or in whole by dynamic or inertia forces. For instance the connecting rod of a reciprocating engine is subjected to inertia forces due to piston acceleration and due to its own acceleration as well as gas pressures on the piston and gravity effects.

Pressure is an effect which occurs when a force is applied on a surface. Pressure is the amount of force acting on a unit area. The symbol of pressure is $\inline P$ .

Load is a Force and is measured in Pounds(lb.) Tons or Newtons.

Stress

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Across any section of a member, the Total Force carried must equal the Load $\inline P$ . This is distributed amongst the internal Forces of cohesion, which we call Stresses. Imagine that the member is cut through at $\inline XX$ . Then each Portion must be in equilibrium under the action of the external load $\inline P$ and the Stresses across $\inline XX$ .

Stress which are Normal to the plane on which they act are called Direct Stresses and they are either tensile or compressive.

The Load transmitted across any Section divided by the cross sectional area is called the Stress $\inline f$ . Where the Load is uniformly distributed across the Section:

$\inline f$ = Load / Area = $\inline \displaystyle\frac{P}{A}$

In some instances the Stress varies throughout the member and the Stress at any point is defined as the limiting ratio of $\inline \displaystyle \frac{\delta P}{\delta A}$ for a small area enclosing the point.

Stress is measured in Load per unit Area and is therefore lb./sq.in. tons/sq.in or Newtons/sq.mm.

Principle Of St. Venant

Principle of St. Venant states that the actual distribution of the Load over the surface of its application will not affect the distribution of Stress or Strain on sections of the body which are at an appreciable distance (relative to the dimensions) away from the Load. Any statically equivalent loading may therefore be substituted for the actual Load distribution, provided that the Stress analysis in the region of the Load are not required.

For example, a rod in simple Tension may have the end Load applied either:

Centrally Concentrated.
Distributed around the circumference of the rod.
Distributed over the end cross-section

These are all statically equivalent but the last is the easiest to deal with analytically and the Principle of St. Venant justifies the choice of the distribution. For points in the rod distant more than three times its greatest width from the area of loading no appreciable error will be introduced.

Strain

Strain $\inline e$ is the measure of the deformation produced in a member by the applied Load.

Direct Stress produces a change in length in the direction of the Stress. If a rod is in tension and the stretch or elongation produced is $\inline x$ then the Direct Stress is defined as the ratio:

Elongation / Original Length. Or $\inline e = \displaystyle\frac{x}{l}}$ .

Normally Tensile Strain is considered Positive and Compressive Strain (i.e. a reduction in length) negative.

Note that as Strain is a Ratio it is Dimensionless.

Hooke's Law. The Principle Of Superposition.

Hooke's Law states that Strain is Proportional to the Stress which Produced it.

This law is obeyed within certain limits by most ferrous alloys and can usualy be assumed to apply with sufficient accuracy to other Engineering Materials such as timber, concrete and non-ferrous alloys.

In general a material is said to be Elastic if it obeys Hooke's Law.

Where a number of Loads are acting together on an Elastic Material, the Principle of Superposition states that the resultant Strain will be the sum of the individual Strains caused by each Load separately.

Young's Modulus Or (modulus Of Elasticity)

Stress is a measure of the internal forces acting within a deformable body. It is a measure of the average force per unit area of a surface within the body on which internal forces act.

Strain is a normalized measure of deformation representing the displacement between particles in the body relative to a reference length.

Young's modulus is a measure of the stiffness of an elastic material and is a quantity used to characterize materials.

The slope of the stress-strain curve at any point is called the tangent modulus. The tangent modulus of the initial, linear portion of a stress-strain curve is called Young's modulus, also known as the Tensile modulus. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds.

i.e. $\inline E$ = Stress / Strain

$E= \frac{f}{e}$

(2)

For a bar of uniform cross-section this can be written as:

$\;\;\;\;\;\;\;E = \frac{P\;l}{A\;x}$

(3)

Thus $\inline E$ is a Constant for a given Material and is usually assumed to be the same for Tension and Compression. For Materials which do not obey Hooke's Law exactly it is often possible to apply an average value for $\inline E$ over a given range of Stress.

Provided that Hooke's Law is obeyed Young's Modulus represents the Strain required to produce Unit Strain. A Stress numerically equal to the Modulus , when applied to a uniform bar, would cause the length to double. For Engineering Materials the Strain will, in fact, rarely exceed $\inline \displaystyle\frac{1}{1000}$ so that the change in length will always be small compared to the original length.

E.G. Mild Steel has a value for $\inline E$ of $\inline 13,400\;tons/sq.in$ and will rarely be stressed above $\inline 10\;tons/sq.in.$ At this value the Strain is

$\inline e = \displaystyle\frac{10}{13,400} = 0.00075$ (from equation (2))

So a Bar of $\inline 10\;inch$ length under a load of $\inline 10\;tons/sq.in.$ will only suffer a change of length of $\inline 0.0075\;inches$ .