**Click to download worked examples**

A bar is a body of which the two cross-sectional dimensions (height and width) are considerably smaller than the third dimension, the length. A bar is one of the most frequently used types of structural members. To understand something about the behaviour of bar type structures, it is necessary to understand the behaviour of a single bar.

In this section, we discuss the case of a bar subject to *extension *when the (straight) bar remains straight after deformation and does not bend.

# The fibre model

In order to imagine the behaviour of a bar, we create a *physical model*. A condition is that the results of the model have to give a sufficiently accurate picture of reality. It is always the experiment that must confirm the correctness of the chosen model and the associated assumptions.

A model that seems to function effectively is the so-called *fibre model*.

The figure above illustrates the fibre model for a bar. The behaviour of the model is described in an *xyz *coordinate system with the x-axis parallel to the fibres and the yz plane perpendicular to the fibres and parallel to the cross-sections.

This model is based on the following assumptions:

- Inspired by the structure of wood, the member is considered to consist of a very large number of parallel fibres in the longitudinal direction. Later we will look at the limiting case in which the number of fibres is so large that the area ΔA of a single fibre approaches zero.
- The fibres are kept together by a very large number of absolutely rigid planes perpendicular to the direction of the fibres. These rigid planes are known as
*cross-sections*. Later we will look at the limiting case in which the number of cross-sections is so large that the distance Δx between two consecutive cross-sections approaches zero. - The plane
*cross-sections*remain plane and normal to the longitudinal fibres of the beam, even after deformation.

The location of a cross-section is defined by its x-coordinate; the location of a fibre is defined by its y and z coordinates.

The following assumptions are made with respect to the *material-behaviour*:

- All the fibres consist of the same material and therefore have the same mechanical properties. In this case, the cross-section of the bar is said to be homogeneous.
- The material behaves
*linear-elastically*and follows*Hooke's law*, with a linear relationship between stess and strain (σ = ε·E).

Note that in a homogeneous cross-section all fibres have the same modulus of elasticity.

If the fibres do not all have the same modulus of elasticity, because they consist of different materials, the cross-section is said to be *inhomogeneous*. In this way, a reinforced concrete beam has an inhomogeneous cross-section, because the "concrete fibres" and the "steel fibres" have different moduli of elasticity.

# The three basic relationships

When investigating the behaviour of a bar, we distinguish three different basic relationships:

- Static or equilibrium relationships.
- Constitutive relationships.
- Kinematic relationships.

Static or equilibrium relationships:

The static relationships link to internal forces and the section forces. They follow from the equilibrium.

Constitutive relationships:

The constitutive relationships link the section forces and the associated deformations. They follow from the behaviour of the material (linear-elastic in this case).

Kinematic relationships:

The kinematic relationships link the deformations and the displacements. They are the results of a permanent cohesion within the bar - holes do not suddenly appear. The kinematic relationships are independent of the material behaviour.

The three basic relationships allow us to link the external forces and the associated displacements.

## The kinematic relationship

In this section we look for the relationship between the deformation and displacement for a bar subject to extension.

Previously we established the deformation quantity strain, ε. For the bar in a tensile test it was defined as: ε = ΔL/L = elongation / original length.

Below, this definition is used also for the strain of individual fibres.

The figure below shows a small segment of a bar subject to extension. The segment has a length Δx, and is bounded by the end-section a and b.

If the bar changes length due to tension or compression, the cross-sections will move with respect to one another. Assume end-section a moves in the x-direction by a distance *u* and end-section b moves by a distance *u+Δu*.

All longitudinal fibres between the end-sections a and b have the same original length, L. This length is equal to the distance Δx between both end-sections. The elongation, ΔL, of the fibres is equal to the difference in displacement, *Δu*, between the end-sections a and b.

For extension, all fibres undergo the same strain, ε

The limit of Δu/Δx as Δx tends to zero is known as the derivative of *u* with respect to *x*:

The strain of the fibres is therefore:

This is the *kinematic relationship* for extension; it provides a link between the deformation quantity *ε* (the strain of the fibres in the bar) and the displacement *u* (of a cross-section in the x-direction).

For a bar segment, the change in length, ΔL, is equal to the difference in displacement between the end-sections:

The total change in length of the bar is found by summing all contributions ε·Δx of the individual segments over the entire length of the bar.

This relationship is the basis for the formulae for calculating the change in length of a bar.

## The constitutive relationship

This section looks at the relationship between deformation and section for for a bar subject to extension. This relationship is dependent on the behaviour of the material, i.e. the *modulus of elasticity*, *E*.

The resultant normal stress, σ, on a fibre *(y,z)* with an area *ΔA *is a small force *ΔN*.

In a linear-elastic material, the fibres follow Hooke's law, σ = ε·E, so that: ΔN = σ·ΔA = ε·E·ΔA

The total normal force, *N*, is found by suming the contributions of all the fibres, i.e. integrating all the forces, *ΔN*, with respect to the cross-section, *A*:

For extension, all fibres undergo the same elongation, and *ε* can be placed outside the integral (as a constant). If the cross-section is homogeneous, all fibres have the same modulus of elasticity and *E* can also be placed outside the integral. Hence:

This is the *constitutive relationship* for extension. It links the normal force, *N*, (a section force) and the strain, *ε*, (a deformation quantity). The constitutive relationship depends on the behaviour (constitution) of the material as it includes the modulus of elasticity, *E*.

*E·A* is known as the *axial stiffness* of the bar. The axial stiffness is a measure of the resistance of the bar to axial deformation, and depends on both the modulus of elasticity, *E*, and the area, *A*, of the cross-section.

## The static relationship

Static or equilibrium relationships link the external forces and the section forces. They follow the equilibrium of a small member segment.

In the figure above, a small segment with length Δx has been isolated from a bar. The bar segment is subject to distributed forces *qx* and qz. The forces act on the bar axis (for clarity this is not drawn as such for *qz*). When the length Δx of the bar segement is sufficiently small, the distributed forces *qx* and *qz* can be considered uniformly distributed.

The (unknown) section forces on the right-hand and left-hand section planes are shown in accordance with their positive directions. The section forces are functions of x, and are generally different in the two section planes. Assume that the forces on the left-hand section plane are *N, V* and *M*. Also assume that these forces increase over distance *Δx* by *ΔN, ΔV* and *ΔM* respectively. The forces on the right-hand section plane are then *(N+ΔN), (V+ΔV)* and *(M+ΔM)*.

The force equilibrium of the bar segment in the x-direction gives:

After fividing by Δx we find:

In the limit Δx → 0, the equation for the force equilibrium for an elementary bar segment changes into the first-order differential equation:

This is the *static relationship* for extension.

Comment: The deriviation is invalid when a concentrated force *Fx* is acting on the bar segment. In that case, there is a step change in the variation of the normal force, *N*. As a function of x, *N* is no longer continuous and differentiable.

# Strain diagram and normal stress diagram

In a bar subject to extension, all fibres undergo the same elongation regardless of the material behaviour.

Using the constitutive relationship, N = E·ε·A, we find a uniform strain over the cross-section:

The figure below shows a uniform strain distribution over a rectangular cross-section in a *strain diagram*. Here, along each fibre (y,z) the value of the associated strain, *ε(y,z)*, is plotted. It is the convention to plot the positive values in the positive x-direction and the negative values in the negative x-direction.

In principle, the strain diagram is a spatial figure. If the strain is independent of the y-coordinate, is in this case, then the figure can be simplified into a plane diagram:

It is common practice to leave out axes and the sign associated with thye strain is placed within the diagram.

In a bar with homogeneous cross-section, all fibres have the same modulus of elasticity, *E*. If such a bar is subject to extension, the fibres are not only subject to the same strain but also the same normal stress:

The figure below shows the uniform stress distribution of the normal stresses in a *normal stress diagram*. Here, in the same way as in the strain diagram, the value of the normal stress, *σ(y,z),* in each fibre, *(y,z)*, is plotted along that fibre.

Like the strain diagram, the stress diagram is a spatial figure. If the stresses are independent of the y-coordinate, it can be simplified into a plane digram.

Here too, the axes are generally omitted and the sign of the stress is placed within the diagram.

Comment: In bars subject to extension, all fibres undergo the same strain *ε*, irrespective whether the cross-section is homogeneous or inhomogeneous. On the other hand, in a bar subject to extension, all fibres have the same normal stress, *σ*, if and only if the cross-section is homogeneous: in an inhomogeneous cross-section, the normal stresses due to extension are no longer uniformly distributed.

# Normal centre and bar axis

This section addresses the location the normal centre, NC, of a cross-section, and by consequency the location of the bar axis. To locate thee NC we must consider bending moments for which we follow a formal approach that can differ from engineering practice.

The resultant of all normal stresses due to extension is the normal force, *N*. For a homogeneous cross-section:

The point of application of the normal centre, *N*, is defined as the *normal force centre* or *normal centre* of the cross-section, indicated by NC. The fibre through the normal centre is defined as the *bar axis*.

Later we shall see that the behaviour of a bar is most easily described in a coordinate system with the x-axis along the bar axis. It is therefore important to know the location of the normal centre, NC. This problem is solved in two ways:

a. in an *xyz* coordinate system with the x-axis through the normal centre, NC, of the cross-section and along the bar axis;

b. in an *x'y'z' *coordinate system with the x'-axis along an arbitrary fibre.

Solution a:

Assume the x-axis passes through the normal centre NC of the cross-section, the point of application of the resultant of all normal stresses due to extension.

The resultant of the normal stress, *σ*, in fibre *(y,z)* with the area ΔA is a small force, *ΔN*:

This small force at fibre *(y,z)* is statically equibalent to an equal small force *ΔNx* at the normal centre, NC, (the origin of the *yz* coordinate system), together with two small bending moments, *ΔMy* and *ΔMz*, acting in the *xy* plane and *xz* plane respectively:

If we sum the contributions of all small forces, *ΔN*, over the entire cross-section, we find the nomral force: