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# Curved Beams

An analysis of stresses and strains in curved beams

**Contents**

- Stress In Bars Of Small Initial Curvature.
- Application To The Design Of A Piston Ring
- Stresses In Bars Of Large Initial Curvature.
- Rectangular Cross-section
- Trapezoidal Cross-section.
- Circular Cross Section
- Deflection Of Curved Beams (direct Method)
- Deflection From Strain Energy ( Castigliano's Theorem)
- Page Comments

## Stress In Bars Of Small Initial Curvature.

Where the radius of curvature is large compared to the dimensions of the cross section, the analysis of stress is similar to that for pure bending. Let be the initial (unstrained) radius of curvature of the neutral surface and the radius of curvature under the action of a pure bending moment . Then the strain in a element at a distance from the neutral axis is given by:**Moment of resistance**is a term in structural engineering. It is found from the moment of inertia and the distance from the outside of the object concerned to its major axis.

Strain =
Since = length along the neutral axis
If is neglected in comparison with and noting from that
Then strain,
Neglecting lateral stress, the normal stress, strain
Substituting in equation (2)
Total normal stress = 0, i.e.
which shows that the neutral axis passes through the centroid of the section.
Moment of resistance,
from equation (4)
Combining equations (3) and (5),
the strain energy of a short length (measured along the neutral surface) under the action og bending moment is:
From equation (4)

## Application To The Design Of A Piston Ring

A

**piston ring**is a split ring that fits into a groove on the outer diameter of a piston in a reciprocating engine such as an internal combustion engine or steam engine.
Suppose it is required to design a split ring so that its outside surface will be circular in both
the stressed and unstressed conditions and the radial pressure exerted will be uniform.
If is the uniform pressure on the outside then the bending moment at is given by:
approx
where is the depth of the ring in the axial direction
integrating
But = a constant for a given condition
i.e. = constant when and
Which is the required variation of thickness.
Using equation (7). The maximum bending stress at any section
which has it's greatest value when i.e.
From which,
which determines the initial radius when values for and are assumed.

## Stresses In Bars Of Large Initial Curvature.

When the radius of curvature is of the same order as the dimensions of the cross section, it is no longer possible to neglect in comparison to and it will be found that the neutral axis does not pass through the centroid. Further the stress is NOT proportional to the distance from the neutral axis where is the strain, is the distance from the neutral axis as before and is the initial radius of the neutral surface. For pure bending the Total normal force on the cross section = . Moment of resistance, But Where is the distance between the neutral axis and the principle axis which is through the centroid ( is positive when the neutral axis is on the same side of the centroid as the centre of curvature) Substituting in equation (9) Rearranging, In this equation is positive measured outwards, a positive bending moment being one that tends to increase the curvature.## Rectangular Cross-section

From equation (8), Let = the distance from the centroid. Also the mean radius of curvature and Then, i.e. Hence, Giving,As is small compared to and , it is difficult to calculate with sufficient accuracy from this equation and the expansion of the log term into a convenient series is of advantage. Then,

Example:

[imperial]

##### Example - Example 1

Problem

A curved bar, initially unstressed, of square cross section, has sides and a mean radius of
curvature of
If a bending moment of is applied to the bar tending to straighten it, find the

**stresses**at the outer and inner faces. andWorkings

But
and
At the inside face,
Thus, Tension
At the outside face,
compression
The actual stress distribution is shown in the diagram.

Solution

- Tension
- Compression

## Trapezoidal Cross-section.

By Moments, By putting and equation (8) becomes i.e. or, From which, And since, can be evaluated from equations (10) and (11).Example:

[imperial]

##### Example - Example 2

Problem

A crane hook whose horizontal cross-section is trapezoidal, wide on the inside and wide
on the outside by thick, carries a vertical load of one ton whose line of action is from the inside edge of this section. The centre of curvature is from the inside edge.
Calculate the

**maximum tensile**and**compressive forces**set up.Workings

Referring to the last figure.
From equation (9)
Direct stress = load / area = in. tensile
Bending stress =
At the inside edge,
(tending to decrease the curvature)
Bending stress = Error: Invalid Equation in tension
The combined stress = tensile.
At the outside edge,
Bending stress =
Combined stress = in compression

d_1 = \frac{2}{3}\left(\frac{2\,+\,2\,\times \,1}{2\,+\,1} \right)} = \frac{8}{9}\,in.

d_2 = \frac{2}{3}\left(\frac{2\,\times \,2\,+ \,1}{2\,+\,1} \right)} = \frac{10}{9}\,in.

## Circular Cross Section

The analysis follows the same method as was used in the previous section on Trapezoidal cross sections.Hence,
And
To evaluate the above expand:
And

## Deflection Of Curved Beams (direct Method)

If the length of an initially curved beam is acted upon by a bending moment it follows from equation (5) that:**Deflection**is a term that is used to describe the degree to which a structural element is displaced under a load.

But is the change of angle subtended by at the centre of curvature and consequently is the angle through which the tangent at one end
of the element rotates relative to the tangent at the other end.
i.e.
The diagram shows a loaded bar which is fixed in direction at and it is required to find the
deflection at the other end .

Due to the action of on at only, the length is rotated through an angle . moves to ', where
The vertical deflection of
The horizontal deflection of
Due to the bending of all the elements along
The vertical deflection at
And the horizontal deflection =

Example:

[imperial]

##### Example - Example 3

Problem

A steel tube having an outside diameter of and a bore of is bent into a quadrant of radius. One end is rigidly attached to a horizontal base plate to which the tangent at that end is perpendicular.
If the free end supports a load of , determine the

**vertical and horizontal deflection**of the free end.Workings

and
Vertical deflection =
Horizontal deflection =

Solution

**Vertical deflection**is**Horizontal deflection**is

## Deflection From Strain Energy ( Castigliano's Theorem)

**Castigliano'**

**s method**is a method for determining the displacements of a linear-elastic system based on the partial derivatives of the strain energy.

**Theorem:**

If is the total strain energy of any structure due to the application of external loads, at in the direction and to the couples then the deflections at in the directions are and and the angular rotations of the couples are , at their applied points.

**Proof**for concentrated loads: If the displacements (in the directions of the loads) produced by gradually applied loads are then, Let alone be increased by then, = increase in external work done Where, are increases in But if the loads were applied gradually from zero, the total strain energy, Subtracting equation (12) and neglecting the products of small quantities, Subtracting equation (13), or Similarly for and and the proof can be extended to incorporate couples.

- 1) In finding the deflection of curved beams and similar problems, only strain energy due to

- 2) Treat all loads as variables initially carry out the partial differentiation and integration

- 3) If the deflection is to be found at a point where, or in a direction there is no load, a load

Example:

[imperial]

##### Example - Example 4

Problem

Obtain an

**expression**for the vertical displacement of in the above diagram. If and find the**displacement**when . andWorkings

The bending moments in the various sections can be written as follows:-
(at ' from )
Constant
(at from )
(at from )
The displacement of the load at vertically
An allowance could be made for the linear extension of
Which is clearly negligible compared to the deflection due to bending.