# 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.##### MISSING IMAGE!

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**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.

## Application To The Design Of A Piston Ring

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**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.

## 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##### MISSING IMAGE!

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## Rectangular Cross-section

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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 - Example 1

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

- Tension
- Compression

## Trapezoidal Cross-section.

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##### Example - Example 2

**maximum tensile**and

**compressive forces**set up.

## Circular Cross Section

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## 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 (4) that:**Deflection**is a term that is used to describe the degree to which a structural element is displaced under a load.

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##### Example - Example 3

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

**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 (11) and neglecting the products of small quantities, Subtracting equation (12), 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 - Example 4

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

**displacement**when . and