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# Curved Beams

An analysis of stresses and strains in curved beams
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## 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 (1)



Total normal stress = 0, i.e.


which shows that the neutral axis passes through the centroid of the section.

Moment of resistance, 

 from equation (3)


Combining equations (2) and (4),

the strain energy of a short length  (measured along the neutral surface) under the action og bending moment  is:




From equation (3)


## 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 (6). 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 (8)


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 (7), 

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.

and
Workings

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 (7) becomes


i.e.  or, 





From which,


And since, 

 can be evaluated from equations (9) and (10).

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 = in tension

The combined stress = tensile.

At the outside edge,

Bending stress =

Combined stress = in compression

## 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 (4) 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.

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.

It is important to stress that  is the total strain energy, expressed in terms of loads and not including statically determinate reactions and the partial derivative with respect to each load in turn (treating the others as constant) gives the deflection at the load points in the direction of the load.

The following principles should be observed in applying the theorem
• 1) In finding the deflection of curved beams and similar problems, only strain energy due to
bending need normally be taken into account (i.e. )
• 2) Treat all loads as variables initially carry out the partial differentiation and integration
and only putting in numerical values at the final stage.
• 3) If the deflection is to be found at a point where, or in a direction there is no load, a load
may be put in where required and given a value of zero in the final reckoning (i.e. )

Generally it will be found that the strain energy method requires less thought in application than the direct method, it being only necessary to obtain an expression for the bending moment; also there is no difficulty over the question of sign as the strain energy is bound to be positive and deflection is positive in the direction of the load. The only disadvantage occurs when a case such as mentioned in note 3 above has to be dealt with in which case the direct method will probably be shorter.
Example:

[imperial]
##### Example - Example 4
Problem

Obtain an expression for the vertical displacement of in the above diagram. If and find the displacement when . and
Workings
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.