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Curved Beams

An analysis of stresses and strains in curved beams

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 \inline R_0 be the initial (unstrained) radius of curvature of the neutral surface and \inline R the radius of curvature under the action of a pure bending moment \inline M.

Then the strain in a \inline n element at a distance \inline y 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 =\inline \;\displaystyle\frac{PQ^{'} - PQ}{PQ} = \displaystyle\frac{(R\,+\,y)(\theta \,+\,\delta \theta)\,-\,(R_0\,+\,y)\theta  }{(R_0\,+y)\theta }
= \frac{R(\theta \,+\,\delta \theta )\,-\,R_0\theta \,+\,y\delta \theta }{(R_)\,+\,y)\theta }=\frac{y\delta \theta }{(R_0\,+\,y)\theta }

Since \inline  R(\theta \,+\,\delta \theta ) = R_0\theta = length along the neutral axis

If \inline y is neglected in comparison with \inline R_0 and noting from \inline R(\theta +\delta \theta ) = R_0 that \inline \delta \theta  = \left(\displaystyle\frac{R_0-R}{R} \right)\theta

Then strain,

Neglecting lateral stress, the normal stress, \inline f = E\times 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, \inline M=\displaystyle\int f\;y\;dA

\inline = E\;I\left(\displaystyle\frac{1}{R} - \displaystyle\frac{1}{R_0} \right)\displaystyle\int f\;y^2\;dA from equation (3)

Combining equations (2) and (4),
\frac{f}{y} = \frac{M}{I} = E\,\left(\frac{1}{R} - \frac{1}{R_0} \right)
the strain energy of a short length \inline \delta ,s (measured along the neutral surface) under the action og bending moment \inline M is:

\delta U = \frac{1}{2}\,M\,\delta \theta
\therefore\;\;\;\;\;\delta \: U = \frac{1}{2}\,M\,\left(\frac{R_0 - R}{R} \right)\theta

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 \inline p is the uniform pressure on the outside then the bending moment at \inline be is given by:

\inline M = \displaystyle\int_{0}^{{\pi  - \theta }}(p\,.\,d\:R\,.\,d\phi )R\,sin\phi approx

where \inline d is the depth of the ring in the axial direction integrating But \inline \displaystyle\frac{M}{I} = E\left(\displaystyle\frac{1}{R} + \displaystyle\frac{1}{R_0}}\right) = a constant for a given condition

i.e. \inline \;\;\;\;\;\;\displaystyle\frac{p\,R^2\;d(1 + cos\theta )}{d\,\displaystyle\frac{t^3}{12}} = constant \inline = \displaystyle\frac{24\,R^2}{t_0^3} when \inline \theta  = 0 and \inline t=t_0

\therefore\;\;\;\;\frac{t}{t_0} = \sqrt[3]{\left(\frac{(1 + cos\,\theta )}{2} \right)}
Which is the required variation of thickness. Using equation (6). The maximum bending stress at any section
= \left(\frac{M}{I} \right)\left(\frac{T}{2} \right) = \left(\frac{6\,p\,R^2}{t^2} \right)\left(1 + cos\theta  \right)= \frac{12\,p\,R^2\.t}{t_0^3}
which has it's greatest value when \inline \theta  = 0 i.e. \inline \hat{f} = \displaystyle\frac{12\,p\,r^2}{t_0^2}

From which, \inline \;\;\;\displaystyle\frac{1}{R} - \displaystyle\frac{1}{R_0} = \displaystyle\frac{f}{E\,y} = \displaystyle\frac{24\,p\,R^2}{E\,t_0^3}
\therefore\;\;\;\frac{1}{R_0} = \left(\frac{1}{R} \right)\left(1 - \frac{2\,\hat{f}\, R}{E\,t_0} \right)
which determines the initial radius when values for \inline t_0 and \inline \hat{f} 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 \inline y in comparison to \inline R 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


f = E\;X\;S = E= \frac{QQ^'}{PQ}= \frac{E\,y\,\delta \theta }{(R_0 + y)\theta }
where \inline S is the strain,\inline y is the distance from the neutral axis as before and \inline R_0 is the initial \inline d radius of the neutral surface.

For pure bending the Total normal force on the cross section = \inline 0. Moment of resistance,

But \inline \displaystyle\int \displaystyle\frac{y^2\;dA}{R_0 + y} = \displaystyle\int\displaystyle\frac{[y(y + R_0) - R_0\;y]}{R_0 + y}\;dA
= \int y\;dA - R_0\;y\int \frac{y\;dA}{r_0 + y}= A\;e - 0

Where \inline e is the distance between the neutral axis and the principle axis which is through the centroid ( \inline e is positive when the neutral axis is on the same side of the centroid as the centre of curvature)

Substituting in equation (8)
M = \left(\frac{E\;\delta \theta }{\theta } \right)A\;e= \left[\int \frac{R_0 + y}{y} \right]\;A\;e

Rearranging, \inline f = \displaystyle\frac{M\,y}{A\,e(R_0\,+\,y)}

In this equation \inline y is positive measured outwards, a positive bending moment being one that tends to increase the curvature.

Rectangular Cross-section


From equation (7), \inline \displaystyle\int \displaystyle\frac{y\;dA}{R_0 + y} = 0

Let \inline z = y - e = the distance from the centroid. Also the mean radius of curvature \inline R_m = R_0 + e and \inline dA \;= \;bdz

Then, \inline \displaystyle\int \displaystyle\frac{z\,+\,e}{R_m\,+\,z}\;bdz = 0

i.e. \inline \displaystyle\int_{-\displaystyle\frac{d}{2}}^{\displaystyle\frac{d}{2}}\displaystyle\frac{R_m + z\;-(R_m - e)}{R_m + z}\;dz = 0
\therefore\;\;\;\int_{-\frac{d}{2}}^{\frac{d}{2}}dz\,-\,(R_m\,-\,e)\int_{-\frac{d}{2}}^{\frac{d}{2}}\frac{dz}{R_m\,+\,z} = 0

Hence, \inline \;\;\;d - (R_m - e)Ln\displaystyle\frac{R_M\;+\displaystyle\frac{d}{2}}{R_m - \displaystyle\frac{d}{2}} = 0

Giving, \inline \;\;\;\;e = R_m - d\;\div \Ln\displaystyle\frac{R_M\;+\displaystyle\frac{d}{2}}{R_m\;-\displaystyle\frac{d}{2}} = 0

As \inline e is small compared to \inline R_m and \inline d, 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.

e = R_m - \frac{d}{2\left[\displaystyle\frac{d}{2R_m}\,+\,\displaystyle\frac{1}{3}(\displaystyle\frac{d}{2R_m})^3 + \displaystyle\frac{1}{5}(\displaystyle\frac{d}{2R_m})^5 + .... \right]}

Example - Example 1
A curved bar, initially unstressed, of square cross section, has \inline 3\;in. sides and a mean radius of curvature of \inline 4.5\;in.

If a bending moment of \inline 30\;tons-in. is applied to the bar tending to straighten it, find the stresses at the outer and inner faces.

\inline R_m = 4.5\;in. and \inline d = 3\;in.
e = R_m - \displaystyle\frac{d}{Ln}\;\frac{R_m\;+\displaystyle\frac{d}{2}}{R_m\;-\displaystyle\frac{d}{2}}

\therefore\;\;\;e = 4.5 - \frac{3}{Ln\;2} = 0.172\;in.

But \inline R_0 = R_m - e = 4.328\;in.

\inline M\;= - 30\;tons.in. and \inline f = \displaystyle\frac{My}{Ae(R_0\;+\;y)}

At the inside face, \inline y\;= - (\displaystyle\frac{d}{2}\;-\;e) = -4.328\; in.

\therefore\;\;\;f\;=\frac{-\,30\;\times \;(-1.328)}{9\;\times \;0.172(4.328\,-1.328)}

Thus, \inline f = 8.6\;tons/sq.in. Tension

At the outside face, \inline y = \displaystyle\frac{d}{2} + e = 1.672\;in.

\therefore\;\;\;f = (-30\,\times \;1.672)\;\div\;[9\;\times \;0.172(4.328\;+\;1.672)]

\inline = 5.45\;tons/sq.in. compression

The actual stress distribution is shown in the diagram.
  • \inline f = 8.6\;tons/sq.in. Tension
  • \inline f= 5.45\;tons/sq.in. Compression

Trapezoidal Cross-section.


By Moments,
d_1 = \frac{(B_1\,+\,2B_2)}{(B_1\,+\,B_2)}\left(\frac{D}{3} \right)
d_2 = \frac{(2B_1\,+\,B_2)}{(B_1\,+\,B_2)}\left(\frac{D}{3} \right)

By putting \inline z = y - e and \inline R_m = R_0 + e equation (7) becomes
\int \frac{z + e}{R_m + z}\;dA = 0

i.e. \inline A - (R_m - e)\displaystyle\int \displaystyle\frac{dA}{(R_m + z)} = 0 or, \inline e = R_m - \displaystyle\frac{A}{\displaystyle\int \displaystyle\frac{dA}{(R_m\,+\,z)}}

= \int_{-d_1}^{d_2} \frac{B_2 + \displaystyle\frac{B_1\,- B_2}{D}d_2 + \displaystyle\frac{B_1-B_2}{D}\times R_m\-\displaystyle\frac{B_1-B_2}{D}(R_m + z)}{R_M + z}\;dz

From which,

And since, \inline A = \left(\displaystyle\frac{B_1\,+\,B_2}{2} \right)\times D

\inline e can be evaluated from equations (9) and (10).

Example - Example 2
A crane hook whose horizontal cross-section is trapezoidal, \inline 2\;in. wide on the inside and \inline 1\;in. wide on the outside by \inline 2\;in. thick, carries a vertical load of one ton whose line of action is \inline 2.5\;in. from the inside edge of this section. The centre of curvature is \inline 2\;in. from the inside edge.

Calculate the maximum tensile and compressive forces set up.
Referring to the last figure.

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.
R_m = 2 + d_1

From equation (9)

\int \frac{dA}{R_m\,+\,z} = \left(1\,+\,\left[\frac{2\,-\,1}{2} \right]\left[ \frac{26}{9}\,+\,\frac{10}{9}  \right]\right)Ln\frac{26\,+\,10}{26\,-\,8}\;-(2\,-\,1)
= 3\,(\ln\,2 )\,-\,1\;=1.0793
A = \left(\frac{2\,+\,1}{2} \right)\times 2 = 3in^2
\therefore\;\;\;e = \frac{26}{9} - \frac{3}{1.0793} = 0.11 in.

Direct stress = load / area = \inline \displaystyle\frac{1}{3}\;ton/sq. in. tensile

Bending stress = \inline \displaystyle\frac{M\;y}{A\;e\;(R_m\;+\;y)}= \displaystyle\frac{M(z\;+\;e)}{A\;e\;(R_m\;+\;z)}

At the inside edge, \inline z = -d_1 = 0.89\,in.

\inline M\;= - (2.5\,+\,d_1)\;= - 3.39\,ton.in. (tending to decrease the curvature)

Bending stress = \inline \left(\displaystyle\frac{- 3.39(- 0.89 + 0.11)}{3\;\times \;0.11(2.89 - 0.89\0}\right)=4.02\;tons/sq.in. in tension

The combined stress = \inline 4.02 + 0.33 = 4.35\;tons/sq.in. tensile.

At the outside edge, \inline z = d_2 = 1.11\;in.

Bending stress = \inline \displaystyle\frac{-\;3.39(1.11\;+\;0.11)}{3\;\times 0.11(2.89\;+\,1.11)}\;= - 3.14\,Tons/sq.in.

Combined stress = \inline 3.14 - 0.33 = 2.81\;tons/sq.in. in compression

Circular Cross Section


The analysis follows the same method as was used in the previous section on Trapezoidal cross sections.

Hence, \inline e = R_m - A\;\div \displaystyle\int \displaystyle\frac{dA}{R_m + z}

And \inline \displaystyle\int \displaystyle\frac{dA}{R_m + z} = 2\displaystyle\int_{-r}^{r}\;\displaystyle\frac{\sqrt{(r^2 - z^2)}}{R_m + z}\;dy= 2\pi \left(R_m - \sqrt{(R_m^2 - r^2)} \right)

\therefore\;\;\;e = R_m - r^2\div 2 \left(R_m - \sqrt{(R_m^2 - r^2)} \right)
To evaluate the above expand:
\sqrt{(R_m^2 - r^2)} = R_m\left[1\,-\,\frac{1}{2}\frac{r^2}{R_m^2}\,-\,\frac{1}{8}\frac{r^4}{R_m^4}\,-\,.... \right]
= \frac{R_m[(\displaystyle\frac{r^2}{R_m^2})\,+\,\frac{1}{2}(\displaystyle\frac{r^4}{R_m^4})\,+\,\frac{1}{4}(\displaystyle\frac{r^6}{16R_m^6})\,+...]}{4\,+\,(\displaystyle\frac{r^2}{R_m^2})\,+\,\frac{1}{2}(\displaystyle\frac{r^4}{R_m^4})\,+\,....}
And \inline f = \displaystyle\frac{M\;y}{A\;e\;(R_0 + y)}

Deflection Of Curved Beams (direct Method)

If the length \inline \delta s of an initially curved beam is acted upon by a bending moment \inline M 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.

\frac{M\,\delta s}{E\,I} = \delta s\left(\frac{1}{R} - \frac{1}{R_0} \right)

But \inline \displaystyle\frac{\delta s}{R} - \displaystyle\frac{\delta s}{R_0} is the change of angle subtended by \inline \delta_s 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. \inline \delta \phi  = \displaystyle\frac{M\;\delta s}{E\;I}


The diagram shows a loaded bar which is fixed in direction at \inline A and it is required to find the deflection at the other end \inline B.

Due to the action of \inline M on \inline \delta s at \inline C only, the length \inline CB is rotated through an angle \inline \delta \phi  = \displaystyle\frac{M\;\delta_s}{E\;I}. \inline B moves to \inline B', where \inline BB' = CB\;\delta \phi

The vertical deflection of \inline B = BB'\;cos\theta= CB\;cos\theta\;\delta \phi  = x\;\delta \phi

The horizontal deflection of \inline B = BB'\;sin\;\theta= y\;\delta \phi

Due to the bending of all the elements along \inline AB

The vertical deflection at \inline B= \displaystyle\int x\;\delta \phi  = \displaystyle\int \displaystyle\frac{M\;x}{E\;I}\;ds

And the horizontal deflection =
\;\int y\,\delta \phi  = \int \frac{M\,y}{E\,I}\;ds

Example - Example 3
A steel tube having an outside diameter of \inline 2\;in. and a bore of \inline 1.5\;in. is bent into a quadrant of \inline 6\;ft. 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 \inline 100\;lb. , determine the vertical and horizontal deflection of the free end.

E = 30\times 10^6\;lb\,in^{-2}

I = \left(\frac{\pi }{64} \right)\left(2^4 - 1.5^4 \right) = 0.537\,in.^4

\inline x = 72\;sin\theta and \inline y = 72(1 - cos\;\theta )

M = 100x = 100\,\times \;72\.sin\theta
\delta s = 72\,\delta \theta

Vertical deflection =\inline \;\displaystyle\int \displaystyle\frac{M\;x}{E\;I}\;ds

= \int_{0}^{\frac{\pi }{2}}\frac{100\times 72^3\,sin^2\theta }{30\times 10^6\times 0.537}\;d\theta = 2.32\;\int_{0}^{\frac{\pi }{2}}\frac{1 - cos\,2\theta }{2}\;d\theta = 1.82 in.

Horizontal deflection = \inline \displaystyle\int \displaystyle\frac{M\;y}{E\;I}ds

= \frac{100\times 72^3}{30\times 10^6\times 0.537}\;\int_{0}^{\frac{\pi }{2}}\frac{1 - cos\,\theta }{2}d\theta
= 2.32\left[-\,cos\,\theta  + cos\,2\theta  \right]{_0^\frac{\pi }{2}} = 1.16\;in.
  • Vertical deflection is \inline 1.82 in.
  • Horizontal deflection is \inline 1.16\;in.

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.

If \inline U is the total strain energy of any structure due to the application of external loads, \inline W_1\;W_2\;... at \inline O_1\;O_2\;... in the direction \inline \;O_1X_1\;O_2X_2 and to the couples \inline M_1\;M_2... then the deflections at \inline O_1\;O_2... in the directions \inline O_1X_1\;O_2X_2... are \inline \displaystyle\frac{\delta U}{\delta W_1} and \inline \displaystyle\frac{\delta U}{\delta W_2} and the angular rotations of the couples are \inline  \displaystyle\frac{\delta\;U}{\delta\;M_1},\inline \displaystyle\;\frac{\delta\;U}{\delta\;M-2} at their applied points.

Proof for concentrated loads:

If the displacements (in the directions of the loads) produced by gradually applied loads \inline W_1\;W_2\;W_3 ... are \inline x_1\;x_2\;x_3 then,

U = \frac{1}{2}W_1x_1\;+\frac{1}{2}\;W_2x_2 + \frac{1}{2}W_3x_3 + ...

Let \inline W_1 alone be increased by \inline \delta\;W_1

then, \inline \delta U = increase in external work done
\delta \,u = W_1\delta x_1 + W_2\delta x_2 + W_3\delta x_3

Where, are increases in \inline x_1\;x_2\;x_3

But if the loads \inline W_1 + \delta W_1\;W_2\;W_3 were applied gradually from zero, the total strain energy,

U + \delta U = \frac{1}{2}(W_1 + \delta W_1) + \frac{1}{2}W_2(x_2 + \delta x_2) + \frac{1}{2}W_3(x_3 + \delta x_3)

Subtracting equation (11) and neglecting the products of small quantities,

\delta U = \frac{1}{2}W_1\delta x_1 + \frac{1}{2}\delta W_1\,x_1 + \frac{1}{2}W_2\delta x_2 + \frac{1}{2}W_3\delta x_3\;+...

Subtracting equation (12), \inline \delta U = \delta W_1x_1 or \inline \displaystyle\frac{\delta U}{\delta W_1} = x_1

Similarly for \inline x_2 and \inline x_3 and the proof can be extended to incorporate couples.

It is important to stress that \inline U 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. \inline \displaystyle\int \displaystyle\frac{M^2}{2\,E\,I}ds)
  • 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. \inline x\;=\left[ \displaystyle\frac{\delta U}{\delta W}\ \right]_{W=0} \right)

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 - Example 4

Obtain an expression for the vertical displacement of \inline A in the above diagram. If \inline a = 2\;in. and \inline t =0.25\;in. find the displacement when \inline W =5\;lb. and \inline E = 30\times 10^6\;lb.in^{-2}
The bending moments in the various sections can be written as follows:-

\inline AB\;\;\;M = Wx' (at \inline x' from \inline A)

\inline BC\;\;\;M = Wa Constant

\inline CD\;\;\;M = Wx' (at \inline x' from \inline D)

\inline DE\;\;\;M = Wx'' (at \inline x'' from \inline D)

U = \int \frac{M^2}{2EI}ds
= \int_{0}^{a}\frac{W^2x^2}{2E\times \displaystyle\frac{t^3}{12}}dx + \int_{0}^{2a}\frac{W^2a^2}{2E\times \displaystyle\frac{(2t)^3}{12}} + \int_{0}^{a}\frac{W^2x'^2}{2E\times\displaystyle\frac{t^3}{12}}dx' + \int_{0}^{1.5a}\frac{W^2x''^2}{2E\times \displaystyle\frac{t^3}{12}}
=\left(\frac{6W^2}{Et^3} \right)\left(\frac{a^3}{3} + \frac{2a^3}{8} + \frac{a^3}{3} + \frac{1.5^3a^3}{3} \right)= \frac{24.5W^2a^3}{2Et^3}

The displacement of the load at \inline A = \displaystyle\frac{\delta U}{\delta\;W}\; = \displaystyle\frac{24.5Wa^3}{Et^3} vertically
= \frac{24.5\times 5\times 2^3}{30\times 10^6\times 0.25^3} = 0.0021\,in.

An allowance could be made for the linear extension of \inline BC
\frac{W\,2a}{2t\,E} = \frac{5\times 2}{0.25\times 30\times 10^6} = \frac{4}{3}\times 10^{-6}
Which is clearly negligible compared to the deflection due to bending.