Problem 2.13. Curved beam – Solution Manual

A rectangular curved beam is subjected to uniform bending moment, M. Height, d and width, h of the cross section are given. Determine and sketch stresses σ_φ and σ_r , if

$\frac{\bar{r}}{d} = 2, 5, 10$

where

a) d = 0.4 m, h = 0.2 m, M = 120 kNm,

b) d = 0.3 m, h = 0.2 m, M = 100 kNm,

c) d = 0.4 m, h = 0.15 m, M = 140 kNm.

Solve Problem

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Check the results for the first set of initial data.

Problem a)

$\frac{\bar{r}}{d} = 2$

$σ_{φ} (r = r_{i}) [N / {mm}^{2}] =$ ×10⁶

$σ_{φ} (r = r_{o}) [N / {mm}^{2}] =$ ×10⁶

$σ_{r} (r = \bar{r}) [N / {mm}^{2}] =$ ×10⁶

Sketch stresses.

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See Figure 2.33.

Calculate all the other results and compare them to the Table hided below.

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Problem a)	$\frac{\bar{r}}{d} = 2$	$\frac{\bar{r}}{d} = 5$	$\frac{\bar{r}}{d} = 10$
$σ_{φ} (r = r_{o}) [N / {mm}^{2}]$	2.706×10⁷	2.411×10⁷	2.328×10⁷
$σ_{φ} (r = r_{i}) [N / {mm}^{2}]$	-1.93×10⁷	-2.11×10⁷	-2.178×10⁷
$σ_{r} (r = \bar{r}) [N / {mm}^{2}]$	2.792×10⁶	1.124×10⁶	5.623×10⁵

Problem b)	$\frac{\bar{r}}{d} = 2$	$\frac{\bar{r}}{d} = 5$	$\frac{\bar{r}}{d} = 10$
$σ_{φ} (r = r_{o}) [N / {mm}^{2}]$	4.01×10⁷	3.572×10⁷	3.448×10⁷
$σ_{φ} (r = r_{i}) [N / {mm}^{2}]$	-2.86×10⁷	-3.126×10⁷	-3.226×10⁷
$σ_{r} (r = \bar{r}) [N / {mm}^{2}]$	4.136×10⁶	1.664×10⁶	8.331×10⁵

Problem c)	$\frac{\bar{r}}{d} = 2$	$\frac{\bar{r}}{d} = 5$	$\frac{\bar{r}}{d} = 10$
$σ_{φ} (r = r_{o}) [N / {mm}^{2}]$	4.21×10⁷	3.751×10⁷	3.621×10⁷
$σ_{φ} (r = r_{i}) [N / {mm}^{2}]$	-3.003×10⁷	-3.282×10⁷	-3.387×10⁷
$σ_{r} (r = \bar{r}) [N / {mm}^{2}]$	4.343×10⁶	1.748×10⁶	8.748×10⁵

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Steps

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Solution for the first set of initial data of Problem a) is presented.

Step 1. Express the radii of the curvature at the edges of the beam from the initial data.

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$d = 0.4 m, \frac{\bar{r}}{d} = 2 \to \bar{r} = 0.8 m r_{o} = \bar{r} + \frac{d}{2} = 1 m r_{i} = \bar{r} - \frac{d}{2} = 0.6 m$

Step 2. Calculate axial stresses, σ_φ at the edges of the cross section.

Check axial stressHide axial stress

Formulas for the calculation are given in Figure 2.33.

$R = (r_{o}^{2} - r_{i}^{2})^{2} - 4 \times r_{o}^{2} r_{i}^{2} {(\ln \frac{r_{o}}{r_{i}})}^{2} = 0.0338 m^{4} σ_{φ} = - \frac{4 M}{R h} (r_{o}^{2} - r_{i}^{2} - \frac{r_{o}^{2} r_{i}^{2}}{r_{}^{2}} \ln \frac{r_{o}}{r_{i}} + r_{o}^{2} \ln \frac{r}{r_{o}} - r_{i}^{2} \ln \frac{r}{r_{i}}) At internal edge σ_{φ} (r = r_{i}) = - \frac{4 M}{R h} (r_{o}^{2} - r_{i}^{2} - r_{o}^{2} \ln \frac{r_{o}}{r_{i}} + r_{o}^{2} \ln \frac{r_{i}}{r_{o}}) = 2.706 \times 10^{4} \frac{kN}{m^{2}} = 27.06 \frac{N}{{mm}^{2}} At external edge σ_{φ} (r = r_{o}) = - \frac{4 M}{R h} (r_{o}^{2} - r_{i}^{2} - r_{i}^{2} \ln \frac{r_{o}}{r_{i}} - r_{i}^{2} \ln \frac{r_{o}}{r_{i}}) = - 1.93 \times 10^{4} \frac{kN}{m^{2}} = - 19.30 \frac{N}{{mm}^{2}}$

Step 3. Calculate radial stresses, σ_r at the middle of the cross section.

Check radial stressHide radial stress

Formulas for the calculation are given in Figure 2.33.

$R = (r_{o}^{2} - r_{i}^{2})^{2} - 4 \times r_{o}^{2} r_{i}^{2} {(\ln \frac{r_{o}}{r_{i}})}^{2} = 0.0338 m^{4} σ_{r} = - \frac{4 M}{R h} (\frac{r_{o}^{2} r_{i}^{2}}{r_{}^{2}} \ln \frac{r_{o}}{r_{i}} + r_{o}^{2} \ln \frac{r}{r_{o}} - r_{}^{2} \ln \frac{r}{r_{i}}) At the middle of the cross section σ_{φ} (r = \bar{r}) = - \frac{4 M}{R h} (\frac{r_{o}^{2} r_{i}^{2}}{r_{}^{2}} \ln \frac{r_{o}}{r_{i}} + r_{o}^{2} \ln \frac{\bar{r}}{r_{o}} - r_{}^{2} \ln \frac{\bar{r}}{r_{i}}) = 2.791 \times 10^{3} \frac{kN}{m^{2}} = 27.91 \frac{N}{{mm}^{2}}$

See Figure 2.33.

Step 4. Sketch stresses.

Compare figureHide figure

Step 5. Calculate all the other results and compare them to the Table hided below.

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Problem a)	$\frac{\bar{r}}{d} = 2$	$\frac{\bar{r}}{d} = 5$	$\frac{\bar{r}}{d} = 10$
$σ_{φ} (r = r_{o}) [N / {mm}^{2}]$	27.06	24.11	23.28
$σ_{φ} (r = r_{i}) [N / {mm}^{2}]$	-19.30	-21.10	-21.78
$σ_{r} (r = \bar{r}) [N / {mm}^{2}]$	2.792	1.124	0.5623

Problem b)	$\frac{\bar{r}}{d} = 2$	$\frac{\bar{r}}{d} = 5$	$\frac{\bar{r}}{d} = 10$
$σ_{φ} (r = r_{o}) [N / {mm}^{2}]$	40.10	35.72	34.48
$σ_{φ} (r = r_{i}) [N / {mm}^{2}]$	-28.6	-31.26	-32.26
$σ_{r} (r = \bar{r}) [N / {mm}^{2}]$	4.136	1.664	0.833

Problem c)	$\frac{\bar{r}}{d} = 2$	$\frac{\bar{r}}{d} = 5$	$\frac{\bar{r}}{d} = 10$
$σ_{φ} (r = r_{o}) [N / {mm}^{2}]$	42.10	37.51	36.21
$σ_{φ} (r = r_{i}) [N / {mm}^{2}]$	-30.03	-32.82	-33.87
$σ_{r} (r = \bar{r}) [N / {mm}^{2}]$	4.343	1.748	0.875

Results

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Solution for the first set of initial data of Problem a) is presented.

The radii of the curvature at the edges of the beam are expressed from the initial data:

$d = 0.4 m, \frac{\bar{r}}{d} = 2 \to \bar{r} = 0.8 m r_{o} = \bar{r} + \frac{d}{2} = 1 m r_{i} = \bar{r} - \frac{d}{2} = 0.6 m$

Axial stresses, σ_φ at the edges of the cross section are

$σ_{φ} = - \frac{4 M}{R h} (r_{o}^{2} - r_{i}^{2} - \frac{r_{o}^{2} r_{i}^{2}}{r_{}^{2}} \ln \frac{r_{o}}{r_{i}} + r_{o}^{2} \ln \frac{r}{r_{o}} - r_{i}^{2} \ln \frac{r}{r_{i}}), where R = (r_{o}^{2} - r_{i}^{2})^{2} - 4 \times r_{o}^{2} r_{i}^{2} {(\ln \frac{r_{o}}{r_{i}})}^{2} = 0.0338 m^{4} At internal edge σ_{φ} (r = r_{i}) = - \frac{4 M}{R h} (r_{o}^{2} - r_{i}^{2} - r_{o}^{2} \ln \frac{r_{o}}{r_{i}} + r_{o}^{2} \ln \frac{r_{i}}{r_{o}}) = 27.06 \frac{N}{{mm}^{2}} At external edge σ_{φ} (r = r_{o}) = - \frac{4 M}{R h} (r_{o}^{2} - r_{i}^{2} - r_{i}^{2} \ln \frac{r_{o}}{r_{i}} - r_{i}^{2} \ln \frac{r_{o}}{r_{i}}) = - 19.30 \frac{N}{{mm}^{2}}$

Radial stresses, σ_r are

$σ_{r} = - \frac{4 M}{R h} (\frac{r_{o}^{2} r_{i}^{2}}{r_{}^{2}} \ln \frac{r_{o}}{r_{i}} + r_{o}^{2} \ln \frac{r}{r_{o}} - r_{}^{2} \ln \frac{r}{r_{i}}), where R = (r_{o}^{2} - r_{i}^{2})^{2} - 4 \times r_{o}^{2} r_{i}^{2} {(\ln \frac{r_{o}}{r_{i}})}^{2} = 0.0338 m^{4} At the middle of the cross section σ_{φ} (r = \bar{r}) = - \frac{4 M}{R h} (\frac{r_{o}^{2} r_{i}^{2}}{r_{}^{2}} \ln \frac{r_{o}}{r_{i}} + r_{o}^{2} \ln \frac{\bar{r}}{r_{o}} - r_{}^{2} \ln \frac{\bar{r}}{r_{i}}) = 2.791 \frac{N}{{mm}^{2}}$

See Figure 2.33.

All the other results are given in the Table below.

Problem a)	$\frac{\bar{r}}{d} = 2$	$\frac{\bar{r}}{d} = 5$	$\frac{\bar{r}}{d} = 10$
$σ_{φ} (r = r_{o}) [N / {mm}^{2}]$	27.06	24.11	23.28
$σ_{φ} (r = r_{i}) [N / {mm}^{2}]$	-19.30	-21.10	-21.78
$σ_{r} (r = \bar{r}) [N / {mm}^{2}]$	2.792	1.124	0.562

Problem b)	$\frac{\bar{r}}{d} = 2$	$\frac{\bar{r}}{d} = 5$	$\frac{\bar{r}}{d} = 10$
$σ_{φ} (r = r_{o}) [N / {mm}^{2}]$	40.10	35.72	34.48
$σ_{φ} (r = r_{i}) [N / {mm}^{2}]$	-28.60	-31.26	-32.26
$σ_{r} (r = \bar{r}) [N / {mm}^{2}]$	4.136	1.664	0.833

Problem c)	$\frac{\bar{r}}{d} = 2$	$\frac{\bar{r}}{d} = 5$	$\frac{\bar{r}}{d} = 10$
$σ_{φ} (r = r_{o}) [N / {mm}^{2}]$	42.10	37.51	36.21
$σ_{φ} (r = r_{i}) [N / {mm}^{2}]$	-30.03	-32.82	-33.87
$σ_{r} (r = \bar{r}) [N / {mm}^{2}]$	4.343	1.748	0.875