Problem 2.14. Curved I beam – Solution Manual

A curved I-beam is subjected to uniform bending moment, M. Central distance between the flanges, d and the thickness of the web t_w are
given. (The difference between d and the height of the beam is neglected.) Three radii are examined: $\frac{\bar{r}}{d} = 2, 5, 10$ . Suppose thin web, so the normal tress, σ_φ in the web is negligible. Determine and sketch stress, σ_r in the web.

a) d = 0.4 m, t_w = 2 mm, M = 120 kNm,
b) d = 0.3 m, t_w = 4 mm, M = 100 kNm,
c) d = 0.4 m, t_w = 3 mm, M = 140 kNm.

Use equilibrium equations, apply the pressure vessel formula.

This solution is not identical to the solution of isotropic plates subjected to in-plane loads, because zero stress is assumed in the hoop direction. The web is cylindrically orthotropic, the stiffness in the hoop direction is negligible compared to the radial stiffness.

Solve Problem

SolveHide solve

Check the results for the first set of initial data.

Problem a)

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

Stress at the top of the web, $σ_{r} (r = r_{o}) [N / {mm}^{2}] =$

Stress at the bottom of the web, $σ_{r} (r = r_{i}) [N / {mm}^{2}] =$

Sketch radial stresses of the web.

Check figureHide figure

Calculate results for all the other set of initial data.

Check tableHide table

	Problem a)			Problem b)			Problem c)
$\bar{r} / d$	2	5	10	2	5	10	2	5	10
d [m]	0.4			0.3			0.4
t_{w [}mm]	2			4			3
M [kNm]	120			100			140
$\bar{r}$ [m]	0.8	2	4	0.6	1.5	3	0.8	2	4
r_o[m]	1	2.2	4.2	0.75	1.65	3.15	1	2.2	4.2
r₁[m]	0.6	1.8	3.8	0.45	1.35	2.85	0.6	1.8	3.8
N [kN]	300			333.33			350
σ_o[N/mm²]	150	68.18	35.71	111.11	50.51	26.45	116.67	53.03	27.78
σ_i[N/mm²]	250	83.33	39.47	185.18	61.73	29.24	194.44	64.82	30.70

Do you need help?

Steps

Step by stepHide steps

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.

Check curvaturesHide curvatures

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

Step 2. Determine the approximate normal force in the flanges.

Check normal forceHide normal force

We assume that moment is carried only by the flanges, normal stresses in the web are neglected:

$N = \frac{M}{d} = \frac{120 \times 10^{6}}{0.4 \times 10^{3}} = \pm 3 \times 10^{5} N$

Step 3. Calculate the radial stresses in the web which arise from the tension and compression of curved flanges.

Check top radial stressHide top radial stress

Apply the pressure vessel formula:

Eq.11-8.

$σ_{o} = \frac{N}{r_{o} t_{w}} = \frac{3 \times 10^{5}}{1000 \times 2} = -150 \frac{N}{{mm}^{2}} (compression)$

Check bottom radial stressHide bottom radial stress

$σ_{i} = \frac{N}{r_{i} t_{w}} = \frac{3 \times 10^{5}}{600 \times 2} = 250 \frac{N}{{mm}^{2}} (tension)$

Step 4. Sketch stresses.

Check figureHide figure

Step 5. Calculate results for all the other set of initial data.

Check tableHide table

	Problem a)			Problem b)			Problem c)
$\bar{r} / d$	2	5	10	2	5	10	2	5	10
d [m]	0.4			0.3			0.4
t_{w [}mm]	2			4			3
M [kNm]	120			100			140
$\bar{r}$ [m]	0.8	2	4	0.6	1.5	3	0.8	2	4
r_o[m]	1	2.2	4.2	0.75	1.65	3.15	1	2.2	4.2
r₁[m]	0.6	1.8	3.8	0.45	1.35	2.85	0.6	1.8	3.8
N [kN]	300			333.33			350
σ_o[N/mm²]	150	68.18	35.71	111.11	50.51	26.45	116.67	53.03	27.78
σ_i[N/mm²]	250	83.33	39.47	185.18	61.73	29.24	194.44	64.82	30.70

Results

Worked out solutionsHide solution

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.

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

We assume that moment is carried only by the flanges, normal stresses in the web are neglected. The approximate normal force in the flanges is

$N = \frac{M}{d} = \frac{120 \times 10^{6}}{0.4 \times 10^{3}} = \pm 3 \times 10^{5} N$

The radial stresses in the web arise from the tension and compression of curved flanges. Applying the pressure vessel formula the top and bottom radial stresses are

Eq.11-8.

$σ_{o} = \frac{N}{r_{o} t_{w}} = \frac{3 \times 10^{5}}{1000 \times 2} = -150 \frac{N}{{mm}^{2}} (compression)$

$σ_{i} = \frac{N}{r_{i} t_{w}} = \frac{3 \times 10^{5}}{600 \times 2} = 250 \frac{N}{{mm}^{2}} (tension)$

Results for the other sets of initial data are

	Problem a)			Problem b)			Problem c)
$\bar{r} / d$	2	5	10	2	5	10	2	5	10
d [m]	0.4			0.3			0.4
t_{w [}mm]	2			4			3
M [kNm]	120			100			140
$\bar{r}$ [m]	0.8	2	4	0.6	1.5	3	0.8	2	4
r_o[m]	1	2.2	4.2	0.75	1.65	3.15	1	2.2	4.2
r₁[m]	0.6	1.8	3.8	0.45	1.35	2.85	0.6	1.8	3.8
N [kN]	300			333.33			350
σ_o[N/mm²]	150	68.18	35.71	111.11	50.51	26.45	116.67	53.03	27.78
σ_i[N/mm²]	250	83.33	39.47	185.18	61.73	29.24	194.44	64.82	30.70