Stress Intensity Calculation for Beam Cracks

[thephysicsman]

Homework Statement

[PLAIN]http://img232.imageshack.us/img232/8514/unbenannthz.png

In this system, a single-span beam is exposed to the following stresses:

$$\sigma_{max}= 255 Mpa$$ at $$F_{max} = 50 kN$$ and
$$\sigma_{min}= 102 MPa$$ at $$F_{min} = 20 kN$$

The stress range is accordingly $$\Delta\sigma = 153 MPa$$

A crack is detected during an inspection of the construction. This runs in the bottom flange of the support, and has a length of 2a = 5 mm. The flange has a thickness of 2t = 10 mm.

[PLAIN]http://img62.imageshack.us/img62/3403/unbenanntcc.png

$$f(\frac{a}{t})$$ = 1.186

a) Calculate the stress intensity K for the maximum and minimum stress!

b) Calculate stress intensity difference $$\Delta K$$!

c) A fracture test of the steel of the beam is carried out. The test specimen has a $$a_{0}/W$$ ratio of 0.5335. The data record below was recorded during the test.

[PLAIN]http://img830.imageshack.us/img830/6829/unbenannthty.png

Determine the fracture toughness $$K_{IC}$$ using the following formula and check if the crack is critical!

$$f(\frac{a_{0}}{W})=2.97$$

$$K_{IC}=\frac{F_{max}*S}{\sqrt{B*B_{N}}W^{1.5}}f(\frac{a_{0}}{W})$$

$$a_0 = 13,87 mm$$, W = 26 mm, B = 13 mm, $$B_N = 10,4 mm$$, S = 104 mm!

Homework Equations

$$K_{component}=\sigma \sqrt{\pi a} f(\frac{a}{t})$$

The Attempt at a Solution

a)

$$K_{component, max}=255 MPa \sqrt{\pi 0.0025 m} 1.186 = 26.80 Mpa \sqrt{m}$$

$$K_{component, min}=102 MPa \sqrt{\pi 0.0025 m} 1.186 = 10.72 Mpa \sqrt{m}$$b)

$$\Delta K = 26.80-10.72 = 16.08 Mpa \sqrt{m}$$

c)

$$K_{IC}=\frac{50 000 N*0.104m}{\sqrt{0.013m*0.0104m}0.026^{1.5}}2.97=317 MPa\sqrt{m}$$Is this correct, or is there something that I have not understood?

 

[anathema]

Your calculations for parts a and b are correct. For part c, your calculation for K_IC is correct, but you also need to consider the stress intensity factor at the critical crack size, which is given by K_c = K_IC / f(a_0/W). This value should be compared to the calculated K_component values to determine if the crack is critical.