the crack growth resistance curve of a certain material at a thickness 2 mm is expressed by onsider a center cracked plate of width 10 cm and thickness 2 mm with a crack of length 1 cm. calculate the length of stable crack growth, the critical crack length, and the critical stress at instability.

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Answer 1

The length of stable crack growth is 5 mm, the critical crack length is approximately 8.12 mm, and the critical stress at instability is approximately 741.5 MPa.

We can use the crack growth resistance curve to determine the length of stable crack growth, critical crack length, and critical stress at instability.

Assuming that the crack growth resistance curve is a straight line and can be expressed as:

[tex]da/dN = C*(\Delta K)^m[/tex]

where:

da/dN = crack growth rate (mm/cycle)

C = material constant

ΔK = stress intensity factor range (MPa√m)

m = material constant

Let's assume the values of C and m as [tex]2.5 * 10^-12[/tex] and 3.0, respectively, for the material in question.

Now, to determine the length of stable crack growth, we can use the Paris Law equation, which is derived from the crack growth resistance curve:

[tex]\Delta a = [(2\Delta K/\pi )C(\Delta K)^m*N]^1/(1-m)[/tex]

where:

Δa = increase in crack length per cycle (mm/cycle)

N = number of cycles.

At the point where the crack starts to grow rapidly, the crack length is equal to the critical crack length [tex](a_c).[/tex]

Thus, we can set Δa equal to [tex]a_c - a_0[/tex],

where [tex]a_0[/tex]is the initial crack length of 1 cm.

Solving for N, we get:

[tex]N = [(a_c - a_0)(1-m)/(2(\Delta K/\pi )*C)]^{1/(m+1)}[/tex]

Let's assume that the critical stress intensity factor for the material is 30 MPa√m.

Using the formula for stress intensity factor, we can find the stress range (Δσ) for a given crack length (a):

ΔK = σ√πa.

where:

σ = stress (MPa)

Assuming that the material is subjected to a tensile stress of 150 MPa, the stress range (Δσ) is 150 MPa.

Therefore, we can calculate the stress intensity factor range (ΔK) for a crack length of 1 cm as:

ΔK = (150 MPa)√(π(1 cm)) ≈ 535.8 MPa√m

Using this value of ΔK in the Paris Law equation, we can calculate the length of stable crack growth as:

[tex]\Delta a = [(2*(535.8 MPa\sqrt{m} )/ \pi )(2.5 x 10^-12)(535.8 MPa\sqrt{m} )^3*N]^1/4[/tex]

Assuming that the length of stable crack growth is 5 mm, we can calculate the critical crack length using the same Paris Law equation:

[tex]a_c = [((5 mm)(1-m)/(2(535.8 MPa\sqrt{m} /\pi )*(2.5 x 10^-12))]^{1/3} \approx 8.12 mm.[/tex]

Finally, we can calculate the critical stress at instability using the formula for stress intensity factor:

[tex]K_Ic = \sigma \sqrt{(\pi a_c)}[/tex]

Solving for σ, we get:

[tex]\sigma = K_Ic/\sqrt{(\pi a_c)} = (30 MPa\sqrt{m} )/\sqrt{(\pi *(8.12 mm))} \approx. 741.5 MPa.[/tex]

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The tributary width for the girder on Grid B between Grids 1 and 2 is most nearly 15 ft. This means that the girder is responsible for carrying the load of the structure over an area that is 15 ft wide.


The tributary area of the column A/3 is most nearly the area of the floor or roof that is supported by the column. To calculate this, we need to determine the distance from the center of the column to the next column or wall in each direction. Assuming that the distance to the next column or wall in each direction is the same, we can calculate the tributary area as follows:
Tributary area = (distance to next column or wall)^2
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So, the tributary area of the column A/3 is most nearly A^2/9.

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