Final answer:
To find the maximum stress induced and bending moment in a steel flat bent into a circular arc, we can use the formula for bending stress. Given the dimensions of the flat, we can calculate the moment of inertia and the distance from the centroid to the extreme fiber. Using the radius of the arc, we can find the bending moment and then solve for the maximum stress.
Step-by-step explanation:
To determine the maximum stress induced and the bending moment that can produce this stress, we can use the formula for bending stress:
σ = (M * c) / I
where σ is the stress, M is the bending moment, c is the distance from the centroid to the extreme fiber, and I is the moment of inertia of the cross-section.
Given that the steel flat has a width of 120 mm and a thickness of 25 mm, we can calculate the moment of inertia as:
I = (w * h^3) / 12
where w is the width and h is the thickness. Plugging in the values, we get:
I = (120 * (25^3)) / 12 = 156,250 mm^4
The distance from the centroid to the extreme fiber can be approximated as half the thickness, so c = 12.5 mm.
Now, let's calculate the bending moment:
M = σ * I / c
Given that the radius of the circular arc is 5 m, we can calculate the circumference of the arc:
C = 2 * π * r = 2 * π * 5 = 31.42 m
The total length of the steel flat is the same as the circumference of the arc, so we can calculate the bending moment:
M = F * d = S * A * d = σ * (w * h) * d = σ * 120 * 25 * 31.42
Since we want to find the maximum stress, we can rearrange the equation to solve for σ:
σ = (M * c) / I = (σ * 120 * 25 * 31.42 * 12.5) / 156250
Simplifying the equation:
σ = (σ * 46875) / 156250
σ = 0.3 * σ
The maximum stress induced will be 0.3σ and the bending moment required to produce this stress will be M = σ * 120 * 25 * 31.42 * 12.5 / 156250.