A steel beam with a rectangular cross section is bent to form an arc of a circle of radius 6 m. Calculate the maximum stress in the beam given that the depth of the beam is 6 mm and the Young's modulusfor steel is 210 MPa
To calculate the maximum stress in the steel beam, we can use the formula for bending stress.
Step 1: Determine the Moment of Inertia
The moment of inertia (I) of a rectangular cross section can be calculated using the formula: I = (b * h^3) / 12, where b is the width and h is the depth of the beam.
In this case, the depth of the beam is given as 6 mm (0.006 m). However, the width of the beam is not given in the question. Let's assume the width as 'w'.
Therefore, I = (w * (0.006)^3) / 12 = (w * 0.000216) / 12 = 0.000018w
Step 2: Calculate the Moment of Force
The moment of force (M) exerted on the beam can be calculated using the formula: M = F * R, where F is the force applied to the beam and R is the radius of the arc.
In this case, the radius is given as 6 m.
Step 3: Calculate the Maximum Stress
The maximum stress (σ) can be calculated using the formula: σ = (M * c) / I, where c is the distance from the centroid of the cross section to the point where the maximum stress occurs. For a beam with rectangular cross section, c is equal to h/2, where h is the depth of the beam.
In this case, c = 0.006 / 2 = 0.003 m.
Substituting the values into the formula: σ = (M * c) / I
However, the force applied to the beam (F) is not provided in the question. To calculate the maximum stress, we need this information.
E=210 GPa (210 MPa isn’t correct)
t=6 mm=6•10⁻³m
R=6 m
Maximum bending stress is
σ(max) = E•t/2R=210•10⁹•6•10⁻³/2•6=105•10⁶ N/m²