The Flexure Formula Learning Goal: To apply the flexure formula to beams under l

Question

The Flexure Formula

Learning Goal:

To apply the flexure formula to beams under load and find unknown stresses, moments, and forces.

max=McI

where M is the magnitude of the internal moment with respect to the neutral axis, c is the perpendicular distance from the neutral axis to the point farthest from the neutral axis, and I is the moment of inertia of the cross-section about the neutral axis. The maximum normal stress will always occur on the top or bottom surface of the beam; in fact, one of these surfaces will experience a maximal tensile stress while the other experiences the same magnitude of stress in compression.

For points not on a surface of the beam, we can use

max/c=/y

to rewrite the flexure formula in the more general form

=MyI.

Part A - Moment Required to Produce a Given Stress

(Figure 1)

Determine the magnitude of the moment M that must be applied to the beam to create a compressive stress ofD=20 MPa at point D. Also calculate the maximum stress developed in the beam. The moment M is applied in the vertical plane about the geometric center of the beam.

Express your answers, separated by a comma, to three significant figures.

Part B - Minimum Allowable Cross-Section

(Figure 2)

Determine the smallest allowable diameter of the rod, d, if x=1.6 m, w0=8.2 kN/m, and the maximum allowable bending stress is allow=185 MPa.

Express your answer to three significant figures and include the appropriate units.

Part C - Absolute Maximum Bending Stress

(Figure 3)

The beam has a square cross-section of 115 mm on each side, is 3 m long, and the initial value of the distributed load isw0=6 kN/m.

Express your answer to three significant figures.

Figure 1of 3

Figure 2

Figure 3

The Flexure Formula

Learning Goal:

To apply the flexure formula to beams under load and find unknown stresses, moments, and forces.

For straight members having a constant cross-section that is symmetrical with respect to an axis with a moment applied perpendicular to that axis, the maximum normal stress in the cross-section can be calculated using the flexure formula:

max=McI

where M is the magnitude of the internal moment with respect to the neutral axis, c is the perpendicular distance from the neutral axis to the point farthest from the neutral axis, and I is the moment of inertia of the cross-section about the neutral axis. The maximum normal stress will always occur on the top or bottom surface of the beam; in fact, one of these surfaces will experience a maximal tensile stress while the other experiences the same magnitude of stress in compression.

For points not on a surface of the beam, we can use

max/c=/y

to rewrite the flexure formula in the more general form

=MyI.

Part A - Moment Required to Produce a Given Stress

The cross-section of a wooden, built-up beam is shown below. The dimensions are L=190 mm and w=35 mm.

(Figure 1)

Determine the magnitude of the moment M that must be applied to the beam to create a compressive stress ofD=20 MPa at point D. Also calculate the maximum stress developed in the beam. The moment M is applied in the vertical plane about the geometric center of the beam.

Express your answers, separated by a comma, to three significant figures.

M=, max= kNm, MPa

Part B - Minimum Allowable Cross-Section

The rod shown in the figure below is supported by smooth journal bearings at A and B that exert only vertical reactions on the shaft.

(Figure 2)

Determine the smallest allowable diameter of the rod, d, if x=1.6 m, w0=8.2 kN/m, and the maximum allowable bending stress is allow=185 MPa.

Express your answer to three significant figures and include the appropriate units.

d=

Part C - Absolute Maximum Bending Stress

Find the absolute maximum bending stress in the beam shown in the figure below.

(Figure 3)

The beam has a square cross-section of 115 mm on each side, is 3 m long, and the initial value of the distributed load isw0=6 kN/m.

Express your answer to three significant figures.

max= MPa

Figure 1of 3

Figure 2

Figure 3

Explanation / Answer

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