Question

The beam is made from three boards nailed together as shown. If the moment acting on the cross section is

M
=
600
N
⋅
m
, determine the maximum bending stress in the beam. Sketch a three dimensional view of the stress distribution and cover the cross section.

Answer 1

To determine the maximum bending stress in the beam, you can use the formula: stress = moment * distance from the neutral axis / moment of inertia.

Step-by-step explanation:

The maximum bending stress in the beam can be determined using the formula: stress = moment * distance from the neutral axis / moment of inertia.

Since the beam is made from three boards nailed together, we need to find the moment of inertia for the cross section. The moment of inertia can be calculated by summing the moment of inertias for each board.

Once we have the moment of inertia, we can then calculate the maximum bending stress using the given moment and the distance from the neutral axis.

Answer 2

The maximum bending stress typically occurs at the top or bottom of the beam, furthest from the neutral axis.

To determine the maximum bending stress in a beam made from three boards nailed together, subjected to a bending moment
`\( M \)` of 600 N·m, we would typically follow these steps:

1. Identify the cross-sectional area of the beam: This includes the width, height, and arrangement of the boards.

2. Calculate the Moment of Inertia (I): For a rectangular cross-section, the moment of inertia can be calculated as
`( I = (1)/(12) b h^3 \),` where
`\( b \)` is the base width of the rectangle, and
`\( h \)` is the height. If the beam is made of three boards, you would either consider them as a single unit (if they are arranged to form a single, wider rectangle) or calculate the moment of inertia for each board and then add them together (if they are stacked on top of each other).

3. Calculate the Neutral Axis: For symmetrical cross-sections, the neutral axis is at the centroid of the section. For non-symmetrical sections or sections with varying material properties, the neutral axis location would have to be calculated taking into account the distribution of material.

4. Determine the Maximum Bending Stress using the formula:

`\[ \sigma = (M y)/(I) \]`

where
`\( \sigma \)`is the stress,
`\( M \)` is the bending moment, is the distance from the neutral axis to the outermost fiber (where the stress is maximum), and
`\( I \)` is the moment of inertia.

5. Identify the location of the maximum stress: The maximum bending stress typically occurs at the top or bottom of the beam, furthest from the neutral axis.

6. Sketch the stress distribution: The stress distribution for a beam under bending is linear, going from compression (negative) on one side to tension (positive) on the other, with zero stress at the neutral axis.