1. Direct and Bending Stresses in Vertical Members, Theory of Structures, CE 4003 (Same as CC 4003)

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Subject: Theory of Structures (CE 4003 Same as CC 4003)

Branch: Civil Engineering 🏗️
Semester: 4th Semester 📚

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1. Direct and Bending Stresses in Vertical Members

When vertical members like columns or shafts are subjected to loads, they experience both direct stresses (axial stresses) and bending stresses. Let's go through these concepts step by step:

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1.1 Introduction to Axial and Eccentric Loads

• Axial Load: A load that acts along the axis of the vertical member. It causes direct stress (compression or tension) along the member's length. The formula to calculate axial stress is:

  $$\sigma = \frac{P}{A}$$

  Where:

  • $P$ = Applied load
  • $A$ = Cross-sectional area of the member

• Eccentric Load: A load that acts off-center or at a distance from the axis of the vertical member. It not only creates direct stress but also causes a bending moment that leads to bending stresses. The formula to calculate the bending stress is:

  $$\sigma_{\text{bending}} = \frac{M}{I} \cdot y$$
  

  Where:

  • $M$ = Bending moment (due to eccentricity)
  • $I$ = Moment of inertia of the cross-section
  • $y$ = Distance from the neutral axis

Example: If you apply a vertical load at the top of a column, it causes both direct stress and bending stress due to the load’s eccentricity.

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1.2 Eccentricity about One Principal Axis Only

In this case, the load is eccentric about one principal axis, either the x-axis or y-axis, and not both. This creates a bending moment that affects the stress distribution across the cross-section.

1.2.1 Nature of Stresses

• The stresses are not uniform throughout the cross-section.
• At the axis of loading, there will be pure axial stress.
• Away from the axis of loading, bending stresses develop, which cause tensile stresses on one side and compressive stresses on the opposite side.

Diagram:
Let’s assume a vertical member with eccentric loading:

     |      |  Load (P)
     |      |
----------------------
        Neutral Axis

In the above diagram, the load is applied off-center, causing both axial and bending stresses.

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1.2.2 Maximum and Minimum Stresses

• Maximum Stress: This occurs at the farthest point from the neutral axis (on the extreme fibers). It combines both axial stress and bending stress.
• Minimum Stress: This occurs at the neutral axis where bending stress is zero, but axial stress still exists.

For a rectangular cross-section, the maximum and minimum stresses at the top and bottom of the section will be:

$$\sigma_{\text{max}} = \sigma_{\text{axial}} \pm \sigma_{\text{bending}}$$ $$\sigma_{\text{min}} = \sigma_{\text{axial}} \mp \sigma_{\text{bending}}$$

Where:

• $\sigma_{\text{axial}}$ is the direct axial stress
• $\sigma_{\text{bending}}$ is the bending stress

Example: If a vertical load is applied at a distance from the center, it will cause the top of the member to be in compression and the bottom in tension.

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1.2.3 Resultant Stresses and Distribution Diagram

• Resultant Stresses: The total stress at any point on the cross-section is a combination of axial stress and bending stress.
• Stress Distribution Diagram: The stress distribution varies across the section, with the maximum compressive stress at one extreme and the maximum tensile stress at the other extreme.

The resultant stress distribution can be represented as:

      Tension |                       | Compression
               |      _______________  |
               |     |                 |    
               |     |                 |
               |     |                 |

Example: For a column under an eccentric load, the stress at the top will be compressive, and at the bottom, it will be tensile.

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1.2.4 Condition for No Tension or Zero Stress at Extreme Fiber

To avoid tension at the extreme fiber, the eccentricity should be controlled. There is a condition where the resultant stress at the extreme fiber will be zero (no tension or compression). This condition occurs when:

$$e = \frac{M}{P}$$

Where:

• $e$ = Eccentricity (distance from the center of the column to the line of action of the load)
• $M$ = Moment due to eccentric load
• $P$ = Axial load

For zero stress at extreme fibers, we need to ensure that the applied load does not cause excessive bending.

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1.2.5 Limit of Eccentricity

There is a limit to how much eccentricity a vertical member can handle before it causes excessive bending and instability. The limit of eccentricity for a column is governed by the slenderness ratio and should be within a safe range to avoid buckling or instability.

• Limit of eccentricity is typically determined by the column's length and cross-sectional area.
• If $e$ exceeds this limit, it can lead to failure due to excessive bending or instability.

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1.2.6 Core of Section for Rectangular and Circular Cross Sections

• Core of the Section: The core is the area inside the section where the load must be applied to avoid producing any tensile stresses.

  • For rectangular sections, the core is defined as the area where the centroidal load can be applied without inducing tension on the section.
  • For circular sections, the core is the circular area centered on the neutral axis.

• Core for Rectangular Section: The core lies within the inner region of the rectangle, avoiding the outer fibers where tension can develop.

• Core for Circular Section: In a circular section, the core is a smaller concentric circle at the center.

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1.2.7 Middle Third Rule

The middle third rule states that for rectangular cross-sections, if the load is applied within the middle third of the section, then no part of the section will experience tensile stress. This rule helps to avoid the development of cracking or failure due to tension.

For a rectangular section:

• The middle third is the central part of the section where the load should be applied to ensure a safe distribution of stresses.

Diagram:

 ------------------------------
|             Middle Third      |
 ------------------------------

This middle third is where the eccentricity should be controlled to avoid excessive bending and ensure stability.

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Question and Answers

1. Question: What happens if a vertical load is applied eccentrically to a column?

   • Answer: The column will experience both axial stress (compression or tension) and bending stress, leading to a non-uniform stress distribution across the section.

2. Question: What is the condition for zero stress at the extreme fiber?

   • Answer: The condition for zero stress at the extreme fiber occurs when the eccentricity is controlled so that the axial stress and bending stress combine to make the stress at the extreme fiber zero.

3. Question: What is the middle third rule for rectangular sections?

   • Answer: The middle third rule states that if the load is applied within the middle third of the section, no part of the section will experience tensile stress, ensuring safe and stable behavior.

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Revision Points 📌

1. Axial Load causes only direct stress, while eccentric load causes both direct stress and bending stress.
2. Maximum and minimum stresses occur at the extreme fibers and the neutral axis, respectively.
3. Resultant stress distribution is a combination of axial and bending stresses, varying across the cross-section.
4. To avoid tension at the extreme fiber, the eccentricity must be controlled.
5. The core of a section is the region where the load should be applied to avoid inducing tension.
6. The middle third rule for rectangular sections ensures that the load application within this region avoids tension stresses.

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