Short Answer:

Resultant stress is the total or combined stress that develops at a point in a material when two or more different types of stresses act simultaneously. It represents the overall effect of normal stresses (tensile or compressive) and shear stresses acting together.

In simple terms, resultant stress is the single stress that would produce the same effect on the material as all the acting stresses combined. It helps engineers to determine the maximum stress condition and to design components safely against failure under complex loading conditions.

Detailed Explanation:

Resultant Stress

Resultant stress is the combined effect of multiple stresses acting on the same element at a given point. In most practical engineering applications, a material or component is rarely subjected to a single type of stress. Instead, it often experiences a combination of direct (normal) and shear stresses at the same time.

For example, a rotating shaft experiences both torsional shear stress due to torque and bending stress due to the weight of pulleys or external forces. Similarly, a pressure vessel wall experiences hoop stress (circumferential), longitudinal stress, and radial stress at the same time.

In such cases, it becomes necessary to determine the single or resultant stress, which represents the actual intensity of stress acting on the material at that point. This resultant stress gives an accurate measure of the total internal resistance that the material must develop to withstand the applied loads safely.

Concept of Resultant Stress

To understand resultant stress, consider a small element of a material that is subjected to a normal stress (σ) and a shear stress (τ) acting simultaneously.

• The normal stress (σ) acts perpendicular to the surface and can be tensile (pulling apart) or compressive (pushing together).
• The shear stress (τ) acts tangentially to the surface and tends to slide one layer of the material over another.

The combination of these stresses produces a resultant stress (σR) on an oblique plane, which is the vector sum of the normal and shear stresses.

Mathematically, the resultant stress can be represented as:

This equation gives the magnitude of the resultant stress acting on a plane, assuming the normal and shear stresses are perpendicular to each other.

Principal and Maximum Shear Stresses

In a material subjected to combined stresses, there exist certain planes where:

• The shear stress becomes zero, and only normal stresses act — these are called principal planes.
• The normal stress becomes zero, and the shear stress reaches its maximum value — these are called maximum shear planes.

The normal stresses acting on the principal planes are known as principal stresses (σ1 and σ2), and they can be calculated using the following formulas:

The maximum shear stress (τmax) is given by:

Once these values are known, the resultant stress can be determined on any plane. The resultant stress helps in identifying the most critical condition, which may cause material yielding or failure.

Illustration of Resultant Stress

Imagine a rectangular element under a tensile stress (σ) of 50 MPa and a shear stress (τ) of 30 MPa. The resultant stress acting on the plane can be calculated as:

Hence, the total or resultant stress acting on that element is 58.3 MPa.
This resultant value represents the true stress condition that the material experiences due to the combination of normal and shear effects.

Importance of Resultant Stress

1. Accurate Stress Analysis:
   It provides the true measure of stress intensity in a material under combined loading conditions.
2. Prevention of Failure:
   By determining the resultant stress, engineers can identify whether the material will fail due to yielding, fracture, or fatigue.
3. Design of Safe Structures:
   Resultant stress helps in designing mechanical components like shafts, beams, and pressure vessels to ensure they operate safely under different load combinations.
4. Understanding Material Behavior:
   It helps in analyzing how a material deforms or fails under multi-axial loading, which is common in real-life conditions.
5. Application in Failure Theories:
   Theories such as Maximum Principal Stress Theory, Maximum Shear Stress Theory, and Von Mises Stress Theory are based on the concept of resultant stresses. These theories are used to predict the safe stress limits of materials.

Applications of Resultant Stress

1. Rotating Shafts:
   Shafts subjected to bending and torsion simultaneously experience resultant stresses that govern their design.
2. Pressure Vessels:
   In cylindrical or spherical pressure vessels, the resultant stress helps to determine the combined effect of hoop, longitudinal, and radial stresses.
3. Beams under Eccentric Loading:
   When a beam carries an off-center load, it produces both direct and bending stresses that combine to form the resultant stress.
4. Bolts and Rivets:
   Fasteners subjected to direct shear and tensile loads are analyzed using resultant stress to prevent joint failure.
5. Machine Elements:
   Components like connecting rods, axles, and brackets are analyzed for resultant stresses to ensure they can sustain operational loads safely.

Conclusion

Resultant stress is the total stress acting on a material when multiple stresses such as normal and shear occur together. It represents the true state of stress at any point in a component and helps determine whether the material will fail or remain safe under combined loading. The concept of resultant stress is essential in mechanical design and analysis because most engineering components experience complex loading rather than a single type of stress. Understanding resultant stress ensures the safe, strong, and efficient design of machine and structural elements.