When a member carries both an axial load ($P$) and a bending moment ($M$), the normal stresses are superimposed.

An eccentric axial load is a load applied parallel to the centroidal axis but offset by a distance (eccentricity, $e$). This creates an equivalent system of an axial load ($P$) acting at the centroid, plus a bending moment ($M = P \times e$).

For materials that are strong in compression but weak or entirely unable to carry tension (like concrete, masonry, or soil under a foundation), it is crucial to ensure that the entire cross-section remains in compression.

To prevent any tensile stress from developing, the combined compressive stress ($P/A$) must be greater than or equal to the maximum tensile flexural stress ($Mc/I$).

When evaluating a critical point on a structural member, we must carefully sum all stresses acting on that point from every internal force:

• Axial Force ($P$): Normal Stress ($\sigma = P/A$)
• Bending Moment ($M$): Flexural Normal Stress ($\sigma = My/I$)
• Shear Force ($V$): Transverse Shear Stress ($\tau = VQ/Ib$)
• Torsion ($T$): Torsional Shear Stress ($\tau = T\rho/J$)

These components are summed algebraically based on their direction to form the final stress state ($\sigma_x, \sigma_y, \tau_{xy}$) for that specific point.

The stress at a point is defined by normal stresses ($\sigma_x, \sigma_y$) and shear stress ($\tau_{xy}$). To find stresses on an inclined plane or the maximum stresses, we use stress transformation equations.

Mohr's Circle

Mohr's Circle is a graphical method to represent the state of stress.

• Center ($C$): $(\frac{\sigma_x + \sigma_y}{2}, 0)$
• Radius ($R$): $\sqrt{(\frac{\sigma_x - \sigma_y}{2})^2 + \tau_{xy}^2}$

The absolute maximum shear stress in the defined plane is equal to the radius of the Mohr's Circle.

While plane stress problems ($\sigma_z = 0, \tau_{xz} = \tau_{yz} = 0$) are commonly solved using a single 2D Mohr's Circle, the true state of stress is triaxial. Even in plane stress, there are actually three principal stresses ($\sigma_1, \sigma_2$, and $\sigma_3 = 0$).

To evaluate the absolute maximum shear stress occurring in any plane (not just the x-y plane), a 3D Mohr's Circle must be constructed. This consists of three intersecting circles drawn between the three principal stresses:

• Circle 1: Between $\sigma_1$ and $\sigma_2$ (the standard 2D in-plane circle).
• Circle 2: Between $\sigma_2$ and $\sigma_3$ (which is $0$ in plane stress).
• Circle 3: Between $\sigma_1$ and $\sigma_3$ (which is $0$ in plane stress).

The Absolute Maximum Shear Stress ($\tau_{abs\_max}$) is the radius of the largest of these three circles.

After combining stresses and determining the principal stresses ($\sigma_1, \sigma_2, \sigma_3$) using Mohr's Circle, the final step in structural design is predicting if the material will fail under this complex multiaxial stress state.

Different materials require different failure theories:

1. Maximum Principal Stress Theory (Rankine's Theory) Used for: Brittle materials (like concrete, cast iron, glass). The material fails when the maximum principal tensile stress reaches the ultimate tensile strength ($\sigma_{ult}$) from a simple tension test, regardless of the other principal stresses.

• Fails if: $\sigma_1 \ge \sigma_{ult}$

2. Maximum Shear Stress Theory (Tresca Criterion) Used for: Ductile materials (like mild steel). The material yields when the absolute maximum shear stress ($\tau_{abs\_max}$) reaches the shear yield strength ($\tau_Y$) from a simple tension test (where $\tau_Y = \sigma_Y / 2$).

• Yields if: $\tau_{abs\_max} = \frac{|\sigma_1 - \sigma_3|}{2} \ge \frac{\sigma_Y}{2}$

3. Maximum Distortion Energy Theory (Von Mises Criterion) Used for: Ductile materials (most accurate and widely used for metals). Yielding occurs when the distortion energy per unit volume equals or exceeds the distortion energy at yield in a simple tension test. The stresses are combined into an "equivalent" or "effective" Von Mises stress ($\sigma_v$):

• Yields if: $\sigma_v \ge \sigma_Y$
• Under plane stress ($\sigma_3 = 0$), the formula is: $\sigma_v = \sqrt{\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2}$