Combined Stresses - Theory & Concepts

Combined Stresses

In real-world structures, members are rarely subjected to a single type of loading. Combined stresses occur when a structural member is subjected to multiple internal forces simultaneously, such as an axial load combined with a bending moment, or a bending moment combined with torsion. These individual stress components must be superimposed or transformed to evaluate the critical state of stress.

Axial and Flexural Combined Stresses

Combined Axial and Flexural Loads

When a member carries both an axial load (

P

) and a bending moment (

M

), the normal stresses are superimposed.

\sigma = \pm \frac{P}{A} \pm \frac{My}{I}

Axial Term ( $P/A$ ): Uniform across the section.
Flexural Term ( $My/I$ ): Varies linearly, zero at centroid.

Sign Convention

Tension: Positive (+)
Compression: Negative (-)
Ensure signs are consistent when adding stresses.

Eccentric Axial Loads and the Core (Kern) of a Section

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

The Core (or Kern)

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

\frac{P}{A} \ge \frac{(P \cdot e) \cdot c}{I}

By setting the edge stress to exactly zero, we can solve for the maximum allowable eccentricity (

e_{max}

). The area defined by this maximum eccentricity in all directions around the centroid is called the Core or Kern. As long as the resultant compressive load acts within this core, the entire section will be under compression.

Rectangular Section ( $b \times h$ ): The core is a rhombus. For a load on the vertical axis, $e \le h/6$ . This is the famous "Middle-Third Rule".
Solid Circular Section (diameter $d$ ): The core is a circle with radius $e \le d/8$ . This is the "Middle-Fourth Rule".

The General State of Stress (3D to 2D)

While structures exist in 3D space, most practical engineering problems (like beams, shafts, and pressure vessels) can be simplified to a Plane Stress condition (2D). This means stresses on one face (usually the z-face) are zero.

Building the Stress Tensor

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.

Stress Transformation and Mohr's Circle

State of Stress and Transformation

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 ( $R$ ): $\sqrt{(\frac{\sigma_x - \sigma_y}{2})^2 + \tau_{xy}^2}$

Principal Stresses

The maximum and minimum normal stresses (where shear stress is zero).

\sigma_{1,2} = \frac{\sigma_x + \sigma_y}{2} \pm R

Maximum In-Plane Shear Stress

\tau_{max} = R

3D 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:

\tau_{abs\_max} = \frac{\sigma_{max} - \sigma_{min}}{2}

If the two in-plane principal stresses (

\sigma_1

and

\sigma_2

) have the same sign (both positive or both negative), the largest circle will actually involve the out-of-plane principal stress

\sigma_3 = 0

. In this case,

\tau_{abs\_max}

will be greater than the in-plane

\tau_{max}

. This distinction is critical for evaluating failure theories.

Failure Theories and Yield Criteria

Failure Theories for Ductile vs. Brittle Materials

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: $\sigma_v = \sqrt{\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2}$

Interactive Tool: 2D Stress Element

Visualize how normal and shear stresses deform a rectangular element, and see how rotating the element changes the stress components acting on its faces. Finding the angle where shear stress becomes zero reveals the Principal Planes.

Eccentrically Loaded Column Visualizer

Combine uniform axial compression with bending stress caused by an eccentric load.

P = 150 kN

e=50

Cross-section: 300mm × 200mm

Compressive Load (

P

): 150 kN

Eccentricity (

e

): 50 mm

Left edge (-150)Centroid (0)Right edge (+150)

Stress Distribution across Width (MPa)

Loading chart...

Axial

Bending

Combined

Stress at Left Face (y = -150)

0.00 MPa

(Compression)

Stress at Right Face (y = +150)

-5.00 MPa

(Compression)

Interactive Tool: Mohr's Circle

Input the stress state (

\sigma_x, \sigma_y, \tau_{xy}

) to visualize the principal stresses and maximum shear stress using Mohr's Circle.

Mohr's Circle for Plane Stress

Input Stresses (MPa)

Normal Stress X (

\sigma_x

): +80

Normal Stress Y (

\sigma_y

): +20

Shear Stress (

\tau_{xy}

): +40

Principal Results

Center (C)

50.0

Radius (R)

50.0

Max Principal (

\sigma_1

)

100.0

Min Principal (

\sigma_2

)

0.0

Max In-Plane Shear (

\tau_{max}

)

50.0

Loading chart...

Red dots indicate principal stresses, max shear stresses, the center, and the original state of stress faces (X and Y).

Key Takeaways

Superposition Principle: For linear elastic materials, stresses from different loads can be added algebraically ( $\sigma = \pm P/A \pm My/I$ ).
Core/Kern of a Section: For materials weak in tension, the resultant eccentric compressive load must act within the "middle-third" (or "middle-fourth" for circles) to ensure the entire section remains entirely in compression.
Principal Stresses: The maximum and minimum normal stresses acting on an element, occurring on planes where shear stress is zero.
Mohr's Circle: A visual tool for evaluating 2D stress transformation, identifying principal stresses ( $C \pm R$ ) and maximum in-plane shear stress ( $R$ ).
3D Mohr's Circle and Absolute Max Shear: Even in plane stress, the out-of-plane stress ( $\sigma_3 = 0$ ) creates three principal circles. If $\sigma_1$ and $\sigma_2$ share the same sign, the absolute maximum shear stress acts on an out-of-plane plane and is larger than the 2D in-plane maximum.
Failure Theories: Once the combined principal stresses are found, they must be compared against material yield strengths using Rankine (Brittle), Tresca (Ductile, conservative), or Von Mises (Ductile, highly accurate) yield criteria.

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