Combined Loading Explained: Axial Load, Bending & Normal Stress Formulas

What Is Combined Loading?

The objective of combined loading analysis is to evaluate the stresses in a structural member when it’s exposed to more than one type of load at the same time. Basically, these are forces that act on a structure in different ways, and understanding how they interact is crucial for designing buildings, bridges, automotive chassis, and internal combustion engines. To get a better understanding of combined loading, it is common to make a cross-section through the area of interest and evaluate the internal loads and moments at the center of the section for equilibrium — the axial and bending components found there combine into a single value called normal stress, which is explained below.

What Is Axial Load?

Axial load is a force applied in a straight line along the longitudinal axis of a structural member, passing through the centroid of its cross-section. There are two types of axial loading: tensile forces, which pull on the structure and tend to stretch it, and compressive forces, which push on it and tend to shorten it. Axial load produces a uniform axial (normal) stress across the cross-section, given by:

σ_axial = P / A

where P is the applied axial force and A is the cross-sectional area. By convention, tensile stress is treated as positive and compressive stress as negative, though some references treat compression as positive instead — always confirm which sign convention a given textbook, code, or software is using before combining results.

What Is Normal Stress? Combining Axial and Bending Stress

Normal stress is any stress that acts perpendicular (normal) to a cross-section, as opposed to shear stress, which acts parallel to it. Axial stress (σ = P/A) and bending stress are both components of normal stress — when a member carries axial load and a bending moment at the same time, the two stresses are added algebraically at any given point on the cross-section. Bending stress itself follows the flexure formula:

σ_bending = M·c / I

where M is the bending moment, c is the distance from the neutral axis to the point of interest, and I is the moment of inertia of the cross-section about the bending axis. Combining the two gives the combined loading formula for normal stress under axial load and bending:

σ = P/A ± M·c/I

The plus or minus sign depends on whether the bending stress adds to or subtracts from the axial stress at the point being checked — bending stress is tensile on one face of the member and compressive on the opposite face, so the two faces must be checked separately to find the maximum combined stress.

Types of Loading in Combined Loading: Axial, Bending, Torsion, and Shear

During combined loading, a part can experience several types of load simultaneously.

Torsional stresses are loads experienced by a part due to twisting. This can be seen in an engine crankshaft as it turns inside an engine — the higher the RPMs, the greater the torsional load on the crankshaft.

Another type of load a part can be subject to is a bending moment. There are two types of bending: positive bending, which causes the top of a structure to compress and the bottom to stretch, and negative bending, which has the opposite effect. Bridges experience bending forces that can be caused by things like wind, earthquakes, and even the weight of the structure itself.

Axial loading, covered above, is simply a force applied in a straight line along the length of a structural element.

Finally, the last type of load is shear load. An example of shear stress is the load required to chip away at a material during a machining operation.

Why Combined Loading and Normal Stress Matter in Structural Design

So why is combined loading important? When a part is subject to multiple loads acting simultaneously, those forces interact in ways that can be difficult to predict. For example, if you had a part subject to both axial and bending stress, such as in a vehicle chassis while off-roading, the bending forces can cause the structural element to twist or buckle, which affects the axial forces — and likewise, the axial forces affect the bending response. When designing a structure, engineers have to calculate the combined forces and stresses to make sure it can withstand everything it’s likely to experience. Repeated combined loading over time is also a major contributor to fatigue failure — see our guide to fatigue analysis for how cyclic stress leads to crack initiation and failure well below a material’s static strength.

There are a few ways to solve combined loading problems. For example, the superposition method is used to determine the combined effect of two or more stresses acting over the cross-section of a member. The superposition method states that you first break the problem into parts where only one type of load acts, solve for the stress resulting from each load individually, and then add the solutions together to solve the complete problem.

Why Is Yield Criteria Important for Combined Loading?

A yield criterion is a theory that establishes the limit of elasticity in an engineering material and the onset of plastic deformation, regardless of the particular combination of stresses involved. Because combined loading rarely produces a simple uniaxial stress state, a yield criterion is what lets engineers compare a complex, multi-axis stress state against a material’s known (uniaxial) yield strength.

Failure Theory Example: Maximum Shear Stress Criterion

Imagine you have a material you want to use for a certain application — you need to make sure it won’t fail under the stresses it’s going to experience. One way to do this is with failure theory, which helps predict when a material is going to fail under the action of external loads. For ductile materials, the maximum shear stress theory (Tresca criterion) states that failure occurs when the shear stress on any plane reaches one-half of the material’s yield stress.

Maximum Distortion Energy Theory (Von Mises Criterion)

One of the most common failure theories is the maximum distortion energy theory, also called the von Mises criterion. It states that yielding is assumed to occur when the energy associated with the change of shape of a body under multiaxial loading equals the distortion energy in a tensile specimen at the point of uniaxial yielding. The distortion energy accounts for both the normal stresses and the shear stresses a material experiences.

What Is Mohr’s Circle Used For?

How do we visualize all these different stresses and predict when a material is going to fail? That’s where Mohr’s circle comes in. Mohr’s circle is a graphical representation of the transformation equations for plane stress. It’s helpful for visualizing the relationships between normal and shear stresses acting on an element at any orientation, so you can predict when and where a material is going to fail. It’s drawn as a circle with points on its edge representing the different stress values; by constructing Mohr’s circle from the calculated stresses, you can determine the maximum and minimum normal stresses, the maximum shear stress, and the orientation at which they occur.

Combined Loading in Practice: Why It Matters for Structural Design

So why does combined loading matter in practice? If you don’t account for combined axial loading and bending, you could end up with a structure that’s not strong enough to withstand the stresses it’s likely to experience — which could lead to collapse. It’s all about predicting when a material is going to fail and making sure you use materials and sections that can withstand the stresses they’re going to experience. By understanding how these forces interact, engineers can push the boundaries and create ambitious designs, while doing it safely and responsibly.

Combined Loading Example 1: Axial Tension and Torque

The section of the pipe at A is under combined loading due to a tensile force P=70 kips and a torque T=6 kip-ft. The pipe has an outside diameter of 4.0 in. and an inside diameter of 3.640 in. Determine the maximum shear stress at point A on the outer surface of the pipe. The radial stress at this point is zero. The yield strength in tension of this pipe is 95 ksi.

Combined Loading Example 2: Axial Load and Bending Moment

A short rectangular steel column has a cross-section 100 mm wide by 150 mm deep. It carries a compressive axial load P = 50 kN applied with an eccentricity e = 20 mm from the centroidal axis (bending about the axis parallel to the 100 mm side). Find the normal stress on each face of the column using the combined loading formula, σ = P/A ± M·c/I.

Step	Calculation	Result
Cross-sectional area, A	100 mm × 150 mm	15,000 mm²
Bending moment, M = P × e	50 kN × 20 mm	1,000 kN·mm (1 kN·m)
Moment of inertia, I = b·h³/12	100 × 150³ / 12	28,125,000 mm⁴
Distance to extreme fiber, c	150 mm / 2	75 mm
Axial stress, σaxial = P/A	50,000 N / 15,000 mm²	3.33 MPa (compression)
Bending stress, σbending = M·c/I	1,000,000 N·mm × 75 / 28,125,000 mm⁴	2.67 MPa
Max. combined stress (near side)	σaxial + σbending	6.00 MPa (compression)
Min. combined stress (far side)	σaxial − σbending	0.67 MPa (compression)

Both faces of this column end up in compression — the eccentric axial load creates enough bending stress to make the stress distribution uneven, but not enough to flip either face into tension, because the load stays within the section’s kern area. If the eccentricity were larger, the far-side combined stress would turn negative in this sign convention, meaning that face would go into tension instead.

Frequently Asked Questions

What is the formula for combined loading?

For a member under combined axial load and bending, the combined normal stress formula is σ = P/A ± M·c/I, where P is the axial force, A is the cross-sectional area, M is the bending moment, c is the distance from the neutral axis to the point being checked, and I is the moment of inertia of the cross-section. The sign depends on whether the bending stress adds to or subtracts from the axial stress at that point.

What is an axial load?

An axial load is a force applied along the longitudinal axis of a member, through the centroid of its cross-section, producing a uniform axial stress of σ = P/A. It can be tensile (pulling, tending to stretch the member) or compressive (pushing, tending to shorten it).

What is the difference between normal stress and shear stress?

Normal stress acts perpendicular to a cross-section (axial and bending stress are both forms of normal stress), while shear stress acts parallel to the cross-section. A member under combined loading typically experiences both at once, which is why failure theories like the maximum shear stress criterion and the von Mises criterion are needed to evaluate the complete stress state rather than either stress alone.

How do you combine axial and bending stress?

Axial and bending stress are combined algebraically at a given point on a cross-section using superposition: calculate the axial stress (P/A) and the bending stress (M·c/I) separately, then add or subtract them depending on whether the bending stress is tensile or compressive at that point. This has to be checked on both the tension and compression faces of the member to find the true maximum and minimum stress.

Why do engineers use Mohr’s circle for combined loading problems?

Mohr’s circle gives a graphical way to find the maximum and minimum normal stresses and the maximum shear stress at a point, along with the orientation at which they occur, directly from the combined normal and shear stresses calculated at that point — without needing to re-derive the stress transformation equations by hand for every angle.

What’s the difference between the maximum shear stress theory and the von Mises criterion?

Both are failure theories for ductile materials under combined loading. The maximum shear stress (Tresca) theory predicts failure once the maximum shear stress reaches half the material’s yield stress, and is simpler but more conservative. The maximum distortion energy (von Mises) theory accounts for the full multiaxial stress state and generally matches experimental data for ductile metals more closely.

For more on the mechanical-properties side of structural design, see our guides to mechanical properties testing, impact testing, and moment of inertia.