Learning topic
Combined Loading
Analyze combined axial load, biaxial bending, eccentric loading, and torsion by superposition, then evaluate critical stresses with an appropriate failure criterion.
Combined loading occurs when several internal force resultants act simultaneously in a member cross-section, such as axial force $N$, bending moments $M_x$ and $M_y$, torque $T$, or shear forces. Within the linear-elastic range, stresses caused by individual actions can often be calculated separately and then combined by superposition.
Principle of superposition
Superposition is applicable when material behavior is linear elastic, deformations are small, and changes in geometry do not significantly alter the loading scheme. For example, an axial force produces a uniform normal-stress component $N/A$, while bending moments produce linearly varying normal stresses.
Biaxial and unsymmetrical bending
If the centroidal axes $x$ and $y$ are principal axes of the cross-section and the bending moment has components $M_x$ and $M_y$, the normal stress is obtained by superposition:
$$\sigma(x,y)=-\frac{M_x}{I_x}y+\frac{M_y}{I_y}x.$$
The neutral-axis equation follows from $\sigma=0$:
$$-\frac{M_x}{I_x}y+\frac{M_y}{I_y}x=0.$$
- $M_x$, $M_y$ — bending-moment components about the principal centroidal axes;
- $I_x$, $I_y$ — principal centroidal second moments of area;
- $x$, $y$ — coordinates of the point in the cross-section.
The signs depend on the adopted axis and moment conventions, which must be used consistently.
Unsymmetrical bending is a typical combined-loading case: the bending moment is not aligned with one principal centroidal axis, so normal stress must include contributions from bending about both principal axes.
Eccentric axial loading
An eccentric axial force $N$, applied with eccentricities $e_x$ and $e_y$ relative to the centroid, can be replaced by a centric force and bending moments. For principal centroidal axes:
$$M_x=Ne_y,\qquad M_y=Ne_x.$$
The normal stress follows from superposition of axial loading and biaxial bending:
$$\sigma(x,y)=\frac{N}{A}-\frac{M_x}{I_x}y+\frac{M_y}{I_y}x.$$
- $N$ — axial force;
- $e_x$, $e_y$ — eccentricities relative to the centroidal axes;
- $A$ — cross-sectional area;
- $M_x$, $M_y$ — bending moments caused by eccentricity;
- $I_x$, $I_y$ — principal centroidal second moments of area.
The signs of $N$, the moments, and the coordinates must follow one consistent convention. When compression is taken as negative, the stress signs directly indicate which parts of the section are in compression or tension.
An eccentrically applied axial force is statically equivalent to a centric axial force plus one or two bending moments. The position of the load determines whether the whole section remains in compression or whether a tensile region appears.
Combined bending and torsion
In power-transmission shafts, bending creates normal stresses while torque creates shear stresses. Because these components act simultaneously at the same material point, strength is assessed using a multiaxial failure criterion.
At a critical point of a circular shaft subjected to bending and torsion, the bending moment produces normal stress $\sigma_b$ and the torque produces shear stress $\tau_t$:
$$\sigma_b=\frac{M_b}{W},\qquad \tau_t=\frac{T}{W_p}.$$
For this plane stress state, the von Mises equivalent stress is:
$$\sigma_{\mathrm{eq,VM}}=\sqrt{\sigma_b^2+3\tau_t^2}.$$
According to the Tresca criterion:
$$\sigma_{\mathrm{eq,T}}=\sqrt{\sigma_b^2+4\tau_t^2}.$$
- $M_b$ — resultant bending moment;
- $T$ — torque;
- $W$ — bending section modulus;
- $W_p$ — polar section modulus.
The resulting equivalent stress is compared with an allowable value appropriate to the material and the adopted design method.
The calculator uses the von Mises criterion for the bending-plus-torsion plane stress state.
General calculation procedure
- Determine the internal force resultants and their critical combinations.
- Select principal centroidal axes and calculate the required section properties.
- Calculate normal and shear stress components produced by each internal action.
- Identify critical points of the section while accounting for stress signs and spatial distribution.
- For a multiaxial state, apply a failure criterion appropriate to the material.
- Where required, separately check stiffness, stability, or fatigue strength.
Limits of the approach
Simple superposition should not be applied automatically to plastic deformation, large displacements, significant geometric nonlinearity, or severe local stress concentrations. Such cases require an appropriately extended mechanical model.
About this topic
Combined loading refers to cases where multiple internal force components act simultaneously within a member's cross-section. This section outlines superposition principles for combining stresses and strains, covering unsymmetrical bending, bending with axial compression/tension, eccentric load application, and combined bending and torsion.