CELE Strength of Materials — Combined Stresses and Mohr's CircleSummary

When multiple loads act on a member, the resultant stress at any point is the algebraic sum of stresses caused by each load acting separately. For example, if an eccentric axial load P acts on a column with eccentricity e, the combined stress at the edge is σ = P/A ± Mc/I, where M = Pe is the induced moment. This principle is fundamental because it allows us to decompose complex loading into simpler components, analyze each, and recombine. The superposition principle is valid only within the elastic range (before yielding) and for small displacements (no geometric nonlinearity).

A general plane-stress state at a point is defined by three components relative to reference axes: σx (normal stress on the x-face), σy (normal stress on the y-face), and τxy (shear stress). These stresses vary with the orientation of the plane selected. The stress transformation equations give the normal stress σx' and shear stress τx'y' on a plane inclined at angle θ (counterclockwise from the x-axis) as: σx' = [(σx + σy)/2] + [(σx − σy)/2]cos(2θ) + τxy·sin(2θ), and τx'y' = −[(σx − σy)/2]sin(2θ) + τxy·cos(2θ). These equations show that normal and shear stresses are functions of plane orientation and follow a predictable pattern. The factor of 2 in the angle is critical and often missed by students.

Principal stresses (σ₁ and σ₂) are the extreme normal stresses (maximum and minimum) at a point. Uniquely, they occur on planes where the shear stress is identically zero. For a 2D plane-stress state, the principal stresses are calculated as: σ₁, σ₂ = [(σx + σy)/2] ± √{[(σx − σy)/2]² + τxy²}. The first term, (σx + σy)/2, is the center of Mohr's circle. The second term (the radical) is the radius. The orientation of the principal planes is given by: tan(2θp) = 2τxy/(σx − σy). The principal planes are perpendicular to each other, 90° apart. Understanding that principal stresses carry zero shear helps explain why they are the governing stresses for failure theories.

The maximum shear stress within the plane of the stress state is τmax = √{[(σx − σy)/2]² + τxy²} = (σ₁ − σ₂)/2. This equals the radius of Mohr's circle. The plane on which τmax acts is always 45° away from the principal planes (or equivalently, 45° from either principal direction). Interestingly, on the plane of maximum shear, the normal stress is not zero—it equals the average stress σavg = (σx + σy)/2. This is why ductile materials often fail by shear on a 45° plane: maximum shear reaches the critical value while normal stress may still be modest.

Mohr's circle is a graphical construction where each point on the circle represents the stress state (σ, τ) on a particular plane. The construction is as follows: (1) Plot the reference stresses as two points: X(σx, τxy) and Y(σy, −τxy). The negative sign on τxy for point Y is a sign convention. (2) Draw a line connecting X and Y; this line is a diameter of the circle. (3) The center of the circle lies at the midpoint: [((σx + σy)/2), 0] on the σ-axis. (4) The radius is R = √{[(σx − σy)/2]² + τxy²}. (5) The circle intersects the σ-axis at σ₁ = center + R and σ₂ = center − R (the principal stresses). (6) The angle 2θ measured on the circle (from the reference diameter XY to the diameter pointing to any point of interest) corresponds to the actual angle θ on the physical element. A crucial feature: a physical rotation of angle θ translates to a rotation of 2θ on Mohr's circle in the same rotational sense.

Shafts are among the most common engineering members, often subjected to both bending moment M (due to transverse loads) and torque T (due to twisting). At the outer surface of a solid circular shaft of diameter d, the bending stress is σ = 32M/(πd³) (acting in the axial direction, σx), and the torsional shear stress is τ = 16T/(πd³) (acting tangentially, τxy). The stress state on the surface is thus σx = 32M/(πd³), σy = 0, τxy = 16T/(πd³). Using Mohr's circle or the principal stress formulas, the principal stresses are found. For design purposes, two equivalent (surrogate) quantities are defined: the equivalent bending moment Me = (1/2)[M + √(M² + T²)] (used with the maximum normal stress theory) and the equivalent torque Te = √(M² + T²) (used with the maximum shear stress theory). These allow the designer to use existing shaft-bending or shaft-torsion formulas without modification.

Three main failure theories are applied in modern design: (1) **Maximum Normal Stress (Rankine)**: failure occurs when the maximum principal stress σ₁ reaches the material's ultimate tensile or compressive strength. Suitable for brittle materials (cast iron, concrete, stone). (2) **Maximum Shear Stress (Tresca)**: failure occurs when τmax reaches half the yield strength of the material, τmax = σy/2. Conservative and widely used for ductile metals like mild steel. (3) **Distortion Energy (von Mises)**: failure occurs when the equivalent von Mises stress σv = √(σ₁² − σ₁σ₂ + σ₂²) reaches the yield strength. Most accurate for ductile materials, and the basis for ASME design codes. Each theory has a different form, and selecting the appropriate one depends on material type (ductile vs. brittle) and loading mode.

Consistent sign conventions are vital in stress transformation to avoid errors. Standard conventions: (1) Normal stress is positive in tension, negative in compression. (2) Shear stress τxy is positive when it acts counterclockwise on a positive (right-facing) face, or equivalently, clockwise on a negative (left-facing) face. (3) The angle θ is measured counterclockwise from the positive x-axis to the plane of interest. (4) On Mohr's circle, the vertical axis represents shear stress τ; positive τ is typically plotted upward (though some texts use downward). When plotting X(σx, τxy) and Y(σy, −τxy), the negative sign on τxy for point Y is essential to the geometry of the circle. Small errors in sign convention propagate through the entire calculation.

A common exam question provides σx, σy, and τxy, then asks for the normal and shear stress on a plane inclined at a given angle θ. The solution uses the transformation equations directly: σx' = [(σx + σy)/2] + [(σx − σy)/2]cos(2θ) + τxy·sin(2θ) and τx'y' = −[(σx − σy)/2]sin(2θ) + τxy·cos(2θ). The result can be verified by confirming that the point (σx', τx'y') lies on Mohr's circle: (σx' − center)² + (τx'y')² = R². This type of problem tests both the transformation equations and the student's understanding of Mohr's circle geometry.

The in-plane shear stress τmax is the maximum shear in the xy-plane. However, if we consider the full 3D stress state and include the third principal stress σ₃ (which is zero for plane stress), the absolute maximum shear stress may be different. For plane stress with σ₁ and σ₂ being the in-plane principals and σ₃ = 0 out-of-plane, the three in-plane/out-of-plane shear stresses are (σ₁ − σ₂)/2, σ₁/2, and σ₂/2. The absolute maximum is the largest of these three. For example, if σ₁ = 100 MPa and σ₂ = 50 MPa, then τabs = 100/2 = 50 MPa (not (100−50)/2 = 25 MPa). This distinction is critical in 3D stress analysis and occasionally tested on the exam.

Superposition of stresses from different loads produces the three components (σx, σy, τxy) that define the plane-stress state. These components form the input data for all subsequent transformation and Mohr's-circle analysis. Without superposition, we could not combine loading effects.

Given a stress state (σx, σy, τxy) in reference axes, the transformation equations deliver the normal and shear stress on any inclined plane. The equations are deterministic and follow from equilibrium of a wedge element. They form the foundation for Mohr's circle.

The transformation equations and Mohr's circle are two representations of the same physical reality. The equations are algebraic; the circle is geometric. Points on the circle satisfy the transformation equations. Many engineers use Mohr's circle to find principal stresses quickly, then verify with equations.

Applying the transformation equations to find where τx'y' = 0 yields the principal stresses. Setting dσx'/dθ = 0 yields the principal-plane angle. The radius of Mohr's circle equals both the maximum shear and the half-difference of principal stresses. These results are interconnected: finding one leads to the others.

A shaft with combined M and T is transformed into an equivalent problem (either M_e or T_e) that fits standard shaft design formulas. The equivalence depends on the failure theory chosen. This chain of logic bridges complex combined loading to simple, practical design rules.

Once principal stresses are calculated, they are inserted into a failure criterion (Rankine, Tresca, or von Mises) to predict whether the material will yield or fracture. The choice of criterion depends on material type and loading. This is the final step connecting stress analysis to engineering design decisions.

Mohr's circle provides geometric intuition for how materials behave. A circle touching the origin on the negative σ-axis (pure compression) has zero τmax if σ₁ and σ₂ have the same sign, which is why hydrostatic stress doesn't cause shear yielding. Visualizing this geometry helps predict failure modes.

Real members are 3D, but in many applications (thin webs, surface analysis, plane regions), the stress perpendicular to the plane is negligible (σ_z ≈ 0). This assumption simplifies the problem to 2D. Understanding when this assumption is valid is crucial for applying the chapter's methods correctly.

A rotating shaft in a mechanical transmission carries both bending stress from weight and gear loads, and torsional shear from power transmission. Using Mohr's circle or the principal-stress formulas, the maximum stresses at the shaft surface are determined. The equivalent bending moment M_e or equivalent torque T_e is then used with AISC 360 or other design standards to size the shaft diameter. Failure to account for the combined effect would under-size the shaft, leading to premature failure.

A pressurized pipe or thin-walled vessel develops hoop stress (circumferential) and longitudinal stress (axial). If the pipe is also bent by gravity or supports, or twisted by imposed torque, these stresses superpose. Mohr's circle determines the maximum normal and shear stresses for failure prediction. Philippine industrial and water-treatment plants extensively use such vessels, and ASME pressure-vessel codes (often referenced alongside NSCP 2015) require combined stress checks.

A column loaded by an off-center (eccentric) axial force P develops both axial stress (P/A) and bending stress (M·c/I, where M = P·e). At the edges of the column, these stresses add or subtract, producing a combined state. Principal stress analysis reveals the maximum tension and compression, which governs whether the column buckles or yields. NSCP 2015 provides combined loading criteria for such cases.

A tall building or tower experiences wind-induced bending moments and, if the building has a helical or spiral geometry, torsional effects. Ground-level members experience the superposition of vertical gravity loads (bending) and lateral wind loads (additional bending in another direction, or torsion). Mohr's circle for biaxial bending states determines the critical stresses for design checks per NSCP 2015.

Welds and bolted connections in steel structures (governed by AISC 360) experience complex stress states due to axial tension/compression and shear from connection forces, plus any bending moments. Mohr's circle analysis determines the maximum principal and shear stresses to compare against the strengths of the weld or bolt material, ensuring safe design.

Although concrete design relies heavily on limit-state methods rather than elastic stress analysis, understanding combined stresses in concrete members (e.g., biaxial bending, shear combined with torsion) is important for predicting crack patterns and failure modes. ACI 318 includes interaction diagrams and combined stress provisions.

Repetitive combined loading (cycling between different stress states) causes fatigue failure at stress levels below the static yield stress. Principal stresses and their mean and alternating components determine the fatigue life using Haigh diagrams or modified Goodman relations. Stress concentrations (notches, holes) amplify local stresses, requiring Mohr's circle analysis at the critical point.

Soil at depth experiences normal stresses (overburden pressure) and shear stresses (from lateral earth pressure, slopes, seismic loading). Mohr's circle in soil mechanics determines if the soil element will slide (fail in shear) based on the Mohr–Coulomb failure criterion. This is foundational to slope stability analysis and retaining-wall design.

Geometric discontinuities (holes, fillets, keyways in shafts) create local stress concentrations where the stress is several times the nominal stress. Under combined loading, Mohr's circle at the stress concentration determines if the material will yield or fracture locally. Design codes provide stress-concentration factors K_t to multiply nominal stresses.