Combined Stresses

Learning Objectives

  • Superimpose compatible elastic stress components at a common point and on a common plane.
  • Evaluate combined axial and bending stress produced by eccentric loading.
  • Locate the neutral axis and identify when an eccentric compressive load causes tension.
  • Apply the middle-third rule to rectangular sections carrying compression-only material systems.
  • Combine beam normal and shear stresses before transforming the resulting stress state when necessary.

Combined Stress

Combined stress is the stress state produced when two or more loading actions contribute normal and/or shear stress at the same material point.

Superposition of Elastic Stresses

Under linear-elastic, small-deformation behavior, compatible stress components can be added algebraically. Add normal stresses to normal stresses and shear stresses to shear stresses using one coordinate and sign convention. Do not add scalar magnitudes that act on different planes without first resolving them to the same stress components.

Axial Plus Bending Stress

Normal stress at a point in a member subjected to axial force and bending moment.

σ=PA±MyI\sigma=\frac{P}{A}\pm\frac{My}{I}
Axial and Bending in One Member

The connected member-and-joint context makes axial and bending actions part of one physical system; use the adjacent formula for the local stress result.

An intact steel wide-flange member spans between two connected frame posts with end joints.

Reading the Member Context

The member is shown without arrows or stress shading on purpose. The raster identifies physical connectivity only; the adjacent formula and point-specific workflow define the combined stress state.

Eccentric Load

An eccentric axial load has a line of action offset from the centroidal axis, producing both a direct axial force and a bending moment.

Equivalent Eccentric Moment

Moment generated by an axial load with eccentricity e.

M=PeM=Pe
Offset Column Cap and Eccentricity

The offset bracket and supported girder seat provide a clear physical cue for an eccentric gravity load path; the adjacent equation supplies the exact moment relationship.

A concrete column with an offset corbel cap bracket supporting a steel beam on an eccentric bearing seat.

Reading the Offset Cap Context

The cap projection is a qualitative cue for offset alignment, not a prescribed proportion, load magnitude, or adequacy claim. Use the eccentric-load formula and interactive stress model for the mechanics.

Interactive Exploration

Vary compressive load and eccentricity in the combined-stress visualizer. Observe how the linear normal-stress distribution shifts and when one edge approaches zero or changes to tension.

Eccentrically Loaded Column Visualizer

Concept and model scope

Combine direct compression with bending from signed eccentricity and inspect the point-by-point linear normal-stress distribution.

Controls

Compressive load

Concentric force magnitude before the eccentric moment Pe is added. Compression is negative in the displayed stress convention.

Range: 20–500 kN. Step: 10 kN.

150 kN

Eccentricity

Signed offset of the compression resultant from the centroid along the section depth. Its sign reverses the bending-stress gradient.

Range: -150–150 mm. Step: 5 mm.

50 mm

Section width

Rectangular section width b. It contributes to area and second moment of area.

Range: 100–400 mm. Step: 10 mm.

200 mm

Section depth

Rectangular section depth h in the eccentricity direction. The middle-third limit is h/6.

Range: 150–500 mm. Step: 10 mm.

300 mm
e = 50 mmP = 150 kN-5.00 MPa0.00 MPastress = 0 axis
Direct stress
-2.50 MPa

P/A with compression negative.

Equivalent moment
7.50 kN·m

M = Pe; sign follows e.

Edge stresses
-5.00 / 0.00 MPa

Top / bottom for the shown axis.

Middle-third limit
±50.0 mm

Both linear edge stresses remain compressive.

Kern

The kern is the region around a section centroid within which a compressive resultant must act to keep the entire cross-section in compression under linear elastic stress distribution.

Middle-Third Condition for a Rectangle

Compression-only eccentricity limit for bending about one centroidal axis of a rectangular section.

∣e∣≤h6|e|\le\frac{h}{6}

Middle-Third Rule

For a rectangular section loaded in compression about one centroidal axis, placing the resultant within the middle third keeps the calculated stress non-tensile across the full section. At ∣e∣=h/6|e|=h/6, one edge reaches zero stress. Beyond that limit, the linear elastic full-contact model predicts tension at one edge and may no longer represent an unreinforced contact interface or masonry-like material accurately.

Eccentric Compression at a Wall Base

The offset wall and footing provide architectural context for eccentric compression and contact assumptions; use the adjacent simulation for the exact kern boundary and stress distribution.

Sectional architectural masonry wall on a broad concrete footing with the wall visibly offset toward one side of the base.

Reading the Wall-Base Context

The wall and footing are shown as an intact connected system with the load-bearing wall offset toward one side of the base. That spatial offset is a qualitative cue for eccentricity, not a prescribed proportion or a proof of adequacy. Use the middle-third condition, the interactive stress distribution, and a suitable contact model when full-area tension-free contact is no longer valid.

Interactive Exploration

Use the middle-third visualizer to move the compressive resultant across the section. Track the kern boundary and the transition from full compression to a stress distribution that includes tension.

Middle-Third Rule

Concept and model scope

Move a compressive resultant across a rectangular section and compare its location with the kern boundary h/6.

Controls

Compressive load

Compression-resultant magnitude. The middle-third location criterion is geometric, while the edge stress magnitudes also depend on this load.

Range: 20–500 kN. Step: 10 kN.

150 kN

Eccentricity

Signed resultant offset from the section centroid along the displayed section length.

Range: -250–250 mm. Step: 5 mm.

0 mm

Section length h

Section dimension in the eccentricity direction. The kern extends from −h/6 to +h/6.

Range: 150–600 mm. Step: 10 mm.

300 mm

Section width b

Perpendicular rectangular-section dimension used in area and second moment of area.

Range: 100–400 mm. Step: 10 mm.

200 mm
e = 0 mmkern = middle third-2.50 MPa-2.50 MPa
Kern limit
±50.0 mm

For the displayed rectangular axis.

Resultant status
Inside kern: full compression

Outside-kern stresses are still shown as the linear full-area prediction, not a no-tension contact solution.

Left edge stress
-2.50 MPa

Compression

Right edge stress
-2.50 MPa

Compression

Combined Stress in Beams

A transversely loaded beam can have normal bending stress σx=−My/I\sigma_x=-My/I and transverse shear stress τxy=VQ/(Ib)\tau_{xy}=VQ/(Ib) at the same point. Their individual maxima generally occur at different depths, so a critical combined state must be evaluated at the actual point of interest before principal stresses are calculated.

Beam-Column Joint Context

The cutaway locates a shared interface where normal and shear components may coexist; evaluate the local stress state at the actual point.

A monolithic concrete beam meets a vertical column at one cutaway joint core.

Reading the Joint Context

The image identifies the shared material region only. It does not provide joint detailing, force magnitudes, stress values, or a failure assessment.

Combined-Stress Workflow

Use a point-based workflow so stresses from different load effects are not incorrectly mixed between different section locations.

Combined Stress Evaluation Workflow
Combined Stress Evaluation WorkflowChoose the material point and coordinate system → Resolve axial force, moments, torque, and shear at the section; Resolve axial force, moments, torque, and shear at the section → Compute normal and shear stress components at that same point; Compute normal and shear stress components at that same point → Superimpose like components with consistent signs; Superimpose like components with consistent signs → Is compression eccentricity/contact behavior important?; Is compression eccentricity/contact behavior important? — Yes → Check neutral axis, kern, and possible loss of full compression; Is compression eccentricity/contact behavior important? — No → Are principal stresses or oriented-plane stresses required?; Check neutral axis, kern, and possible loss of full compression → Are principal stresses or oriented-plane stresses required?; Are principal stresses or oriented-plane stresses required? — Yes → Apply stress transformation or Mohr's Circle; Are principal stresses or oriented-plane stresses required? — No → Report governing point, stress state, and assumptions; Apply stress transformation or Mohr's Circle → Report governing point, stress state, and assumptions

Choose the material point and coordinate system → Resolve axial force, moments, torque, and shear at the section; Resolve axial force, moments, torque, and shear at the section → Compute normal and shear stress components at that same point; Compute normal and shear stress components at that same point → Superimpose like components with consistent signs; Superimpose like components with consistent signs → Is compression eccentricity/contact behavior important?; Is compression eccentricity/contact behavior important? — Yes → Check neutral axis, kern, and possible loss of full compression; Is compression eccentricity/contact behavior important? — No → Are principal stresses or oriented-plane stresses required?; Check neutral axis, kern, and possible loss of full compression → Are principal stresses or oriented-plane stresses required?; Are principal stresses or oriented-plane stresses required? — Yes → Apply stress transformation or Mohr's Circle; Are principal stresses or oriented-plane stresses required? — No → Report governing point, stress state, and assumptions; Apply stress transformation or Mohr's Circle → Report governing point, stress state, and assumptions

  • Choose the material point and coordinate system: terminator
  • Resolve axial force, moments, torque, and shear at the section: process
  • Compute normal and shear stress components at that same point: process
  • Superimpose like components with consistent signs: process
  • Is compression eccentricity/contact behavior important?: decision
  • Check neutral axis, kern, and possible loss of full compression: process
  • Are principal stresses or oriented-plane stresses required?: decision
  • Apply stress transformation or Mohr's Circle: process
  • Report governing point, stress state, and assumptions: terminator

Full-Contact Assumption

The middle-third result comes from a linear stress distribution over the full area. Once an interface cannot carry tension and contact lifts off, the effective compression zone changes and the simple full-area formula must be replaced by an appropriate contact model.

Rectangular Footing Contact Context

The cutaway emphasizes the connected footing, bearing layer, and ground interface; use the middle-third and contact models for the actual contact region.

A rectangular concrete footing rests on a granular soil layer beneath an offset square pier.

Reading the Footing Interface

The visible underside is a physical interface, not a pressure diagram. No contact width, pressure distribution, code limit, or adequacy conclusion can be inferred from the raster.

Key Takeaways
  • Combine stress components only at the same material point and in the same coordinate system.
  • Eccentric axial load is equivalent to a centroidal axial load plus the moment PePe.
  • For a rectangular compression section, the middle-third rule identifies the full-compression kern limit.
  • Maximum bending stress and maximum transverse shear stress usually occur at different depths.
  • Use stress transformation only after the local combined stress components have been established.

References