Thermal Deformation and Bearing/Contact Stress
Key Takeaways
- A uniform temperature change produces free deformation ΔL = αLΔT when movement is unrestrained but no thermal force in the ideal free member
- Full restraint converts the prevented thermal strain into stress σ = −EαΔT, while finite restraint stiffness produces an intermediate force
- Temperature gradients can create curvature and warping even when average temperature change is small
- Average bearing stress uses the effective engaged contact area, not automatically the gross member, plate, or support area
- Eccentric bearing can create nonuniform pressure or loss of contact, requiring a compression-only contact model and local deformation checks
Thermal Deformation and Bearing/Contact Stress
Ask what can move: Temperature change creates strain. It creates force only to the extent that supports, framing, friction, or connections prevent the corresponding free movement.
Uniform Temperature Change
For a member with constant coefficient of thermal expansion α, length L, and uniform temperature change ΔT, the free length change is
ΔL_T = αLΔT.
Heating gives expansion when α and ΔT are positive; cooling gives contraction. An ideal unrestrained member develops deformation but no thermal stress. A perfectly restrained linear-elastic member has total axial strain zero:
0 = σ/E + αΔT,
so
σ_T = -EαΔT.
Under the stated tension-positive convention, restrained heating produces compression and restrained cooling produces tension. The corresponding force magnitude is AEα|ΔT|. This ideal result assumes uniform temperature, full axial restraint, linear elasticity, and no slip, yielding, cracking, creep, or support flexibility.
| Condition | Movement | Ideal axial force |
|---|---|---|
| Free end | αLΔT | Zero |
| Full restraint | Zero | -AEαΔT |
| Finite elastic restraint | Between free and zero | Between zero and full-restraint force |
| Temperature gradient | Curvature/warping possible | Depends on compatibility and restraint |
Use the coefficient, temperature range, installation temperature, and combinations specified by the current reference or problem. For 2026, that means the PE Civil handbook active for the test date and, where applicable, AASHTO 8th with its May 2018 errata, AISC 15th, and ACI 318-14—not the April 2027 edition set.
Worked Free, Full, and Partial Restraint
A steel member has L = 30 ft = 360 in, A = 8.0 in², E = 29,000 ksi, and problem-given α = 6.5 × 10⁻⁶/°F. It heats uniformly by 60°F.
If free,
ΔL_T = 6.5 × 10⁻⁶(360)(60) = 0.1404 in.
If fully restrained,
σ_T = -29,000(6.5 × 10⁻⁶)(60) = -11.31 ksi
and
P_T = A|σ_T| = 8(11.31) = 90.5 kips compression.
Now model the external restraint as an axial spring with k_r = 1,000 kip/in. The member stiffness is
k_m = AE/L = 8(29,000)/360 = 644.4 kip/in.
The restrained thermal force follows compatibility: the free movement is absorbed by elastic shortening of the member plus movement of the restraint. Thus,
P = ΔL_T / (1/k_m + 1/k_r)
P = [k_m k_r/(k_m + k_r)]ΔL_T = 55.0 kips compression.
The member stress is 55.0/8 = 6.88 ksi, between zero and the 11.31-ksi full-restraint value. Limiting checks confirm the expression: as k_r → 0, force tends to zero; as k_r → ∞, force tends to k_mΔL_T = AEαΔT.
Differential Temperature
A temperature gradient through a beam depth, slab thickness, wall, or bridge deck makes one fiber seek a different strain from another. The free response can include curvature, not just axial length change. Continuity, diaphragms, bearings, and support fixity can restrain that curvature and create moment, shear, or local force. Do not replace a gradient with its average temperature unless the requested effect truly depends only on average strain.
Expansion bearings and joints are intended to accommodate specified movement and rotation. A seized bearing, closed joint, misplaced setting, or excessive friction can turn expected free movement into restraint force. Conversely, assuming full fixity when a bearing can move may grossly overstate demand. Trace thermal actions through the actual support degrees of freedom.
Effective Bearing Area
Average contact stress is
q_avg = P/A_eff,
where A_eff is the area actually engaged in load transfer. It is not automatically the gross beam area, base-plate plan area, masonry unit area, or support footprint. Gaps, grout extent, holes, edge geometry, plate flexibility, construction tolerance, and eccentricity can reduce contact. For pin or bolt bearing, the relevant projected contact dimensions and code-defined strength model control; for a plate on concrete, use only effective-area enhancement explicitly permitted by ACI 318-14.
Worked eccentric contact
A rigid 12-in-wide seat bears over an effective contact length of 6 in, so A_eff = 12(6) = 72 in². It carries P = 360 kips with eccentricity e = 1.0 in along the 6-in dimension. The average pressure is
q_avg = 360/72 = 5.0 ksi.
For a linear full-contact check, the contact-area inertia is
I = 12(6³)/12 = 216 in⁴,
and M = Pe = 360 kip-in. At c = 3 in,
q = P/A ± Mc/I = 5.0 ± 360(3)/216
q_max = 10.0 ksi, q_min = 0.
This is the boundary of full compression contact: e = 6/6 = 1.0 in, the middle-third limit for a rectangular interface with no tensile contact capacity. A larger eccentricity would make the linear full-area expression predict tension. The interface cannot provide tensile bearing merely to preserve the formula; solve a reduced compression-only contact block and recheck local plate/support deformation. Using the gross 12 × 10 = 120 in² seat area would report only 3.0 ksi and conceal the actual contact condition.
Bearing and Thermal Workflow
- Separate uniform temperature change from through-depth gradient.
- Calculate free movement first; then identify each restrained translation and rotation.
- Use compatible member/support stiffness to obtain force when restraint is partial.
- Carry movement and force to joints, bearings, anchors, adjacent members, and foundations.
- Draw the real contact interface and identify effective dimensions and load eccentricity.
- Compute average and edge pressures; if tension appears where contact cannot take it, revise the contact zone.
- Check the applicable AASHTO, AISC, or ACI bearing resistance plus local yielding, crushing, splitting, bending, and deformation.
Keep force divided by contact area as pressure units. Never divide by gross member area merely because it is the easiest dimension shown.
What free thermal expansion occurs in the worked 30-ft steel member for α = 6.5 × 10⁻⁶/°F and ΔT = 60°F?
If the same worked steel member is perfectly restrained during the 60°F temperature increase, what axial force develops under the ideal elastic assumptions?
For the worked 12-in by 6-in effective contact area with P = 360 kips and e = 1 in, which pressure result is correct?