Structural analysis, deformation, stability and serviceability

Combine the three ingredients of an elastic analysis

Equilibrium relates external loads to reactions and internal actions. Compatibility requires connected deformations to fit the restraints and each other. A constitutive relation connects force or stress to deformation, such as an assumed elastic modulus. In a statically determinate model, equilibrium can determine reactions directly; redundant supports usually require deformation information as well.

Choose degrees of freedom and boundary conditions deliberately. A simple support restrains specified motion while permitting rotation in the idealization. A fixed end restrains rotation as well. Actual connections have finite stiffness and can slip. A calculation using the wrong support model may be internally consistent while predicting the wrong force distribution.

Separate stiffness from strength and usability

Flexural rigidity EI combines material modulus E with the cross-section’s second moment of area I. Under small-rotation, slender-beam assumptions, moment is related to curvature through EI. Integrating with appropriate boundary conditions gives deflection. This model omits effects such as large rotations, significant shear deformation or changing cracked stiffness unless they are added explicitly.

Serviceability concerns performance during use: deflection can disturb partitions or drainage, and vibration can affect occupants or equipment. These requirements are separate from resistance to collapse. An applicable criterion and operating condition must be supplied before a deflection can be called acceptable. A magnitude alone does not establish a pass or failure.

Ask whether the equilibrium remains stable

A compressed member’s lateral displacement can add moment because its axial force now acts through an offset. This geometric effect is often called a second-order effect. A perfectly straight elastic pin-ended column has ideal Euler critical force π²EI/L². End restraint, imperfections, material nonlinearity and local buckling change what that model can establish for a real member.

Computer models assemble equilibrium and compatibility over many elements. Refine a mesh to examine numerical convergence, but also examine restraints, load introduction and material assumptions. Two meshes can converge to nearly the same result for an unsuitable physical model. Check reactions, units, deformed shape and a simpler limiting case before trusting a visually detailed contour plot.

One load account, two bending stiffnesses

Supply a simply supported 6 m beam with uniform line load 2 kN/m and constant EI=6 MN m². Reactions are 6 kN each and the maximum bending moment is wL²/8=9 kN m. The slender linear beam model gives maximum downward deflection 5 wL⁴/(384 EI)=0.005625 m, or 5.625 mm.

The plotted function is v(x)=w x(L³−2 Lx²+x³)/(24 EI), the downward magnitude for x in metres and consistent newton units. It is zero at both ends. Doubling EI gives 2.8125 mm at midspan but leaves the reactions and moments unchanged in this determinate case. No allowable deflection or strength has been supplied, so neither beam is qualified by this comparison.

Try the reasoning

1. If an ideal pin-ended column’s length doubles with EI unchanged, what happens to its Euler critical force?

   Show the answer and reasoning

   It becomes one-quarter because the ideal expression varies with 1/L². That scaling is not a real-column capacity approval.

2. An axial force of 100 kN acts through a 10 mm lateral offset. What additional moment magnitude results?

   Show the answer and reasoning

   Pδ=100×0.010=1 kN m. A second-order analysis must account for how that offset itself changes under load.

3. Does a mesh change below 1% prove a correct support assumption?

   Show the answer and reasoning

   No. It indicates limited numerical sensitivity for that chosen model, not physical validation of its restraints.

What to carry forward

Check equilibrium, deformation, stability and use separately, and connect each result to the assumptions that produced it.