Deflections of bent bars by hand
You will not integrate w″ = M/EI twice for a bent bar — the axis changes direction, so there is no single w(x) to integrate. The practical hand method is the unit-load method (Castigliano’s second theorem in its engineering form), and it is worth knowing because it explains where a deflection comes from.
The recipe
To find the displacement δ of a point in a chosen direction:
- Solve the real structure and write the internal actions M(s), N(s), T(s) for every member.
- Remove the real load, apply a unit force (1 N) at the point, in the direction whose displacement you want, and solve again → M̄(s), N̄(s), T̄(s).
- Integrate the products over the whole developed length:
Each integral is one physical mechanism: bending, stretching, twisting. For a rotation instead of a displacement, apply a unit moment. Because the integrands are usually a product of two straight lines or a line and a parabola, the integrals are almost always done from a table rather than by calculus.
Which term actually matters
For slender bars, bending dominates. A quick order-of-magnitude comparison for the L-bracket of chapter 7 with a 1 m arm and a 100 mm deep section:
| Mechanism | Relative contribution | When it matters |
|---|---|---|
| bending (M·M̄/EI) | 100 % | always |
| torsion (T·T̄/GJ) | 0 % in plane, up to several hundred % out of plane for an open section | out-of-plane loads |
| axial (N·N̄/EA) | typically well under 1 % | short stocky members, arches, trusses |
| shear | a few % for L/h < 10 | short deep members — not included here |
FrameLab includes bending, axial and torsional flexibility in full (it is a finite-element solution, so all of them come out automatically). It does not include shear deformation — the Euler–Bernoulli assumption. For L/h ratios above about 10, the error from that is under a per cent.
A hand check you can always do
For the L-bracket with a vertical tip load P, arm a, column b:
The middle term is the rotation of the column top (caused by the constant moment P·a along it) multiplied by the lever arm a. Run the first example in chapter 1 and compare — the numbers agree to three digits.