Beam with varying stiffness

A linear elastic analysis of a continuous beam whose stiffness EI changes along its length: haunches, cover plates over a support, lapped members, a cracked zone, a stiffer member in one span. It gives the shear, moment and deflection envelopes and the reactions under the ASCE 7-22 load combinations or one load set, with pattern live load. No code sets this method: it is beam theory, solved by finite elements. Size the member in its material calculation with these forces.

Inputs of the beam with varying stiffness #

• Base stiffness: E and I entered, or an AISC shape (Shapes Database v16.0, Ix about the strong axis; its self-weight can be added as dead load). This is the stiffness wherever no segment is entered. E = 29,000 ksi is prefilled (steel); enter the modulus of your material and the gross, effective or cracked I you judge right.
• Stiffness segments: each row gives E and I from x0 to x1, measured from the left end of the beam (overhang included). A segment replaces the base stiffness where it lies: for a cover-plated zone enter the I of the member with its plates. Where two segments overlap, the lower row in the table wins.
• Linear haunch: enter I at x1 as well. I then varies linearly from I at x0 to I at x1, modelled as the number of steps of constant I you choose (8 prefilled), each at its midpoint value. A haunch whose depth varies linearly has an I that varies roughly with the cube of the depth: if that matters, enter it as several segments that follow your I.
• Spans and supports: the left support and each span with the support at its right end (pin, fixed or none), and the left and right overhangs, as in the continuous beam analysis.
• Loads: line loads (uniform, partial or linearly varying), point loads and concentrated moments, each with its load type (D, L, Lr, S, R, W, E) and where it acts. From and To are measured from the start of the span or overhang chosen; From = To = 0 loads all of it. Downward and clockwise positive.
• Load combinations: ASCE 7-22 2.3.1 (LRFD) or 2.4.1 (ASD) by the method switch, or one unfactored load set.
• Pattern live load: ASCE 7-22 4.3.3: live load on all spans, on each span alone, on alternate spans and on each pair of adjacent spans, enveloped.
• Deflection limits: span / n for live load and for total load, your choice; the IBC 2021 Table 1604.3 floor values (360 and 240) are prefilled and 0 turns a check off. For an overhang or cantilever the span is twice its length.

The beam is an analysis: it has no auto-design.

Checks performed #

A stiffer zone attracts moment: over an interior support a cover plate raises the hogging moment there and lowers the span moments. The analysis shows how much.

Limitations of the beam with varying stiffness #

• Euler-Bernoulli, first-order, linear elastic: no shear deformation (it counts in deep or short members), no second-order effects, no cracking that follows the load (enter the cracked I where you judge it applies).
• Stiffness is constant in each piece. A linear haunch is approximated in steps; add steps to see the result settle.
• Up to 6 spans and 20 segments. Supports are rigid; no springs, no settlement.
• No member strength checks, no connection or haunch detailing: the drawing shows the stiffness entered, not plates or welds.

Verification examples #

Verification status: 13 values compared in 5 examples (1 published, 4 hand calculations from the cited equations): 13 within tolerance, 0 recorded deviations. Thin: only 1 published example. Code basis: the provisions it uses were read in the target edition.

Udoeyo's example is a published worked example with members of different I; the others are closed forms worked by hand from the equations named. A published example of a haunched or cover-plated continuous beam (for example from a bridge design text) is not on hand; the haunch is checked against the exact integral of the linear variation.