Qualitative Diagrams, Influence Lines, and Critical Placement

Qualitative Diagrams, Influence Lines, and Critical Placement

Use the current NCEES PE Civil Reference Handbook for mechanics formulas and AASHTO LRFD Bridge Design Specifications, 8th Edition (2017) with the May 2018 errata when a July 2026 bridge question invokes prescribed vehicle or lane loading. The mechanics of an influence line are general, but actual bridge load patterns, multiple-presence rules, and dynamic allowance must come from that exact supplied edition.

Two diagrams with different independent variables

A response diagram shows an internal action along a member for one fixed load arrangement. For example, M(x) plots bending moment at every beam coordinate x while a truck remains in one position. Its horizontal axis is the observation location.

An influence line shows one selected response at one fixed location while a unit load moves. For example, the influence line for moment at section C plots M_C(z) as the unit-load position z travels across the structure. Its horizontal axis is load position. Each ordinate answers: “How much response at C is caused by a unit load here?”

The drawings may look similar for a simple beam, but they are not interchangeable. Moving the observation section creates a response diagram; moving the load while holding the response definition fixed creates an influence line. Write the fixed response beside every influence line, including its sign convention.

Qualitative response diagrams

Before calculating ordinates, use relationships and boundary conditions:

• with downward w, dV/dx = -w and dM/dx = V;
• a point load jumps shear but not moment;
• an applied couple jumps moment but not shear;
• unloaded intervals have constant shear and linear moment;
• simple supports and internal hinges have zero moment unless an external couple acts there;
• a smooth moment extremum occurs where shear crosses zero.

Also use deformation sense as a reasonableness check: positive sagging moment bends a conventional simply supported beam concave upward in the common structural sign sketch. For continuous members, moments can change sign at points of contraflexure. Do not force the entire diagram to be positive merely because all applied loads point downward.

Simple-beam influence lines

Consider a simply supported beam of span L. A downward unit load is at coordinate z from the left support. Equilibrium gives reaction influence ordinates

R_A(z) = (L - z)/L and R_B(z) = z/L.

Thus the influence line for R_A decreases linearly from 1 at A to 0 at B; the line for R_B does the reverse.

At a fixed section C, located a from A, the positive moment influence line is triangular:

• if 0 ≤ z ≤ a, M_C(z) = z(L - a)/L;
• if a ≤ z ≤ L, M_C(z) = a(L - z)/L.

It is zero at both supports and reaches a(L-a)/L when the unit load is at C. The ordinate has units of length because a force multiplied by it produces moment.

For shear immediately to the right of C, using the left segment,

• if the unit load is left of C, V_C(z) = -z/L;
• if the unit load is right of C, V_C(z) = (L-z)/L.

The shear influence line jumps by 1 as the unit load crosses the section. This discontinuity is not a plotting error: placing a point load infinitesimally on opposite sides changes which side of the cut contains it. Label C- and C+ when needed.

The Müller-Breslau principle provides a powerful shape check: release the restraint associated with the desired response and impose a positive unit displacement or rotation. The resulting displaced shape is proportional to the influence line. For determinate beams this quickly recovers straight segments; for indeterminate structures it reveals curved qualitative shapes even when exact ordinates require analysis.

Convert ordinates into moving-load response

For concentrated loads P_i at positions z_i, the selected response is

Response = Σ P_i y(z_i),

where y is the appropriate influence ordinate. Preserve each load's spacing and sign. For a uniform moving load w covering a region, the response is w times the signed area under the influence line over that region. Load positive regions to maximize a positive response; loading a negative region reduces it unless the specified loading must occupy that region.

Worked axle placement

A 30-ft simple beam carries a two-axle group: 12 kip and 20 kip separated by 10 ft. Find the larger of two natural placements for positive moment at midspan C, where a = 15 ft. The peak influence ordinate is

15(30-15)/30 = 7.5 ft.

Place the 20-kip axle at midspan and the 12-kip axle at z = 5 ft (or symmetrically at 25 ft). At 5 ft, the ordinate is 5(15)/30 = 2.5 ft. Therefore

M_C = 20(7.5) + 12(2.5) = 180 kip-ft.

If the 12-kip axle is at midspan and the 20-kip axle is 10 ft away,

M_C = 12(7.5) + 20(2.5) = 140 kip-ft.

The first candidate governs. For a real multi-axle train, do not assume the heaviest axle always sits exactly at the section; translate the intact train through candidate positions because the sum of all load-ordinate products controls.

If a 1.5 kip/ft uniform load instead covers the whole 30-ft beam, the triangular midspan influence-line area is 0.5(30)(7.5) = 112.5 ft², giving 1.5(112.5) = 168.75 kip-ft. For maximum shear, the influence line has positive and negative regions, so selective placement differs from moment placement.

Exam workflow

State the fixed response, sketch and sign the influence line, place loads without changing their spacing, multiply point loads by ordinates or uniform load by signed area, and compare translations. Then check units: a dimensionless reaction ordinate produces force, while a moment ordinate in feet produces force-feet. That sequence prevents a familiar error—reading the fixed-load bending-moment diagram as though it predicted every moving-load position.