CIV-24

Point-load influence (simple beam)

Ma = P a b / L at a section from the left support. Influence ordinate η = a b / L for P = 1.

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InfluenceMüller-Breslau

Governing equation

M=\dfrac{Pab}{L}

$a \in \left[0.2, 19\right]$

where

P: Point load (kN)
a: Distance from left (m)
L: Span (m)
M: Section moment (kN·m)
\eta: Influence ordinate (m)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

M=\dfrac{Pab}{L}

$a \in \left[0.2, 19\right]$

$a = 0.2$ $M = 15.68$

Along the curve

②

Domain of the variable — blue

a_{\min} = 0.2,\quad a_{\max} = 19

$a \in \left[0.2, 19\right]$

$a = 0.2$ $a = 0.2$

Along the curve

③

Response curve — green

M=\dfrac{Pab}{L}

$a \in \left[0.2, 19\right]$

$a = 0.2$ $M = 15.68$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}M}{\mathrm{d}a}

$a \in \left[0.2, 19\right]$

$a = 0.318$ $M^{\prime} = 74.91$

Along the curve

Lecture brief

Historical brief

From Euler’s 1744 elastica and Navier’s beam theory to Mohr’s circle and transformed-section RC, structural mechanics grew as a closed-form craft before finite elements. The lab keeps those governing lines for buckling, flexure, joints and influence. This sheet (CIV-24 — Point-load influence (simple beam)) is the form associated with Müller-Breslau. Working symbols: $P$ , $a$ , $L$ $\rightarrow$ $M$ , $\eta$ . Müller-Breslau: the influence line is the deflected shape when the restraint is released a unit amount.

Purpose

Purpose: compute

M

\eta

from

P

a

L

in Structural & civil via

M=\dfrac{Pab}{L}

Ma = P a b / L at a section from the left support. Influence ordinate η = a b / L for P = 1. Use it when a real structural & civil question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

P = 80.000\,\mathrm{kN}

a = 4.000\,\mathrm{m}

L = 10.000\,\mathrm{m}

, the governing relation

M=\dfrac{Pab}{L}

yields

M = 192.00\,\mathrm{kN·m}

\eta = 2.400\,\mathrm{m}

. A simple beam, a travelling load, a moment peak. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Point load P80 kNDistance from left a4 mSpan L10 m

Outputs

Section moment M192.00 kN·m
Influence ordinate \eta2.400 m

Reading speed

Watch on YouTube

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Narration of this film

A simple beam, a travelling load, a moment peak.

Müller-Breslau: the influence line is the deflected shape when the restraint is released a unit amount.

Reading speed

Watch on YouTube