V-M-D-Diagrams - Beam Loading and Moment/Deflection Coefficients

Oct 25, 2026

Uploaded by dabgoudw

V-M-D diagrams compile coefficient tables and formula references for analyzing beams under concentrated and uniform loading. Content includes concentrated load equivalents for simple spans, fixed-end and pinned-end reactions, maximum positive and negative moments, deflection coefficients, and maximum deflection expressions. Additional sections provide cantilevered beam diagram formulas with moment and reaction coefficients across different numbers of equal load spaces, plus continuous beam shear and moment coefficient data for equal spans under uniform and concentrated loads.

LanguageEnglish

CategoryTechnology

FormatPDF

Pages19

Reading Time48 min

AccessFree to download

Concentrated Load Equivalents

Cantilevered Beams: Beam Diagrams and Formulas

Equal Loads, Equally Spaced

Continuous Beams: Moments and Shear Coefficients

Equal Spans, Equally Loaded

Concentrated Loads

| Concentrated Load Equivalents | | | | | | | --- | --- | --- | --- | --- | --- | | | | | Simple
Beam | Beam Fixed One
End,Supported
at Other | Beam Fixed
Both Ends | | n | Loading | Coeff. | ▲ | | | | 0 | | a | 0.125 | 0.070 | 0.042 | | | | b | 一 | 0.125 | 0.083 | | | | c | 0.500 | 0.375 | 一 | | | | d
e | 一
0.013 | 0.625
0.005 | 0.500
0.003 | | | | f | 1.000 | 1.000 | 0.667 | | | | g | 1.000 | 0.415 | 0.300 | | 2 | | a | 0.250 | 0.156 | 0.125 | | | | b | 一 | 0.188 | 0.125 | | | | C | 0.500 | 0.313 | 一 | | | | d | 一 | 0.688 | 0.500 | | | | e | 0.021 | 0.009 | 0.005 | | | | f
g | 2.000
0.800 | 1.500
0.477 | 1.000 | | | | | | | 0.400 | | 3 | | a | 0.333 | 0.222 | 0.111 | | | | D
c
d | 一
1.000
一 | 0.333
0.667 | 0.222
一 | | | | | | | | | | | | | 1.333 | 1.000 | | | | e | 0.036 | 0.015 | 0.008 | | | | f | 2.667 | 2.667 | 1.778 | | | | g | 1.022 | 0.438 | 0.333 | | 4 | | a | 0.500 | 0.266 | 0.188 | | | | b | 一 | 0.469 | 0.313 | | | | C | 1.500 | 1.031 | 一 | | | | d | 一 | 1.969 | 1.500 | | | | e | 0.050 | 0.021 | 0.010 | | | | f | 4.000 | 3.750 | 2.500 | | | | g | 0.950 | 0.428 | 0.320 | | 5 | | a
b
c | 0.600
一
2.000 | 0.360 | 0.200 | | | | | | 0.600
1.400 | 0.400 | | | | | | | 一 | | | | d | 一 | 2.600 | 2.000 | | | | e | 0.063 | 0.027 | 0.013 | | | | f | 4.800 | 4.800 | 3.200 | | | | g | 1.008 | 0.424 | 0.312 | | Equivalent simple span uniform load (kips):fP
Maximum positive moment(kip-ft):aPL
Maximum negative moment(kip-ft):bPL
Deflection coeffcient for equivalent simple span uniform load:g
Pinned end reaction (kips):cP
Number of equal load spaces:n
Fixed end reaction(kips):dP
Span of beam(ft):L
Span of beam(in.):1
Maximum deflection(in.):ePI³/EI | | | | | | | Cantilevered Beams
Beam Diagrams and Formulas
—Equal Loads,Equally Spaced | | | | | | | | --- | --- | --- | --- | --- | --- | --- | | No.Spans | | System | | | | | | 2 | | | | | | | | 3 | | b
b
台
台
5 | | | | | | 4 | | | | | | | | 5 | | M2 | | | | | | ≥6
(even) | | | | | | | | ≥7
(odd) | | 台 | | | | | | n | | | 2 | 3 | 4 | 5 | | Typical
Span
Loading | | | P2
P2'
P | P
P
P2
P2 | PG
P
P2 | 2
ND | | Moments | M₁
M₂
M₃
M₄
M₅ | 0.086×PL
0.096×PL
0.063×PL
0.039×PL
0.051×PL | 0.167×PL
0.188×PL
0.125×PL
0.083×PL
0.104×PL | 0.250×PL
0.278×PL
0.167×PL
0.083×PL
0.139×PL | 0.333×PL
0.375×PL
0.250×PL
0.167×PL
0.208×PL | 0.429×PL
0.480×PL
0.300×PL
0.171×PL
0.249×PL | | Reactions | ABCDEFGH | 0.414×P
1.172×P
0.438×P
1.063×P
1.086×P
1.109×P
0.977×P
1.000×P | 0.833×P
2.333×P
0.875×P
2.125×P
2.167×P
2.208×P
1.958×P
2.000×P | 1.250×P
3.500×P
1.333×P
3.167×P
3.250×P
3.333×P
2.917×P
3.000×P | 1.667×P
4.667×P
1.750×P
4.250×P
4.333×P
4.417×P
3.917×P
4.000×P | 2.071×P
5.857×P
2.200×P
5.300×P
5.429×P
5.557×P
4.871×P
5.000×P | | Dimensions
Cantilever | a
b
C
d
e
f | 0.172×L
0.125×L
0.220×L
0.204×L
0.157×L
0.147×L | 0.250×L
0.200×L
0.333×L
0.308×L
0.273×L
0.250×L | 0.200×L
0.143×L
0.250×L
0.231×L

FAQs

What is included in the V-M-D diagram tables for simple beams?

The tables provide concentrated-load equivalents, reaction coefficients (pinned and fixed ends), and maximum positive/negative moment coefficients, along with deflection coefficients and maximum deflection expressions.

How are cantilevered beams handled in the document?

Cantilever sections list beam diagram formulas and provide moment and reaction coefficients across different numbers of spans or equal load spaces, along with dimensional coefficients in terms of beam length.

What additional coefficients are provided for continuous beams?

For continuous beams with equal spans, the document provides shear and moment coefficients for uniform loading and for concentrated loads (including values given for different load placements such as center or third points).

Oct 25, 2026

Uploaded by dabgoudw