Portal:Logic/Selected article/3: Difference between revisions

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In civil engineering and structural analysis Clapeyron's theorem of three moments is a relationship among the bending moments at three consecutive supports of a horizontal beam.

Let A,B,C be the three consecutive points of support, and denote by l the length of AB by $l^{'}$ the length of BC, by w and $w^{'}$ the weight per unit of length in these segments. Then^[1] the bending moments $M_{A}, M_{B}, M_{C}$ at the three points are related by:

M_{A} l + 2 M_{B} (l + l^{'}) + M_{C} l^{'} = \frac{1}{4} w l^{3} + \frac{1}{4} w^{'} (l^{'})^{3} .

This equation can also be written as ^[2]

M_{A} l + 2 M_{B} (l + l^{'}) + M_{C} l^{'} = \frac{6 a_{1} x_{1}}{l} + \frac{6 a_{2} x_{2}}{l^{'}}

where a₁ is the area on the bending moment diagram due to vertical loads on AB, a₂ is the area due to loads on BC, x₁ is the distance from A to the center of gravity for the b.m. diagram for AB, x₂ is the distance from C to the c.g. for the b.m. diagram for BC.

The second equation is more general as it does not require that the weight of each segment be distributed uniformly.

Figure 01-Sample continuous beam section

Derivation of Three Moments Equations

Mohr's Theorem^[3] can be used to derive the Three Moment Theorem^[4] (TMT).

Mohr's First Theorem

The change in slope of a deflection curve between two points of a beam is equal to the area of the M/EI diagram between those two points.(Figure 02)

Mohr's Second Theorem

Consider two points k1 and k2 on a beam. The deflection of k1 and k2 relative to the point of intersection between tangent at k1 and k2 and vertical through k1 is equal to the M/EI diagram between k1 and k2 about k1.(Figure 03)

Three Moment Equation expresses the relation between bending moments at three successive supports of a continuous beam, subject to a loading on a two adjacent span with or without settlement of the supports.

The Sign Convention

According to the Figure 04,

The moment M1, M2, and M3 be positive if they cause compression in the upper part of the beam. ( Sagging positive)
The deflection downward positive. (Downward settlement positive)
Let ABC is a continuous beam with support at A,B, and C. Then moment at A,B, and C are M1, M2, and M3, respectively.
Let A' B' and C' be the final positions of the beam ABC due to support settlements.

Derivation of Three Moment Theorem

PB'Q is a tangent drawn at B' for final Elastic Curve A'B'C' of the beam ABC. RB'S is a horizontal line drawn through B'. Consider, Triangles RB'P and QB'S.

\frac{P R}{R B^{'}} = \frac{S Q}{B^{'} S},

Template:NumBlk Template:NumBlk Template:NumBlk

From (1), (2), and (3),

\frac{Δ B - Δ A + P A^{'}}{L 1} = \frac{Δ C - Δ B - Q C^{'}}{L 2}

Template:NumBlk

Draw The M/EI diagram to find the PA' and QC'.

From Mohr's Second Theorem
PA' = First moment of area of M/EI diagram between A and B about A.

P A^{'} = (\frac{1}{2} \times \frac{M_{1}}{E_{1} I_{1}} \times L_{1}) \times L_{1} \times \frac{1}{3} + (\frac{1}{2} \times \frac{M_{2}}{E_{2} I_{2}} \times L_{1}) \times L_{1} \times \frac{2}{3} + \frac{A_{1} X_{1}}{E_{1} I_{1}}

QC' = First moment of area of M/EI diagram between B and C about C.

Q C^{'} = (\frac{1}{2} \times \frac{M_{3}}{E_{2} I_{2}} \times L_{2}) \times L_{2} \times \frac{1}{3} + (\frac{1}{2} \times \frac{M_{2}}{E_{2} I_{2}} \times L_{2}) \times L_{2} \times \frac{2}{3} + \frac{A_{2} X_{2}}{E_{2} I_{2}}

Substitute in PA' and QC' on equation (a), Three Moment Theorem (TMT)can be obtain.

Three Moment Equation

\frac{M_{1} L_{1}}{E_{1} I_{1}} + 2 M_{2} (\frac{L_{1}}{E_{1} I_{1}} + \frac{L_{2}}{E_{2} I_{2}}) + \frac{M_{3} L_{2}}{E_{2} I_{2}} = 6 [\frac{Δ A - Δ B}{L_{1}} + \frac{Δ C - Δ B}{L_{2}}] - 6 [\frac{A_{1} X_{1}}{E_{1} I_{1} L_{1}} + \frac{A_{2} X_{2}}{E_{2} I_{2} L_{2}}]

Notes

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External links

CodeCogs: Continuous beams with more than one span

↑ J. B. Wheeler: An Elementary Course of Civil Engineering, 1876, Page 118 [1]
↑ Srivastava and Gope: Strength of Materials, page 73
↑ Template:Cite web
↑ Template:Cite web

[1] J. B. Wheeler: An Elementary Course of Civil Engineering, 1876, Page 118 [1]

[2] Srivastava and Gope: Strength of Materials, page 73

[3] Template:Cite web

[4] Template:Cite web

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+In [[civil engineering]] and [[structural analysis]] [[Benoît Paul Émile Clapeyron|Clapeyron]]'s '''theorem of three moments''' is a relationship among the bending moments at three consecutive supports of a horizontal beam.
+Let ''A,B,C'' be the three consecutive points of support, and denote by ''l'' the length of ''AB'' by <math>l'</math> the length of ''BC'', by ''w'' and <math>w'</math> the weight per unit of length in these segments.  Then<ref>J. B. Wheeler: An Elementary Course of Civil Engineering, 1876, Page 118 [http://books.google.com/books?id=IRhDAAAAIAAJ&pg=PA118&lpg=PA118#v=onepage&q&f=false]</ref> the bending moments <math>M_A,\, M_B,\, M_C</math> at the three points are related by:
+:<math>M_A l + 2 M_B (l+l') +M_C l' = \frac{1}{4} w l^3 + \frac{1}{4} w' (l')^3.</math>
+This equation can also be written as <ref>[http://books.google.com/books?id=W9ZuLZWldUoC&pg=PA73&lpg=PA73#v=onepage&q&f=false Srivastava and Gope: Strength of Materials, page 73]</ref>
+:<math>M_A l + 2 M_B (l+l') +M_C l' = \frac{6 a_1 x_1}{l} + \frac{6 a_2 x_2}{l'}</math>
+where ''a''<sub>1</sub> is the area on the [[Shear and moment diagram|bending moment diagram]] due to vertical loads on AB, ''a''<sub>2</sub> is the area due to loads on BC, ''x''<sub>1</sub> is the distance from A to the center of gravity for the b.m. diagram for AB, ''x''<sub>2</sub> is the distance from C to the c.g. for the b.m. diagram for BC.
+The second equation is more general as it does not require that the weight of each segment be distributed uniformly.
+[[File:Figure 01 - Sample continuous beam section.png|thumb|Figure 01-Sample continuous beam section]]
+==Derivation of Three Moments Equations ==
+Mohr's Theorem<ref>{{cite web|title=Mohr's Theorem|url=http://www.colincaprani.com/files/notes/SAIII/Mohrs%20Theorems.pdf}}</ref> can be used to derive the Three Moment Theorem<ref>{{cite web|title=Three Moment Theorem|url=http://www.duke.edu/~hpgavin/ce130/three-moment.pdf}}</ref>  (TMT).
+===Mohr's First Theorem===
+The change in [[slope]] of a [[deflection (physics)|deflection]] curve between two points of a beam is equal to the area of the M/EI diagram between those two points.(Figure 02)
+[[File:Mohr's First Theorem.png|thumb|Figure 02-Mohr's First Theorem]]
+===Mohr's Second Theorem===
+Consider two points k1 and k2 on a [[beam (structure)|beam]]. The [[deflection (physics)|deflection]] of k1 and k2 relative to the point of intersection between tangent at k1 and k2 and vertical through k1 is equal to the M/EI diagram between k1 and k2 about k1.(Figure 03)
+[[File:Mohr's Second Theorem.png|thumb|Figure03-Mohr's Second Theorem]]
+Three Moment Equation expresses the relation between [http://bending%20moments bending moments] at three successive supports of a [http://continuous%20beam continuous beam], subject to a loading on a two adjacent span with or without [[settlement (structural)|settlement]] of the supports.
+===The Sign Convention===
+According to the Figure 04,
+# The moment M1, M2, and M3 be positive if they cause [[Compression (physical)|compression]] in the upper part of the beam. ( [[Sagging]] positive)
+# The [[deflection (physics)|deflection]] downward positive. (Downward settlement positive)
+# Let ABC is a [[:wikt:continuous|continuous]] beam with support at A,B, and C. Then moment at A,B, and C are M1, M2, and M3, respectively.
+# Let A' B' and C' be the final positions of the beam ABC due to support [[settlement (structural)|settlements]].
+[[File:Deflection Curve of a Continuous Beam.png|thumb|Figure 04-Deflection Curve of a Continuous Beam Under Settlement]]
+===Derivation of Three Moment Theorem===
+PB'Q is a tangent drawn at B' for final [[Elasticity (physics)|Elastic]] Curve A'B'C' of the [[beam (structure)|beam]] ABC. RB'S is a horizontal line drawn through B'.
+Consider, Triangles RB'P and QB'S.
+:<math>\dfrac{PR}{RB'} = \dfrac{SQ}{B'S},</math>
+{{NumBlk|:|<math>\dfrac{PR}{L1} = \dfrac{SQ}{L2}</math>|{{EquationRef|1}}}}
+{{NumBlk|:|<math>PR = \Delta B - \Delta A + PA'</math>|{{EquationRef|2}}}}
+{{NumBlk|:|<math>SQ = \Delta C - \Delta B - QC'</math>|{{EquationRef|3}}}}
+From (1), (2), and (3),
+:<math>\dfrac{\Delta B - \Delta A + PA'}{L1} = \dfrac{\Delta C - \Delta B - QC'}{L2}</math>
+{{NumBlk|:|<math>\dfrac{PA'}{L1} + \dfrac{QC'}{L2} = \dfrac{\Delta A -\Delta B}{L1} + \dfrac{\Delta C -\Delta B}{L2}</math>|{{EquationRef|a}}}}
+Draw The [http://M/EI%20diagram M/EI diagram] to find the PA' and QC'.
+[[File:Moment Diagram.png|thumb|Figure 05 - M / EI Diagram]]
+'''From Mohr's Second Theorem''' <br />
+PA' = First moment of area of M/EI diagram between A and B about A.
+:<math>PA' = \left(\frac{1}{2} \times \frac{M_1}{E_1 I_1} \times L_1\right)\times L_1\times \frac{1}{3} + \left(\frac{1}{2} \times \frac{M_2}{E_2 I_2} \times L_1\right)\times L_1\times\frac{2}{3}+ \frac{A_1 X_1}{E_1 I_1}</math>
+QC' = First moment of area of M/EI diagram between B and C about C.
+:<math>QC' = \left(\frac{1}{2} \times \frac{M_3}{E_2 I_2} \times L_2\right)\times L_2\times\frac{1}{3} + \left(\frac{1}{2} \times \frac{M_2}{E_2 I_2} \times L_2\right)\times L_2\times\frac{2}{3}+ \frac{A_2 X_2}{E_2 I_2}</math>
+Substitute in PA' and QC' on equation (a), [http://Three%20Moment%20Theorem Three Moment Theorem] (TMT)can be obtain.
+==Three Moment Equation==
+<br />
+:<math>\frac{M_1 L_1}{E_1 I_1}+ 2M_2\left(\frac{L_1}{E_1 I_1} + \frac{L_2}{E_2 I_2}\right)+\frac{M_3 L_2}{E_2 I_2} = 6 [\frac{\Delta A - \Delta B}{L_1} + \frac{\Delta C - \Delta B}{L_2}] - 6 [\frac{A_1 X_1}{E_1 I_1 L_1} + \frac{A_2 X_2}{E_2 I_2 L_2}]</math>
+==Notes==
+{{reflist}}
+==External links==
+*[http://www.codecogs.com/library/engineering/materials/beams/multiple-continuous-beams.php CodeCogs: Continuous beams with more than one span]
+{{DEFAULTSORT:Theorem Of Three Moments}}
+[[Category:Structural analysis]]
+[[Category:Continuum mechanics]]
+[[Category:Physics theorems]]

Portal:Logic/Selected article/3: Difference between revisions

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Latest revision as of 06:13, 20 October 2012

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