Td 30 C⋅:= Design Temperature

NL 100 N⋅ 103

⋅:= Installation Preload

SY 450 MPa⋅:= Yield Stress

EE 207 109

⋅ N⋅ m2−

⋅:= Beam Elastic Modulus

α 11.6 106−

⋅1

C⋅:= Thermal Expansion Coefficient

mvp 7850 kg⋅ m3−

⋅:= Beam Density

mvi 730 kg⋅ m3−

⋅:= Internal Fluid Density

mve 1027 kg⋅ m3−

⋅:= External Fluid Density

The effective length is used for soft ends, eg beams on soil. For hard ends fe = 1

The bending moment and deflection tend to infinity as the compressive axial load tends to ther buckling load

Buckling will occur in the orientation with the lowest buckling load

The selected orientation for bending might not be the orientation for buckling, depending on constraints etc.

Depending on beam geometry and axial load:

The maximum stress can occur either at the top of the beam or at the base of the beam

PIPENG.COM : Circular Beam Bending Combined Load

This example should be used with the Pipeng Toolbox :

http://pipeng.com/index.php/ts/itdmotbeam003a : Pipeng beam bending calculators

Copyright Pipeng Ltd : www.pipeng.com : Creative Commons attribution license

Data Values

OD 0.3 m⋅:= Beam Diameter

tn 0.015 m⋅:= Beam Wall Thickness

Lo 10.5 m⋅:= Nominal Length

fe 1.05:= Effective Length Factor

Le fe Lo⋅:= Le 1.1025 101

× m= Effective Length

x 6.5 m⋅:= Distance From Left End

Ts 19.5 C⋅:= Installation Temperature

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Point Load

wad 1100N

m⋅:= Distributed Load Start

wld 1200N

m⋅:= Distributed Load End

dm 2800− N⋅ m⋅:= Concentrated Moment

do 0.0015− rad⋅:= Angular Displacement

dy 0.015− m⋅:= Lateral Displacement

dt 35 C⋅:= Temperature Difference

waw 732N

m⋅:= Weight Load Start

For beams with axial load calculated from temperature, and also with a temperature difference load:

The design temperature is asumed to be the average temperature across the beam (along the whole beam).

Note: Large displacements with slope greater than 20-25 degrees or 0.4-0.5 radians are not valid.

Combined loads:

p 3 -4000 point load -4000 N (upwards) at a = 3 m

d 2 1100 1200 partial distributed load 1100 N/m downwards at 2 m 1200 N/m downwards at end

m 4 -2800 concentrated moment at 4 m 2800 N.m anti clockwise

a 4 -0.0015 angular displacement at 4 m left end rotated anti clockwise 0.0015 rad

y 3 -0.015 lateral displacement at 3 m right side displaced downwards 0.015 m

t 2 35 uniform delta temperature of 35 C from 2 m

w 732 weight load 732 N/m downwards

ap 3 m⋅:= Point Load Location

ad 2 m⋅:= Distributed load Start

am 4 m⋅:= Concentrated Moment Location

ao 1.5 m⋅:= Angular Displacement Location

ay 2.5 m⋅:= Lateral Displacement Location

at 3.5 m⋅:= Temperature Load Start

aw 0 m⋅:= Weight Load Start

wp 4000− N⋅:=

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lor 9.910589 103−

× m2

= Slenderness ratio

yaOD

2:= ya 1.5 10

1−× m= Outer Fiber Distance

ybOD

2:= yb 1.5 10

1−× m= Outer Fiber Distance

ZMI

ya:= ZM 9.115822 10

4−× m

3= Section Modulus

h ya yb+:= h 3 101−

× m= Section Height In Plane Of Bending

Mass And Weight

MP AX mvp⋅:= MP 1.054279 102

×kg

m= Beam Unit Mass

MC AC mvi⋅:= MC 4.179653 101

×kg

m= Contents Fluid Unit Mass

MD AD mve⋅:= Displaced Fluid Unit Mass

ML MP MC+:= ML 1.472245 102

×kg

m= Unit Mass Full

waw g MP MC+ MD−( )⋅:= waw 7.318713 102

×N

m= Unit Weight Full Submerged

Cross Section Area : Hollow Section

ID OD 2 tn⋅−:= ID 2.7 101−

× m= Internal Diameter

AXπ

4OD

2ID

2−( )⋅:= AX 1.343031 10

2−× m

2= Beam Cross Section Area

ADπ

4OD

2( )⋅:= AD 7.068583 102−

× m2

= Displaced Fluid Cross Section Area

ACπ

4ID

2( )⋅:= AC 5.725553 102−

× m2

= Contents Cross Section Area

Iπ

64OD

4ID

4−( )⋅:= I 1.367373 10

4−× m

4= Beam Second Area Moment

rrI

AX:= rr 1.009022 10

1−× m= Radius Of Gyration

EA EE AX⋅:= EA 2.780074 109

× N= Axial Stiffness

EAA EE α⋅ AX⋅:= EAA 3.224886 104

×N

C= Thermal Expansion Modulus

EI EE I⋅:= EI 2.830463 107

× N m2

⋅= Bending Stiffness

lorL

rr:=

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The Johnson buckling load is assumed to be >= 0

Buckling Load From LengthFB 1( ) 2.298264 106

× N=FB cc( ) lt LBT cc( )←

FBE cc( ) Le lt≥if

FBJ cc( ) otherwise

:=

Johnson Buckling Load From LengthFBJ 1( ) 2.07047 106

× N=FBJ cc( ) max 0 SY AX⋅SY AX⋅ Le⋅

2 π⋅

21

cc EI⋅⋅−,

:=

Euler Buckling Load From LengthFBE 1( ) 2.298264 106

× N=FBE cc( )cc π

2⋅ EI⋅

Le2

:=

Transition Buckling LengthLBT 1( ) 9.61489 100

× m=LBT cc( )2 π

2⋅ cc⋅ EI⋅

SY AX⋅:=

UX 3 1, 2,( ) 4=

Unit Step FunctionUX 1 2, 0,( ) 0=UX xx aa, n,( )

0 n 0=if

aa aa−( )n

otherwise

xx aa<if

xx aa−( )n

otherwise

:=

Functions

Axial StressSX 1.776675− 101

× MPa=SXNA

AX:=

Axial LoadNA 2.38613− 105

× N=NA NL EAA Td Ts−( )⋅−:=

Fully Restrained Axial Load

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YA0

3.799698 102−

× m= Deflection At A

RB0

wp:= RB0

4− 103

× N= Reaction At B

MB0

wp− Le ap−( )⋅:= MB0

3.21 104

× N m⋅= Moment At B

OB0

0 rad⋅:= OB0

0 100

×= Angle At B

YB0

0 m⋅:= YB0

0 100

× m= Deflection At B

RX0

RA0

wp UX x ap, 0,( )⋅−:= RX0

4 103

× N= Shear At X

MX0

MA0

RA0

x⋅+ wp UX x ap, 1,( )⋅−:= MX0

1.4 104

× N m⋅= Moment At X

OX0

OA0

MA0

x⋅

EI+

RA0

x2

⋅

2 EI⋅+

wp

2 EI⋅UX x ap, 2,( )⋅−:=

OX0

3.684954− 103−

×= Slope At X

YX0

YA0

OA0

x⋅+MA

0x2

⋅

2 EI⋅+

RA0

x3

⋅

6 EI⋅+

wp

6 EI⋅UX x ap, 3,( )⋅−:=

YX0

9.42834 103−

× m= Deflection At X

Beam Bending - Free-Fix

kk 0.25:= Buckling Load Factor

lt LBT kk( ):= lt 4.807445 100

× m= Buckle Transition Length

fbe FBE kk( ):= fbe 5.745661 105

× N= Euler Buckling Load

fbj FBJ kk( ):= fbj 0 100

× N= Johnson Buckling Load

fb FB kk( ):= fb 5.745661 105

× N= Buckling Load

Free-Fix : Point Load

RA0

0 N⋅:= RA0

0 100

× N= Reaction At A

MA0

0 N⋅ m⋅:= MA0

0 100

× N m⋅= Moment At A

OA0

wp

2 EI⋅Le ap−( )

2⋅:= OA

04.550537− 10

3−×= Angle At A

YA0

wp−

6 EI⋅2 Le

3⋅ 3 Le

2⋅ ap⋅− ap

3+( )⋅:=

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OB1

0 rad⋅:= OB1

0 100

×= Angle At B

YB1

0 m⋅:= YB1

0 100

× m= Deflection At B

RX1

RA1

wad UX x ad, 1,( )⋅−wld wad−

2 Le ad−( )⋅UX x ad, 2,( )⋅−:=

RX1

5.062188− 103

× N= Shear At X

MX1

MA1

RA1

x⋅+wad

2UX x ad, 2,( )⋅−

wld wad−

6 Le ad−( )⋅UX x ad, 3,( )⋅−:=

MX1

1.130578− 104

× N m⋅= Moment At X

OX1

OA1

MA1

x⋅

EI+

RA1

x2

⋅

2 EI⋅+

wad

6 EI⋅UX x ad, 3,( )⋅−

wld wad−

24 EI⋅ Le ad−( )⋅UX x ad, 4,( )⋅−:=

OX1

4.272593 103−

×= Slope At X

YX1

YA1

OA1

x⋅+MA

1x2

⋅

2 EI⋅+

RA1

x3

⋅

6 EI⋅+

wad

24 EI⋅UX x ad, 4,( )⋅−

wld wad−

120 EI⋅ Le ad−( )⋅UX x ad, 5,( )⋅−:=

YX1

1.176657− 102−

× m= Deflection At X

Free-Fix : Distributed Load

RA1

0 N⋅:= RA1

0 100

× N= Reaction At A

MA1

0 N⋅ m⋅:= MA1

0 100

× N m⋅= Moment At A

OA1

wad

6 EI⋅Le ad−( )

3⋅

wld wad−

24 EI⋅Le ad−( )

3⋅+:= OA

14.869512 10

3−×= Angle At A

YA1

wad−

24 EI⋅Le ad−( )

3⋅ 3 Le⋅ ad+( )⋅

wld wad−

120 EI⋅Le ad−( )

3⋅ 4 Le⋅ ad+( )⋅−:=

YA1

4.274837− 102−

× m= Deflection At A

RB1

wad wld+

2Le ad−( )⋅:= RB

11.037875 10

4× N= Reaction At B

MB1

wad−

2Le ad−( )

2⋅

wld wad−

6Le ad−( )

2⋅−:=

MB1

4.615535− 104

× N m⋅= Moment At B

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OB2

0 rad⋅:= OB2

0 100

×= Angle At B

YB2

0 m⋅:= YB2

0 100

× m= Deflection At B

RX2

RA2

:= RX2

0 100

× N= Shear At X

MX2

MA2

RA2

x⋅+ dm UX x am, 0,( )⋅+:= MX2

2.8− 103

× N m⋅= Moment At X

OX2

OA2

MA2

x⋅

EI+

RA2

x2

⋅

2 EI⋅+

dm

EIUX x am, 1,( )⋅+:=

OX2

4.4763 104−

×= Slope At X

YX2

YA2

OA2

x⋅+MA

2x2

⋅

2 EI⋅+

RA2

x3

⋅

6 EI⋅+

dm

2 EI⋅UX x am, 2,( )⋅+:=

YX2

1.012763− 103−

× m= Deflection At X

Free-Fix : Concentrated Moment

RA2

0 N⋅:= RA2

0 100

× N= Reaction At A

MA2

0 N⋅ m⋅:= MA2

0 100

× N m⋅= Moment At A

OA2

dm−

EILe am−( )⋅:= OA

26.949394 10

4−×= Angle At A

YA2

dm

2 EI⋅Le

2am

2−( )⋅:= YA

25.220732− 10

3−× m= Deflection At A

RB2

0 N⋅:= RB2

0 100

× N= Reaction At B

MB2

dm:= MB2

2.8− 103

× N m⋅= Moment At B

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OB3

0 rad⋅:= OB3

0 100

×= Angle At B

YB3

0 m⋅:= YB3

0 100

× m= Deflection At B

RX3

RA3

:= RX3

0 100

× N= Shear At X

MX3

MA3

RA3

x⋅+:= MX3

0 100

× N m⋅= Moment At X

OX3

OA3

MA3

x⋅

EI+

RA3

x2

⋅

2 EI⋅+ do UX x ao, 0,( )⋅+:=

OX3

0 100

×= Slope At X

YX3

YA3

OA3

x⋅+MA

3x2

⋅

2 EI⋅+

RA3

x3

⋅

6 EI⋅+ do UX x ao, 1,( )⋅+:=

YX3

0 100

× m= Deflection At X

Free-Fix : Angular Displacement

RA3

0 N⋅:= RA3

0 100

× N= Reaction At A

MA3

0 N⋅ m⋅:= MA3

0 100

× N m⋅= Moment At A

OA3

do−:= OA3

1.5 103−

×= Angle At A

YA3

do ao⋅:= YA3

2.25− 103−

× m= Deflection At A

RB3

0 N⋅:= RB3

0 100

× N= Reaction At B

MB3

0 N⋅ m⋅:= MB3

0 100

× N m⋅= Moment At B

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OB4

0 rad⋅:= OB4

0 100

×= Angle At B

YB4

0 m⋅:= YB4

0 100

× m= Deflection At B

RX4

RA4

:= RX4

0 100

× N= Shear At X

MX4

MA4

RA4

x⋅+:= MX4

0 100

× N m⋅= Moment At X

OX4

OA4

MA4

x⋅

EI+

RA4

x2

⋅

2 EI⋅+:=

OX4

0 100

×= Slope At X

YX4

YA4

OA4

x⋅+MA

4x2

⋅

2 EI⋅+

RA4

x3

⋅

6 EI⋅+ dy UX x ay, 0,( )⋅+:=

YX4

0 100

× m= Deflection At X

Free-Fix : Lateral Displacement

RA4

0 N⋅:= RA4

0 100

× N= Reaction At A

MA4

0 N⋅ m⋅:= MA4

0 100

× N m⋅= Moment At A

OA4

0 rad⋅:= OA4

0 100

×= Angle At A

YA4

dy−:= YA4

1.5 102−

× m= Deflection At A

RB4

0 N⋅:= RB4

0 100

× N= Reaction At B

MB4

0 N⋅ m⋅:= MB4

0 100

× N m⋅= Moment At B

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OB5

0 rad⋅:= OB5

0 100

×= Angle At B

YB5

0 m⋅:= YB5

0 100

× m= Deflection At B

RX5

RA4

:= RX5

0 100

× N= Shear At X

MX5

MA5

RA5

x⋅+:= MX5

0 100

× N m⋅= Moment At X

OX5

OA5

MA5

x⋅

EI+

RA5

x2

⋅

2 EI⋅+

α

hdt⋅ UX x at, 1,( )⋅+:=

OX5

6.123833− 103−

×= Slope At X

YX5

YA5

OA5

x⋅+MA

5x2

⋅

2 EI⋅+

RA5

x3

⋅

6 EI⋅+

α

2 h⋅dt⋅ UX x at, 2,( )⋅+:=

YX5

1.385517 102−

× m= Deflection At X

Free-Fix : Delta Temperature

RA5

0 N⋅:= RA5

0 100

× N= Reaction At A

MA5

0 N⋅ m⋅:= MA5

0 100

× N m⋅= Moment At A

OA5

α−

hdt⋅ Le at−( )⋅:= OA

51.018383− 10

2−×= Angle At A

YA5

α

2 h⋅dt⋅ Le

2at

2−( )⋅:= YA

57.396009 10

2−× m= Deflection At A

RB5

0 N⋅:= RB5

0 100

× N= Reaction At B

MB5

0 N⋅ m⋅:= MB5

0 100

× N m⋅= Moment At B

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OB6

0 rad⋅:= OB6

0 100

×= Angle At B

YB6

0 m⋅:= YB6

0 100

× m= Deflection At B

RX6

RA6

waw UX x aw, 1,( )⋅−:= RX6

4.757164− 103

× N= Shear At X

MX6

MA6

RA6

x⋅+waw

2UX x aw, 2,( )⋅−:= MX

61.546078− 10

4× N m⋅= Moment At X

OX6

OA6

MA6

x⋅

EI+

RA6

x2

⋅

2 EI⋅+

waw

6 EI⋅UX x aw, 3,( )⋅−:=

OX6

4.591637 103−

×= Slope At X

YX6

YA6

OA6

x⋅+MA

6x2

⋅

2 EI⋅+

RA6

x3

⋅

6 EI⋅+

waw

24 EI⋅UX x aw, 4,( )⋅−:=

YX6

1.213794− 102−

× m= Deflection At X

Free-Fix : Weight Load

RA6

0 N⋅:= RA6

0 100

× N= Reaction At A

MA6

0 N⋅ m⋅:= MA6

0 100

× N m⋅= Moment At A

OA6

waw

6 EI⋅Le( )

3⋅:= OA

65.775131 10

3−×= Angle At A

YA6

waw−

24 EI⋅Le( )

3⋅ 3 Le⋅( )⋅:= YA

64.775311− 10

2−× m= Deflection At A

RB6

waw( ) Le( )⋅:= RB6

8.068881 103

× N= Reaction At B

MB6

waw−

2Le( )

2⋅:= MB

64.447971− 10

4× N m⋅= Moment At B

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obt

0

6

n

OBn∑

=

:= obt 0 100

×= Total Angle At B

ybt

0

6

n

YBn∑

=

:= YB6

0 100

× m= Total Deflection At B

rxt

0

6

n

RXn∑

=

:= rxt 5.819352− 103

× N= Total Shear At X

mxt

0

6

n

MXn∑

=

:= mxt 1.556656− 104

× N m⋅= Total Moment At X

oxt

0

6

n

OXn∑

=

:= oxt 4.969273− 104−

×= Total Slope At X

yxt

0

6

n

YXn∑

=

:= yxt 1.633756− 103−

× m= Total Deflection At X

Free-Fix : Total Load

rat

0

6

n

RAn∑

=

:= rat 0 100

× N= Total Reaction At A

mat

0

6

n

MAn∑

=

:= mat 0 100

× N m⋅= Total Moment At A

oat

0

6

n

OAn∑

=

:= oat 1.894788− 103−

×= Total Angle At A

yat

0

6

n

YAn∑

=

:= yat 2.898486 102−

× m= Total Deflection At A

rbt

0

6

n

RBn∑

=

:= rbt 1.444763 104

× N= Total Reaction At B

mbt

0

6

n

MBn∑

=

:= mbt 6.133506− 104

× N m⋅= Total Moment At B

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Longitudinal Stress BaseSLB 4.697183− 101

× MPa=SLB SX SBB+:=

Longitudinal Stress TopSLA 1.143832 101

× MPa=SLA SX SBA+:=

Maximum Bending Stress TopSBB 2.920507− 107

× Pa=SBBmxa yb⋅

I:=

Maximum Bending Stress TopSBA 2.920507 107

× Pa=SBAmxa− ya⋅

I:=

Max Bending Moment Axial Loadmxa 2.662282− 104

× N m⋅=mxamxt

1NA

fb+

:=

Free-Fix : Stress Check

SBAmxt− ya⋅

I:= SBA 1.707642 10

7× Pa= Maximum Bending Stress Top

SBBmxt yb⋅

I:= SBB 1.707642− 10

7× Pa= Maximum Bending Stress Top

SLA SX SBA+:= SLA 6.903308− 101−

× MPa= Longitudinal Stress Top

SLB SX SBB+:= SLB 3.484318− 101

× MPa= Longitudinal Stress Base

yxayxt

1NA

fb+

:= yxa 2.794142− 103−

× m= Max Displacement Axial Load

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YA0

1.291215 102−

× m= Deflection At A

RB0

wp:= RB0

4− 103

× N= Reaction At B

MB0

wp− Le2

ap2

−( )⋅

2 Le⋅:= MB

02.041735 10

4× N m⋅= Moment At B

OB0

0 rad⋅:= OB0

0 100

×= Angle At B

YB0

0 m⋅:= YB0

0 100

× m= Deflection At B

RX0

RA0

wp UX x ap, 0,( )⋅−:= RX0

4 103

× N= Shear At X

MX0

MA0

RA0

x⋅+ wp UX x ap, 1,( )⋅−:= MX0

2.317347 103

× N m⋅= Moment At X

OX0

OA0

MA0

x⋅

EI+

RA0

x2

⋅

2 EI⋅+

wp

2 EI⋅UX x ap, 2,( )⋅−:=

OX0

1.817273− 103−

×= Slope At X

YX0

YA0

OA0

x⋅+MA

0x2

⋅

2 EI⋅+

RA0

x3

⋅

6 EI⋅+

wp

6 EI⋅UX x ap, 3,( )⋅−:=

YX0

5.202712 103−

× m= Deflection At X

Beam Bending - Guide-Fix

kk 1.0:= Buckling Load Factor

lt LBT kk( ):= lt 9.61489 100

× m= Buckle Transition Length

fbe FBE kk( ):= fbe 2.298264 106

× N= Euler Buckling Load

fbj FBJ kk( ):= fbj 2.07047 106

× N= Johnson Buckling Load

fb FB kk( ):= fb 2.298264 106

× N= Buckling Load

Guide-Fix : Point Load

RA0

0 N⋅:= RA0

0 100

× N= Reaction At A

MA0

wp Le ap−( )2

⋅

2 Le⋅:= MA

01.168265− 10

4× N m⋅= Moment At A

OA0

0 rad⋅:= OA0

0 100

×= Angle At A

YA0

wp−

12 EI⋅Le ap−( )

2⋅ Le 2 ap⋅+( )⋅:=

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OB1

0 rad⋅:= OB1

0 100

×= Angle At B

YB1

0 m⋅:= YB1

0 100

× m= Deflection At B

RX1

RA1

wad UX x ad, 1,( )⋅−wld wad−

2 Le ad−( )⋅UX x ad, 2,( )⋅−:=

RX1

5.062188− 103

× N= Shear At X

MX1

MA1

RA1

x⋅+wad

2UX x ad, 2,( )⋅−

wld wad−

6 Le ad−( )⋅UX x ad, 3,( )⋅−:=

MX1

1.19578 103

× N m⋅= Moment At X

OX1

OA1

MA1

x⋅

EI+

RA1

x2

⋅

2 EI⋅+

wad

6 EI⋅UX x ad, 3,( )⋅−

wld wad−

24 EI⋅ Le ad−( )⋅UX x ad, 4,( )⋅−:=

OX1

2.273995 103−

×= Slope At X

YX1

YA1

OA1

x⋅+MA

1x2

⋅

2 EI⋅+

RA1

x3

⋅

6 EI⋅+

wad

24 EI⋅UX x ad, 4,( )⋅−

wld wad−

120 EI⋅ Le ad−( )⋅UX x ad, 5,( )⋅−:=

YX1

7.244737− 103−

× m= Deflection At X

Guide-Fix : Distributed Load

RA1

0 N⋅:= RA1

0 100

× N= Reaction At A

MA1

wad

6 Le⋅Le ad−( )

3 wld wad−

24 Le⋅Le ad−( )

3⋅+:= MA

11.250156 10

4× N m⋅= Moment At A

OA1

0 rad⋅:= OA1

0 100

×= Angle At A

YA1

wad−

24 EI⋅Le ad−( )

3⋅ Le ad+( )⋅

wld wad−

240 EI⋅Le ad−( )

3⋅ 3 Le⋅ 2 ad⋅+( )⋅−:=

YA1

1.590518− 102−

× m= Deflection At A

RB1

wad wld+

2Le ad−( )⋅:= RB

11.037875 10

4× N= Reaction At B

MB1

wad−

6 Le⋅Le ad−( )

2⋅ 2 Le⋅ ad+( )⋅

wld wad−

24 Le⋅Le ad−( )

2⋅ 3 Le⋅ ad+( )⋅−:=

MB1

3.365379− 104

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 15 Of 58

OB2

0 rad⋅:= OB2

0 100

×= Angle At B

YB2

0 m⋅:= YB2

0 100

× m= Deflection At B

RX2

RA2

:= RX2

0 100

× N= Shear At X

MX2

MA2

RA2

x⋅+ dm UX x am, 0,( )⋅+:= MX2

1.015873− 103

× N m⋅= Moment At X

OX2

OA2

MA2

x⋅

EI+

RA2

x2

⋅

2 EI⋅+

dm

EIUX x am, 1,( )⋅+:=

OX2

1.624054 104−

×= Slope At X

YX2

YA2

OA2

x⋅+MA

2x2

⋅

2 EI⋅+

RA2

x3

⋅

6 EI⋅+

dm

2 EI⋅UX x am, 2,( )⋅+:=

YX2

3.674423− 104−

× m= Deflection At X

Guide-Fix : Concentrated Moment

RA2

0 N⋅:= RA2

0 100

× N= Reaction At A

MA2

dm−

LeLe am−( )⋅:= MA

21.784127 10

3× N m⋅= Moment At A

OA2

0 rad⋅:= OA2

0 100

×= Angle At A

YA2

dm am⋅

2 EI⋅Le am−( )⋅:= YA

21.389879− 10

3−× m= Deflection At A

RB2

0 N⋅:= RB2

0 100

× N= Reaction At B

MB2

dm am⋅

Le:= MB

21.015873− 10

3× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 16 Of 58

OB3

0 rad⋅:= OB3

0 100

×= Angle At B

YB3

0 m⋅:= YB3

0 100

× m= Deflection At B

RX3

RA3

:= RX3

0 100

× N= Shear At X

MX3

MA3

RA3

x⋅+:= MX3

3.85097 103

× N m⋅= Moment At X

OX3

OA3

MA3

x⋅

EI+

RA3

x2

⋅

2 EI⋅+ do UX x ao, 0,( )⋅+:=

OX3

6.156463− 104−

×= Slope At X

YX3

YA3

OA3

x⋅+MA

3x2

⋅

2 EI⋅+

RA3

x3

⋅

6 EI⋅+ do UX x ao, 1,( )⋅+:=

YX3

1.3929 103−

× m= Deflection At X

Guide-Fix : Angular Displacement

RA3

0 N⋅:= RA3

0 100

× N= Reaction At A

MA3

do− EI⋅

Le:= MA

33.85097 10

3× N m⋅= Moment At A

OA3

0 rad⋅:= OA3

0 100

×= Angle At A

YA3

do aoLe

2−

⋅:= YA3

6.01875 103−

× m= Deflection At A

RB3

0 N⋅:= RB3

0 100

× N= Reaction At B

MB3

do EI⋅

Le:= MB

33.85097− 10

3× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 17 Of 58

OB4

0 rad⋅:= OB4

0 100

×= Angle At B

YB4

0 m⋅:= YB4

0 100

× m= Deflection At B

RX4

RA4

:= RX4

0 100

× N= Shear At X

MX4

MA4

RA4

x⋅+:= MX4

0 100

× N m⋅= Moment At X

OX4

OA4

MA4

x⋅

EI+

RA4

x2

⋅

2 EI⋅+:=

OX4

0 100

×= Slope At X

YX4

YA4

OA4

x⋅+MA

4x2

⋅

2 EI⋅+

RA4

x3

⋅

6 EI⋅+ dy UX x ay, 0,( )⋅+:=

YX4

0 100

× m= Deflection At X

Guide-Fix : Lateral Displacement

RA4

0 N⋅:= RA4

0 100

× N= Reaction At A

MA4

0 N⋅ m⋅:= MA4

0 100

× N m⋅= Moment At A

OA4

0 rad⋅:= OA4

0 100

×= Angle At A

YA4

dy−:= YA4

1.5 102−

× m= Deflection At A

RB4

0 N⋅:= RB4

0 100

× N= Reaction At B

MB4

0 N⋅ m⋅:= MB4

0 100

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 18 Of 58

OB5

0 rad⋅:= OB5

0 100

×= Angle At B

YB5

0 m⋅:= YB5

0 100

× m= Deflection At B

RX5

RA5

:= RX5

0 100

× N= Shear At X

MX5

MA5

RA5

x⋅+:= MX5

2.614509− 104

× N m⋅= Moment At X

OX5

OA5

MA5

x⋅

EI+

RA5

x2

⋅

2 EI⋅+

α

hdt⋅ UX x at, 1,( )⋅+:=

OX5

1.944074− 103−

×= Slope At X

YX5

YA5

OA5

x⋅+MA

5x2

⋅

2 EI⋅+

RA5

x3

⋅

6 EI⋅+

α

2 h⋅dt⋅ UX x at, 2,( )⋅+:=

YX5

4.398468 103−

× m= Deflection At X

Guide-Fix : Delta Temperature

RA5

0 N⋅:= RA5

0 100

× N= Reaction At A

MA5

α− EI⋅

h Le⋅dt⋅ Le at−( )⋅:= MA

52.614509− 10

4× N m⋅= Moment At A

OA5

0 rad⋅:= OA5

0 100

×= Angle At A

YA5

α at⋅

2 h⋅dt⋅ Le at−( )⋅:= YA

51.782171 10

2−× m= Deflection At A

RB5

0 N⋅:= RB5

0 100

× N= Reaction At B

MB5

MA5

:= MB5

2.614509− 104

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 19 Of 58

OB6

0 rad⋅:= OB6

0 100

×= Angle At B

YB6

0 m⋅:= YB6

0 100

× m= Deflection At B

RX6

RA6

waw UX x aw, 1,( )⋅−:= RX6

4.757164− 103

× N= Shear At X

MX6

MA6

RA6

x⋅+waw

2UX x aw, 2,( )⋅−:= MX

66.342122− 10

2× N m⋅= Moment At X

OX6

OA6

MA6

x⋅

EI+

RA6

x2

⋅

2 EI⋅+

waw

6 EI⋅UX x aw, 3,( )⋅−:=

OX6

2.221345 103−

×= Slope At X

YX6

YA6

OA6

x⋅+MA

6x2

⋅

2 EI⋅+

RA6

x3

⋅

6 EI⋅+

waw

24 EI⋅UX x aw, 4,( )⋅−:=

YX6

6.775155− 103−

× m= Deflection At X

Guide-Fix : Weight Load

RA6

0 N⋅:= RA6

0 100

× N= Reaction At A

MA6

waw

6 Le⋅Le( )

3⋅:= MA

61.482657 10

4× N m⋅= Moment At A

OA6

0 rad⋅:= OA6

0 100

×= Angle At A

YA6

waw−

24 EI⋅Le( )

3⋅ Le( )⋅:= YA

61.59177− 10

2−× m= Deflection At A

RB6

waw( ) Le( )⋅:= RB6

8.068881 103

× N= Reaction At B

MB6

waw−

3Le( )

2⋅:= MB

62.965314− 10

4× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 20 Of 58

obt

0

6

n

OBn∑

=

:= obt 0 100

×= Total Angle At B

ybt

0

6

n

YBn∑

=

:= YB6

0 100

× m= Total Deflection At B

rxt

0

6

n

RXn∑

=

:= rxt 5.819352− 103

× N= Total Shear At X

mxt

0

6

n

MXn∑

=

:= mxt 2.043108− 104

× N m⋅= Total Moment At X

oxt

0

6

n

OXn∑

=

:= oxt 2.80752 104−

×= Total Slope At X

yxt

0

6

n

YXn∑

=

:= yxt 3.393255− 103−

× m= Total Deflection At X

Guide-Fix : Total Load

rat

0

6

n

RAn∑

=

:= rat 0 100

× N= Total Reaction At A

mat

0

6

n

MAn∑

=

:= mat 4.864513− 103

× N m⋅= Total Moment At A

oat

0

6

n

OAn∑

=

:= oat 0 100

×= Total Angle At A

yat

0

6

n

YAn∑

=

:= yat 1.853984 102−

× m= Total Deflection At A

rbt

0

6

n

RBn∑

=

:= rbt 1.444763 104

× N= Total Reaction At B

mbt

0

6

n

MBn∑

=

:= mbt 7.390151− 104

× N m⋅= Total Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 21 Of 58

Longitudinal Stress BaseSLB 4.277606− 101

× MPa=SLB SX SBB+:=

Longitudinal Stress TopSLA 7.242555 100

× MPa=SLA SX SBA+:=

Maximum Bending Stress TopSBB 2.500931− 107

× Pa=SBBmxa yb⋅

I:=

Maximum Bending Stress TopSBA 2.500931 107

× Pa=SBAmxa− ya⋅

I:=

Max Bending Moment Axial Loadmxa 2.279804− 104

× N m⋅=mxamxt

1NA

fb+

:=

Guide-Fix : Stress Check

SBAmxt− ya⋅

I:= SBA 2.241276 10

7× Pa= Maximum Bending Stress Top

SBBmxt yb⋅

I:= SBB 2.241276− 10

7× Pa= Maximum Bending Stress Top

SLA SX SBA+:= SLA 4.64601 100

× MPa= Longitudinal Stress Top

SLB SX SBB+:= SLB 4.017952− 101

× MPa= Longitudinal Stress Base

yxayxt

1NA

fb+

:= yxa 3.786368− 103−

× m= Max Displacement Axial Load

itdhebeam003201 9/09/2019 9:28 AM Page 22 Of 58

YA0

0 100

× m= Deflection At A

RB0

wp ap⋅ 3 Le2

⋅ ap2

−( )⋅

2 Le3

⋅

:= RB0

1.592357− 103

× N= Reaction At B

MB0

wp− ap⋅ Le2

ap2

−( )⋅

2 Le2

⋅

:= MB0

5.555741 103

× N m⋅= Moment At B

OB0

0 rad⋅:= OB0

0 100

×= Angle At B

YB0

0 m⋅:= YB0

0 100

× m= Deflection At B

RX0

RA0

wp UX x ap, 0,( )⋅−:= RX0

1.592357 103

× N= Shear At X

MX0

MA0

RA0

x⋅+ wp UX x ap, 1,( )⋅−:= MX0

1.649677− 103

× N m⋅= Moment At X

OX0

OA0

MA0

x⋅

EI+

RA0

x2

⋅

2 EI⋅+

wp

2 EI⋅UX x ap, 2,( )⋅−:=

OX0

3.12227− 104−

×= Slope At X

YX0

YA0

OA0

x⋅+MA

0x2

⋅

2 EI⋅+

RA0

x3

⋅

6 EI⋅+

wp

6 EI⋅UX x ap, 3,( )⋅−:=

YX0

1.140781 103−

× m= Deflection At X

Beam Bending - Pin-Fix

kk 2.05:= Buckling Load Factor

lt LBT kk( ):= lt 1.376643 101

× m= Buckle Transition Length

fbe FBE kk( ):= fbe 4.711442 106

× N= Euler Buckling Load

fbj FBJ kk( ):= fbj 4.105508 106

× N= Johnson Buckling Load

fb FB kk( ):= fb 4.105508 106

× N= Buckling Load

Pin-Fix : Point Load

RA0

wp

2 Le3

⋅

Le ap−( )2

⋅ 2 Le⋅ ap+( )⋅:= RA0

2.407643− 103

× N= Reaction At A

MA0

0 N⋅ m⋅:= MA0

0 100

× N m⋅= Moment At A

OA0

wp− ap⋅

4 EI⋅ Le⋅Le ap−( )

2⋅:= OA

06.191207 10

4−×= Angle At A

YA0

0 m⋅:=

itdhebeam003201 9/09/2019 9:28 AM Page 23 Of 58

OB1

0 rad⋅:= OB1

0 100

×= Angle At B

YB1

0 m⋅:= YB1

0 100

× m= Deflection At B

RX1

RA1

wad UX x ad, 1,( )⋅−wld wad−

2 Le ad−( )⋅UX x ad, 2,( )⋅−:=

RX1

2.353479− 103

× N= Shear At X

MX1

MA1

RA1

x⋅+wad

2UX x ad, 2,( )⋅−

wld wad−

6 Le ad−( )⋅UX x ad, 3,( )⋅−:=

MX1

6.300829 103

× N m⋅= Moment At X

OX1

OA1

MA1

x⋅

EI+

RA1

x2

⋅

2 EI⋅+

wad

6 EI⋅UX x ad, 3,( )⋅−

wld wad−

24 EI⋅ Le ad−( )⋅UX x ad, 4,( )⋅−:=

OX1

4.781192 104−

×= Slope At X

YX1

YA1

OA1

x⋅+MA

1x2

⋅

2 EI⋅+

RA1

x3

⋅

6 EI⋅+

wad

24 EI⋅UX x ad, 4,( )⋅−

wld wad−

120 EI⋅ Le ad−( )⋅UX x ad, 5,( )⋅−:=

YX1

2.442678− 103−

× m= Deflection At X

Pin-Fix : Distributed Load

RA1

wad

8 Le3

⋅

Le ad−( )3

⋅ 3 Le⋅ ad+( )⋅wld wad−

40 Le3

⋅

Le ad−( )3

⋅ 4 Le⋅ ad+( )⋅+:=

RA1

2.708709 103

× N= Reaction At A

MA1

0 N⋅ m⋅:= MA1

0 100

× N m⋅= Moment At A

OA1

wad−

48 EI⋅ Le⋅Le ad−( )

3⋅ Le 3 ad⋅+( )⋅

wld wad−

240 EI⋅ Le⋅Le ad−( )

3⋅ 2 Le⋅ 3 ad⋅+( )⋅−:=

OA1

9.46592− 104−

×= Angle At A

YA1

0 m⋅:= YA1

0 100

× m= Deflection At A

RB1

wad wld+

2Le ad−( )⋅ RA

1−:= RB

17.670041 10

3× N= Reaction At B

MB1

RA1

Le⋅wad

2Le ad−( )

2⋅−

wld wad−

6Le ad−( )

2⋅−:=

MB1

1.629183− 104

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 24 Of 58

OB2

0 rad⋅:= OB2

0 100

×= Angle At B

YB2

0 m⋅:= YB2

0 100

× m= Deflection At B

RX2

RA2

:= RX2

3.308067 102

× N= Shear At X

MX2

MA2

RA2

x⋅+ dm UX x am, 0,( )⋅+:= MX2

6.497564− 102

× N m⋅= Moment At X

OX2

OA2

MA2

x⋅

EI+

RA2

x2

⋅

2 EI⋅+

dm

EIUX x am, 1,( )⋅+:=

OX2

1.577796− 105−

×= Slope At X

YX2

YA2

OA2

x⋅+MA

2x2

⋅

2 EI⋅+

RA2

x3

⋅

6 EI⋅+

dm

2 EI⋅UX x am, 2,( )⋅+:=

YX2

1.25936 104−

× m= Deflection At X

Pin-Fix : Concentrated Moment

RA2

3− dm⋅

2 Le3

⋅

Le2

am2

−( )⋅:= RA2

3.308067 102

× N= Reaction At A

MA2

0 N⋅ m⋅:= MA2

0 100

× N m⋅= Moment At A

OA2

dm

4 EI⋅ Le⋅Le am−( )⋅ 3 am⋅ Le−( )⋅:= OA

21.536431− 10

5−×= Angle At A

YA2

0 m⋅:= YA2

0 100

× m= Deflection At A

RB2

3 dm⋅

2 Le3

⋅

Le2

am2

−( )⋅:= RB2

3.308067− 102

× N= Reaction At B

MB2

dm

2 Le2

⋅

3 am2

⋅ Le2

−( )⋅:= MB2

8.471439 102

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 25 Of 58

OB3

0 rad⋅:= OB3

0 100

×= Angle At B

YB3

0 m⋅:= YB3

0 100

× m= Deflection At B

RX3

RA3

:= RX3

1.425691 102

× N= Shear At X

MX3

MA3

RA3

x⋅+:= MX3

9.266992 102

× N m⋅= Moment At X

OX3

OA3

MA3

x⋅

EI+

RA3

x2

⋅

2 EI⋅+ do UX x ao, 0,( )⋅+:=

OX3

1.997168− 104−

×= Slope At X

YX3

YA3

OA3

x⋅+MA

3x2

⋅

2 EI⋅+

RA3

x3

⋅

6 EI⋅+ do UX x ao, 1,( )⋅+:=

YX3

4.907497 104−

× m= Deflection At X

Pin-Fix : Angular Displacement

RA3

3− do⋅ EI⋅ ao⋅

Le3

:= RA3

1.425691 102

× N= Reaction At A

MA3

0 N⋅ m⋅:= MA3

0 100

× N m⋅= Moment At A

OA3

do− 13 ao⋅

2 Le⋅−

⋅:= OA3

1.193878 103−

×= Angle At A

YA3

0 m⋅:= YA3

0 100

× m= Deflection At A

RB3

RA3

−:= RB3

1.425691− 102

× N= Reaction At B

MB3

3− do⋅ EI⋅ ao⋅

Le2

:= MB3

1.571824 103

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 26 Of 58

OB4

0 rad⋅:= OB4

0 100

×= Angle At B

YB4

0 m⋅:= YB4

0 100

× m= Deflection At B

RX4

RA4

:= RX4

9.504607− 102

× N= Shear At X

MX4

MA4

RA4

x⋅+:= MX4

6.177994− 103

× N m⋅= Moment At X

OX4

OA4

MA4

x⋅

EI+

RA4

x2

⋅

2 EI⋅+:=

OX4

1.331445 103−

×= Slope At X

YX4

YA4

OA4

x⋅+MA

4x2

⋅

2 EI⋅+

RA4

x3

⋅

6 EI⋅+ dy UX x ay, 0,( )⋅+:=

YX4

3.271664− 103−

× m= Deflection At X

Pin-Fix : Lateral Displacement

RA4

3 EI⋅ dy⋅

Le3

:= RA4

9.504607− 102

× N= Reaction At A

MA4

0 N⋅ m⋅:= MA4

0 100

× N m⋅= Moment At A

OA4

3− dy⋅

2 Le⋅:= OA

42.040816 10

3−×= Angle At A

YA4

0 m⋅:= YA4

0 100

× m= Deflection At A

RB4

RA4

−:= RB4

9.504607 102

× N= Reaction At B

MB4

3 EI⋅ dy⋅

Le2

:= MB4

1.047883− 104

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 27 Of 58

OB5

0 rad⋅:= OB5

0 100

×= Angle At B

YB5

0 m⋅:= YB5

0 100

× m= Deflection At B

RX5

RA5

:= RX5

4.686411− 103

× N= Shear At X

MX5

MA5

RA5

x⋅+:= MX5

3.046167− 104

× N m⋅= Moment At X

OX5

OA5

MA5

x⋅

EI+

RA5

x2

⋅

2 EI⋅+

α

hdt⋅ UX x at, 1,( )⋅+:=

OX5

4.410877 104−

×= Slope At X

YX5

YA5

OA5

x⋅+MA

5x2

⋅

2 EI⋅+

RA5

x3

⋅

6 EI⋅+

α

2 h⋅dt⋅ UX x at, 2,( )⋅+:=

YX5

2.276333− 103−

× m= Deflection At X

Pin-Fix : Delta Temperature

RA5

3− α EI⋅

2 h⋅ Le3

⋅

dt⋅ Le2

at2

−( )⋅:= RA5

4.686411− 103

× N= Reaction At A

MA5

0 N⋅ m⋅:= MA5

0 100

× N m⋅= Moment At A

OA5

α

4 h⋅ Le⋅dt⋅ Le at−( )⋅ 3 at⋅ Le−( )⋅:= OA

51.212361− 10

4−×= Angle At A

YA5

0 m⋅:= YA5

0 100

× m= Deflection At A

RB5

RA5

−:= RB5

4.686411 103

× N= Reaction At B

MB5

RA5

Le⋅:= MB5

5.166768− 104

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 28 Of 58

OB6

0 rad⋅:= OB6

0 100

×= Angle At B

YB6

0 m⋅:= YB6

0 100

× m= Deflection At B

RX6

RA6

waw UX x aw, 1,( )⋅−:= RX6

1.731333− 103

× N= Shear At X

MX6

MA6

RA6

x⋅+waw

2UX x aw, 2,( )⋅−:= MX

64.207116 10

3× N m⋅= Moment At X

OX6

OA6

MA6

x⋅

EI+

RA6

x2

⋅

2 EI⋅+

waw

6 EI⋅UX x aw, 3,( )⋅−:=

OX6

3.529261 104−

×= Slope At X

YX6

YA6

OA6

x⋅+MA

6x2

⋅

2 EI⋅+

RA6

x3

⋅

6 EI⋅+

waw

24 EI⋅UX x aw, 4,( )⋅−:=

YX6

1.722463− 103−

× m= Deflection At X

Pin-Fix : Weight Load

RA6

waw

8 Le3

⋅

Le( )3

⋅ 3⋅ Le⋅:= RA6

3.02583 103

× N= Reaction At A

MA6

0 N⋅ m⋅:= MA6

0 100

× N m⋅= Moment At A

OA6

waw−

48 EI⋅Le( )

3⋅:= OA

67.218914− 10

4−×= Angle At A

YA6

0 m⋅:= YA6

0 100

× m= Deflection At A

RB6

waw( ) Le( )⋅ RA6

−:= RB6

5.043051 103

× N= Reaction At B

MB6

RA6

Le⋅waw

2Le( )

2⋅−:= MB

61.111993− 10

4× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 29 Of 58

obt

0

6

n

OBn∑

=

:= obt 0 100

×= Total Angle At B

ybt

0

6

n

YBn∑

=

:= YB6

0 100

× m= Total Deflection At B

rxt

0

6

n

RXn∑

=

:= rxt 7.65595− 103

× N= Total Shear At X

mxt

0

6

n

MXn∑

=

:= mxt 2.750445− 104

× N m⋅= Total Moment At X

oxt

0

6

n

OXn∑

=

:= oxt 2.075857 103−

×= Total Slope At X

yxt

0

6

n

YXn∑

=

:= yxt 7.955672− 103−

× m= Total Deflection At X

Pin-Fix : Total Load

rat

0

6

n

RAn∑

=

:= rat 1.836598− 103

× N= Total Reaction At A

mat

0

6

n

MAn∑

=

:= mat 0 100

× N m⋅= Total Moment At A

oat

0

6

n

OAn∑

=

:= oat 2.048731 103−

×= Total Angle At A

yat

0

6

n

YAn∑

=

:= yat 0 100

× m= Total Deflection At A

rbt

0

6

n

RBn∑

=

:= rbt 1.628423 104

× N= Total Reaction At B

mbt

0

6

n

MBn∑

=

:= mbt 8.158356− 104

× N m⋅= Total Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 30 Of 58

Longitudinal Stress BaseSLB 4.980079− 101

× MPa=SLB SX SBB+:=

Longitudinal Stress TopSLA 1.426728 101

× MPa=SLA SX SBA+:=

Maximum Bending Stress TopSBB 3.203404− 107

× Pa=SBBmxa yb⋅

I:=

Maximum Bending Stress TopSBA 3.203404 107

× Pa=SBAmxa− ya⋅

I:=

Max Bending Moment Axial Loadmxa 2.920166− 104

× N m⋅=mxamxt

1NA

fb+

:=

Pin-Fix : Stress Check

SBAmxt− ya⋅

I:= SBA 3.017221 10

7× Pa= Maximum Bending Stress Top

SBBmxt yb⋅

I:= SBB 3.017221− 10

7× Pa= Maximum Bending Stress Top

SLA SX SBA+:= SLA 1.240546 101

× MPa= Longitudinal Stress Top

SLB SX SBB+:= SLB 4.793897− 101

× MPa= Longitudinal Stress Base

yxayxt

1NA

fb+

:= yxa 8.446589− 103−

× m= Max Displacement Axial Load

itdhebeam003201 9/09/2019 9:28 AM Page 31 Of 58

YA0

0 100

× m= Deflection At A

RB0

wp ap2

⋅

Le3

3 Le⋅ 2 ap⋅−( )⋅:= RB0

7.273361− 102

× N= Reaction At B

MB0

wp− ap2

⋅

Le2

Le ap−( )⋅:= MB0

2.376787 103

× N m⋅= Moment At B

OB0

0 rad⋅:= OB0

0 100

×= Angle At B

YB0

0 m⋅:= YB0

0 100

× m= Deflection At B

RX0

RA0

wp UX x ap, 0,( )⋅−:= RX0

7.273361 102

× N= Shear At X

MX0

MA0

RA0

x⋅+ wp UX x ap, 1,( )⋅−:= MX0

9.144086− 102

× N m⋅= Moment At X

OX0

OA0

MA0

x⋅

EI+

RA0

x2

⋅

2 EI⋅+

wp

2 EI⋅UX x ap, 2,( )⋅−:=

OX0

1.168937− 104−

×= Slope At X

YX0

YA0

OA0

x⋅+MA

0x2

⋅

2 EI⋅+

RA0

x3

⋅

6 EI⋅+

wp

6 EI⋅UX x ap, 3,( )⋅−:=

YX0

4.628768 104−

× m= Deflection At X

Beam Bending - Fix-Fix

kk 4:= Buckling Load Factor

lt LBT kk( ):= lt 1.922978 101

× m= Buckle Transition Length

fbe FBE kk( ):= fbe 9.193058 106

× N= Euler Buckling Load

fbj FBJ kk( ):= fbj 5.050347 106

× N= Johnson Buckling Load

fb FB kk( ):= fb 5.050347 106

× N= Buckling Load

Fix-Fix : Point Load

RA0

wp

Le3

Le ap−( )2

⋅ Le 2 ap⋅+( )⋅:= RA0

3.272664− 103

× N= Reaction At A

MA0

wp− ap⋅

Le2

Le ap−( )2

⋅:= MA0

6.357906 103

× N m⋅= Moment At A

OA0

0 rad⋅:= OA0

0 100

×= Angle At A

YA0

0 m⋅:=

itdhebeam003201 9/09/2019 9:28 AM Page 32 Of 58

OB1

0 rad⋅:= OB1

0 100

×= Angle At B

YB1

0 m⋅:= YB1

0 100

× m= Deflection At B

RX1

RA1

wad UX x ad, 1,( )⋅−wld wad−

2 Le ad−( )⋅UX x ad, 2,( )⋅−:=

RX1

1.030922− 103

× N= Shear At X

MX1

MA1

RA1

x⋅+wad

2UX x ad, 2,( )⋅−

wld wad−

6 Le ad−( )⋅UX x ad, 3,( )⋅−:=

MX1

5.176656 103

× N m⋅= Moment At X

OX1

OA1

MA1

x⋅

EI+

RA1

x2

⋅

2 EI⋅+

wad

6 EI⋅UX x ad, 3,( )⋅−

wld wad−

24 EI⋅ Le ad−( )⋅UX x ad, 4,( )⋅−:=

OX1

1.794683 104−

×= Slope At X

YX1

YA1

OA1

x⋅+MA

1x2

⋅

2 EI⋅+

RA1

x3

⋅

6 EI⋅+

wad

24 EI⋅UX x ad, 4,( )⋅−

wld wad−

120 EI⋅ Le ad−( )⋅UX x ad, 5,( )⋅−:=

YX1

1.406209− 103−

× m= Deflection At X

Fix-Fix : Distributed Load

RA1

wad

2 Le3

⋅

Le ad−( )3

⋅ Le ad+( )⋅wld wad−

20 Le3

⋅

Le ad−( )3

⋅ 3 Le⋅ 2 ad⋅+( )⋅+:=

RA1

4.031266 103

× N= Reaction At A

MA1

wad−

12 Le2

⋅

Le ad−( )3

Le 3 ad⋅+( )⋅wld wad−

60 Le2

⋅

Le ad−( )3

⋅ 2 Le⋅ 3 ad⋅+( )⋅−:=

MA1

9.720792− 103

× N m⋅= Moment At A

OA1

0 rad⋅:= OA1

0 100

×= Angle At A

YA1

0 m⋅:= YA1

0 100

× m= Deflection At A

RB1

wad wld+

2Le ad−( )⋅ RA

1−:= RB

16.347484 10

3× N= Reaction At B

MB1

RA1

Le⋅ MA1

+wad

2Le ad−( )

2⋅−

wld wad−

6Le ad−( )

2⋅−:=

MB1

1.143144− 104

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 33 Of 58

OB2

0 rad⋅:= OB2

0 100

×= Angle At B

YB2

0 m⋅:= YB2

0 100

× m= Deflection At B

RX2

RA2

:= RX2

3.522734 102

× N= Shear At X

MX2

MA2

RA2

x⋅+ dm UX x am, 0,( )⋅+:= MX2

6.680031− 102

× N m⋅= Moment At X

OX2

OA2

MA2

x⋅

EI+

RA2

x2

⋅

2 EI⋅+

dm

EIUX x am, 1,( )⋅+:=

OX2

2.062542− 105−

×= Slope At X

YX2

YA2

OA2

x⋅+MA

2x2

⋅

2 EI⋅+

RA2

x3

⋅

6 EI⋅+

dm

2 EI⋅UX x am, 2,( )⋅+:=

YX2

1.427591 104−

× m= Deflection At X

Fix-Fix : Concentrated Moment

RA2

6− dm⋅ am⋅

Le3

Le am−( )⋅:= RA2

3.522734 102

× N= Reaction At A

MA2

dm−

Le2

Le2

4 am⋅ Le⋅− 3 am2

⋅+( )⋅:= MA2

1.577799− 102

× N m⋅= Moment At A

OA2

0 rad⋅:= OA2

0 100

×= Angle At A

YA2

0 m⋅:= YA2

0 100

× m= Deflection At A

RB2

RA2

−:= RB2

3.522734− 102

× N= Reaction At B

MB2

dm

Le2

3 am2

⋅ 2 am⋅ Le⋅−( )⋅:= MB2

9.260339 102

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 34 Of 58

OB3

0 rad⋅:= OB3

0 100

×= Angle At B

YB3

0 m⋅:= YB3

0 100

× m= Deflection At B

RX3

RA3

:= RX3

1.525489− 103

× N= Shear At X

MX3

MA3

RA3

x⋅+:= MX3

2.344549 103

× N m⋅= Moment At X

OX3

OA3

MA3

x⋅

EI+

RA3

x2

⋅

2 EI⋅+ do UX x ao, 0,( )⋅+:=

OX3

1.769531 104−

×= Slope At X

YX3

YA3

OA3

x⋅+MA

3x2

⋅

2 EI⋅+

RA3

x3

⋅

6 EI⋅+ do UX x ao, 1,( )⋅+:=

YX3

8.164837− 104−

× m= Deflection At X

Fix-Fix : Angular Displacement

RA3

6 EI⋅ do⋅

Le3

Le 2 ao⋅−( )⋅:= RA3

1.525489− 103

× N= Reaction At A

MA3

2 do⋅ EI⋅

Le2

3 ao⋅ 2 Le⋅−( )⋅:= MA3

1.226023 104

× N m⋅= Moment At A

OA3

0 rad⋅:= OA3

0 100

×= Angle At A

YA3

0 m⋅:= YA3

0 100

× m= Deflection At A

RB3

RA3

−:= RB3

1.525489 103

× N= Reaction At B

MB3

2 do⋅ EI⋅

Le2

Le 3 ao⋅−( )⋅:= MB3

4.558291− 103

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 35 Of 58

OB4

0 rad⋅:= OB4

0 100

×= Angle At B

YB4

0 m⋅:= YB4

0 100

× m= Deflection At B

RX4

RA4

:= RX4

3.801843− 103

× N= Shear At X

MX4

MA4

RA4

x⋅+:= MX4

3.75432− 103

× N m⋅= Moment At X

OX4

OA4

MA4

x⋅

EI+

RA4

x2

⋅

2 EI⋅+:=

OX4

1.975325 103−

×= Slope At X

YX4

YA4

OA4

x⋅+MA

4x2

⋅

2 EI⋅+

RA4

x3

⋅

6 EI⋅+ dy UX x ay, 0,( )⋅+:=

YX4

5.506251− 103−

× m= Deflection At X

Fix-Fix : Lateral Displacement

RA4

12 EI⋅ dy⋅

Le3

:= RA4

3.801843− 103

× N= Reaction At A

MA4

6− EI⋅ dy⋅

Le2

:= MA4

2.095766 104

× N m⋅= Moment At A

OA4

0 rad⋅:= OA4

0 100

×= Angle At A

YA4

0 m⋅:= YA4

0 100

× m= Deflection At A

RB4

RA4

−:= RB4

3.801843 103

× N= Reaction At B

MB4

MA4

−:= MB4

2.095766− 104

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 36 Of 58

OB5

0 rad⋅:= OB5

0 100

×= Angle At B

YB5

0 m⋅:= YB5

0 100

× m= Deflection At B

RX5

RA5

:= RX5

4.517022− 103

× N= Shear At X

MX5

MA5

RA5

x⋅+:= MX5

3.060565− 104

× N m⋅= Moment At X

OX5

OA5

MA5

x⋅

EI+

RA5

x2

⋅

2 EI⋅+

α

hdt⋅ UX x at, 1,( )⋅+:=

OX5

4.028375 104−

×= Slope At X

YX5

YA5

OA5

x⋅+MA

5x2

⋅

2 EI⋅+

RA5

x3

⋅

6 EI⋅+

α

2 h⋅dt⋅ UX x at, 2,( )⋅+:=

YX5

2.143586− 103−

× m= Deflection At X

Fix-Fix : Delta Temperature

RA5

6− α⋅ at⋅ EI⋅

h Le3

⋅

dt⋅ Le at−( )⋅:= RA5

4.517022− 103

× N= Reaction At A

MA5

α EI⋅

h Le2

⋅

dt⋅ Le at−( )⋅ 3 at⋅ Le−( )⋅:= MA5

1.245004− 103

× N m⋅= Moment At A

OA5

0 rad⋅:= OA5

0 100

×= Angle At A

YA5

0 m⋅:= YA5

0 100

× m= Deflection At A

RB5

RA5

−:= RB5

4.517022 103

× N= Reaction At B

MB5

α− EI⋅

h Le2

⋅

dt⋅ Le at−( )⋅ 3 at⋅ Le+( )⋅:= MB5

5.104517− 104

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 37 Of 58

OB6

0 rad⋅:= OB6

0 100

×= Angle At B

YB6

0 m⋅:= YB6

0 100

× m= Deflection At B

RX6

RA6

waw UX x aw, 1,( )⋅−:= RX6

7.227229− 102

× N= Shear At X

MX6

MA6

RA6

x⋅+waw

2UX x aw, 2,( )⋅−:= MX

63.349798 10

3× N m⋅= Moment At X

OX6

OA6

MA6

x⋅

EI+

RA6

x2

⋅

2 EI⋅+

waw

6 EI⋅UX x aw, 3,( )⋅−:=

OX6

1.251685 104−

×= Slope At X

YX6

YA6

OA6

x⋅+MA

6x2

⋅

2 EI⋅+

RA6

x3

⋅

6 EI⋅+

waw

24 EI⋅UX x aw, 4,( )⋅−:=

YX6

9.320301− 104−

× m= Deflection At X

Fix-Fix : Weight Load

RA6

waw Le⋅

2:= RA

64.034441 10

3× N= Reaction At A

MA6

waw−

12Le

2⋅:= MA

67.413285− 10

3× N m⋅= Moment At A

OA6

0 rad⋅:= OA6

0 100

×= Angle At A

YA6

0 m⋅:= YA6

0 100

× m= Deflection At A

RB6

waw Le⋅ RA6

−:= RB6

4.034441 103

× N= Reaction At B

MB6

RA6

Le⋅ MA6

+waw

2Le

2⋅−:= MB

67.413285− 10

3× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 38 Of 58

obt

0

6

n

OBn∑

=

:= obt 0 100

×= Total Angle At B

ybt

0

6

n

YBn∑

=

:= YB6

0 100

× m= Total Deflection At B

rxt

0

6

n

RXn∑

=

:= rxt 1.051839− 104

× N= Total Shear At X

mxt

0

6

n

MXn∑

=

:= mxt 2.507138− 104

× N m⋅= Total Moment At X

oxt

0

6

n

OXn∑

=

:= oxt 2.722234 103−

×= Total Slope At X

yxt

0

6

n

YXn∑

=

:= yxt 1.019892− 102−

× m= Total Deflection At X

Fix-Fix : Total Load

rat

0

6

n

RAn∑

=

:= rat 4.699038− 103

× N= Total Reaction At A

mat

0

6

n

MAn∑

=

:= mat 2.103893 104

× N m⋅= Total Moment At A

oat

0

6

n

OAn∑

=

:= oat 0 100

×= Total Angle At A

yat

0

6

n

YAn∑

=

:= yat 0 100

× m= Total Deflection At A

rbt

0

6

n

RBn∑

=

:= rbt 1.914667 104

× N= Total Reaction At B

mbt

0

6

n

MBn∑

=

:= mbt 9.210302− 104

× N m⋅= Total Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 39 Of 58

Longitudinal Stress BaseSLB 4.663378− 101

× MPa=SLB SX SBB+:=

Longitudinal Stress TopSLA 1.110027 101

× MPa=SLA SX SBA+:=

Maximum Bending Stress TopSBB 2.886702− 107

× Pa=SBBmxa yb⋅

I:=

Maximum Bending Stress TopSBA 2.886702 107

× Pa=SBAmxa− ya⋅

I:=

Max Bending Moment Axial Loadmxa 2.631466− 104

× N m⋅=mxamxt

1NA

fb+

:=

Fix-Fix : Stress Check

SBAmxt− ya⋅

I:= SBA 2.750315 10

7× Pa= Maximum Bending Stress Top

SBBmxt yb⋅

I:= SBB 2.750315− 10

7× Pa= Maximum Bending Stress Top

SLA SX SBA+:= SLA 9.73639 100

× MPa= Longitudinal Stress Top

SLB SX SBB+:= SLB 4.52699− 101

× MPa= Longitudinal Stress Base

yxayxt

1NA

fb+

:= yxa 1.070469− 102−

× m= Max Displacement Axial Load

itdhebeam003201 9/09/2019 9:28 AM Page 40 Of 58

YA0

0 m⋅:= YA0

0 100

× m= Deflection At A

RB0

wp ap⋅

Le:= RB

01.088435− 10

3× N= Reaction At B

MB0

0 N⋅ m⋅:= MB0

0 100

× N m⋅= Moment At B

OB0

wp ap⋅

6 EI⋅ Le⋅Le

2ap

2−( )⋅:= OB

07.213431− 10

4−×= Angle At B

YB0

0 m⋅:= YB0

0 100

× m= Deflection At B

RX0

RA0

wp UX x ap, 0,( )⋅−:= RX0

1.088435 103

× N= Shear At X

MX0

MA0

RA0

x⋅+ wp UX x ap, 1,( )⋅−:= MX0

4.92517− 103

× N m⋅= Moment At X

OX0

OA0

MA0

x⋅

EI+

RA0

x2

⋅

2 EI⋅+

wp

2 EI⋅UX x ap, 2,( )⋅−:=

OX0

3.276549− 104−

×= Slope At X

YX0

YA0

OA0

x⋅+MA

0x2

⋅

2 EI⋅+

RA0

x3

⋅

6 EI⋅+

wp

6 EI⋅UX x ap, 3,( )⋅−:=

YX0

2.670264 103−

× m= Deflection At X

Beam Bending - Pin Pin

kk 1:= Buckling Load Factor

fb FB kk( ):= fb 2.298264 106

× N= Buckling Load

lt LBT kk( ):= lt 9.61489 100

× m= Buckle Transition Length

fbe FBE kk( ):= fbe 2.298264 106

× N= Euler Buckling Load

fbj FBJ kk( ):= fbj 2.07047 106

× N= Johnson Buckling Load

fb FB kk( ):= fb 2.298264 106

× N= Buckling Load

Pin-Pin : Point Load

RA0

wp

LeLe ap−( )⋅:= RA

02.911565− 10

3× N= Reaction At A

MA0

0 N⋅ m⋅:= MA0

0 100

× N m⋅= Moment At A

OA0

wp− ap⋅

6 EI⋅ Le⋅2 Le⋅ ap−( )⋅ Le ap−( )⋅:= OA

09.797922 10

4−×= Angle At A

itdhebeam003201 9/09/2019 9:28 AM Page 41 Of 58

OB1

wad

24 EI⋅ Le⋅Le

2ad

2−( )2⋅

wld wad−

360 EI⋅ Le⋅Le ad−( )

2⋅ 8 Le

2⋅ 9 ad⋅ Le⋅+ 3 ad

2⋅+( )⋅+:=

OB1

2.11529 103−

×= Angle At B

YB1

0 m⋅:= YB1

0 100

× m= Deflection At B

RX1

RA1

wad UX x ad, 1,( )⋅−wld wad−

2 Le ad−( )⋅UX x ad, 2,( )⋅−:=

RX1

8.757617− 102

× N= Shear At X

MX1

MA1

RA1

x⋅+wad

2UX x ad, 2,( )⋅−

wld wad−

6 Le ad−( )⋅UX x ad, 3,( )⋅−:=

MX1

1.590599 104

× N m⋅= Moment At X

OX1

OA1

MA1

x⋅

EI+

RA1

x2

⋅

2 EI⋅+

wad

6 EI⋅UX x ad, 3,( )⋅−

wld wad−

24 EI⋅ Le ad−( )⋅UX x ad, 4,( )⋅−:=

OX1

5.233604 104−

×= Slope At X

YX1

YA1

OA1

x⋅+MA

1x2

⋅

2 EI⋅+

RA1

x3

⋅

6 EI⋅+

wad

24 EI⋅UX x ad, 4,( )⋅−

wld wad−

120 EI⋅ Le ad−( )⋅UX x ad, 5,( )⋅−:=

YX1

6.927783− 103−

× m= Deflection At X

Pin-Pin : Distributed Load

RA1

wad

2 Le⋅Le ad−( )

2⋅

wld wad−

6 Le⋅Le ad−( )

2⋅+:=

RA1

4.186427 103

× N= Reaction At A

MA1

0 N⋅ m⋅:= MA1

0 100

× N m⋅= Moment At A

OA1

wad−

24 EI⋅ Le⋅Le ad−( )

2⋅ Le

22 ad⋅ Le⋅+ ad

2−( )⋅

wld wad−

360 EI⋅ Le⋅Le ad−( )

2⋅ 7 Le

2⋅ 6 ad⋅ Le⋅+ 3 ad

2⋅−( )⋅−:=

OA1

2.004237− 103−

×= Angle At A

YA1

0 m⋅:= YA1

0 100

× m= Deflection At A

RB1

wad wld+

2Le ad−( )⋅ RA

1−:= RB

16.192323 10

3× N= Reaction At B

MB1

0 N⋅ m⋅:= MB1

0 100

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 42 Of 58

OB2

dm

6 EI⋅ Le⋅Le

23 am

2⋅−( )⋅:= OB

21.09991− 10

4−×= Angle At B

YB2

0 m⋅:= YB2

0 100

× m= Deflection At B

RX2

RA2

:= RX2

2.539683 102

× N= Shear At X

MX2

MA2

RA2

x⋅+ dm UX x am, 0,( )⋅+:= MX2

1.149206− 103

× N m⋅= Moment At X

OX2

OA2

MA2

x⋅

EI+

RA2

x2

⋅

2 EI⋅+

dm

EIUX x am, 1,( )⋅+:=

OX2

1.813041− 105−

×= Slope At X

YX2

YA2

OA2

x⋅+MA

2x2

⋅

2 EI⋅+

RA2

x3

⋅

6 EI⋅+

dm

2 EI⋅UX x am, 2,( )⋅+:=

YX2

3.591529 104−

× m= Deflection At X

Pin-Pin : Concentrated Moment

RA2

dm−

Le:= RA

22.539683 10

2× N= Reaction At A

MA2

0 N⋅ m⋅:= MA2

0 100

× N m⋅= Moment At A

OA2

dm−

6 EI⋅ Le⋅2 Le

2⋅ 6 am⋅ Le⋅− 3 am

2⋅+( )⋅:= OA

23.963119 10

5−×= Angle At A

YA2

0 m⋅:= YA2

0 100

× m= Deflection At A

RB2

dm

Le:= RB

22.539683− 10

2× N= Reaction At B

MB2

0 N⋅ m⋅:= MB2

0 100

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 43 Of 58

OB3

do ao⋅

Le:= OB

32.040816− 10

4−×= Angle At B

YB3

0 m⋅:= YB3

0 100

× m= Deflection At B

RX3

RA3

:= RX3

0 100

× N= Shear At X

MX3

MA3

RA3

x⋅+:= MX3

0 100

× N m⋅= Moment At X

OX3

OA3

MA3

x⋅

EI+

RA3

x2

⋅

2 EI⋅+ do UX x ao, 0,( )⋅+:=

OX3

2.040816− 104−

×= Slope At X

YX3

YA3

OA3

x⋅+MA

3x2

⋅

2 EI⋅+

RA3

x3

⋅

6 EI⋅+ do UX x ao, 1,( )⋅+:=

YX3

9.234694 104−

× m= Deflection At X

Pin-Pin : Angular Displacement

RA3

0 N⋅:= RA3

0 100

× N= Reaction At A

MA3

0 N⋅ m⋅:= MA3

0 100

× N m⋅= Moment At A

OA3

do−

LeLe ao−( )⋅:= OA

31.295918 10

3−×= Angle At A

YA3

0 m⋅:= YA3

0 100

× m= Deflection At A

RB3

RA3

−:= RB3

0 100

× N= Reaction At B

MB3

0 N⋅ m⋅:= MB3

0 100

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 44 Of 58

OB4

OA4

:= OB4

1.360544 103−

×= Angle At B

YB4

0 m⋅:= YB4

0 100

× m= Deflection At B

RX4

RA4

:= RX4

0 100

× N= Shear At X

MX4

MA4

RA4

x⋅+:= MX4

0 100

× N m⋅= Moment At X

OX4

OA4

MA4

x⋅

EI+

RA4

x2

⋅

2 EI⋅+:=

OX4

1.360544 103−

×= Slope At X

YX4

YA4

OA4

x⋅+MA

4x2

⋅

2 EI⋅+

RA4

x3

⋅

6 EI⋅+ dy UX x ay, 0,( )⋅+:=

YX4

6.156463− 103−

× m= Deflection At X

Pin-Pin : Lateral Displacement

RA4

0 N⋅:= RA4

0 100

× N= Reaction At A

MA4

0 N⋅ m⋅:= MA4

0 100

× N m⋅= Moment At A

OA4

dy−

Le:= OA

41.360544 10

3−×= Angle At A

YA4

0 m⋅:= YA4

0 100

× m= Deflection At A

RB4

RA4

−:= RB4

0 100

× N= Reaction At B

MB4

0 N⋅ m⋅:= MB4

0 100

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 45 Of 58

OB5

α

2 h⋅ Le⋅dt⋅ Le

2at

2−( )⋅:= OB

56.708398 10

3−×= Angle At B

YB5

0 m⋅:= YB5

0 100

× m= Deflection At B

RX5

RA5

:= RX5

0 100

× N= Shear At X

MX5

MA5

RA5

x⋅+:= MX5

0 100

× N m⋅= Moment At X

OX5

OA5

MA5

x⋅

EI+

RA5

x2

⋅

2 EI⋅+

α

hdt⋅ UX x at, 1,( )⋅+:=

OX5

5.845648 104−

×= Slope At X

YX5

YA5

OA5

x⋅+MA

5x2

⋅

2 EI⋅+

RA5

x3

⋅

6 EI⋅+

α

2 h⋅dt⋅ UX x at, 2,( )⋅+:=

YX5

1.650033− 102−

× m= Deflection At X

Pin-Pin : Delta Temperature

RA5

0 N⋅:= RA5

0 100

× N= Reaction At A

MA5

0 N⋅ m⋅:= MA5

0 100

× N m⋅= Moment At A

OA5

α−

2 h⋅ Le⋅dt⋅ Le at−( )

2⋅:= OA

53.475435− 10

3−×= Angle At A

YA5

0 m⋅:= YA5

0 100

× m= Deflection At A

RB5

RA5

−:= RB5

0 100

× N= Reaction At B

MB5

RA5

Le⋅:= MB5

0 100

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 46 Of 58

OB6

waw

24 EI⋅Le

3⋅:= OB

61.443783 10

3−×= Angle At B

YB6

0 m⋅:= YB6

0 100

× m= Deflection At B

RX6

RA6

waw UX x aw, 1,( )⋅−:= RX6

7.227229− 102

× N= Shear At X

MX6

MA6

RA6

x⋅+waw

2UX x aw, 2,( )⋅−:= MX

61.076308 10

4× N m⋅= Moment At X

OX6

OA6

MA6

x⋅

EI+

RA6

x2

⋅

2 EI⋅+

waw

6 EI⋅UX x aw, 3,( )⋅−:=

OX6

3.838053 104−

×= Slope At X

YX6

YA6

OA6

x⋅+MA

6x2

⋅

2 EI⋅+

RA6

x3

⋅

6 EI⋅+

waw

24 EI⋅UX x aw, 4,( )⋅−:=

YX6

4.783754− 103−

× m= Deflection At X

Pin-Pin : Weight Load

RA6

waw

2Le⋅:= RA

64.034441 10

3× N= Reaction At A

MA6

0 N⋅ m⋅:= MA6

0 100

× N m⋅= Moment At A

OA6

waw−

24 EI⋅Le

3⋅:= OA

61.443783− 10

3−×= Angle At A

YA6

0 m⋅:= YA6

0 100

× m= Deflection At A

RB6

waw Le⋅ RA6

−:= RB6

4.034441 103

× N= Reaction At B

MB6

0 N⋅ m⋅:= MB6

0 100

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 47 Of 58

obt

0

6

n

OBn∑

=

:= obt 1.05926 102−

×= Total Angle At B

ybt

0

6

n

YBn∑

=

:= YB6

0 100

× m= Total Deflection At B

rxt

0

6

n

RXn∑

=

:= rxt 2.56081− 102

× N= Total Shear At X

mxt

0

6

n

MXn∑

=

:= mxt 2.05947 104

× N m⋅= Total Moment At X

oxt

0

6

n

OXn∑

=

:= oxt 2.302408 103−

×= Total Slope At X

yxt

0

6

n

YXn∑

=

:= yxt 3.041544− 102−

× m= Total Deflection At X

Pin-Pin : Total Load

rat

0

6

n

RAn∑

=

:= rat 5.563271 103

× N= Total Reaction At A

mat

0

6

n

MAn∑

=

:= mat 0 100

× N m⋅= Total Moment At A

oat

0

6

n

OAn∑

=

:= oat 3.247569− 103−

×= Total Angle At A

yat

0

6

n

YAn∑

=

:= yat 0 100

× m= Total Deflection At A

rbt

0

6

n

RBn∑

=

:= rbt 8.88436 103

× N= Total Reaction At B

mbt

0

6

n

MBn∑

=

:= mbt 0 100

× N m⋅= Total Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 48 Of 58

Longitudinal Stress BaseSLB 7.442839 100

× MPa=SLB SX SBB+:=

Longitudinal Stress TopSLA 4.297635− 101

× MPa=SLA SX SBA+:=

Maximum Bending Stress TopSBB 2.520959 107

× Pa=SBBmxa yb⋅

I:=

Maximum Bending Stress TopSBA 2.520959− 107

× Pa=SBAmxa− ya⋅

I:=

Max Bending Moment Axial Loadmxa 2.298062 104

× N m⋅=mxamxt

1NA

fb+

:=

Pin-Pin : Stress Check

SBAmxt− ya⋅

I:= SBA 2.259225− 10

7× Pa= Maximum Bending Stress Top

SBBmxt yb⋅

I:= SBB 2.259225 10

7× Pa= Maximum Bending Stress Top

SLA SX SBA+:= SLA 4.035901− 101

× MPa= Longitudinal Stress Top

SLB SX SBB+:= SLB 4.8255 100

× MPa= Longitudinal Stress Base

yxayxt

1NA

fb+

:= yxa 3.393911− 102−

× m= Max Displacement Axial Load

itdhebeam003201 9/09/2019 9:28 AM Page 49 Of 58

YA0

5.6752 102−

× m= Deflection At A

RB0

wp:= RB0

4− 103

× N= Reaction At B

MB0

0 N⋅ m⋅:= MB0

0 100

× N m⋅= Moment At B

OB0

wp

2 EI⋅Le

2ap

2−( )⋅:= OB

07.952807− 10

3−×= Angle At B

YB0

0 m⋅:= YB0

0 100

× m= Deflection At B

RX0

RA0

wp UX x ap, 0,( )⋅−:= RX0

4 103

× N= Shear At X

MX0

MA0

RA0

x⋅+ wp UX x ap, 1,( )⋅−:= MX0

1.81− 104

× N m⋅= Moment At X

OX0

OA0

MA0

x⋅

EI+

RA0

x2

⋅

2 EI⋅+

wp

2 EI⋅UX x ap, 2,( )⋅−:=

OX0

6.506003− 103−

×= Slope At X

YX0

YA0

OA0

x⋅+MA

0x2

⋅

2 EI⋅+

RA0

x3

⋅

6 EI⋅+

wp

6 EI⋅UX x ap, 3,( )⋅−:=

YX0

3.380419 102−

× m= Deflection At X

Beam Bending - Guide-Pin

kk 0.25:= Buckling Load Factor

lt LBT kk( ):= lt 4.807445 100

× m= Buckle Transition Length

fbe FBE kk( ):= fbe 5.745661 105

× N= Euler Buckling Load

fbj FBJ kk( ):= fbj 0 100

× N= Johnson Buckling Load

fb FB kk( ):= fb 5.745661 105

× N= Buckling Load

Guide-Pin : Point Load

RA0

0 N⋅:= RA0

0 100

× N= Reaction At A

MA0

wp Le ap−( )⋅:= MA0

3.21− 104

× N m⋅= Moment At A

OA0

0 rad⋅:= OA0

0 100

×= Angle At A

YA0

wp−

6 EI⋅Le ap−( )⋅ 2 Le

2⋅ 2 ap⋅ Le⋅+ ap

2−( )⋅:=

itdhebeam003201 9/09/2019 9:28 AM Page 50 Of 58

OB1

wad

6 EI⋅Le ad−( )

2⋅ 2 Le⋅ ad+( )⋅

wld wad−

24 EI⋅Le ad−( )

2⋅ 3 Le⋅ ad+( )⋅+:=

OB1

1.310857 102−

×= Angle At B

YB1

0 m⋅:= YB1

0 100

× m= Deflection At B

RX1

RA1

wad UX x ad, 1,( )⋅−wld wad−

2 Le ad−( )⋅UX x ad, 2,( )⋅−:=

RX1

5.062188− 103

× N= Shear At X

MX1

MA1

RA1

x⋅+wad

2UX x ad, 2,( )⋅−

wld wad−

6 Le ad−( )⋅UX x ad, 3,( )⋅−:=

MX1

3.484957 104

× N m⋅= Moment At X

OX1

OA1

MA1

x⋅

EI+

RA1

x2

⋅

2 EI⋅+

wad

6 EI⋅UX x ad, 3,( )⋅−

wld wad−

24 EI⋅ Le ad−( )⋅UX x ad, 4,( )⋅−:=

OX1

1.00024 102−

×= Slope At X

YX1

YA1

OA1

x⋅+MA

1x2

⋅

2 EI⋅+

RA1

x3

⋅

6 EI⋅+

wad

24 EI⋅UX x ad, 4,( )⋅−

wld wad−

120 EI⋅ Le ad−( )⋅UX x ad, 5,( )⋅−:=

YX1

5.438838− 102−

× m= Deflection At X

Guide-Pin : Distributed Load

RA1

0 N⋅:= RA1

0 100

× N= Reaction At A

MA1

wad

2Le ad−( )

2 wld wad−

6Le ad−( )

2⋅+:= MA

14.615535 10

4× N m⋅= Moment At A

OA1

0 rad⋅:= OA1

0 100

×= Angle At A

YA1

wad−

24 EI⋅Le ad−( )

2⋅ 5 Le

2⋅ 2 ad⋅ Le⋅+ ad

2−( )⋅

wld wad−

120 EI⋅Le ad−( )

2⋅ 9 Le

2⋅ 2 ad⋅ Le⋅+ ad

2−( )⋅−:=

YA1

8.816615− 102−

× m= Deflection At A

RB1

wad wld+

2Le ad−( )⋅:= RB

11.037875 10

4× N= Reaction At B

MB1

0 N⋅ m⋅:= MB1

0 100

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 51 Of 58

OB2

dm− am⋅

EI:= OB

23.95695 10

4−×= Angle At B

YB2

0 m⋅:= YB2

0 100

× m= Deflection At B

RX2

RA2

:= RX2

0 100

× N= Shear At X

MX2

MA2

RA2

x⋅+ dm UX x am, 0,( )⋅+:= MX2

0 100

× N m⋅= Moment At X

OX2

OA2

MA2

x⋅

EI+

RA2

x2

⋅

2 EI⋅+

dm

EIUX x am, 1,( )⋅+:=

OX2

3.95695 104−

×= Slope At X

YX2

YA2

OA2

x⋅+MA

2x2

⋅

2 EI⋅+

RA2

x3

⋅

6 EI⋅+

dm

2 EI⋅UX x am, 2,( )⋅+:=

YX2

1.79052− 103−

× m= Deflection At X

Guide-Pin : Concentrated Moment

RA2

0 N⋅:= RA2

0 100

× N= Reaction At A

MA2

dm−:= MA2

2.8 103

× N m⋅= Moment At A

OA2

0 rad⋅:= OA2

0 100

×= Angle At A

YA2

dm am⋅

2 EI⋅2 Le⋅ am−( )⋅:= YA

23.571148− 10

3−× m= Deflection At A

RB2

0 N⋅:= RB2

0 100

× N= Reaction At B

MB2

0 N⋅ m⋅:= MB2

0 100

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 52 Of 58

OB3

do:= OB3

1.5− 103−

×= Angle At B

YB3

0 m⋅:= YB3

0 100

× m= Deflection At B

RX3

RA3

:= RX3

0 100

× N= Shear At X

MX3

MA3

RA3

x⋅+:= MX3

0 100

× N m⋅= Moment At X

OX3

OA3

MA3

x⋅

EI+

RA3

x2

⋅

2 EI⋅+ do UX x ao, 0,( )⋅+:=

OX3

1.5− 103−

×= Slope At X

YX3

YA3

OA3

x⋅+MA

3x2

⋅

2 EI⋅+

RA3

x3

⋅

6 EI⋅+ do UX x ao, 1,( )⋅+:=

YX3

6.7875 103−

× m= Deflection At X

Guide-Pin : Angular Displacement

RA3

0 N⋅:= RA3

0 100

× N= Reaction At A

MA3

0 N⋅ m⋅:= MA3

0 100

× N m⋅= Moment At A

OA3

0 rad⋅:= OA3

0 100

×= Angle At A

YA3

do− Le ao−( )⋅:= YA3

1.42875 102−

× m= Deflection At A

RB3

0 N⋅:= RB3

0 100

× N= Reaction At B

MB3

0 N⋅ m⋅:= MB3

0 100

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 53 Of 58

OB4

0 rad⋅:= OB4

0 100

×= Angle At B

YB4

0 m⋅:= YB4

0 100

× m= Deflection At B

RX4

RA4

:= RX4

0 100

× N= Shear At X

MX4

MA4

RA4

x⋅+:= MX4

0 100

× N m⋅= Moment At X

OX4

OA4

MA4

x⋅

EI+

RA4

x2

⋅

2 EI⋅+:=

OX4

0 100

×= Slope At X

YX4

YA4

OA4

x⋅+MA

4x2

⋅

2 EI⋅+

RA4

x3

⋅

6 EI⋅+ dy UX x ay, 0,( )⋅+:=

YX4

0 100

× m= Deflection At X

Guide-Pin : Lateral Displacement

RA4

0 N⋅:= RA4

0 100

× N= Reaction At A

MA4

0 N⋅ m⋅:= MA4

0 100

× N m⋅= Moment At A

OA4

0 rad⋅:= OA4

0 100

×= Angle At A

YA4

dy−:= YA4

1.5 102−

× m= Deflection At A

RB4

0 N⋅:= RB4

0 100

× N= Reaction At B

MB4

0 N⋅ m⋅:= MB4

0 100

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 54 Of 58

OB5

α

hdt⋅ Le at−( )⋅:= OB

51.018383 10

2−×= Angle At B

YB5

0 m⋅:= YB5

0 100

× m= Deflection At B

RX5

RA5

:= RX5

0 100

× N= Shear At X

MX5

MA5

RA5

x⋅+:= MX5

0 100

× N m⋅= Moment At X

OX5

OA5

MA5

x⋅

EI+

RA5

x2

⋅

2 EI⋅+

α

hdt⋅ UX x at, 1,( )⋅+:=

OX5

4.06 103−

×= Slope At X

YX5

YA5

OA5

x⋅+MA

5x2

⋅

2 EI⋅+

RA5

x3

⋅

6 EI⋅+

α

2 h⋅dt⋅ UX x at, 2,( )⋅+:=

YX5

3.222667− 102−

× m= Deflection At X

Guide-Pin : Delta Temperature

RA5

0 N⋅:= RA5

0 100

× N= Reaction At A

MA5

0 N⋅ m⋅:= MA5

0 100

× N m⋅= Moment At A

OA5

0 rad⋅:= OA5

0 100

×= Angle At A

YA5

α−

2 h⋅dt⋅ Le at−( )

2⋅:= YA

53.831667− 10

2−× m= Deflection At A

RB5

0 N⋅:= RB5

0 100

× N= Reaction At B

MB5

MA5

:= MB5

0 100

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 55 Of 58

OB6

waw

6 EI⋅2⋅ Le

3⋅:= OB

61.155026 10

2−×= Angle At B

YB6

0 m⋅:= YB6

0 100

× m= Deflection At B

RX6

RA6

waw UX x aw, 1,( )⋅−:= RX6

4.757164− 103

× N= Shear At X

MX6

MA6

RA6

x⋅+waw

2UX x aw, 2,( )⋅−:= MX

62.901893 10

4× N m⋅= Moment At X

OX6

OA6

MA6

x⋅

EI+

RA6

x2

⋅

2 EI⋅+

waw

6 EI⋅UX x aw, 3,( )⋅−:=

OX6

9.031023 103−

×= Slope At X

YX6

YA6

OA6

x⋅+MA

6x2

⋅

2 EI⋅+

RA6

x3

⋅

6 EI⋅+

waw

24 EI⋅UX x aw, 4,( )⋅−:=

YX6

4.831452− 102−

× m= Deflection At X

Guide-Pin : Weight Load

RA6

0 N⋅:= RA6

0 100

× N= Reaction At A

MA6

waw

2Le

2⋅:= MA

64.447971 10

4× N m⋅= Moment At A

OA6

0 rad⋅:= OA6

0 100

×= Angle At A

YA6

waw−

24 EI⋅5⋅ Le

4⋅:= YA

67.958852− 10

2−× m= Deflection At A

RB6

waw Le⋅:= RB6

8.068881 103

× N= Reaction At B

MB6

0 N⋅ m⋅:= MB6

0 100

× N m⋅= Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 56 Of 58

obt

0

6

n

OBn∑

=

:= obt 2.578555 102−

×= Total Angle At B

ybt

0

6

n

YBn∑

=

:= YB6

0 100

× m= Total Deflection At B

rxt

0

6

n

RXn∑

=

:= rxt 5.819352− 103

× N= Total Shear At X

mxt

0

6

n

MXn∑

=

:= mxt 4.57685 104

× N m⋅= Total Moment At X

oxt

0

6

n

OXn∑

=

:= oxt 1.548312 102−

×= Total Slope At X

yxt

0

6

n

YXn∑

=

:= yxt 9.612841− 102−

× m= Total Deflection At X

Guide-Pin : Total Load

rat

0

6

n

RAn∑

=

:= rat 0 100

× N= Total Reaction At A

mat

0

6

n

MAn∑

=

:= mat 6.133506 104

× N m⋅= Total Moment At A

oat

0

6

n

OAn∑

=

:= oat 0 100

×= Total Angle At A

yat

0

6

n

YAn∑

=

:= yat 1.23603− 101−

× m= Total Deflection At A

rbt

0

6

n

RBn∑

=

:= rbt 1.444763 104

× N= Total Reaction At B

mbt

0

6

n

MBn∑

=

:= mbt 0 100

× N m⋅= Total Moment At B

itdhebeam003201 9/09/2019 9:28 AM Page 57 Of 58

Longitudinal Stress BaseSLB 6.810141 101

× MPa=SLB SX SBB+:=

Longitudinal Stress TopSLA 1.036349− 102

× MPa=SLA SX SBA+:=

Maximum Bending Stress TopSBB 8.586816 107

× Pa=SBBmxa yb⋅

I:=

Maximum Bending Stress TopSBA 8.586816− 107

× Pa=SBAmxa− ya⋅

I:=

Max Bending Moment Axial Loadmxa 7.827589 104

× N m⋅=mxamxt

1NA

fb+

:=

Max Displacement Axial Loadyxa 1.644043− 101−

× m=yxayxt

1NA

fb+

:=

Longitudinal Stress BaseSLB 3.2441 101

× MPa=SLB SX SBB+:=

Longitudinal Stress TopSLA 6.797451− 101

× MPa=SLA SX SBA+:=

Maximum Bending Stress TopSBB 5.020776 107

× Pa=SBBmxt yb⋅

I:=

Maximum Bending Stress TopSBA 5.020776− 107

× Pa=SBAmxt− ya⋅

I:=

Guide-Pin : Stress Check

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