pipeng.compipeng.com/check/itdhebeam003201.pdf · Point Load wad 1100 N m:= ⋅ Distributed Load...
Transcript of pipeng.compipeng.com/check/itdhebeam003201.pdf · Point Load wad 1100 N m:= ⋅ Distributed Load...
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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