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AnalysisofVerticalTrussColumn
Bergel, Guy L ARE 320L
1
ContentsIntroduction .................................................................................................................................................. 2
Analysis Checks ............................................................................................................................................. 3
Wind Load Calculations ................................................................................................................................. 4
Layout............................................................................................................................................................ 6
Column and Bracing Sizing ............................................................................................................................ 7
Window System Analysis ............................................................................................................................ 11
Nonlinear Analysis ...................................................................................................................................... 13
Connection to the Main Structure .............................................................................................................. 14
Roof Analysis ............................................................................................................................................... 15
Other Necessary Checks ............................................................................................................................. 17
Conclusion ................................................................................................................................................... 17
2
Introduction
The vertical truss columns are primarily used to resist wind loads. These columns are located on the
southern and western wall of the lobby as shown below. For more detail on the layout, see structural
floor plan documents.
I am assuming that the west side of
the wind column wall will control the
design since this wall has a larger
surface area. Therefore, only these
wind columns will be analyzed using
maximum expected easterly winds.
3
AnalysisChecks
Using Autodesk Robot/Revit, I will check the following items and change them accordingly:
1. Column and bracing sizing
I will determine whether the initial HSS 4X4X1/2 columns and the initial HSS 2X2X1/8 braces will
exceed the maximum permitted deflection of length/360.
2. Maximum deflections of glass/mullions system
I will determine if the glass/mullion system on the roof of the lobby will deflect excessively.
Mullions are HSS 2X4X1/4. Since glass is sensitive to deflections, I will use length/480 as the
governing deflection requirement.
3. Nonlinear analysis
Nonlinear analysis will determine if any of the materials will exhibit nonlinear relations between
stress and strain. Second order analysis will determine the effect of axial load on a deformed
beam/column. When the beam/column deflects, the magnitude of axial load determines the
additional moment and shear that the beam or column will feel, thus decreasing its stiffness.
I will determine if second‐order effects significantly increase column deflections or create any
instabilities in the system. I will assume an unstable system if tangent stiffness iterations do not
converge.
4. Connection to the main building structure
I will determine if additional lateral reinforcement is necessary in the main structure (where
classroom and museum are located) in order to effectively transfer forces from the lobby to the
main structure.
5. Roof Loads
I will determine the deflection of the roof system due to dead loads only.
Assumptions:
1. I will neglect self‐weight of steel and concrete since these forces are relatively small.
2. Horizontal mullions connecting to wind column do not bend, and simply transfer forces to
columns.
4
WindLoadCalculations
The following factors were obtained from tables that may be found in ASCE 7‐10:
Building Category: II
Class B surface and exposure
Wind speed(V): 115 mph
Wind Directional Factor (D): 0.85
Kzt=1
Gust Factor(G): 0.85
Enclosure Category: Enclosed
Internal Pressure(Cpi: ‐0.18
α=7
Zg=1200
Cp,z=0.8
Z: height above ground
5
Sample Calculations:
1. Kz(less than 15’)=2.01*(15/Zg)2/α=2.01*(15/1200)(2/7)=0.57
2. Kz( 16’ or greater)=2.01*(Z/Zg)2/α=2.01*(16/1200)(2/7)=0.58 (Z=16’ for this example)
3. qz=0.00256KzKztDV2, qz(1’)=0.00256*0.57*1*0.85*115
2=16.54
4. qh=qz at roof elevation=23.7
5. p= qzGCp,z‐qhGCpi, p(1’)=16.53*0.85*0.8‐23.7*0.85*‐0.18=14.87 psf
6. point load=p*Tributary Area=14.87 psf * (12’*2’)*1k/1000lb=0.36 k/ft
Height (ft) Kz qz Cp,z p(psf) Load (k/f) South Edge Column (k/ft)
1 0.57472 16.53905 0.8 14.87588 0.36 0.28
3 0.57472 16.53905 0.8 14.87588 0.36 0.28
5 0.57472 16.53905 0.8 14.87588 0.36 0.28
7 0.57472 16.53905 0.8 14.87588 0.36 0.28
9 0.57472 16.53905 0.8 14.87588 0.36 0.28
11 0.57472 16.53905 0.8 14.87588 0.36 0.28
13 0.57472 16.53905 0.8 14.87588 0.36 0.28
15 0.57472 16.53905 0.8 14.87588 0.36 0.28
17 0.595644 17.14121 0.8 15.28535 0.37 0.29
19 0.614877 17.69468 0.8 15.66171 0.38 0.30
21 0.632713 18.20797 0.8 16.01075 0.38 0.30
23 0.649374 18.68743 0.8 16.33678 0.39 0.31
25 0.66503 19.13798 0.8 16.64315 0.40 0.32
27 0.679816 19.56346 0.8 16.93248 0.41 0.32
29 0.693838 19.96699 0.8 17.20688 0.41 0.33
31 0.707185 20.3511 0.8 17.46808 0.42 0.33
33 0.719931 20.7179 0.8 17.7175 0.43 0.34
35 0.732137 21.06914 0.8 17.95634 0.43 0.34
37 0.743854 21.40633 0.8 18.18563 0.44 0.35
39 0.755127 21.73074 0.8 18.40623 0.44 0.35
41 0.765994 22.04347 0.8 18.61889 0.45 0.35
43 0.776489 22.34549 0.8 18.82426 0.45 0.36
45 0.786641 22.63763 0.8 19.02292 0.46 0.36
47 0.796475 22.92064 0.8 19.21537 0.46 0.37
49 0.806015 23.19518 0.8 19.40205 0.47 0.37
51 0.815281 23.46182 0.8 19.58337 0.47 0.37
53 0.82429 23.7211 0.8 19.75968 0.24 0.19
6
LayoutThe structural layout of the lobby was laid out in Autodesk Revit. The global coordinate system is shown
on the bottom left:
The Revit model was transferred to Autodesk Robot:
Wind Loads: Dead Loads:
7
ColumnandBracingSizingColumn numbering:
478
479
480
190
191
192
127
128
129
64
65
66
1
2
3
y
x
8
Original Deflections of Columns:
U_x(in) U_y(in) U_z(in)
1 0.0117 2.0263 ‐0.3345
2 ‐0.0062 2.0937 0.0496
3 ‐0.0056 ‐0.0474 1.974
64 0.0114 2.0256 ‐0.3307
65 ‐0.006 2.0907 0.0493
66 ‐0.0054 ‐0.0471 1.9755
127 0.0124 2.287 ‐0.4178
128 ‐0.0062 2.3124 0.0469
129 ‐0.0062 ‐0.0443 2.1955
190 0.0124 2.4045 ‐0.4422
191 ‐0.0061 2.4902 ‐0.0667
192 ‐0.0063 0.0902 2.2881
478 0.0104 1.8895 ‐0.3552
479 ‐0.0051 1.9656 ‐0.0474
480 ‐0.0052 0.0643 1.7954
Results:
Current bar sizes: HSS 4X4X1/2 columns, HSS 2X2X1/8 diagonals
Maximum Deflection=2.49”
Maximum Permitted Deflection=56’*12”/1’*1/360=1.76”
The column and bracing sizes must be increased. I will try HSS 6X6X3/8 columns with HSS
2X2X1/4 diagonal members.
9
New Deflections of Columns:
U_x(in) U_y(in) U_z(in)
1 0.0108 0.9554 ‐0.1759
2 ‐0.0057 1.0003 0.0425
3 ‐0.0051 ‐0.0387 0.9061
64 0.0105 0.9595 ‐0.1716
65 ‐0.0057 1.0041 0.0433
66 ‐0.0049 ‐0.0397 0.9126
127 0.0107 1.0966 ‐0.219
128 ‐0.0055 1.1211 0.0407
129 ‐0.0051 ‐0.0359 1.0258
190 0.0106 1.1713 ‐0.233
191 ‐0.0054 1.2204 ‐0.0378
192 ‐0.0052 0.0548 1.0859
478 0.0089 0.9153 ‐0.1884
479 ‐0.0046 0.9582 0.0293
480 ‐0.0043 0.0399 0.8458
Results:
Maximum Deflection=1.22”
Maximum Permitted Deflection=56’*12”/1’*1/360=1.76”
The column deflection is within the acceptable limits.
10
Deflection Diagrams Due to Wind Loads:
11
WindowSystemAnalysisBeam Numbering:
I will analyze one section of mullions.
U_x(in) U_y(in) U_z(in) Total U_Z(in)
546_DL 0 0.0055 ‐0.0013 ‐0.0103
546_wind 0 ‐0.0119 ‐0.009
561_DL 0 ‐0.0003 ‐0.0101 ‐0.006
561_wind 0 ‐0.0025 0.0041
572_DL 0 0.0001 ‐0.0148 ‐0.0158
572_wind 0 ‐0.0014 ‐0.001
583_DL 0 0.0003 ‐0.0054 ‐0.0079
583_wind ‐0.0001 ‐0.0017 ‐0.0025
594_DL 0 ‐0.0002 ‐0.0153 ‐0.0139
594_wind 0 ‐0.0013 0.0014
605_DL 0 ‐0.0002 ‐0.0103 ‐0.0023
605_wind 0 ‐0.0017 0.008
616_DL 0 ‐0.0002 ‐0.0007 0.007
616_wind 0 0.0028 0.0077
546 594 561 583 572 605 616
y
x
12
Note: all deflections are in local coordinate systems
Current bar sizes: HSS 8X4X1/2, Mullions are HSS 6X2X1/2
Deflection Permitted: length/480=5’*12”/1’/480=0.125” All mullions meet this criteria
Angular distortions must also be calculated in order to determine if mullions distort too
much for glass panels
I will assume that there is a barrier between mullion and glass in order to prevent any
loads from transferring to glass
I will also assume that the mullions are separated from the adjacent roofing system.
This will prevent shear forces from transferring from the mullions to the roofing
membrane.
R_x(rad) R_y(rad) R_z(rad)
546_a 0.001 0.004 ‐0.001
546_b 0 0.004 ‐0.001
561_a 0 0.004 ‐0.001
561_b 0 0.002 ‐0.001
572_a 0 0.002 ‐0.001
572_b 0 0 ‐0.001
583_a 0 0 ‐0.001
583_b 0.001 ‐0.001 ‐0.001
594_a 0.001 ‐0.001 ‐0.001
594_b 0.001 ‐0.003 ‐0.001
605_a 0.001 ‐0.003 ‐0.001
605_b 0.001 ‐0.003 ‐0.001
616_a 0.001 ‐0.003 ‐0.001
616_b 0.001 ‐0.002 ‐0.001
13
NonlinearAnalysis
New Deflections Difference between second and First Order results
Beam U_x(in) U_y(in) U_z(in) Beam U_x(in) U_y(in) U_z(in)
1 0.0088 0.9578 ‐0.1773 1 ‐0.002 0.0024 ‐0.0014
2 ‐0.0087 1.0026 0.0417 2 ‐0.003 0.0023 ‐0.0008
3 ‐0.0076 ‐0.0391 0.908 3 ‐0.0025 ‐0.0004 0.0019
64 0.0087 0.9624 ‐0.1729 64 ‐0.0018 0.0029 ‐0.0013
65 ‐0.0086 1.0069 0.0425 65 ‐0.0029 0.0028 ‐0.0008
66 ‐0.0073 ‐0.0402 0.9152 66 ‐0.0024 ‐0.0005 0.0026
127 0.0089 1.0989 ‐0.2204 127 ‐0.0018 0.0023 ‐0.0014
128 ‐0.0085 1.1236 0.0403 128 ‐0.003 0.0025 ‐0.0004
129 ‐0.0075 ‐0.0366 1.0279 129 ‐0.0024 ‐0.0007 0.0021
190 0.0088 1.1735 ‐0.2348 190 ‐0.0018 0.0022 ‐0.0018
191 ‐0.0081 1.2224 ‐0.0385 191 ‐0.0027 0.002 ‐0.0007
192 ‐0.0075 0.0558 1.0879 192 ‐0.0023 0.001 0.002
478 0.0076 0.9165 ‐0.1894 478 ‐0.0013 0.0012 ‐0.001
479 ‐0.0064 0.9591 0.0291 479 ‐0.0018 0.0009 ‐0.0002
480 ‐0.0059 0.0407 0.8467 480 ‐0.0016 0.0008 0.0009
The results indicate a slight variation between first and second order deflections of columns
The slight difference means these columns did not lose a significant amount of stiffness due to
wind loading and behave linearly.
14
ConnectiontotheMainStructureColumn Numbering:
Column U_x(in) U_y(in) U_z(in)
473 0.0013 ‐0.4001 0.0589
474 0.0012 ‐0.3852 0.0589
475 0.0011 ‐0.3597 0.0661
476 0.001 ‐0.3323 0.0653
477 0.0008 ‐0.2948 0.0689
Concrete column size: 12X36, 3.5ksi concrete
By connecting the vertical truss columns to concrete columns (as opposed to girders), higher
lateral resistance is provided on the upper portions of the truss columns
Concrete Columns experience highest deflection in the y‐direction.
Maximum deflections are lower than maximum permitted deflection of 1.77” (same as truss
column)
Reinforcement layout must be determined in accordance to ACI
473
474
475
476
477
y
x
15
RoofAnalysisRoof Normal Stresses:
16
Roof Shear Stress:
High normal stresses are felt in bottom center column. This is most likely due to the fact that it
is connected to a beam that connects to the mullions thus transferring stresses from one
portion of the lobby to the other. In order to prevent high stress concentrations, it is
recommended to separate this column from the roof membrane.
Highest shear stresses occur at all column‐roof connections. This suggests that shear stress on
the roofing membrane does not occur due to transfer of forces from the mullions.
17
OtherNecessaryChecks1. Check stresses on each bar and compare to yield strength.
S max (ksi)
S min (ksi)
S max (ksi)
S max(Mz) (ksi)
S min(My) (ksi)
S min(Mz) (ksi)
Fx/Ax (ksi)
MAX 34.48 1.85 25.28 16.12 0 0 3.46
Bar 210 132 208 128 384 1 195
Node 209 129 211 129 262 1 192
Case 3 3 3 3 3 2 3
Min
Min ‐0.54 ‐34.74 0 0 ‐25.28 ‐16.12 ‐3.3
Bar 410 205 389 384 208 128 127
Node 274 208 130 262 211 129 127
Case 3 3 1 1 3 3 3
Data suggest that some bars are approaching yield strength. Their size or thickness may need to be
increased in order to maintain a linear elastic behavior
2. Design strength must be checked‐ Safety factors must be applied to bar forces and compared to
required strength to determine if the design strength of each member is satisfied.
3. Glass strength must be checked‐It needs to be determined whether the glass curtain wall will
withstand angular distortions of mullions without shattering.
Conclusion HSS 4X4X1/2 columns, HSS 2X2X1/8 diagonals were changed to HSS 6X6X3/8 columns with HSS
2X2X1/4 diagonal members in order to maintain minimum deflection requirements
Mullion system fit deflection criteria and do not need to be resized. However to prevent any
excessive stresses, the mullions must be separated from both the wind columns, and the glass.
This will prevent forces from transferring to the glass thus causing it to break.
It is safe to assume that truss columns behave linearly and elastically.
12X36 rectangular concrete columns are required on the opposite side of the lobby to resist
wind forces. This will prevent excessive deflection of the top of the truss columns
Beams and columns in glass roof portion should be disconnected from beams and columns in
concrete roof portion. This will prevent excessive normal stresses transferring between the two
sections and causing high stress concentrations.
Recommended