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FROM: TO: All structural design teams of ME104Q FROM: Prof. R. Perucchio DATE: 18 Nov 2020 – Revised 30 Nov 2020 – Due Thursday, 3 December at 11 p.m. EST RE: STRUCTURAL Design of the new timber bridge

across the danube, 104 a.D. Introduction We have been requested by our associated firm ARCHI-MELIORA of Roma, Italia, 104 A.D., to provide the structural design of the modular pine timber truss for the new imperial bridge to be erected across the river Danube near the city of Pontes. The order issued by the Emperor Trajan calls for a twenty-pier bridge 1040 m long and 12 m wide, to replace the original bridge built by Apollodorus of Damascus, destroyed during the recent war with Dacia, see Fig. 1 below. Note that this was the longest bridge ever built in the Roman Empire. According to the most modern building techniques in the Roman Empire, 104 A.D., the truss will be made with seasoned pine timber from the nearby mountains. The construction is scheduled to begin in 105 A.D. The Emperor will personally review the proposed designs.

Each bridge module spans 52 m center to center of the piers and consists of six identical

planar trusses equally spaced at 2.4 m through the width of the bridge and laterally braced to each other. Each design team is required to provide the structural design for the typical planar truss according to the specifications below.

The picture can't be displayed.

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Planar truss specifications Material: The truss is made with straight bars of pine timber. The timber specifications are: • Maximum length available is 8 m. Any length equal or less than 8 m can be used. • Cross sections available are (all dimensions are cm): 5x5, 10x10, 20x20, 30x30, 40X40:

Different cross sections can be used in the same truss design. • The mechanical properties of pine are :

elastic modulus 𝐸 = 10𝐺𝑃𝑎, strength in tension 𝜎!" = 65𝑀𝑃𝑎, in compression 𝜎#$ = 27𝑀𝑃𝑎, density of the wood is 𝜌 = 350 𝑘𝑔/𝑚%.

Geometrical and kinematic (support) constraints: With reference to Fig. 2 (which shows the vertical space occupied by half module, delimited to the left by the symmetry line at the center of a pier and to the right by the symmetry line at center of the span between two piers): • The entire truss must fit into the yellow area and must support the bridge deck at joints a-e.

Thus joints are required at locations a, b , c, d, and e. • All joints resting on the pier (that is, located on line ABC) must be pin connected to the pier so

that joints do not move (but the attached bars can rotate.) • Joints along the pier symmetry line AH and GF are connected with the other (symmetric)

portions of the truss. Thus these joints must be free to move vertically (along the line of symmetry), but must be constrained in the horizontal direction (perpendicular to the line of symmetry). This condition is represented by a roller. Note that the truss must have at least one joint along line AH and also along center-span line GF so that it can be properly connected with

Figure 1. A section of the old bridge over the river Danube as built by Apollodorus. Notice the modular nature of the timber truss structure.

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the symmetrical trusses at the left and at the right. Figure 3 shows the bridge structural symmetry simulated by the rollers.

Loads: The live load acting on the truss consists of vertical forces (acting downward) applied simultaneously to joints a-e according to the following table.

Table 1 – Live loads joint Force [kN]

a 16 b 32 c 32 d 32 e 16

The dead load due to the mass of the truss bars must be taken into account. For timber trusses, the imperial building code requires that a factor of safety be used to design the truss bars both in tension (strength failure) and in compression (crushing or buckling failure). Structural stiffness: the imperial building code requires that the differential vertical deformation between center pier and midspan of the bridge measured at the pavement level be less than 0.05% of the span. Design objectives

We need to design a truss that (a) satisfies all the above requirements and (b) reduces the weight as much as possible. WORK EXPECTED:

Students will continue to work with the teams already set up for the previous projects. Members of each design team are expected to work in collaboration to perform the design tasks specified below. Each team will prepare a single written report (one report per team!) Truss design

Each team will perform two design iterations on the truss problem. In the first iteration, the team selects a truss configuration that satisfies all the requirements defined above. This implies the following steps (the modeling is done in Dr.Frame): 1) Select the truss geometry and enter the bar elements. Apply boundary conditions and material

properties. Follow the geometrical, support, symmetry, and length requirements (Fig. 2.) As indicated above, only half of the modular span needs to be modeled since the identical trusses to

f.s. = 5.0

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the left and to the right are automatically simulated by the roller boundary conditions (Fig. 3) 2) Select the cross-sectional area of each bar. 3) Compute the joint loads due to the mass of each bar (dead load), apply loads to joints, and print

out forces on each truss element and reactions at supports. 4) Compute the joint loads due to the live load, apply loads to the appropriate joints, and print out

forces on each truss element and reactions at supports. 5) For the elements in tension: compute due to the sum of the loads in (3) and (4) and verify

that . Tabulate the results as shown in the Appendix.

6) For the elements in compression: compute and due to the sum of the loads in (3)

and (4) and verify that and . Tabulate the results as shown

in the Appendix. 7) To verify the truss stiffness, compute , where and denote the total downward

displacement at points e and a, respectively (see Fig. 2). Tabulate the results as shown in the Appendix. If D< 26mm the stiffness of the truss is verified.

8) If the truss passes verification, record the weight of the truss (this is equal to the sum of the vertical reactions computed in (3).) If the truss does not pass verification, make the appropriate changes in step (1) and (2) and repeat the process. The current design iteration is not completed until results are verified.

In the second design iteration the geometry and, where feasible, the element cross-sectional areas of the first truss are modified in order to reduce as much as possible the weight of the truss. This will represent your optimized truss design. This iteration follows exactly the same step (1) – (8) used for the first iteration. Project report The typewritten report must include 1) Title page 2) Abstract (.5 page): give a brief summary of the report including the weight of the truss for your

optimized design. 3) Introduction (1 page): define the design problem. List all design constraints (dimensions,

materials, etc.). 4) Truss design (4-page max.): a section on your optimized truss design (i.e., your second

iteration). Begin with a paragraph describing the approach followed to refine your design from the first to the second iteration (the first is described in the appendix to the report, see (6) below). Add three separate figures taken from Dr.Frame showing the truss geometry (with element dimensions and joint labels), the element forces and the support reactions for the live loads, and the element forces and the support reactions for the dead loads. Include a table showing the results for each element (follow the structure of Tables A1, A2, and A3 in the Appendix) and

σapplied

σapplied ≤ σ allow. =σ st

f.s.Papplied σapplied

Papplied ≤Pcriticalf.s.

σapplied ≤ σ allow. =σ cr

f.s.

€

D = ve − va

€

ve

€

va

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the total weight of this truss. IMPORTANT: submit via email to Prof. Perucchio rlp@me.rochester.edu the following files (make sure that names of all team members are indicated in the accompanying message): a) Two Dr.Frame (drf) files for the optimized truss (one with the live loads and the other with

the dead loads) , and b) One Excel file with worksheets for load table, verification table – forces, and verification

table – displacements for the optimized truss (follow the structure of Tables A1, A2, and A3 in the Appendix.)

5) References: list the references consulted in the course of the project. 6) Appendix (4 pages max.): give the results for the first truss design. Provide the same material

as in (4) including files submission, but do not repeat the description of the design approach. As always, make sure that you use correct English grammar and syntax. Your reports must read easily, convincingly, and well!

The reports must be submitted on Gradescope by Thursday, December 3, by 11 pm EST. No late submissions accepted. The Dr.Frame (drf) files and Excel files (for iterations 1 and 2)must also be emailed to Prof. Perucchio rlp@me.rochester.edu by the same deadline. PowerPoint presentation

A video zoom PowerPoint presentation session will be held on Saturday, December 5. • Eachgroupwillpresenttheprojectseparatelyandthepresentationisnotopentothe

public.• Instructionsonhowtopreparethepresentationwillbepostedshortlytogetherwith

theschedulecontainingtheorderinwhichteamswillpresent.• Prof. Perucchio and the TAs will evaluate the presentations and question the

presentersatthistime.• Everyteammemberisexpectedtoattend!Theabsenceofdesignteammemberswill

countheavilyagainsttheindividualpresentationgrade.

line of symmetrybridge deck(street level)

stone/brick pier

A B

C

D

E F

G

line of symmetry

Figure 2. Design constraints for bridge trussGeometrical constraints : truss must fit into yellow area and must support bridge deck at joints a-e.Support constraints: all joints resting on the pier (line ABC) must be pin-connected (joint does not move).Symmetry constraints: joints along symmetry lines AH and GF must be free to move on the line of symmetry (and constrained in the direction perpendicular to the symmetry line). Truss must have at least one joint along each of these two lines.Length constraint: bars must be 8 m long or less.6

10 m 5 m 7.5 m

26 m3

m 5 m

12 m

4.5

m

6.5 m 6.5 m 6.5 m 6.5 m

H

a b c d e

3 m

4.5

m

(a)

(b)Figure 3. Structural symmetry: (a) truss design: notice roller boundary conditions on symmetry planes. (b) Modular structure simulated by the boundary conditions applied in (a).7

line of symmetrybridge deck(street level)

line of symmetry

Figure 3. Structural symmetry: (a) truss design: notice roller boundary conditions on symmetry lines. (b) Modular structure simulated by the boundary conditions applied in (a).

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APPENDIX Example of truss design

Note that the attached example refers to a bridge design subjected to a distributed live service load different from the one assigned. Use this example only as a guide for creating the various tables needed for setting up the model, tabulating element forces, and performing structural verification.

For the first design iteration, we select a simple truss geometry, see Fig. A1. Notice that, on

purpose, this configuration violates the geometrical constraints (some joints are outside of the yellow area, some elements exceed the required length, and the positioning of the joints on the bridge deck is incorrect) and thus cannot be used for the first design iteration. Only half of the truss is modeled. To represent the symmetry condition, joints D, I, and J are attached to rollers, which allow motion in the vertical direction only. Joints A and C are pin-connected to the surface of the pier.

In this example we use pine timber with mechanical properties , ,

, and density . The element properties are listed in Table A1. For this iteration, all bars have the same 40cm x 40cm cross section. The applied dead loads and live loads are also listed in Table A1. The element forces and support reactions due to dead loads are shown in Fig. A2. Those due to live loads are in Fig. A3. The total applied stress, the allowable stress, and the critical load (for elements in compression) are given in Table A2. All applied stresses are smaller (in magnitude) then the allowable stresses (in tension or compression). Also, for the elements in compression, the element forces are below The verification of the displacements is given in Table A3. Thus, the truss is verified. The verification of the displacements is given in Table A3. Thus, the truss is verified.

Design considerations

The results suggest several approaches for modifying the truss for weight reduction: • reduce the cross section of bars that are under-stressed in tension; • try to remove bars that are almost unloaded; • reduce the cross section of bars in compression that carry forces well below the critical loads. Most importantly, different truss configurations should also be tested and compared to the present one, in order to provide different avenues for weight minimization

E = 10GPa σst = 100MPaσcr = 27MPa ρ = 0.35 ×103kg/m3

Pcritical / f.s.

R. PERUCCHIO 11/25/02

7.5 m

12.5 m12.5 m

7.5 m

10.0 m

7.4 m

8.6 m

7.0 m

2.5 m 5.0 m

5.6 m

5.0 m

2.5 m

4.0 m

4.0 m

4.7 m

2.5 m

A

B

C

D

E

F

G

H

I

J

Figure A1. Bridge Truss Example : Geometry and Load Table

TABLE A1 - BRIDGE TRUSS EXAMPLE LOAD TABLE

R. PERUCCHIO 11/25/02

elementside

[m]]area

[m^2]2nd Area

mom. [m^4] length [m]dead load

[kN]live load

[kN] joint dead load

[kN]live load

[kN]

AD 0.4 0.16 0.0021 7.5 4.12 A 5.49AB 0.4 0.16 0.0021 12.5 6.87 B 12.20 41.65DC 0.4 0.16 0.0021 12.5 6.87 C 7.86DB 0.4 0.16 0.0021 1 0 5.49 49.00 D 8.24 24.50BC 0.4 0.16 0.0021 7.5 4.12 E 6.45CE 0.4 0.16 0.0021 8.6 4.72 F 5.52 29.40BE 0.4 0.16 0.0021 7.4 4.07 G 4.70BF 0.4 0.16 0.0021 7 3.85 34.30 H 4.45 22.05FE 0.4 0.16 0.0021 2.5 1.37 I 3.08FH 0.4 0.16 0.0021 5 2.75 24.50 J 1.79 9.80FG 0.4 0.16 0.0021 5.6 3.08EG 0.4 0.16 0.0021 5 2.75 Total 59.77 127.40HG 0.4 0.16 0.0021 2.5 1.37GI 0.4 0.16 0.0021 4 2.20HI 0.4 0.16 0.0021 4.7 2.58HJ 0.4 0.16 0.0021 4 2.20 19.60IJ 0.4 0.16 0.0021 2.5 1.37

Total 108.80 59.77 127.409

-1.8 kN9.2 kN

-29.0 kN

7.7 kN

-9.3 kN

-21.3 kN

31.3 kN

-20.3 kN

-19.5 kN

-38.1 kN

6.8 kN

11.4 kN

-16.0 kN

17.5 kN

-2.4 kN0.0 kN-6.8 kN

15.5 kN

29.0 kN15.5 kN

12.3 kN

29.1 kN

47.5 kN

J

I

H

G

F

E

D

C

B

A

8.2 kN 12.2 kN 5.5 kN 4.5 kN 1.8 kN

5.5 kN 7.9 kN

6.4 kN 4.7 kN 3.1 kN

Figure A2. Bridge Truss Example : Dead Loads

17.5 kN

-2.4 kN

0.0 kN

-6.8 kN

15.5 kN

8.2 kN

5.5 kN

2.0 kN1.5 kN

6.8 kN

-9.8 kN18.5 kN

-63.2 kN

16.1 kN

-31.9 kN

-47.5 kN

71.2 kN

-47.6 kN

-61.3 kN

-89.7 kN

16.2 kN

27.0 kN

-53.6 kN

37.7 kN

-7.3 kN4.8 kN-20.1 kN

31.8 kN

63.2 kN31.9 kN

67.2 kN

110.2 kN

3.9 kN17.2 kN

J

I

H

G

F

E

D

C

B

A

24.5 kN41.6 kN 29.4 kN 22.0 kN 9.8 kN

Figure A3. Bridge Truss Example : Live Loads

37.7 kN

-7.3 kN

4.8 kN

-20.1 kN

31.9 kN

110.2 kN

3.9 kN17.2 kN

D

A

24.5 kN41.6 kN

37.7 kN

5.8 kN4.4 kN

3.9 kN2.9 kN

3.9 kN2.9 kN20.1 kN10

TABLE A2. BRIDGE TRUSS EXAMPLE VERIFICATION TABLE - FORCES

R. PERUCCHIO

element el. F

orce

(dea

d lo

ad)

[kN

]

el. F

orce

(liv

e lo

ad)

[kN

]

el.

Forc

e to

tal

[kN

]

appl

ied

stre

ss

[Mpa

]

allo

w.

stre

ss [

Mpa

]

Pcrit

ical

[kN

]

Pcri

t./s.

f. [k

N]

buck

ling

failu

re

crus

h fa

ilure

stre

ng. f

ailu

re

AD -6.8 -20.1 -26.9 -0.17 -5.4 -3743 -749 safe safeAB 0 4.8 4.8 0.03 2 0 safeDC -2.4 -7.3 -9.7 -0.06 -5.4 -1348 -270 safe safeDB 17.5 37.7 55.2 0.35 2 0 safeBC -16 -53.6 -69.6 -0.44 -5.4 -3743 -749 safe safeCE -38.1 -89.7 -127.8 -0.80 -5.4 -2847 -569 safe safeBE 11.4 2 7 38.4 0.24 2 0 safeBF 6.8 16.2 2 3 0.14 2 0 safeFE -19.5 -61.2 -80.7 -0.50 -5.4 -33688 -6738 safe safeFH -21.3 -47.5 -68.8 -0.43 -5.4 -8422 -1684 safe safeFG 31.3 71.2 102.5 0.64 2 0 safeEG -20.3 -47.6 -67.9 -0.42 -5.4 -8422 -1684 safe safeHG -9.3 -31.8 -41.1 -0.26 -5.4 -33688 -6738 safe safeGI 7.7 16.1 23.8 0.15 2 0 safeHI 9.2 18.5 27.7 0.17 2 0 safeHJ -29 -63.2 -92.2 -0.58 -5.4 -13159 -2632 safe safeIJ -1.8 -9.8 -11.6 -0.07 -5.4 -33688 -6738 safe safe

TABLE A3. BRIDGE TRUSS EXAMPLE VERIFICATION TABLE - DISPLACEMENTS

R. PERUCCHIO

joint v. d

ispl

. -liv

e lo

ad [

mm

]

v. d

ispl

. -de

ad

load

[m

m]

v. d

ispl

. tot

al

[mm

]

V_J

- V_

D

[mm

]

AB 0.25 0.08 0.33 3.82CD 0.09 0.03 0.12EF 1.08 0.45 1.53GH 2.38 1.02 3.4IJ 2.75 1.19 3.94

total truss weight 59.77 [kN]total vertical support 187.17 [kN]diff. vertical deformation 3.82 [mm]TRUSS PASSES VERIFICATION IN TENSION AND COMPRESSION

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