07-Clase 07-1

Designing A Concrete Arch Bridge?VThis is the famousSchwandbach bridge inSwitzerland, designed byRobert Maillart in 1933. Itspans 37.4 meters (122 feet)and was designed using thesame graphical methods thatwill be demonstrated in thislesson.

To proceed with this lesson,click on the Next button hereor at the top of any page.When you are done with thislesson, click on the Contentsbutton here or at the top of anypage to return to the list oflessons.

Designing A Concrete Arch Bridge?VThe Problem: This is apartially finished form diagramof a bridge that will span99 feet. The roadway slopesat 7%. Eight walls spaced11 feet apart bring total loadsof 120 kips each from the stiffdeck to a concrete arch below.The arch is a concrete slabthat is 8 inches thick and16 feet wide throughout thespan. The allowable axialstress in the arch is 800 lb/in2.Shape the arch in such a waythat segment oe is parallel tothe bridge deck as shown.

20 ft

oe

99 ft11 ft 11 ft

7100

Form Diagram

Designing A Concrete Arch Bridge?VStep 1: What is themaximum allowable axialforce in the arch?The given parameters are anarch thickness of 8 inches, anarch width of 16 feet, and anallowable stress of 800 lb/in2.

The cross sectional area of the arch is its width times its thickness:Area = (width)(thickness) = (16 ft)(12 in./ft)(8 in.) = 1536 in2

The allowable axial force in the arch is the area of the arch times the allowable stress:Force = (Area)(allowable stress) = (1536 in2)(800 lb/in2) = 1,228,800 lb = 1229 kips

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oe

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Form Diagram

Designing A Concrete Arch Bridge?VStep 2: Construct a loadingdiagram and apply intervalnotation.Working clockwise from theupper left, place externalgravity loads on the formdiagram. The spaces orintervals between forces arelabeled with the uppercaseletters of the alphabet.In this diagram, the first120 kip load is placed overthe leftmost arch wall. LettersA and B are assigned to theintervals on either side of theload. This is load AB.

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Loading Diagram

BA

120 k

oe

Form Diagram

Designing A Concrete Arch Bridge?VThe second load is placedover the next arch wall. LetterC is added to the diagram,and this becomes load BC.With each load, a vertical lineof action is extended downthrough the area of the formdiagram where the arch willbe constructed.

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Loading Diagram

CBA

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Form Diagram

Designing A Concrete Arch Bridge?VThe process continues untilall the loads on the arch havebeen constructed and labeled.This is load CD.

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Loading Diagram

DCBA

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oe

Form Diagram

Designing A Concrete Arch Bridge?VThis is load DE.

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Loading Diagram

EDCBA

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oe

Form Diagram

Designing A Concrete Arch Bridge?VThis is load EF.

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Loading Diagram

FEDCBA

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oe

Form Diagram

Designing A Concrete Arch Bridge?VThis is load FG.

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Loading Diagram

GFEDCBA

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oe

Form Diagram

Designing A Concrete Arch Bridge?VThis is load GH.

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Loading Diagram

HGFEDCBA

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Form Diagram

Designing A Concrete Arch Bridge?VThis is load HI.The loading diagram is nowcomplete. All of the externalgravity loads acting on thearch have been placed on theloading diagram.Vertical lines of action havebeen drawn for each load,extending down through thearea of the form diagramwhere the arch will beconstructed.

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Loading Diagram

IHGFEDCBA

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Form Diagram

Designing A Concrete Arch Bridge?VStep 3: Construct a loadline to any convenient scale.The Load Line is a graphicalsummation of the loads actingon the structure. On the LoadLine, the magnitude of theforces from the loadingdiagram are drawn to scale.Beginning with force AB, weplot a segment of the loadline, parallel to AB, labeledab. The length of ab scales to120 kips, the magnitude of theforce.

Loading Diagram

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Form Diagram

ba

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kips

Designing A Concrete Arch Bridge?VOnce again, we workclockwise around the FormDiagram, plotting each loadonto the Load Line in a tip-to-tail fashion.Although the Load Line iscomposed of vectors, theends of the force segmentsare marked with horizontal tickmarks rather than arrowheads. This helps to keep thediagram legible and accurate.

Loading Diagram

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IHGFEDCBA

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oe

Form Diagramcba

400 kipsLoad Line

Vectors arrangedtip-to-tail

Designing A Concrete Arch Bridge?VSince all of the loads on ourarch are of equal magnitude,each segment of the LoadLine will be equal in length.When the loads on a structurevary in magnitude, or are notall strictly vertical in direction,the segments of the Load Linewill vary in length anddirection as well.

Loading Diagram

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IHGFEDCBA

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Form Diagram dcba

400 kipsLoad Line

Designing A Concrete Arch Bridge?VWe continue across theLoading Diagram, plotting allthe loads on the arch onto theLoad Line.Notice how intervals on theLoading Diagram correspondto points on the Load Line.

Loading Diagram

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Form Diagramedcba

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Designing A Concrete Arch Bridge?VWe continue across theLoading Diagram...

Loading Diagram

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Form Diagram

fedcba

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Loading Diagram

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oe

Form Diagram

gfedcba

400 kipsLoad Line


Loading Diagram

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oe

Form Diagram

hgfedcba

400 kipsLoad Line

Designing A Concrete Arch Bridge?VThe Load Line is nowcomplete.The total length of the LoadLine is equal to the total loadon the arch, in this case960 kips.

Loading Diagram

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IHGFEDCBA

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oe

Form Diagram

hi

gfedcba

400 kipsLoad Line

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kips

020

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010

0012

00 k

ips

Designing A Concrete Arch Bridge?VStep 4: Construct ray oe ofknown direction butunknown length.Ray oe is a vector whose lengthis equal to the magnitude of theforce in arch segment oe, andwhose direction is parallel tothat force.Construct ray oe throughpoint e on the load line, parallelto arch segment oe. Althoughwe dont yet know the locationof point o, the left end of rayoe, it must occur somewherealong this line.

Loading Diagram

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IHGFEDCBA

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oe

Form Diagram

oe

ihgfedcba

400 kipsLoad Line

Designing A Concrete Arch Bridge?VStep 5: Construct thelongest ray and find point o.Even before we know thelocation of point o, we cansketch the general layout ofthe rays of the Force Polygon.We can see from this sketchthat, for any point o that lieson ray oe, ray oa will be thelongest ray, corresponding tothe greatest force in the arch.Thus we must limit ray oa toa length of 1229 kips.

Here is the equation we used in Step 1 to find the maximumallowable force in the arch:

Force = (Area)(allowable stress) = (1536 in2)(800 lb/in2)= 1,228,800 lb = 1229 kips

Loading Diagram

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IHGFEDCBA

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oe

Form Diagramodoc

og

Ray oa is

the long

est ray

400 kipsLoad Line

ofoe

oboa

ihg

d

ba

oh

fe

c

Designing A Concrete Arch Bridge?VFrom point a, we use acompass to strike an arc ofradius 1229 kips, themaximum force allowable inthe arch. Where this arcintersects with ray oe ispoint o.The Load Line has nowbecome part of a ForcePolygon. Point o is the Pole ofthe diagram.

Here is the equation we used in Step 1 to find the maximumallowable force in the arch:

Force = (Area)(allowable stress) = (1536 in2)(800 lb/in2)= 1,228,800 lb = 1229 kips

Loading Diagram

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IHGFEDCBA

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oe

Form Diagramradius =

1229 kips

oe

ray oa

ihgfedcba

400 kipsForce Polygon

Pole o

Designing A Concrete Arch Bridge?VBy scaling the length of rayoe on the Force Polygon, wecan now determine themagnitude of the force insegment oe of the arch.Loading Diagram

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oe

Form Diagramradius =

1229 kips

oe

oa

o

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Designing A Concrete Arch Bridge?VStep 6: Through the Pole o,construct the remainingrays of the Force Polygon,beginning at the center andworking toward the ends.As each ray is drawn,construct parallel to it thecorresponding segment of theconcrete arch on the FormDiagram.Scale the length of each rayto find the magnitude of theforce in the correspondingsegment of the arch.

Loading Diagram

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oeod

Form Diagram

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o

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Designing A Concrete Arch Bridge?VThis is ray oc.

Loading Diagram

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oeodoc

Form Diagram

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Designing A Concrete Arch Bridge?VThis is ray ob.

Loading Diagram

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oeodocob

Form Diagram

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ob

o

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Designing A Concrete Arch Bridge?VThis is ray oa.

Loading Diagram

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Form Diagram

oeodoc

oboa

o

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Designing A Concrete Arch Bridge?VThis is ray of.

Loading Diagram

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IHGFEDCBA

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ofoeodocob

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Form Diagram

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odoc

oboa

o

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Designing A Concrete Arch Bridge?VThis is ray og.

Loading Diagram

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IHGFEDCBA

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Form Diagram

ofoeodoc

oboa

o

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Designing A Concrete Arch Bridge?VThis is ray oh.

Loading Diagram

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IHGFEDCBA

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ohogofoeodoc

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Form Diagram

ofoeodoc

oboa

o

hgfedcba

ogoh1132 kips


Designing A Concrete Arch Bridge?VThis is ray oi.

Loading Diagram

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IHGFEDCBA

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oiohog

ofoeodocob

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Form Diagram

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ofoeodoc

oboa

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oi1168 kips


Designing A Concrete Arch Bridge?VThis is the bridge that youhave designed.The force polygon is agraphical tool that allows youto find the forces in astructure, and to findsimultaneously theappropriate form for thestructure.

kipskips

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Maximum

force in a

rch

fe

c

Designing A Concrete Arch Bridge?VUsing the same method thatwe used to find the form andforces for the arch, we canfind the form of a suspensionbridge whose cable can safelycarry a given maximum force.The pole of the Force Polygonlies to the right of the loadline, rather than to the left.The forces in the cable aretensile, rather thancompressive.

Click on the Contents button to begin anew lesson.Click on the image of the Schwandbach Bridgeto return to the beginning of this lesson.

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Loading Diagram

IHGFEDCBA

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oa ob oc od oe ofog

ohoi

Form Diagram20 ft

400 kipsForce Polygoni

hgfedcba oa

obocodoeofogohoi

o

Schwandbach BridgeThe ProblemAllowable ForceLoading DiagramLoad LineRay 'oe'Point 'o'Form & ForcesThe SolutionSuspension Bridge

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07-Clase 07-1

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