Designing A Concrete Arch Bridge ? V This is the famous Schwandbach bridge in Switzerland, designed by Robert Maillart in 1933. It spans 37.4 meters (122 feet) and was designed using the same graphical methods that will be demonstrated in this lesson. To proceed with this lesson, click on the Next button here or at the top of any page. When you are done with this lesson, click on the Contents button here or at the top of any page to return to the list of lessons.
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Designing A Concrete Arch Bridge?V
This 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?V
The 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?V
Step 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
20 ft
oe
99 ft11 ft 11 ft
7100
Form Diagram
Designing A Concrete Arch Bridge?V
Step 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.
20 ft
Loading Diagram
BA
120 k
oe
Form Diagram
Designing A Concrete Arch Bridge?V
The 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.
20 ft
Loading Diagram
CBA
120 k120 k
oe
Form Diagram
Designing A Concrete Arch Bridge?V
The process continues untilall the loads on the arch havebeen constructed and labeled.This is load CD.
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Loading Diagram
DCBA
120 k120 k120 k
oe
Form Diagram
Designing A Concrete Arch Bridge?V
This is load DE.
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Loading Diagram
EDCBA
120 k120 k120 k120 k
oe
Form Diagram
Designing A Concrete Arch Bridge?V
This is load EF.
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Loading Diagram
FEDCBA
120 k120 k120 k120 k120 k
oe
Form Diagram
Designing A Concrete Arch Bridge?V
This is load FG.
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Loading Diagram
GFEDCBA
120 k120 k120 k120 k120 k120 k
oe
Form Diagram
Designing A Concrete Arch Bridge?V
This is load GH.
20 ft
Loading Diagram
HGFEDCBA
120 k120 k120 k120 k120 k120 k120 k
oe
Form Diagram
Designing A Concrete Arch Bridge?V
This 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.
20 ft
Loading Diagram
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oe
Form Diagram
Designing A Concrete Arch Bridge?V
Step 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
20 ft
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oe
Form Diagram
ba
400 kipsLoad Line
120
kips
Designing A Concrete Arch Bridge?V
Once 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
20 ft
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oe
Form Diagramcba
400 kipsLoad Line
Vectors arrangedtip-to-tail
Designing A Concrete Arch Bridge?V
Since 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
20 ft
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oe
Form Diagram dcba
400 kipsLoad Line
Designing A Concrete Arch Bridge?V
We 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
20 ft
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oe
Form Diagramedcba
400 kipsLoad Line
Designing A Concrete Arch Bridge?V
We continue across theLoading Diagram...
Loading Diagram
20 ft
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oe
Form Diagram
fedcba
400 kipsLoad Line
Designing A Concrete Arch Bridge?V
We continue across theLoading Diagram...
Loading Diagram
20 ft
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oe
Form Diagram
gfedcba
400 kipsLoad Line
Designing A Concrete Arch Bridge?V
We continue across theLoading Diagram...
Loading Diagram
20 ft
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oe
Form Diagram
hgfedcba
400 kipsLoad Line
Designing A Concrete Arch Bridge?V
The Load Line is nowcomplete.The total length of the LoadLine is equal to the total loadon the arch, in this case960 kips.
Loading Diagram
20 ft
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oe
Form Diagram
hi
gfedcba
400 kipsLoad Line
960
kips
020
040
060
080
010
0012
00 k
ips
Designing A Concrete Arch Bridge?V
Step 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 don’t yet know the locationof point o, the left end of rayoe, it must occur somewherealong this line.
Loading Diagram
20 ft
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oe
Form Diagram
oe
ihgfedcba
400 kipsLoad Line
Designing A Concrete Arch Bridge?V
Step 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
20 ft
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oe
Form Diagramodoc
og
Ray oa is the longest ray
400 kipsLoad Line
ofoe
oboa
ihg
d
ba
oh
fe
c
Designing A Concrete Arch Bridge?V
From 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
20 ft
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oe
Form Diagramradius = 1229 kips
oe
ray oa
ihgfedcba
400 kipsForce Polygon
Pole o
Designing A Concrete Arch Bridge?V
By scaling the length of rayoe on the Force Polygon, wecan now determine themagnitude of the force insegment oe of the arch.Loading Diagram
20 ft
kips1098
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oe
Form Diagramradius = 1229 kips
oe
oa
o
ihgfedcba
1098 kips
0 200 400 600 800 1000 1200
400 kipsForce Polygon
Designing A Concrete Arch Bridge?V
Step 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
20 ft
kips1098
kips1113
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oeod
Form Diagram
oeod
oa
o
ihgfedcba
1113 kips
400 kipsForce Polygon
Designing A Concrete Arch Bridge?V
This is ray oc.
Loading Diagram
20 ft
kips1098
kips1113
kips1141
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oeodoc
Form Diagram
oeodoc
oa
o
ihgfedcba
1141 kips
400 kipsForce Polygon
Designing A Concrete Arch Bridge?V
This is ray ob.
Loading Diagram
20 ft
kips1098
kips1113
kips1141
kips1180
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oeodocob
Form Diagram
oa
oeodoc
ob
o
ihgfedcba
1180 kips
400 kipsForce Polygon
Designing A Concrete Arch Bridge?V
This is ray oa.
Loading Diagram
20 ft
1229oa kips
1098kips1113
kips1141
kips1180
kips
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oeodocob
Form Diagram
oeodoc
oboa
o
ihgfedcba
1229 kips
400 kipsForce Polygon
Designing A Concrete Arch Bridge?V
This is ray of.
Loading Diagram
20 ft
kips1097
kips1098
kips1113
kips1141
kips1180
kips1229
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
ofoeodocob
oa
Form Diagram
oeof
odoc
oboa
o
ihgfedcba
1097 kips
400 kipsForce Polygon
Designing A Concrete Arch Bridge?V
This is ray og.
Loading Diagram
20 ft
kips1108kips
1097kips
1098kips1113
kips1141
kips1180
kips1229
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
ogofoeodocob
oa
Form Diagram
ofoeodoc
oboa
o
ihgfedcba
og1108 kips
400 kipsForce Polygon
Designing A Concrete Arch Bridge?V
This is ray oh.
Loading Diagram
20 ft
kips1132kips
1108kips1097
kips1098
kips1113
kips1141
kips1180
kips1229
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
ohogofoeodocob
oa
Form Diagram
ofoeodoc
oboa
o
hgfedcba
ogoh1132 kips
400 kipsForce Polygon
Designing A Concrete Arch Bridge?V
This is ray oi.
Loading Diagram
20 ft
kipskips1132kips
1108kips1097
kips1098
kips1113
kips1141
kips1180
kips1229
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oiohogofoeodoc
oboa
Form Diagram
1168
ofoeodoc
oboa
o
ihgfedcba
ogoh
oi1168 kips
400 kipsForce Polygon
Designing A Concrete Arch Bridge?V
This 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
kips1132kips
1108kips1097
kips1098
kips1113
kips1141
kips11801229
oiohogofoeodoc
oboa 1168
20 ftForm Diagram
IHGFEDCBA
oi
400 kipsForce Polygon
ofoeodoc
oboa
o
ihg
d
ba
ogoh
Maximum force in arch
fe
c
Designing A Concrete Arch Bridge?V
Using 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.
kips1229
kips1168
kips1132
kips1108
kips1097
kips1098 kips
1113 kips1141 kips1180
Loading Diagram
IHGFEDCBA
120 k120 k120 k120 k120 k120 k120 k120 k
oa ob oc od oe of ogoh oi
Form Diagram20 ft
400 kipsForce Polygoni
hgfedcba oa
obocodoeofogohoi
o
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