Bridge Constructor in Concrete

The site is converted from Bridge Constructor under the assumption that 2.5 meters (the width of a square on the default grid) is equivalent to 8 feet. (In reality, 2.5 m ≈ 8.202 ft.)

The bridge is composed entirely of plain (unreinforced) class A concrete, with density (ρ) 145 lb/ft³, compressive strength (f_c′) 4 klb/in², tensile strength 0 klb/ft², shear strength 2(f_c′ ⋅ 1 lb/in²)^1/2, and modulus of elasticity (E) 1820(f_c′ ⋅ 1 klb/in²)^1/2. The cross section of the bridge has width (b) 12 ft.

If a member of the bridge is directly beneath the riding surface of the bridge, then the member is a filled member. A filled member is separated into a fill part and a load&h;bearing part. The fill part&a;s upper profile and lower profile are defined by two separate functions. The load&h;bearing part&a;s upper profile is identical to the fill part&a;s lower profile, but the load&h;bearing part is designed as if its centerline profile were identical to the fill part&a;s lower profile. The load&h;bearing part has a rectangular cross section with height h. It is assumed that axial load and flexural load are resisted by only the load&h;bearing part, while shear load is resisted by both parts.

If a member of the bridge is not a filled member, then it is an unfilled member. An unfilled member has a rectangular cross section with height h, and its centerline profile is defined by a function. It has buckling strength of π²EI ÷ ℓ², where I (the cross section&a;s second moment of area) is min(b, h)³max(b, h) ÷ 12 and ℓ is the length of the member.

The live load on the riding surface of the bridge consists of: cars, represented with a uniform load of 640 lb/ft; and a truck, represented with a point load of 72 klb.

Camatuga

Island 1

(Westlands)

Site 3

The road passes: from a point at (0 ft, 0 ft); over points at (48 ft, −24 ft) and (112 ft, −24 ft); to a point at (160 ft, 0 ft).

ABC, CDE, and EFG are three&h;hinge arches.

Rotational equilibrium of ABC about A

F_{B, x, BC}(y_C − y_A) + F_{B, y, BC}(x_C − x_A) = ρb(∫x_Ax_B((h_ABℓ_ABx_B − x_A + y_A − (y_A + y_B − y_Ax_B − x_A(x − x_A)))(x − x_A)dx) + ∫x_Bx_C((h_BCℓ_BCx_C − x_B + y_A − (y_B + y_C − y_Bx_C − x_B(x − x_B)))(x − x_A)dx)) + σ_cars∫x_Ax_C((x − x_A)dx) + F_truck(x_truck − x_A)(x_truck ∈ (x_A, x_C))

F_{B, x, BC}(y_C − y_A) + F_{B, y, BC}(x_C − x_A) = ρb(∫0 ftx_B − x_A((h_ABℓ_ABx_B − x_A + y_A − (y_A + y_B − y_Ax_B − x_Ax))xdx) + ∫x_B − x_Ax_C − x_A((h_BCℓ_BCx_C − x_B + y_A − (y_B + y_C − y_Bx_C − x_B(x − x_B + x_A)))xdx)) + σ_cars∫0 ftx_C − x_A(xdx) + F_truck(x_truck − x_A)(x_truck ∈ (x_A, x_C))

F_{B, x, BC}(y_C − y_A) + F_{B, y, BC}(x_C − x_A) = ρb(∫0 ftx_B − x_A((h_ABℓ_ABx_B − x_A − y_B − y_Ax_B − x_Ax)xdx) + ∫x_B − x_Ax_C − x_A((h_BCℓ_BCx_C − x_B − (y_B − y_A) − y_C − y_Bx_C − x_B(x − (x_B − x_A)))xdx)) + σ_cars∫0 ftx_C − x_A(xdx) + F_truck(x_truck − x_A)(x_truck ∈ (x_A, x_C))

F_{B, x, BC}(y_C − y_A) + F_{B, y, BC}(x_C − x_A) = ρb(∫0 ftx_B − x_A((h_ABℓ_ABx_B − x_A − y_B − y_Ax_B − x_Ax)xdx) + ∫x_B − x_Ax_C − x_A((h_BCℓ_BCx_C − x_B − (y_B − y_A) − y_C − y_Bx_C − x_Bx + (y_C − y_B)(x_B − x_A)x_C − x_B)xdx)) + σ_cars∫0 ftx_C − x_A(xdx) + F_truck(x_truck − x_A)(x_truck ∈ (x_A, x_C))

F_{B, x, BC}(y_C − y_A) + F_{B, y, BC}(x_C − x_A) = ρb(∫0 ftx_B − x_A((h_ABℓ_ABx_B − x_A − y_B − y_Ax_B − x_Ax)xdx) + ∫x_B − x_Ax_C − x_A((h_BCℓ_BC − (y_C − y_B)(x_B − x_A)x_C − x_B − (y_B − y_A) − y_C − y_Bx_C − x_Bx)xdx)) + σ_cars∫0 ftx_C − x_A(xdx) + F_truck(x_truck − x_A)(x_truck ∈ (x_A, x_C))

F_{B, x, BC}(y_C − y_A) + F_{B, y, BC}(x_C − x_A) = ρb((h_ABℓ_AB2(x_B − x_A)x² − y_B − y_A3(x_B − x_A)x³)0 ftx_B − x_A + (h_BCℓ_BC − (y_C − y_B)(x_B − x_A) − (y_B − y_A)(x_C − x_B)2(x_C − x_B)x² − y_C − y_B3(x_C − x_B)x³)x_B − x_Ax_C − x_A) + σ_cars(12x²)0 ftx_C − x_A + F_truck(x_truck − x_A)(x_truck ∈ (x_A, x_C))

F_{B, x, BC}(y_C − y_A) + F_{B, y, BC}(x_C − x_A) = ρb(h_ABℓ_AB(x_B − x_A)2 − (y_B − y_A)(x_B − x_A)²3 + h_BCℓ_BC − (y_C − y_B)(x_B − x_A) − (y_B − y_A)(x_C − x_B)2(x_C − x_B)((x_C − x_A)² − (x_B − x_A)²) − y_C − y_B3(x_C − x_B)((x_C − x_A)³ − (x_B − x_A)³)) + σ_cars2(x_C − x_A)² + F_truck(x_truck − x_A)(x_truck ∈ (x_A, x_C))

F_{B, x, BC}(y_C − y_A) + F_{B, y, BC}(x_C − x_A) = ρb(h_ABℓ_AB(x_B − x_A)2 − (y_B − y_A)(x_B − x_A)²3 + h_BCℓ_BC − (y_C − y_B)(x_B − x_A) − (y_B − y_A)(x_C − x_B)2(x_C − x_B)(x_C² − 2x_Cx_A + x_A² − x_B² + 2x_Bx_A − x_A²) − y_C − y_B3(x_C − x_B)((x_C − x_A)³ − (x_B − x_A)³)) + σ_cars2(x_C − x_A)² + F_truck(x_truck − x_A)(x_truck ∈ (x_A, x_C))

Rotational equilibrium of BC about B

Horizontal equilibrium of BC

Vertical equilibrium of BC

Site 4

The road passes from pin 0 at (0 ft, 0 ft) to pin 1 at (128 ft, 0 ft), over pin 2 at (88 ft, −7.5 ft).

Site 5

The road passes from pin 0 at (0 ft, 0 ft) to pin 1 at (192 ft, 0 ft), over pin 2 at (80 ft, −64 ft).

Island 2

(Central Mainland)

Site 1

The road passes from pin 0 at (0 ft, 0 ft) to pin 1 at (192 ft, 0 ft), over pin 2 at (64 ft, −32 ft) and pin 3 at (128 ft, −32 ft).

Site 3

The road passes from pin 0 at (0 ft, 0 ft) to pin 1 at (192 ft, 0 ft), over pin 2 at (56 ft, −64 ft) and pin 3 at (136 ft, −64 ft).

Site 5

The road passes from pin 0 at (0 ft, 0 ft) to pin 1 at (224 ft, 0 ft), over pin 2 at (128 ft, −16 ft).

Site 6

The road passes from point A at (0 ft, 0 ft) to point B at (160 ft, 0 ft). Arch CED passes through pin C at (0 ft, −16 ft), pin E at (80 ft, y_E), and pin D at (160 ft, −16 ft), where y_E ∈ (−16 ft, 0 ft).

Circle at pin C

(0 ft − 80 ft)² + (−16 ft − y_CED)² = r_CED²

y_CED² + 32 ft ⋅ y_CED + 6656 ft² = r_CED²

Circle at pin E

(80 ft − 80 ft)² + (y_E − y_CED)² = r_CED²

y_CED² − 2y_Ey_CED + y_E² = r_CED²

y_CED² + 32 ft ⋅ y_CED + 6656 ft² = y_CED² − 2y_Ey_CED + y_E²

2y_Ey_CED + 32 ft ⋅ y_CED = y_E² − 6656 ft²

y_CED = y_E² − 6656 ft²2y_E + 32 ft

r_CED = |y_E − y_CED|

y = y_CED + (r_CED² − (x − 80 ft)²)^1/2

Rotational equilibrium of bridge about pin C

160 ft ⋅ F_{D, y} = ∫0 ft160 ft((640 lb/ft − 145 lb/ft³ ⋅ 12 ft ⋅ (y_CED + (r_CED² − (x − 80 ft)²)^1/2))x ⋅ dx) + x_truck ⋅ 72 klb

160 ft ⋅ F_{D, y} = 640 lb/ft ⋅ ∫0 ft160 ft(x ⋅ dx) − 1.74 klb/ft² ⋅ (y_CED∫0 ft160 ft(x ⋅ dx) + ∫0 ft160 ft((r_CED² − (x − 80 ft)²)^1/2x ⋅ dx)) + x_truck ⋅ 72 klb

160 ft ⋅ F_{D, y} = 640 lb/ft ⋅ (x²/2)0 ft160 ft − 1.74 klb/ft² ⋅ (y_CED(x²/2)0 ft160 ft − ((r_CED² − (x − 80 ft)²)^1/2(r_CED² − x² + 40 ft ⋅ x + 3200 ft²)/3 + 40 ft ⋅ r_CED²atan(80 ft − x(r_CED² − (x − 80 ft)²)^1/2))0 ft160 ft) + x_truck ⋅ 72 klb

160 ft ⋅ F_{D, y} = 8192 klb⋅ft − 1.74 klb/ft² ⋅ (y_CED ⋅ 12800 ft² − ((r_CED² − (x − 80 ft)²)^1/2(r_CED² − x² + 40 ft ⋅ x + 3200 ft²)/3 + 40 ft ⋅ r_CED²atan(80 ft − x(r_CED² − (x − 80 ft)²)^1/2))0 ft160 ft) + x_truck ⋅ 72 klb

Island 3

(Ridge)

Site 1

The road passes from pin 0 at (0 ft, 0 ft) to pin 1 at (256 ft, 0 ft), over pin 2 at (32 ft, −64 ft) and pin 3 at (224 ft, −64 ft).

Site 2

The road passes from pin 0 at (0 ft, 0 ft) to pin 1 at (192 ft, 0 ft), over pin 2 at (96 ft, −96 ft).

Site 5

The road passes from pin 0 at (0 ft, 0 ft) to pin 1 at (384 ft, 0 ft), over pin 2 at (0 ft, −48 ft), pin 3 at (192 ft, −16 ft), and pin 4 at (384 ft, −48 ft).

Site 6

The road passes from pin 0 at (0 ft, 0 ft) to pin 1 at (224 ft, 0 ft), over pin 2 at (8 ft, −64 ft).

Site 8

The road passes from pin 0 at (0 ft, 0 ft) to pin 1 at (320 ft, 0 ft), over pin 2 at (32 ft, −96 ft) and pin 3 at (288 ft, −96 ft).

Island 4

(Eastern Mainland)

Site 1

The road passes from pin 0 at (0 ft, 0 ft) to pin 1 at (96 ft, 0 ft), over pin 2 at (96 ft, −16 ft).