# Three-Hinged Arches - Continuous and Point Loads

## Support reactions and bending moments

### Three-Hinged Arch - Continuous Load

#### Bending Moment

Mm = (q L2 / 8) (4 (xm / L - (xm / L)2) - ym / yc)                               (1a)

where

Mm = moment at m (Nm, lbf ft)

q = continuous load (N/m, lbf/ft)

xm = x-coordinate for m (m, ft)

ym = y-coordinate for m (m, ft)

yc = y-coordinate for center hinge (m, ft)

L = horizontal distance between the supports (m, ft)

Cartesian coordinates related to a center located in the hinge of support no. 1.

#### Support Reactions

R1y = R2y

= q L / 2                                (1b)

where

R = support force (N, lbf)

R1x = R2x

= q L2 / (8 yc)                               (1c)

### Three-Hinged Arch - Half Continuous Load

#### Bending Moment

Mm = (q L2 / 16) (8 (xm / L - (xm / L)2) - 2 xm / L  - ym / yc)                               (2a)

#### Support Reactions

R1y = 3 q L / 8                                (2b)

R2y = q L / 8                                (2c)

R1x R2x

= q L2 / (16 yc)                               (2d)

### Three-Hinged Arch - Horizontal Continuous Load

#### Bending Moment

Mm = (q L2 / 2) (xm / L - 3 xm / L + (xm / L)2)                                (3a)

Mk = (q L2 / 4) (2 (L - xk) / L - yk / yc)                                (3b)

where

Mk = moment at k (Nm, lbf ft)

yk = y-coordinate for k (m, ft)

xk = x-coordinate for k (m, ft)

#### Support Reactions

R1y = - q yc2 / (2 L)                                (3c)

R2y = q yc2 / (2 L)                                (3d)

R1x = - 3 q yc / 4                               (3e)

R2x = q yc / 4                               (3f)

### Three-Hinged Arch - Eccentric Point Load

#### Bending Moment

Mm = (F a / 2) (2 (b / a) (xm / L) - ym / yc)                                (4a)

Mk = (F a /2) (2 (L - xk) / L - yk / yc)                                (4b)

#### Support Reactions

R1y = F b / L                                (4c)

R2y = F a / L                                 (4d)

R1x = R2x

= F a / (2 yc)                               (4f)

## Related Topics

• Mechanics - Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more
• Statics - Loads - force and torque, beams and columns
• Beams and Columns - Deflection and stress, moment of inertia, section modulus and technical information of beams and columns

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