# Three-Hinged Arches - Continuous and Point Loads

## Support reactions and bending moments

### Three-Hinged Arch - Continuous Load

Bending Moment

*M _{m} = (q L^{2} / 8) (4 (x_{m} / L - (x_{m} / L)^{2}) - y_{m} / y_{c}) (1a)*

*where*

*M _{m} = moment at m (Nm, lb_{f} ft)*

*q = continuous load (N/m, lb _{f}/ft)*

*x _{m} = x-coordinate for m (m, ft)*

*y _{m} = y-coordinate for m (m, ft)*

*y _{c} = 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

*R _{1y} = R_{2y} *

* =* q L / 2 (1b)

*where *

*R = support force (N, lb _{f})*

*R _{1x} = R_{2x} *

* =* q L^{2} / (8 y_{c}) (1c)

### Three-Hinged Arch - Half Continuous Load

Bending Moment

*M _{m} = (q L^{2} / 16) (8 (x_{m} / L - (x_{m} / L)^{2}) - 2 x_{m} / L - y_{m} / y_{c}) (2a)*

#### Support Reactions

*R _{1y} = 3 q L / 8 (2b)*

*R _{2y} =* q L / 8 (2c)

*R _{1x} = R_{2x} *

* =* q L^{2} / (16 y_{c}) (2d)

### Three-Hinged Arch - Horizontal Continuous Load

Bending Moment

*M _{m} = (q L^{2} / 2) (x_{m} / L - 3 x_{m} / L + (x_{m} / L)^{2}) (3a)*

*M _{k} = (q L^{2} / 4) (2 (L - x_{k}) / L - y_{k} / y_{c}) (3b)*

*where *

*M _{k} = moment at k (Nm, lb_{f} ft)*

*y _{k} = y-coordinate for k (m, ft)*

*x _{k} = x-coordinate for k (m, ft)*

#### Support Reactions

*R _{1y} = - q y_{c}^{2} / (2 L) (3c)*

*R _{2y} =*

*q y*(3d)

_{c}^{2}/ (2 L)*R _{1x} = - 3 q y_{c }/ 4 (3e)*

*R _{2x} =* q y

_{c}/ 4 (3f)

### Three-Hinged Arch - Eccentric Point Load

Bending Moment

*M _{m} = (F a / 2) (2 (b / a) (x_{m} / L) - y_{m} / y_{c}) (4a)*

*M _{k} = (F a /2) (2 (L - x_{k}) / L - y_{k} / y_{c}) (4b)*

#### Support Reactions

*R _{1y} = F b / L (4c)*

*R _{2y} =* F a / L (4d)

*R _{1x} = R_{2x} *

* =* F a / (2 y_{c}) (4f)