A three-hinged arch bridge carries an unsymmetrical distributed load. How does the horizontal reaction at the abutments specifically relate to the internal shear and bending moment at the arch crown?

A three-hinged arch bridge is a specific type of arch structure that has three pin hinges: one at each abutment and one at the crown. Abutments are the supports at the ends of the arch that resist both vertical and horizontal forces from the bridge. This three-hinged configuration makes the structure statically determinate, meaning all reaction forces and internal forces can be calculated using only the equations of static equilibrium. The horizontal reaction, denoted as H, is the horizontal thrust exerted by the arch onto its abutments, which the abutments resist. Due to the arch action, this horizontal thrust is a primary characteristic of arches, significantly reducing bending moments throughout the structure compared to a simply supported beam. The arch crown refers to the highest point of the arch, where the third hinge is located. Internal shear force (V) is the force acting perpendicular to the cross-section of the arch, tending to cause sliding. Internal bending moment (M) is the rotational force acting within the arch's cross-section, tending to cause it to bend. An unsymmetrical distributed load means the applied load is not evenly balanced about the center of the arch.

Regarding the relationship at the arch crown:

1. Internal Bending Moment (M) at the Arch Crown: For a three-hinged arch, the presence of a hinge at the crown fundamentally dictates that the internal bending moment at that specific point is always zero (M_crown = 0). A pin hinge, by definition, cannot transmit a bending moment. Therefore, the horizontal reaction (H) does not cause or determine a non-zero bending moment at the arch crown; the moment is zero irrespective of the value of H. The horizontal reaction H is, however, crucial in the overall equilibrium equations that lead to this zero moment at the crown when considering one half of the arch.

2. Internal Shear Force (V) at the Arch Crown: Under an unsymmetrical distributed load, the internal shear force at the arch crown (V_crown) is generally not zero. To understand its relationship with the horizontal reaction (H), it is important to first define the orientation of forces. Assuming the tangent to the arch at its crown is horizontal (which is the standard case for most arch designs where the crown is the apex):
The horizontal reaction (H) does *notdirectly equal or cause the internal shear force at the crown. Instead, the horizontal reaction (H) is precisely equal to the *axial compressive force(also known as normal force) acting within the arch's cross-section at the crown. This means H directly represents the squeezing force along the arch axis at its highest point.
The internal shear force at the crown (V_crown) is determined by the net vertical forces acting on one side of the arch up to the crown. It is the algebraic sum of the vertical reaction at one abutment and all the vertical components of the distributed loads acting on that same half of the arch up to the crown. Because the load is unsymmetrical, the net vertical force on one side will not be balanced by the other side, resulting in a non-zero shear force at the crown. The horizontal reaction (H), while essential for the overall stability and internal force distribution throughout the arch, primarily contributes to the axial force at the crown, not the shear force, when the crown's tangent is horizontal.