Introduction:
An influence line is a graphical representation of the variation in a structure's response due to a unit load moving along the structure. In the case of a two-hinged parabolic arch, the influence line for horizontal thrust is the graphical representation of the variation in horizontal thrust due to a unit load moving along the arch.

Influence Line for Horizontal Thrust:
The influence line for horizontal thrust in a two-hinged parabolic arch is parabolic. This is because the horizontal thrust in a two-hinged parabolic arch is directly proportional to the vertical displacement of the arch under a unit load. Since the vertical displacement of a two-hinged parabolic arch is parabolic, the influence line for horizontal thrust must also be parabolic.

Explanation:
The horizontal thrust in a two-hinged parabolic arch is generated due to the horizontal component of the reaction force at the hinges. When a unit load moves along the arch, it causes the arch to deflect vertically. This vertical deflection generates a horizontal reaction force at the hinges, which is proportional to the vertical deflection.

The vertical deflection of a two-hinged parabolic arch is given by the equation y = a(x^2), where y is the vertical deflection at a point x along the arch, and a is a constant that depends on the span and rise of the arch.

The horizontal thrust at a point x along the arch is given by the equation H = (w/2)(y/x), where w is the unit load and y/x is the slope of the arch at that point. Substituting the equation for y into the equation for H, we get H = (w/2)a(x), which is a linear equation in x.

Therefore, the influence line for horizontal thrust in a two-hinged parabolic arch is a parabolic curve that passes through the hinges and has a maximum value at the midpoint of the arch. The shape of the influence line is identical to the shape of the arch's vertical deflection curve.

Conclusion:
In summary, the influence line for horizontal thrust in a two-hinged parabolic arch is parabolic. This is because the horizontal thrust is directly proportional to the arch's vertical deflection, which is parabolic in shape. Understanding the influence line for horizontal thrust is important in designing and analyzing two-hinged parabolic arches.