Understanding Temperature Effects on Three-Hinged Arches

A three-hinged arch is a statically determinate structure, which means its internal forces and reactions can be calculated using only the equations of static equilibrium. It has three hinges: two at the supports and one typically at the crown. This structural configuration provides flexibility, allowing it to accommodate movements like thermal expansion or settlement without inducing internal stresses, unlike statically indeterminate structures.

Analyzing Statement 1: Temperature Stresses in a Three-Hinged Arch

Statement 1 says: "No stresses are produced in a three-hinged arch due to temperature change alone."

This statement is correct for an unrestrained three-hinged arch. Because it is statically determinate and has hinges allowing rotation and expansion/contraction, a uniform change in temperature will cause the arch segments to expand or contract freely. The arch will simply change its shape and size slightly, but without developing internal stresses if no external loads are present and supports allow the resulting movements (like outward spread at supports and upward movement at the crown). The hinges prevent the build-up of bending moments or axial stresses purely from thermal expansion.

Analyzing Statement 2 & 3: Temperature Effect on Horizontal Thrust

Statement 2 says: "There is a decrease in horizontal thrust due to a rise in temperature."

Statement 3 says: "There is an increase in horizontal thrust due to a rise in temperature."

These statements refer to a "load-carrying" three-hinged arch, implying the horizontal thrust ($H$) exists due to external loads like vertical dead or live loads. This thrust is a horizontal reaction component at the supports that balances the outward thrust created by the arch action under vertical loads.

When the temperature rises, the material of the three-hinged arch expands. This thermal expansion causes the arch to attempt to spread outwards horizontally at the supports and rise upwards at the crown. The original shape of the arch changes. For a given vertical load, the effect of this expansion is to effectively increase the 'rise' of the arch relative to its span or to move the support points outwards. Both of these geometric changes lead to a decrease in the horizontal thrust required to support the vertical loads in equilibrium.

Think of it this way: the horizontal thrust is proportional to the load and the span, and inversely proportional to the rise. A rise in temperature increases the span and/or the effective rise. Therefore, to maintain equilibrium under the same vertical load, the horizontal thrust must decrease.

Hence, Statement 2 is correct, and Statement 3 is incorrect.

Identifying Correct Statements on Temperature Effects

Based on the analysis:

• Statement 1 is correct: No stresses from temperature change alone (in an unrestrained arch).
• Statement 2 is correct: Horizontal thrust decreases with a rise in temperature (in a load-carrying arch).
• Statement 3 is incorrect: Horizontal thrust increases with a rise in temperature.

Therefore, the correct statements are 1 and 2.

Revision Table: Three-Hinged Arch Properties and Temperature Effects

Additional Information: Arch Types and Thermal Analysis

Structures react differently to temperature changes based on their static determinacy and material properties. Arches are commonly used to span large distances and convert vertical loads into outward thrust, which is resisted by abutments or tie rods.

• Two-Hinged Arch: Statically indeterminate to the first degree. Temperature change induces horizontal thrust and bending moments. A rise in temperature increases horizontal thrust in a two-hinged arch if the supports are fixed horizontally.
• Fixed Arch (No Hinges): Statically indeterminate to the third degree. Temperature change induces axial forces, shear forces, and bending moments, as well as horizontal and vertical reactions. A rise in temperature significantly increases horizontal thrust and bending moments.
• Coefficient of Thermal Expansion: This material property ($\alpha$) determines how much a material expands or contracts per degree of temperature change. The length change ($\Delta L$) is given by $\Delta L = L_0 \alpha \Delta T$, where $L_0$ is the original length and $\Delta T$ is the temperature change.

Understanding the static determinacy is crucial for analyzing the effect of temperature changes. Statically determinate structures can deform freely under temperature changes without stress, while indeterminate structures develop internal stresses and altered reactions because external restraints prevent free deformation.