91Ó°ÊÓ

Flying Buttress. (a) A symmetric building has a roof sloping upward at \(35.0^{\circ}\) above the horizontal on each side. If each side of the uniform roof weighs \(10,000 \mathrm{N}\) , find the horizontal force that this roof exerts at the top of the wall, which tends to push out the walls. Which type of building would be more in danger of collapsing: one with tall walls or one with short walls? Explain. (b) As you saw in part (a), tall walls are in danger of collapsing from the weight of the roof. This problem plagued the ancient builders of large structures. A solution used in the great Gothic cathedrals during the 1200 s was the flying buttress, a stone support running between the walls and the ground that helped to hold in the walls. A Gothic church has a uniform roof weighing a total of \(20,000 \mathrm{N}\) and rising at \(40^{\circ}\) above the horizontal at each wall. The walls are 40 \(\mathrm{m}\) tall, and a flying buttress meets each wall 10 \(\mathrm{m}\) below the base of the roof. What horizontal force must this flying buttress apply to the wall?

Step by step solution

01

Understand Components of Force

The weight of the roof can be resolved into vertical and horizontal components. The vertical component is the weight itself, and the horizontal component is responsible for the outward push on the walls. In part (a), each side of the roof produces such a horizontal component.

02

Calculate the Horizontal Component of Force for Part (a)

The horizontal force \( F_h \) exerted by one side of the roof is given by \( F_h = F \cdot \sin(\theta) \), where \( F = 10,000 \, \text{N} \) (weight of one side of the roof) and \( \theta = 35.0^{\circ} \). Therefore:\[ F_h = 10,000 \times \sin(35^{\circ}) \approx 5,735.764 \, \text{N} \].

03

Discuss Wall Height Impact

Tall walls are more susceptible to collapse due to greater torque induced by the same horizontal force, compared to short walls. This is because the lever arm (distance from the base to the point of force application) is larger in taller walls.

04

Understand the Role of Flying Buttresses in Part (b)

To prevent walls from collapsing, flying buttresses apply a force to counteract the horizontal component of the roof's force. The problem states that the flying buttress meets the wall 10m below the roof base, at 30m height.

05

Calculate Horizontal Force Required by Buttress

For the roof weighing \(20,000 \, \text{N}\) with a slope of \(40^{\circ}\), find the total horizontal component of force exerted by both sides. Let \( F_t = 20,000 \, \text{N} \cdot \sin(40^{\circ}) \). Solving for one side gives:\[ F_{h,b} = 10,000 \cdot \sin(40^{\circ}) \approx 6,427.876 \, \text{N} \].Since the buttress needs to counteract half this force,\[ F_{buttress} = F_{h,b} = 6,427.876 \, \text{N} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horizontal Force

In building structures, understanding horizontal force is critical, particularly when dealing with sloped roofs. When a roof has a slope, like in Gothic architecture, it exerts force not only in the vertical direction (due to gravity) but also horizontally. This horizontal component tends to push the walls outward. To determine this force, we use trigonometry.

The horizontal force (\( F_h \)) can be calculated using the formula:\[ F_h = F \cdot \sin(\theta) \]where \( F \) is the weight of one side of the roof and \( \theta \) is the angle of the roof slope. This mathematical operation resolves the force vector into components we can analyze effectively. In the context of the exercise, for a roof weighing 10,000 N and sloping at 35°, the horizontal force is approximately 5,735.76 N.

• Horizontal forces are significant because they impact wall stability.
• They depend on the slope and weight of the roof.

Recognizing the influence of horizontal force helps in planning architectural solutions like flying buttresses to maintain stability.

Wall Stability

Wall stability is an important consideration in architectural design, especially for structures with sloped roofs like in the original exercise. The key concern is how horizontal forces from the roof can destabilize walls, particularly more so in taller walls due to increased leverage.

This happens because the height of a wall becomes a lever arm when forces apply. Taller walls have longer lever arms, meaning the torque (rotational force) is greater, increasing the risk of walls collapsing. This is why tall walls, without adequate support, are more prone to instability under the same horizontal force. Short walls, with smaller lever arms, are comparatively more stable. Thus, architects must consider:

• The height of the walls and their susceptibility to torque.
• The need for supports or countermeasures, like flying buttresses, to mitigate horizontal forces.

Flying buttresses are crucial in directing some of this force towards the ground, holding the walls intact, and allowing for the beautiful tall structures seen in Gothic architecture.

Gothic Architecture

Gothic architecture is renowned for its majestic tall structures and intricate designs, notably in cathedrals built during the 12th to 16th centuries. One of the hallmark solutions to the design challenges presented by these impressive edifices is the flying buttress.

Originally implemented to address specific structural challenges like wall stability under the influence of sloped roofs, flying buttresses provided a groundbreaking solution. These external supports effectively countered the horizontal forces exerted by ornate roofs, allowing architects to construct taller and more light-filled interiors. This innovation:

• Helped distribute the weight of the roofs more evenly.
• Allowed for larger windows and thus more natural light inside.
• Enabled the walls to be thinner yet maintain stability.

Thanks to flying buttresses, the architects of the Gothic period could build their vision of reaching toward the heavens, characterized by immense facades and soaring towers, which still evoke awe and admiration today.