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Chapter 11: Problem 75

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 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?

Short Answer

Expert verified
The horizontal force is 5,736 N. Tall walls are more at risk of collapse. The flying buttress needs to apply 12,856 N.

Step by step solution

01

Understanding the Problem

The problem involves calculating the horizontal force exerted by a sloped roof on the walls of a building and determining which type of building is more susceptible to wall collapse. Additionally, it includes using flying buttresses as support to prevent wall collapse in Gothic architecture.
02

Analyzing Part (a) for the Symmetric Building

Each side of the roof is subject to gravitational force acting vertically downward with a magnitude of 10,000 N. The weight of the roof can be resolved into two components: one perpendicular to the roof surface and the other along the roof surface. The horizontal component that pushes out the walls is the one along the roof surface.
03

Calculating Horizontal Force for Part (a)

The horizontal component of the force exerted by the roof is given by \( F_{\text{horizontal}} = W \sin(\theta) \), where \( W \) is the weight of one side of the roof (10,000 N) and \( \theta \) is the angle of the roof slope (35.0 degrees). \[F_{\text{horizontal}} = 10,000 \sin(35.0^{\circ}) = 10,000 \times 0.5736 = 5,736 \, \text{N}\]
04

Interpreting the Results of Part (a)

Tall walls are more susceptible to collapse under the horizontal force since they require greater structural support to prevent bending or breaking. The taller the wall, the more leverage the horizontal force has, increasing the risk of collapse.
05

Analyzing Part (b) for Gothic Architecture

In Gothic architecture, a solution to prevent the collapse was the use of flying buttresses. We need to find the horizontal force required by the flying buttress to hold the wall. The total weight of the roof is 20,000 N, with the angle of the roof being 40 degrees, and the buttress meets the wall 10 meters below the roof.
06

Calculating Horizontal Force for Part (b)

The horizontal force exerted by the roof is calculated by resolving the total weight of the roof into its horizontal component:\[F_{\text{horizontal}} = W \sin(\theta) = 20,000 \sin(40^{\circ})\]\[F_{\text{horizontal}} = 20,000 \times 0.6428 = 12,856 \, \text{N}\]This horizontal force must be counteracted by the flying buttress.

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Key Concepts

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

Statics
In the world of mechanics, statics is the branch that deals with analyzing forces in a state of rest. This means when an object is not moving, the sum of all forces acting on it equals zero.

For buildings, it's crucial to understand these concepts to ensure they withstand different forces without collapsing. By examining these forces and making sure they balance out, architects and engineers can design structures that stand the test of time. A prime example is a roof's weight pressing down and creating a horizontal force that could push walls outward. Without proper design, such forces might lead to structural failure.
Forces
Forces in mechanics dictate how objects move or stay still. These forces can act vertically, horizontally, or at an angle.

The intriguing part is breaking down these forces into components to analyze how they interact. In our problem, the roof's weight represents a force acting downward due to gravity.

Understanding that this weight can be broken into two components:
  • One parallel to the roof slope, which is the cause of the threatening horizontal force.
  • The other perpendicular to the roof surface.
This horizontal component is key because it acts outward, pushing on the walls of the building. Calculating this horizontal force helps in understanding the type of support the structure needs.
Equilibrium
Equilibrium, a state of balance, is where all forces cancel each other out, resulting in no net movement. In architecture, achieving equilibrium is vital for a building's stability.

When you see a roof pushing against walls, the walls must offer enough resistance to balance that force, keeping everything in place.

Equilibrium ensures that each push has a counteracting pull, enabling the structure to maintain its integrity. In our problem, focusing on equilibrium allows us to calculate precisely how much force supports, like flying buttresses, need to apply to maintain balance and prevent collapse.
Architectural Engineering
Architectural engineering combines principles of engineering and architecture to design buildings that are not just aesthetically pleasing but also safe and resilient.

It involves understanding the forces acting on a structure and ensuring they are properly managed. This knowledge becomes especially vital when designing features like the Gothic flying buttress, an innovative solution devised during the medieval period to counteract the horizontal forces generated by soaring roofs.

Flying buttresses are structures that redistribute the forces, enhancing the stability and longevity of the building. Architects use these engineering principles to craft not only beautiful but enduring structures.

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Most popular questions from this chapter

CP Blo Stress on the Shin Bone. The compressive strength of our bones is important in everyday life. Young's modulus for bone is about \(1.4 \times 10^{10}\) Pa. Bone can take only about a 1.0\(\%\) change in its length before fracturing. (a) What is the maximum force that can be applied to a bone whose minimum cross- sectional area is 3.0 \(\mathrm{cm}^{2} ?\) (This is approximately the cross-sectional area of a tibia, or shin bone, at its narrowest point.) (b) Estimate the maximum height from which a 70 -kg man could Jump and not fracture the tibia. Take the time between when he first touches the floor and when he has stopped to be 0.030 s, and assume that the stress is distributed equally between his legs.

Two people carry a heavy electric motor by placing it on a light board 2.00 \(\mathrm{m}\) long. One person lifts at one end with a force of \(400 \mathrm{N},\) and the other lifts the opposite end with a force of 600 \(\mathrm{N}\) . (a) What is the weight of the motor, and where along the board is its center of gravity located? (b) Suppose the board is not light but weighs \(200 \mathrm{N},\) with its center of gravity at its center, and the two people each exert the same forces as before. What is the weight of the motor in this case, and where is its center of gravity located?

A nonuniform fire escape ladder is 6.0 m long when extended to the icy alley below. It is held at the top by a frictionless pivot, and there is negligible frictional force from the icy surface at the bottom. The ladder weighs \(250 \mathrm{N},\) and its center of gravity is 2.0 \(\mathrm{m}\) along the ladder from its bottom. A mother and child of total weight 750 \(\mathrm{N}\) are on the ladder 1.5 \(\mathrm{m}\) from the pivot. The ladder makes an angle \(\theta\) with the horizontal. Find the magnitude and direction of (a) the force exerted by the icy alley on the ladder and (b) the force exerted by the ladder on the pivot. (c) Do your answers in parts (a) and (b) depend on the angle \(\theta\) ?

BIO Supporting a Broken Leg. A therapist tells a 74 -kg patient with a broken leg that he must have his leg in a cast suspended horizontally. For minimum discomfort, the leg should be supported by a vertical strap attached at the center of mass of the leg-cast system. (Fig. P11.55). In order to comply with these instructions, the patient consults a table of typical mass distributions and finds that both upper legs (thighs) together typically account for 21.5\(\%\) of body weight and the center of mass of each thigh is 18.0 cm from the hip joint. The patient also reads that the two lower legs (including the feet) are 14.0\(\%\) of body weight, with a center of mass 69.0 \(\mathrm{cm}\) from the hip joint. The cast has a mass of \(5.50 \mathrm{kg},\) and its center of mass is 78.0 \(\mathrm{cm}\) from the hip joint. How far from the hip joint should the supporting strap be attached to the cast?

BIO Leg Raises. In a simplified version of the musculature action in leg raises, the abdominal muscles pull on the femur (thigh bone) to raise the leg by pivoting it about one end (Fig. P11.57. When you are lying. horizontally, these muscles make an angle of approximately \(5^{\circ}\) with the femur, and if you raise your legs, the muscles remain approximately horizontal, so the angle \(\theta\) increases. We shall assume for simplicity that these muscles attach to the femur in only one place, 10 \(\mathrm{cm}\) from the hip joint (although, in reality, the situation is more complicated). For a certain \(80-\mathrm{kg}\) person having a leg 90 \(\mathrm{cm}\) long, the mass of the leg is 15 \(\mathrm{kg}\) and its center of mass is 44 \(\mathrm{cm}\) from his hip joint as measured along the leg. If the person raises his leg to \(60^{\circ}\) above the horizontal, the angle between the abdominal muscles and his femur would also be about \(60^{\circ} .\) (a) With his leg raised to \(60^{\circ},\) find the tension in the abdominal muscle on each leg. As usual, begin your solution with a free-body diagram. (b) When is the tension in this muscle greater: when the leg is raised to \(60^{\circ}\) or when the person just starts to raise it off the ground? Why? (Try this yourself to check your answer.) (c) If the abdominal muscles attached to the femur were perfectly horizontal when a person was lying down, could the person raise his leg? Why or why not?
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