(Aktu Btech) Structural Analysis Important Unit-5 Analysis of Arches

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Q1. Discuss what types of straining actions are sustained by an arch ?

Ans.

• 1. Consider a cross-section PQ of the arch Fig.(c). Let T be the resultant thrust acting through D along the linear arch. 
• 2. Thrust T is neither normal to the cross-section nor does it act through the centre C of the cross-section. 

• 3. Resultant thrust T can be resolved normal and tangential to the section PQ. 
• 4. Let N be the normal component and F be the tangential component. 
• 5. The tangential component F will cause shear force at the section PQ and normal component N acts eccentrically, the eccentricity being equal to CD. 
• 6. Thus the action of N acting at D is two fold. 
  • i. A normal thrust Nat C 
  • ii. A bending moment, M = N. e at C. 
• 7. Hence unlike beams a section of arch is subjected to three straining actions. 
  • i. Shear force, F
  • ii. Bending moment, M 
  • iii. Normal thrust, N 
• Sometimes shear force F is also known as radial shear. 

Q2. Show that the parabolic shape is a funicular shape for a three hinged arch subjected to a uniformly distributed load over its entire span. 

Ans. 1. Let          L = Span of the arch, 

h = Central rise, and

w = UDL applied on the arch.

2. From symmetry we have,

3. For horizontal thrust, taking moment about C,

4. Let us now consider any section at distance x from A

Equation of parabola is given by,

5. The value of bending moment at any section of the arch, 

6. A parabolic arch therefore prone to a UDL The bending moment is zero at any point over its whole length. For this reason, a funicular shape is a parabolic shape for a three-hinged arch.

Q3. With a neat sketch describe about spandrel braced arch. 

Ans. Spandrel Braced Arch: 

• 1. A three hinged spandrel braced arch is shown in the Fig.  
• 2. The spandrel of the arch in a spandrel braced arch refers to the area above the arch rib and below the level of the crown.

Types of Spandrel Braced Arch: The spandrel braced arch can be of following types: 

• 1. Closed Spandrel Arch: The area above the arch rib and below the level of the crown in a closed spandrel arch can be filled.
• 2. Open Spandrel Arch: In an open spandrel arch, a truss-like system can be used to fill the space above the arch rib and below the level of the crown.
• Note: The spandrel is often filled with dirt in the case of concrete and masonry arch bridges with short spans.

Q4. A three hinged parabolic arch hinged at the supports and at the crown has a span of 24 m and a central rise of 4 m. It carries concentrated load of 50 kN at 18 m from the left support and UDL of 30 kN/m over the left portion. Determine the normal thrust, radial shear at a section 6 m from the left hand support.

Ans. Given: Span of arch, L = 24 m, Central rise of arch, h = 4m  

Concentrated load, W= 50 kN, Intensity of UDL, w = 30 kN/m

Distance of section = 6m

To Find: Normal thrust and Radial shear.

1. The arch is shown in Fig. Taking moment about support B, we get  

2. Taking moment about crown C (Right part), 

Q5. A three hinged parabolic arch of 50 m span and a rise of 10 m is subjected to a uniformly distributed load of 20 kN/m intensity over its left half portion and point load of 100 kN at right quarter span. Calculate the bending moment, normal thrust and radial shear at a section 10 m from the left support.

Ans. Given: Span of arch, L = 50 m, Rise of arch, h = 10 mm

Intensity of UDL, w = 20 kN/m, Point load, W = 100 kN

To Find: BM, Normal thrust and Radial shear.

1. Reactions at supports :

ii. Taking moment about support A,

iii. Taking moment about C,

2. Bending Moment at Section D:

i. Let a section X-X at a distance 10 m from A. 

Rise of arch at section D,

ii. Bending moment at D, 

3. Normal Thrust and Radial Shear at Section D: 

i. Consider the equilibrium of the part AD of arch. 

iii. Normal thrust at D,

iv. Radial shear at D, 

Q6. A circular arch to span 25 m with a central rise 5 m is hinged at the crown and springing. It carries a point load of 100 kN at 6 m from the left support. Calculate: 

i. The reactions at the supports. 

ii. The reactions at crown. 

Ans. Given: Span of arch, L = 25 m, Central rise, h = 5 m, Load = 100 kN

To Find: V_A, V_B and V_C

3. Considering right part (CB part) of arch,

Taking moment about point C.

5. Angle between plane of section and vertical plane passing through the crown of circular arch,

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