Mechanical eng
MECHANICS LAB AM 317
EXP 9 TWO-PINNED ARCH
I. OBJECTIVES
I.1 To explore the relationship between the loads and the horizontal reaction force in a two-pinned arch.
I.2 To determine the horizontal thrust force and reaction influence lines for a point load moving across a two-pinned arch.
I.3 To determine the horizontal thrust force for a uniformly distributed load on a two-pinned arch.
Figure 1 St. Louis Arch
II. INTRODUCTION AND BACKGROUND
The main advantage an arch has over a beam, is that it can carry a much larger load. Historically arches were important because they could be constructed using small, easily carried blocks of brick or stone rather than using a massive, monolithic stone beam or lentil. Romans used the semicircular arch in bridges, aqueducts, and in large-scale architecture. In most cases they did not use mortar, relying simply on the precision of their stone finish. When an arch is loaded by gravity forces, the pressure acts downward on the arch and has the effect of compressing it together instead of pulling it apart. A free-body diagram of the arch supports shows the arch requires both horizontal and vertical reaction
R. Ehrgott (Created) 2/7 03/31/01 T. Hao (Revised) 08/28/16
forces (Figure 2). This horizontal reaction force, called thrust, can cause the arch to collapse if it is not properly restrained.
HA
HB A B
VA VB
Figure 2 Free-Body Diagram of Two Hinged Arch.
One of the disadvantages of the arch resisting loads in compression is the possibility that the arch may buckle. Any practical arch design would include analysis involving stress, deflection and buckling. To perform such analysis, reaction forces, shear and moment diagrams are needed for a given load case. Since the arch is indeterminate, having more unknown reactions forces than equilibrium equations, methods such as the flexibility method are required to determine the reaction forces.
III. EQUIPMENT LIST
Structures test frame
Digital force display and power supply
Aluminum arch
Hangers and weights
Scale
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Figure 3 Test Frame and Two Hinged Arch Set-Up
IV. PROCEDURE
• Visually inspect all parts of the test frame (including electrical leads) for damage or wear.
• Check that electrical connections are correct and secure.
• Check that all components are secured correctly and fastenings are sufficiently tight.
• Check the four securing thumbscrews are in the position shown in Figure 3 and the rolling pivot on the right is gently resting against the load cell.
• Make sure the digital force display is on and the force transducer is connected from the socket marked ‘Force Output’ on the right support, to the Digital Force Display ‘Force Input 1’.
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• Carefully zero the force meter using the dial on the right-hand side support. Gently apply a small load with a finger to the crown of the arch and release. Zero the meter again if necessary.
NEVER apply excessive loads to any part of the equipment.
PART A
The first experiment is to measure the horizontal thrust force HB for a single load that is placed at increasing distances from the left support. A theoretical equation for the horizontal thrust force for a given load P at location x, is given below:
)2( 8 5 233
3 LxxLrL Px
H B −+= 9.1
where: HB = the horizontal thrust reaction at B (N) P = the load (N) L = the span of the arch (m) x = the load location, distance from the left-hand side support (m) r = the Rise of the arch (m)
IV.1A Measure the necessary dimensions of the two-pinned arch and record the
data.
IV.2A Adjust the “set zero” control on the right support so that the digital force reads zero.
IV.3A Apply a 100 gm load to the left most hanger and record the thrust force. Moving the 100 gm load to the remaining hangers and record the thrust force for each location. Repeat the process for a 500 gm load. The horizontal thrust shown on the digital force display has units of Newton.
PART B
The second part of the experiment involves determining the horizontal thrust force for a uniform or distributed load w. The equation for the thrust force is given in Eq. (9.2).
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r wL
H B 8
2
= 9.2
where: HB = the horizontal thrust reaction at B (N) w = the intensity of uniform distributed loads (N/m) L = the span of the arch (m) r = the Rise of the arch (m)
IV.1B Adjust the “set zero” control on the right support so that the digital force
reads zero.
IV.2B Apply 60 gm on each of the nine hangers (total of 540 gm) and record the thrust force. Repeat the procedure with 120 gm on each hanger (1080 gm total).
V. REPORT
V.1 Plot the experimental and theoretical thrust force determined in Part A of the experiment.
V.2 Compare the theoretical and experimental thrust force for the uniformly loaded arch.
VI. SELECTED REFERENCES
Megson, T.H.G., Structural and Stress Analysis, 2nd edition, Butterworth- Heinmann, 2005, pp. 133-149.
Timoshenko, S.P., Young, D.H., Theory of Structures, 2nd edition, McGraw Hill Co., New York, 1965.
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Table I Arch data
Rise of arch, r (m) Length of arch, L (m)
Table II Weight data
Added mass 100 gm 500 gm Hanger mass
Total mass
Total weight (N)
Table III Data for a point load moving across a two-pinned arch
Load location x (mm)
Measured thrust force for 100 gm load (N)
Measured thrust force for 500 gm load (N)
0 0 0 50
100
150
200
250
300
350
400
450
500 0 0
Table IV Data for a uniformly distributed load
Total mass (gm)
Total mass of hangers
(gm)
Uniform distributed load (N/m)
Measured thrust force
(N)
Calculated thrust force
(N) % Error
540
1080
R. Ehrgott (Created) 7/7 03/31/01 T. Hao (Revised) 08/28/16