1. THE FOUR-HINGE METHOD OF
ANALYSIS OF MASONRY ARCHES
STUDENT: PAVEL TROFIMOV
D08118499 DT024/4

2. SUMMARY OF WORK CARRIED
OUT
I. Preliminary design
Figure 3 Segmental masonry arch
Figure 1 Force diagram Figure 2 Line of thrust
Table 1 Stresses in the arch
Due to self-weight
fm.min = 0.035 (N/mm2
) , Tension
fm.max = 0.068 (N/mm2
) , Compression
Allowable
ft.bond.MIN = 0.091 (N/mm2
), Tension
fm.k = 10.7 (N/mm2
), Compression

3. SUMMARY OF WORK CARRIED
OUT
II. Theoretical analysis
Results
Model 1 (Unreinforced) P = 2.4 kN
αB=64.97˚
βC=70.44˚
Model 2 (Reinforced) P = 68.4 kN
αB=64.97˚
βC=57.39˚
Figure 6 Angles to hinges B&C
Figure 5 Collapse load Model 2Figure 4 Collapse load Model 1

4. SUMMARY OF WORK CARRIED
OUT
III. Experimental
Figure 7 Experimental setup
Figure 8 Arch contruction
Figure 9 Abutment model 2 Figure 11 Model 1 test setupFigure 10 Abutment model

5. PROJECT FINDINGS
Figure 12 Failure mode, Model 1 Figure 13 Failure mode, Model 2
Table 2 Collapse load
Name kN
Model 1Exp. 9.58
Model 2Exp. 35.88
Model 1Theor. 2.36
Model 2Theor. 68.42
Figure 15 Tension zone
Figure 16 Compression zoneFigure 14 Load vs. displacement

6. SUMMARY OF WORK CARRIED
OUT
IV. LUSAS Finite Element Analysis
Applied actions:
PMODEL 1 = 5.7 kN
PMODEL 2 = 20.0 kN
Figure 17 Isometric view, LUSAS model Figure 18 Maximum principal stresses (Model 2)
Table 3 Stresses LUSAS vs. allowable, Compression ‘-‘, Tension ‘+’
Name N/mm2
ShearEpoxy ShearMasonry FlexureMasonry TensionSteel TensionEpoxy
Model 1LUSAS 0.24 (B) 0.78 / -0.92 (B)
Model 2LUSAS 2.91 (A-B) 0.84 (B) 2.70 / -3.50 (B) 8.99 (B) 0.01 (B)
Allowable 2.03 1.82 0.091 / -10.7 154 0.12

7. PROJECT FINDINGS
• The formation of four-hinge mechanism does not necessarily lead to ultimate collapse but rather to SLS failure in
unreinforced masonry arches.
• In masonry arches reinforced with the near surface reinforcement, as examined, it’s likely to get a failure via debonding of
reinforcement.
• Strengthening of the arch on the soffit with near surface reinforcement proves to provide a stiffer structure. It reduces
deflections and, prevents crack formation and propogation.
• This form of strenghtening of masonry arches proves to inefficient, as it difficult to utilize the tensile properties of
reinforcement at the hinges where it expected to be in tension.
• The four-hinge mechanism of failure is the mode of failure most likely to occur in the arch with a quarter-span point load,
however, it has been shown in leterature that in 90% of the cases the presence of the fill responsible for the stability of the
arch, thus it can result in a different mode of failure.
• Based on the limited models tested, the method of analysis to predict the collapse load for masonry arches must be treated
with caution as it heavily relies on the assumptions such as: abutments are rigid and the four-hinge mechanism of failure is
the only mode of failure allowed to develop.
Figure 19 Debonded reinforcement, tension zone. Figure 20 Debonded reinforcement, compression zone.

8. SUGGESTIONS FOR THE FUTURE
RESEARCH
• The assumptions that abutments are rigid must be strictly adhered to, this can be done by ensuring no rotation is allowed
in the reaction frame e.g. by providing spikes welded to the reaction frame to prevent the rotation of concrete abutment
inside the casing.
• To investigate the behaviuor and the mode of failure of the arches further, presence of the fill must be considered.
• Different method of reinforcing the arch may be concidered in aim to achieve a full composite action between steel and
masonry. This can be done by cutting grooves in the masonry and embedding steel bars inside the grooves which can
provide for a larger contact area between steel and brick. Also, mechanical ancors may be considered either insitu or
drilled to hold the steel in place.
• Provision of strain gauges on the reinforcing steel to record the stresses in the reinforcement during the testing. The
attempt must be made to place the gauges where hinges are expected to form on the intrados.
• Different finite element modelling techniques may by investigated to approximate the real structure. Finite element model
to simulate the cracked section can give more accurate results on the behaviour of the arch under load.
Figure 21 Abutment rotation.

9. THANK YOU

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