Stress-Strain Behaviour of Concrete
A typical relationship between stress and strain for normal strength concrete is presented in Figure 1. After an initial linear portion lasting up to about 30 – 40% of the ultimate load, the curve becomes non-linear, with large strains being registered for small increments of stress. The non-linearity is primarily a function of the coalescence of microcracks at the paste-aggregate interface. The ultimate stress is reached when a large crack network is formed within the concrete, consisting of the coalesced microcracks and the cracks in the cement paste matrix. The strain corresponding to ultimate stress is usually around 0.003 for normal strength concrete. The stress-strain behaviour in tension is similar to that in compression.
The descending portion of the stress-strain curve, or in other words, the post-peak response of the concrete, can be obtained by a displacement or a strain controlled testing machine. In typical load controlled machines, a constant rate of load is applied to the specimen. Thus any extra load beyond the ultimate capacity leads to a catastrophic failure of the specimen. In a displacement controlled machine, small increments of displacement are given to the specimen. Thus, the decreasing load beyond the peak load can also be registered. The strain at failure is typically around 0.005 for normal strength concrete, as shown in Figure 2. The post peak behaviour is actually a function of the stiffness of the testing machine in relation to the stiffness of the test specimen, and the rate of strain. With increasing strength of concrete, its brittleness also increases, and this is shown by a reduction in the strain at failure.
Figure 1. Stress-strain relationship for ordinary concrete
Figure 2. Complete stress-strain curve including post-peak response
It is interesting to note that although cement paste and aggregates individually have linear stress-strain relationships, the behaviour for concrete is non-linear. This is due to the mismatch and microcracking created at the interfacial transition zone.
Understanding the post peak response of concrete
Concrete belongs to a class of materials that
can be called ‘Strain – softening’, indicating a reduction
in stress beyond the peak value with an increase in the deformation (as
against the strain hardening behaviour commonly exhibited by metals like
steel). Figure 3 shows different types of material behaviour.
Figure 3. Different types of material behaviour (post peak response)
Although the ductility of concrete is several orders of magnitude lower than steel, it still exhibits considerable deformation before failure. In conventional testing machines, where the test is performed under control of loading rate, a sudden failure of the specimen occurs as soon as the maximum load level is reached – the machine gives small increments of load to the specimen and the resultant deformation is measured, as a result, when the incremental load goes over the maximum level, the specimen fractures suddenly. This is depicted in Figure 4. In order to obtain the entire stress-strain graph, inclusive of the post peak region, deformation or strain controlled test must be performed.
Figure 4. Modes of testing – Green indicates load control, red indicates displacement control
A displacement controlled test is possible using a machine with a servo valve, in a closed loop. As shown in the schematic diagram in Figure 5, the machine compresses the concrete specimen at a constant displacement rate of the specimen – the LVDT on the specimen provides feedback to the controller, which then indicates to the servo valve the degree of piston movement to be provided (to keep the specimen displacement constant). In this way, the load response of the specimen is continuously studied as it undergoes incremental displacements. Failure occurs when the cracks in the specimen grow to an ‘unstable’ size.
Figure 5. Closed loop servo controlled test system
Prof. Jason Weiss, School of Civil Engineering, Purdue University.
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