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Concept# Shear modulus

Summary

In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:
where
= shear stress
is the force which acts
is the area on which the force acts
= shear strain. In engineering , elsewhere
is the transverse displacement
is the initial length of the area.
The derived SI unit of shear modulus is the pascal (Pa), although it is usually expressed in gigapascals (GPa) or in thousand pounds per square inch (ksi). Its dimensional form is M1L−1T−2, replacing force by mass times acceleration.
The shear modulus is one of several quantities for measuring the stiffness of materials. All of them arise in the generalized Hooke's law:
Young's modulus E describes the material's strain response to uniaxial stress in the direction of this stress (like pulling on the ends of a wire or putting a weight on top of a column, with the wire getting longer and the column losing height),
the Poisson's ratio ν describes the response in the directions orthogonal to this uniaxial stress (the wire getting thinner and the column thicker),
the bulk modulus K describes the material's response to (uniform) hydrostatic pressure (like the pressure at the bottom of the ocean or a deep swimming pool),
the shear modulus G describes the material's response to shear stress (like cutting it with dull scissors).
These moduli are not independent, and for isotropic materials they are connected via the equations
The shear modulus is concerned with the deformation of a solid when it experiences a force parallel to one of its surfaces while its opposite face experiences an opposing force (such as friction). In the case of an object shaped like a rectangular prism, it will deform into a parallelepiped. Anisotropic materials such as wood, paper and also essentially all single crystals exhibit differing material response to stress or strain when tested in different directions.

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The load-carrying capacity of many reinforced concrete structures is governed by shear failures, occurring before reaching the flexural capacity of the member. For redundant systems, such as slabs subjected to concentrated loads, local shear failures (typically initiated at locations with highest shear forces) can however occur after redistributions of internal forces due to the propagation of the shear cracks. Such process can depend upon the development of shear strains and the softening response of the member and can be stable or unstable. A suitable understanding and modelling of the complete shear response of reinforced concrete, including its deformations both for its pre-and post-peak branches, is thus instrumental for a consistent and comprehensive analysis of the shear response and strength of redundant elements.Such topic has received little attention in the past and analyses of redistributions of internal forces in concrete structures are often performed on the basis of refined flexural models, but coarse considerations for shear strains (typically elastic laws). This situation is a consequence of the lack of consistent experimental measurements on the shear deformations of reinforced members both before and after reaching the maximum shear capacity. Currently, however, the advent of refined measurements techniques such as Digital Image Correlation allows for an accurate tracking of the shear strains and for a fundamental understanding of its development. In this paper, taking advantage of such techniques, a comprehensive approach for determining the shear strains and their distribution across the depth of a section is presented. This approach allows reproducing accurately the development of shear strains and to predict the load-carrying capacity of redundant systems. The model is validated with selected test data and is considered as an effort to contribute to future numerical implementations of reinforced concrete shell models with realistic out-of-plane responses.

Reinforced concrete planar members, as slabs and shells, are structural elements commonly used in the construction technique, which are typically designed without the arrangement of shear reinforcement. Despite the fact that this solution allows for fast and economic construction, the absence of shear reinforcement can give rise to the potential localization of strains within a critical shear crack and eventually to the shear failure of the member. In the case of redundant systems, most research on the mechanics of shear failures has been devoted to the strength of the member, neglecting in many cases the development of shear deformations due to inclined cracking as well as the redistributions of internal forces, which are instrumental for the analysis of the response of these members. The contribution to the state-of-the-art includes a series of theoretical works explaining the observed responses for a series of experimental programmes. These experimental campaigns comprise tests in tension, shear tests in one- and two-way slabs as well as punching tests. For their instrumentation, in addition to classical measurement devices, Fibre-Optic Measurements and Digital Image Correlation were intensively used.This thesis starts by revisiting the basis of the interaction between reinforcement and concrete. A series of bond tests show the stress concentrations occurring near the ribs and its complex transfer of forces with the surrounding concrete. In addition, tests on beams failing in shear show a complex interaction between bond stresses and kinking on the reinforcement due to the development of dowel action. These phenomena are normally neglected for concrete design due to the ductile nature of reinforcement, but may be relevant for fatigue and negative tension-stiffening effects.An important step in the knowledge is performed on the understanding of the shear response with respect to the characterization of the deformations in concrete members. Based on a series of test results, a complete description of the deformation field (including shear strains) is presented. On that basis, a rational model is proposed, consistent with the mechanical model of the Critical Shear Crack Theory. This model allows for a precise description of the response and also to describe the through-thickness distribution of the shear deformation.A general frame for modelling of reinforced concrete slabs is thus presented accounting for the redistribution of internal forces during propagation of the shear crack. This approach is used to investigate a testing programme performed on three wide slabs, analysing in a scientific manner the influence of the width of the member on the shear resistance. The detailed experimental data allow to capture the crack propagation and internal forces redistributions. Clear conclusions and answers are obtained, showing the influence of the shape of the failure surface and of its propagation on the load-carrying capacity.The research ends with a final investigation on the dowelling action of compression reinforcement, with an application to slabs failing in punching. Based on a large testing programme including eleven axisymmetric punching tests, an analytical approach is developed to estimate the contribution of the dowel action on the load-carrying capacity. This approach is formulated within the frame of the Critical Shear Crack Theory, and is incorporated in a consistent and efficient manner for design purposes.

Katrin Beyer, Bastian Valentin Wilding, Michele Godio

When designing unreinforced masonry buildings, the wall stiffness and, consequently, the masonry elastic and shear modulus E and G are essential parameters. Current codes provide empirical estimates of the masonry elastic modulus and a ratio between the shear and elastic modulus, G/E. This ratio, commonly taken as 0.4, is not based on scientific evidence and there appears to be no consensus concerning its value and influencing parameters, meaning that current code standards might not accurately portray the shear deformations of masonry elements. To give the choice of the G/E ratio a theoretical foundation, this paper presents closed-form expressions for the G/E ratio of running-bond masonry that capture the effects of finite joint thickness, finite wall thickness and orthotropic block properties. Based on the geometry of blocks and joints as well as their elastic parameters, a validation of the developed expression using 3D finite element analyses shows good performance. For modern masonry typologies with hollow clay bricks, a G/E ratio of 0.20-0.25 is obtained. For historical masonry typologies, such as dry-stacked or mortared stone masonry, as well as solid clay brick masonry, ratios between 0.30 and 0.40 are computed.

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