8.9: Resulting Forces and Mohr Circles

Last updated
Save as PDF

Page ID: 34845

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Screen Shot 2020-08-23 at 1.49.24 PM.png — Figure 8-49: Horizontal cutting force: α**=60^o, UCS=100 MPa,** φ**=20^o, h**_i=0.1 m & w=0.1 m.

Screen Shot 2020-08-23 at 1.50.24 PM.png — Figure 8-50: Vertical cutting force: α**=60^o, UCS=100 MPa,** φ**=20^o, h**_i=0.1 m & w=0.1 m.

Figure 8-49 and Figure 8-50 show the horizontal and vertical cutting forces. The transition tensile failure/shear failure occurs at a tensile strength of about -8.5 MPa, so 8.5% of the UCS value. From a tensile strength of -8.5 MPa (8.5% UCS) to -20 MPa (20% UCS), both tensile failure and shear failure are possible. Below a tensile strength of -20 MPa (20% UCS) only shear failure is possible. Figure 8-51 and Figure 8-52 show the Mohr circles for tensile strengths of -8.5 MPa (8.5% UCS) and -20 MPa (20% UCS). One can see that with a tensile strength of -8.5 MPa both tensile failure and shear failure are possible, with a tensile strength of -20 MPa also both tensile failure and shear failure are possible. It should be mentioned here that the forces shown are peak forces, so average forces may reduce to 50%-60%. So the two limiting cases are shown.

Screen Shot 2020-08-23 at 1.51.48 PM.png — Figure 8-51: Mohr circles tensile strength -8.5 MPA.

Screen Shot 2020-08-23 at 1.53.00 PM.png — Figure 8-52: Mohr circles tensile strength -20 MPa.

Screen Shot 2020-08-23 at 1.53.47 PM.png — Figure 8-53: The tensile vs. shear failure range based on the BTS to cohesion ratio.

Screen Shot 2020-08-23 at 1.54.31 PM.png — Figure 8-54 The tensile vs. shear failure range based on the UCS to BTS ratio.