3.3 Analytical investigation of the shear behaviour of bolted joints
3.3.5 Analytical models based on the equilibrium of the forces acting on the
In Holmberge’s study (1991), an analytical mechanical model was proposed about the maximum shear resistance and maximum deformation of a bolted joint. As shown in Figure 3.14, the loading condition of the deformed grouted rock bolt is considered to be in four stages:
Bolt and surrounding medium in an elastic state;
Bolt in elastic and surrounding medium in a yielding state;
Bolt and surrounding medium yielded;
Ultimate bent condition as shown in Figure 3.14 d.
Figure 3.14 Four stages of grouted rock bolt subject to shear loading (Holmberge,1991) The failure mode of cable bolts was determined by the angle between the bolt and the joint. This theory helps understand the relationship between bolt and grout and the axially and laterally contribution of bolt to shear resistance.
In the elastic stage, the relationships between the axial bolt load, the lateral bolt load, the axial deformation and the lateral deformation of a bolt, are as follow:
71 is the lateral bolt deformation at the joint intersection.
Under the elastic state, the failure of bolts may occur at the bolt-joint intersection due to a combination of the axial force and the shear force in the bolt. Thus substituting , , into the following failure criterion and yields a detailed failure criterion as follow:
+ = 1 (3.72)
+ = 1 (3.73)
In the elastic bolt at yield stage, the expressions of lateral load, lateral deformation and axial load are
= (3.74)
= + + − ∙ + + + (3.75)
= + (3.76)
Where is the yielding length of the subgrade.
The yield criterion at plastic hinges and the failure criterion at intersection are
+ = 1 (3.77)
+ + = 1 (3.78)
When the combination of axial load and bending moment exceeds the yield strength of the bolt, the bolt reaches to yield stage. When the shear load continues to apply on the bolt, the ultimate yield stage is reached, it was assumed that
= (3.79)
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= (3.80)
Thus the corresponding slope ( ) is
= − (3.81)
Hence:
= (3.82)
Additionally, the mean extension strain is
= 1 + + + 1 + − 1 (3.83)
Where is the length of yielding bolt
Pellet and Egger (1996) proposed a theoretically model for reinforced rock joints and the contribution of fully grouted rock bolts to the shear strength of joints being investigated. In their analysis, the bolt behaviour was divided into elastic stage and plastic stage as shown in Figure 3.15. These yield and failure criterions as well as stress state of bolts in shear are presented in Figure 3.16.
a) elastic zone; (b) plastic zone
Figure 3.15 Force components and deformation of a bolt
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The bolt contribution is divided into additional cohesion effect and confining effect, which are related to the parallel force component and the normal force component to the joint. That is:
= + ∆ + ( + ∆ ) ∅ (3.84)
Where is the joint cohesion; ∆ is the additional cohesion provided by the bolt;
is the initial confining stress on the joint; ∆ is the additional confining stress provided by the bolt; ∅ is the joint friction angle.
Figure 3.16 Shear force versus axial force in the bolt
The shear forces and corresponding displacements of the bolt at the end of the elastic stage and plastic region were expressed as:
= − (3.85)
= 1 − 16 (3.86)
= (3.87)
= ∆
( ∆ ) (3.88)
∆ = ± (1 − ) (3.89)
Where is shear force acting at point O (Figure 3.16)at the yield stress of the bolt;
o
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is the bearing capacity of the grout or rock; is dimeter of the bolt; is yield stress of the bolt; is axial force acting on shear plane at the yield stress of the bolt; is shear force acting at shear plane at failure of the bolt; is failure stress of the bolt;
is axial force acting at shear plane at failure of the bolt; is axial displacement at point O; E is bolt elasticity; = tan ; is shear displacement at point O. ∆ is the increment of plastic rotation; l is distance between point O and point A; l is the length of the part O-A at failure as indicated in Figure 3.16.
Based on the numerical and laboratory studies of reinforced rock joints, Ferrero (1995) proposed a calculation model which was applicable to bolts installed perpendicular to the joint surfaces. Rock material properties (strong rock > 50 and weak rock < 50 ) decided the mechanisms of reinforcement failure in this model. The overall reinforcement system shear resistance contains two parts: the friction effect and the dowel effect and the global joint resistance can be expressed as:
= + + ( − ) (3.90)
Where T is load induced in the bolt; α is angle between the joint and the dowel axis; Q is force due to dowel effect;φ is Joint friction angle. Figure 3.17 shows the loading state of the bar and its element.
(a) (b)
Figure 3.17 (a) Considered static scheme and (b) forces acting on a bar element When the surrounding rock behaves in the plastic condition, equilibrium equations at both parallel and perpendicular directions to the bar axis can be depicted as:
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= +
= −
= −
(3.91)
Where is the bar curvature, is the static moment on the bar.
With some assumptions, the relationship between the bar tension and its curvature can be derived as follow:
= (3.92)
For the first yielding mechanism in stiff and strong rock condition, the broken bars were measured and their shapes were approximated with a parabolic equation. The curvature equation along the bar can be expressed as
= / == / (3.93)
Jalalifar (2006) proposed analytical models and evaluated the bolt behaviour subjected to shearing. According to beam theory, the transversely loaded bolt behaviour is shown in Figure 3.18.
Figure 3.18 Deformed shape, shear force, bending moment and shear displacement diagrams (Jalalifar, 2006)
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The axial force and the shear force at bolt-joint intersection are calculated as:
= 1 − 4( ) (3.94)
= 1 − ( ) (3.95)
Where A is bolt cross section area; σ is failure stress at bolt material ; Q is shear force acting at point C in elastic limit; Q is shear force acting at point C at failure of the bolt material.
Axial force and shear force at the point of maximum bending moment are calculated as:
= 1 − 16
. (3.96)
= . 1 − ( ) (3.97)
The hinge point distance to the shear joint can be expressed as a following equation.
= (3.98) Where p is reaction force; σ is elastic yield stress
Li (2016) following the research of Jalalifar (2006) proposed a mathematical solution for the behaviour of cable bolts. Figure 3.19 shows a typical loading state of a bolted concrete joint subject to shearing, based on the theory of beam on elastic foundation.
Clearly, the contribution of bolts to the joint shear resistance is attributed to the induced forces in bolts and across the joint plane. Thus, the global induced forces at the joint could be determined only by the induced forces in bolts. The global contribution of a bolt to the joint is calculated with the following equation:
= cos( − ) + sin( − ) + [ sin( − ) − cos( − )] ∅ (3.99)
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Figure 3.19 Forces induced at the bolted joint plane (Li, 2016)
In the elastic stage, there are three redundant reactions for transforming the original problem into to a statically determinate beam by removing all redundant reactions.
Considering half of the beam, a combination of axial force , shear force , and bending moment can be used to represent the restraint of the other half as shown in Figure 3.20.
Figure 3.20 Loading state of a statically determinate beam
Based on the force method of statically indeterminate structures, three compatibility equations can be written in a matrix form,
+
∆
∆
∆
=
∆
∆
∆
(3.100)
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Thus, the flexibility matrix and the deformation matrix induced by external loads can be obtained based on beam theory as:
Where is the plastic hinge distance from the joint plane to hinge; is a concentration coefficient of the distribution of shear stress at the cable cross section, which is equal to 4/3 for a solid cross section and determined using the inner and outer radius for a hollow cross section (Gere and Timoshenko, 1990). Because of the symmetry of the deflecting beam, it is reasonable to assume = 0 . Thus substituting equations (3.101) and
Solving equations (3.103) in reference to ∆ produces:
= ∆ (3.104)
= ( )( )∙ ∆ (3.105)
= ( )( )∙ ∆ (3.106)
∆ = ∙∆ ∙ ∆ (3.107)
In similar manner as used in the elastic stage, the redundant reactions can be written in the plastic stage as follows:
= ∆ (3.108)
= ∆ (3.109)
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= ∆ (3.110)
Since the deflection angle of bolts in the elastic stage of the host material is very small, in the plastic stage the deflection angle is assumed to start from zero. Thus the deformation compatibility relationship in the plastic stage is the same as in the elastic stage. With the shear force ( ) and the host medium reaction strength ( ), the corresponding contributions can be described as follows:
= (3 − ) + (3.111)
= ( − 4 − 4 ) + (3.112)
Then, the actual deformation curve at failure is given by:
( ) = − (3.113)