Nominal Shear Capacity of Bolts

Nominal Shear Capacity of Bolts

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The shear capacity of a bolt is dependent on the cross sectional area of the bolt and its material properties. This page presents as an illustrative method for determining the nominal shear capacity of a steel bolt for design in accordance with Eurocode 3. The page is divided into the following sections:

  1. Introduction
  2. Shear Area
  3. Shear Stress
  4. Shear Capacity in Accordance with EN 1993-1-8
  5. Other Standards
  6. Single, Double or Multiple Shear?
Double shear lap connection

References

  1. EN 1993-1-8:2005 – Eurocode 3 – Design of Steel Structures – Part 1-8 – Design of Joints
  2. EN 1993-1-1:2005 – Eurocode 3 – Design of steel structures. General rules and rules for buildings
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Proof Load / Tensile Strength of BoltsBearing Stress and Bolts

1. Introduction

Bolts should never experience a significant shear load unless the design engineer has specifically intended for them to be used bearing type connection. However, there are many reasons why one may want to specify a bearing type connection, in which case it is necessary to know the shear capacity (strength) of the bolts. There are varying approaches that can be used to determine the shear capacity of an individual fastener, and these vary between design code or standards. However, the shear capacity of a bolt across its cross section is always less than the tensile capacity of the bolt normal to the same section. This is due to the mechanical behaviour of steel in pure shear. Fundamentally, the yielding of typical structural steels under pure shear loading starts to occur when the shear stress is equal to the magnitude of material yield stress (or 0.2% proof stress) divided by {\sqrt{3} . The relationship between pure shear stress and normal stresses at yield is described by the von Mises and Tresca failure criteria. These are not discussed here; however I might one day produce an article on it after I remind myself of the basic of Mohr’s Circle…

2. Shear Area

In general, when a bolt experiences shear across the threaded portion, the shear area can be considered equal to the nominal tensile stress area. This approach for shear across the threaded portion is adopted in many common standards, for example Eurocode 3, EN 1993-1-8: 2005 [1]. Therefore, for shearing forces, where the shear plane crosses the threaded portion of a bolt, the shear area is given by:

(1)   \begin{equation*}  \begin{split} A_{\tau} & = \frac{\pi }{4}\left(\frac{d_{2}+d_{3}}{2} \right)^{2}\\  & = A_{\sigma} \end{split} \end{equation*}

If you wanted to be very conservative you could use the slightly smaller cross sectional area prescribed by the minor (root) diameter; however this approach is not normally used. In this case, the shear area would be equal to:

(2)   \begin{equation*}  \begin{split} A_{\tau}=\frac{\pi }{4}\left {(d_{1})^{2}} \end{split} \end{equation*}

In cases where the bolt experiences the shearing force across the unthreaded (shank) portion of the bolt, the designer can consider the full cross sectional area prescribed by the major diameter. This approach is permitted in Table 3.4 of Eurocode EN 1993-1-8: 2005 – Design of Joints [1]: begin{equation} A_{tau}=frac{pi }{4}left {d^{2}} end{equation}

In cases where the bolt experiences the shearing force across the unthreaded (shank) portion of the bolt, the designer can consider the full cross sectional area prescribed by the major diameter. This approach is permitted in Table 3.4 of Eurocode EN 1993-1-8: 2005 – Design of Joints [1]: begin{equation} A_{tau}=frac{pi }{4}left {d^{2}} end{equation}

3. Shear Stress

In general, the permissible shear stress in a bolt should be limited to the magnitude of shear stress at the point of yielding under pure shear, as predicted by the von Mises Yield Criterion. Many codes and standards use this as the basis for determining bolt shear capacity. The yield stress under pure shear in accordance with the von Mises Yield Criterion is given by: begin{equation} label{eq2} begin {split} f_{tau} & = frac{f_{yb}}{sqrt{3}} & = 0.58 f_{yb} end{split} end{equation} It should be noted that Table 3.4 of Eurocode EN 1993-1-8: 2005 [1] actually uses the UTS of the bolt material rather than the yield (proof) stress, which results in a higher shear capacity. This is explained in the following section.

4. Shear Capacity in Accordance with EN 1993-1-8

4.1 Across the threaded portion

Table 3.4 of Eurocode EN 1993-1-8: 2005 [1] provides the following formula for determining shear capacity of a single fastener when the shearing plane crosses the threaded portion of the bolt, based on the UTS of the bolt material rather than the yield (proof) stress: begin{equation} F_{v,Rd}= alpha_{v} f_{ub} frac{A_{S}}{gamma_{M2}}  end{equation} Where:

{A_{S}=the tensile stress area of the bolt (mm2)
{f_{ub}=the ultimate tensile stress of the bolt (N.mm-2)
 {gamma_{M2}=1.25, the partial safety factor for bolt resistance to fracture when the bolt is placed in tension; as per Table 2.1 of EN 1993-1-8, which refers to Clause 6.1 of EN 1993-1-1 [2].
{alpha_{v}is dependent on the bolt material classification and is presented in Table 1

Table 1 – Alpha factor for bolts, depending on classification

Bolt Classification4.64.85.65.86.88.810.9
αv0.60.50.60.50.50.60.5

4.2 Across the shank

Table 3.4 of Eurocode EN 1993-1-8: 2005 [1] provides the following formula for determining shear capacity of a single fastener when the shearing plane crosses the shank of the bolt. begin{equation} F_{v,Rd}= 0.6 f_{ub} frac{A_{S}}{gamma_{M2}}  end{equation} Where:

{A}=the gross cross section area of the bolt, based on the major diameter
{f_{ub}=the ultimate tensile stress of the bolt
 {gamma_{M2}=1.25, the partial safety factor for bolt resistance to fracture when the bolt is placed in tension; as per Table 2.1 of EN 1993-1-8, which refers to Clause 6.1 of EN 1993-1-1 [2].

5. Other Standards

The shear capacity of bolts are defined differently across various standards. In the future I may extend this page to include a discussion of other standards.

6. Single, Double or Multiple Shear?

As described on the page discussing bolt bearing type connections, bolts can be loaded across single or multiple shear planes. In typical mechanical and structural engineering design it is common to encounter single and double shear lap joints. A single shear connection is one in which the bolt, pin, rivet, dowel or similar is only able to resist the separation forces across a single shear plane. A very simple illustration of a single-shear lap joint connection is presented in Figure 1, where: (a) illustrates a single-shear lap joint under equilibrium loading; (b) illustrates the failure mode in which the rivet is sheared (separated) across the shearing plane; and (c) illustrates failure of the connection due plastic bending of the splice plates. A double-shear connection, is one in which the bolt, pin, rivet, dowel or similar is able to resist the separation forces across two shear planes.  A very simple illustration of a double-shear lap joint connection is presented in Figure 2: (a) illustrates a double-shear lap joint under equilibrium loading; and (b) illustrates the failure mode where the rivet is sheared across the two shearing planes.

Single shear lap connection - shear capacity

Figure 1 – Single shear lap connection

Double shear lap connection - shear capacity

Figure 2 – Double shear lap connection

6.1. Cross-Sectional Area to be Considered

    \begin{flalign*} &R_{1} &=& \dfrac{wl}{2} &\qquad&\\ &R_{2} &=& R{_{1}} &\qquad&\\ &V_{x} &=& w\left(\dfrac{l}{2}-x\right) &\qquad&\\ &V_{max} &=& R{_{1}}=R{_{2}} &\qquad&\text{(at}\;R_{1}\;\text{and}\;R_{2}\text{)} \\ &M_{max} &=&\dfrac{wl^{2}}{8} &\qquad&\text{(at centre)}\\ &M_{x} &=&\dfrac{wx}{2}\left(l-x\right) &\qquad&\\ &\delta_{max} &=&\dfrac{5wl^{4}}{384EI} &\qquad&\text{(at centre)}\\ &\delta_{x} &=&\dfrac{wx}{24EI}\left(l^{3}-2lx^{2}+x^{3}\right) &\qquad& \end{flalign*}

When calculating the shear stress in a bolt it is necessary to determine the average shear stress across all the shear planes. For a single lap joint (Figure 1) there is only one shear plane across the fastener. The shear stress induced across a single shear plane is simply: begin{equation} begin {split} tau & =frac{F}{A} & = frac{4F} {pi d^{2}} end{split} end{equation} For a double lap joint (Figure 2) we can immediately see that the shear stress is halved as we have doubled the shear area (two shear planes). The shear capacity of the fastener under this loading configuration is therefore doubled. The shear stress induced across each shear plane of the fastener is given by: begin{equation} begin {split} tau & =frac{F}{2A} & = frac{2F} {pi d^{2}} end{split} end{equation} It is essential to make sure that the number of shear planes across the bolt is taken into account when determining the utilisation factor for fasteners in a joint.