# Shear (6.2.6)

Generally, shear is not a problem in steel design, because the webs of rolled
shapes are capable to resist large shearing forces. The shear might play
important role under some conditions:

#### Large concentrated loads are placed near beam supports

Tall buildings: if columns in the upper floors are offset with the respect to
the columns below them. A horizontal beam than carries weight of the upper beam
and a great shear might be present.

#### Columns and beams are rigidly connected together

#### Beam is coped or notched

1—Large concentrated loads are placed near beam supports, e.g. columns are offset

2—Columns and beams are rigidly connected together: moment M is replaced by the forces T (shear);

3—Beam is coped

From mechanics of material, shear is a stress or the forces
*resisting* shear or attempts to shear a member. Since shear is a stress, its average value over the
cross-section is $\tau = V/A$. But most of the shear is carried close to normal
axis. At any point along the $y$ axis:

$$
\begin{equation}
\tau(y) = \frac{V S_x}{I_x b} \label{shear:eq1}
\end{equation}
$$

where $V$ is shear force (load), $S_x$ is statical moment of portion area at considered
point against centroidal axis, $I_x$ moment of inertia of the whole section, $b$ is
thickness/width of the section at the considered point (discontinuities of width of
sections cause discontinuities of the shear forces along the height).

1—shear is a stress resisting attempts to shear
a member; in the case of a rectangular section the maximum is in the middle and
according to $(\ref{shear:eq1})$ magnitude of shear stress goes parabolic over the section

2—Shear is present also during bending; shear is then a stress bonding these "slices" together

3—Shear on I-section

The assessment according to the Eurocode is

$$
\begin{equation}
V_{Ed} \le V_{c, Rd},
\end{equation}
$$

where $V_{Ed}$ is the design shear force from load and $V_{c, Rd}$ is design shear
resistance. For $V_{c, Rd}$ is either $V_{pl,Rd}$ taken if plastic design is
considered, or $V_{c,Rd}$ when elastic design is requested.

## Resistance for plastic design

$$
\begin{equation}
V_{pl,Rd} = \frac{A_{V}f_y / \sqrt{3}}{\gamma_{M0}},
\end{equation}
$$

assuming no torsion is involved, $A_V$ is the shear area. $A_{V}$ should be
taken according to the code (complicated): for commonly used profiles the parts of
sections parallel to the force can be taken.

For commonly used profiles the parts of sections
parallel to the force can be taken as $A_V$ when resistance to shear is being
determined

## Resistance for elastic design

If no shear buckling needs to be considered (chapter 5 of EN 1993-1-5), then
for verifying the design elastic shear resistance $V_{c,Rd}$, the assessment is being made by
comparing stresses at the critical point of the cross-section:

$$
\frac{\tau_{Ed}}{f_y / (\sqrt{3}\cdot \gamma_{M0})} \le 1.0
$$

where $\tau_{Ed}$ is determined for $V_{Ed}$ from $(\ref{shear:eq1})$. For
I- or H- sections the shear stress in the web might be taken as

$$
\begin{equation}
\tau_{Ed} = \frac{V_{Ed}}{A_w}
\end{equation}
$$

if area of flanges $A_f = b_f t_f$ is large compared to area $A_w = h_w t_w$ of the web: $A_f / A_w \ge 0.6$.

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