Shear Center

What happens when loads act in a plane that is not a plane of symmetry? -Loads must be applied at particular point in the cross section, called shear center, if the beam is to bend without twisting. Shear Stress distribution 

Constrained by the shape of the cross section

o

Its resultant acts at the shear center



Not necessarily the centroid

Shear Center 

A lateral load acting on a beam will produce bending without twisting only if it acts through the shear center



The shear center

o

Is a property of the cross section like the centroid

o

It lies on an axis of symmetry



For a doubly symmetric section S and C coincide

Why a property of the cross section?

Locating Shear Centre

Unsymmetric Loading of Thin-Walled Members



Beam loaded in a vertical plane of symmetry deforms in the symmetry plane without twisting.



Beam without a vertical plane of symmetry bends and twists under loading.

Unsymmetric Loading of Thin-Walled Members



If the shear load is applied such that the beam does not twist, then the shear stress distribution satisfies



F and F’ indicate a couple Fh and the need for the application of a torque as well as the shear load.



When the force P is applied at a distance e to the left of the web centerline, the member bends in a vertical plane without twisting.

Shear stress distribution strategy 1. Determine location of centroid and Iyy, Izz and Iyz as needed - (symmetric sections subject to Vy needs only Izz) 1. Divide section into elements according to geometry (change in slope) 2. Start with a vector s following element center line from a free end 3. Calculate first moment of area(s). This determines the shear flow distribution - Negative shear value indicate direction of shear flow opposite to assumed vector s 4. Calculate first moment of area(s). This determines the shear flow distribution o

For symmetric sections subject to bending about one axis



Elements parallel to bending axis-Linear distribution



Elements normal to bending axis-Parabolic distribution

For unsymmetric sections shear flow in all elements is parabolic 5. When moving from one element to another the end value of shear in one element equals the initial value for the subsequent element (from equilibrium) Shear Centres for Some Other Sections

Shear Center 

How to locate Shear Center?



Doubly symmetric cross sections- Coincides with centroid



Singly symmetric cross sections- Lies on the axis of symmetry

o

Thin-walled open sections



Opposite side of open part

Doubly or singly symmetric section

A point on a line parallel to the axis of a beam through which any transverse force must be applied to avoid twisting of the section. A beam section will rotate when the resultant of the internal shearing forces is not collinear with the externally applied force. The shear center may be determined by locating the line of action of the resultant of the internal shear forces. A rolled wide flange beam section has two axes of symmetry, and therefore the shear center coincides with the geometric center or centroid of the section. When such a beam member is loaded transversely in the plane of the axes, it will bend without twisting. However, a channel section with one axis of symmetry is subject to twisting unless the externally applied load passes through the shear center of the section (Fig. 1). The shear center for a channel bending about the strong axis (x–x) is located a distance e from the center of the web and lies behind the web. An expression may be derived to determine this eccentricity for different sizes of channels of uniform thickness, such that e = B2h2t/4Ix, where B is the flange width, h is the depth of section, t is the thickness of web and flange, and Ix is the moment of inertia about the x axis.

Fig. 1 Shear center under P at distance e from centerline web, on symmetrical axis. Add to 'My Saved Images'

Other structural sections with one axis of symmetry will twist unless properly restrained or loaded through the shear center. Figure 2a illustrates the loading of a channel through the shear center causing bending about the nonsymmetrical axis and no twist; Fig. 2b illustrates the condition which will cause twisting; and Fig. 2c indicates a condition which will resist twisting. See also: Loads, transverse

Fig. 2 Three conditions of channel section under load. Channel (a) resists bending, (b) twists, and (c) resists twisting.