Mechanics of Solids: Shear Stresses in Beams and Related Problems
Mechanics of Solids: Shear Stresses in Beams and Related Problems
Mechanics of Solids: Shear Stresses in Beams and Related Problems
Chapter 7
Shear Stresses in Beams and Related Problems
Part A- Shear Stresses in Beams
• If a shear and bending moment are present at one section through a
beam, a different bending moment will exist at an adjoining section,
although the shear may remain constant.
𝑑𝑀 = 𝑉 𝑑𝑥
• Consider the shear and bending moment diagrams shown in Fig. 1
• The change in the bending moment in a distance 𝑑𝑥 is P 𝑑𝑥.
• Shear stresses on mutually perpendicular planes of an infinitesimal
element are considered equal.
Fig. 1: Shear and bending moment diagrams for the loading shown
Fig. 2: Shear flow model of an I-beam. (a) beam segment with bending stresses simulated by blocks.
(b) Shear force transmitting through a dowel. (c) for determining the force on a dowel only a change in moment
Is needed. (d) The longitudinal shear force divided by the area of the imaginary cut yields shear stress.
(e) Horizontal cut below the flange for determining the shear stress. (f) Vertical cut through the flange for
determining the shear stress.
Shear Flow
• Consider an elastic beam made from several continuous longitudinal
planks whose cross section is shown in Fig. 3(a).
• As, 𝑑𝑀 𝑑𝑥 = 𝑉 and 𝑞 = 𝑉𝑄 𝐼.
• Here, 𝑉 is the total shear force at the section, I is the moment of inertia
of the whole cross sectional area about the neutral axis.
• 𝑄 is the statical moment around the neutral axis of the partial area of the
cross section to one side of the imaginary longitudinal cut, and 𝑦 is the
distance from the neutral axis to the centroid of the partial area 𝐴𝑓𝑔ℎ𝑗 . 𝑡
is the width of the imaginary longitudinal cut, usually equal to the
thickness or width of the member.
• The proper sectioning of some cross-sectional areas of beams is shown in
Figs. 6(a), (b),(d), and (e). The use of inclined plane should be avoided
unless the section is made across a small thickness.
Fig. 6: Sectioning for partial areas of cross sections for computing shear stresses
Warpage of Plane Sections Due to Shear
• The maximum shear stress, hence, maximum shear strain, occurs at
𝑦 = 0. and no shear strain takes place at 𝑦 = ± ℎ 2.
• This behavior warps the initially plane sections through a beam, as
shown in Fig. 7.
• However, warpage of the sections is important only for very short
members and is negligibly small for small members. This can be
substantiated by the two-dimensional finite-element studies.
Fig. 11: Shear stress for an equal Fig. 12: Shear center for the sections
leg angle is at 𝑆 shown is at 𝑆
• Such an element of the rod viewed from the top is shown in Fig. 15.
• At both ends of the element, the torques are equal to 𝐹 𝑟 acting in the
direction shown. The component of these vectors toward the axis of
the spring 𝑂, resolved at the point of intersection of the vectors,
2𝐹 𝑟 𝑑Φ 2 = 𝐹 𝑟 𝑑Φ
• This vector component opposes the couple developed by the vertical
shears 𝑉 = 𝐹, which are 𝑟 𝑑Φ apart.
• The nominal direct shear stress for any point on the cross section is 𝜏 =
𝐹 𝐴. Superposition of this nominal direct and the torsional shear stress
at 𝐸 gives the maximum combined shear stress.
• Thus, since 𝑇 = 𝐹 𝑟, 𝑑 = 2𝑐, and 𝐽 = 𝜋𝑑4 32
𝐹 𝑇𝑐 𝑇𝑐 𝐹𝐽 16𝐹 𝑟 𝑑
𝜏𝑚𝑎𝑥 = + = +1 = +1
𝐴 𝐽 𝐽 𝐴𝑇𝑐 𝜋𝑑 3 4𝑟