CE 416

Prestress Concrete Design Sessional

Course Teacher

Ms. Sabreena Nasrin

Mr. Galib Muktadir Ratul

Department of Civil Engineering

Ahsanullah University of Science and Technology

Presentation TopicShear Stress

Presented byAyesha Binta Ali

Id No.: 10.01.03.0334th Year, 2nd Semester

Department of Civil Engineering

Presentation Outline

Definition

Sign Convention

Calculating Shear Stress

Other Form of Shear

Stress

Testing Machine of Shear Stress

Horizontal and Vertical Shear Stress

Mohr Circle

Shear Force vs Shear

Stress

Shear Stress on Beam

Shear Stress in Steel

Shear Stress in Concrete

Shear Stress Distribution

Definition A shear stress, denoted (Greek

: tau), is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section.

It is the form of stress that subjects an object to which force is applied to skew, tending to cause shear strain.

Definition(Cont.)

Shear Stress Parallel to the Cross Section (Horizontal)

Shear Stress Parallel to the Cross Section (Inclined)

Shear Stress in 2D View

Shear Stress in 3D View

Definition(Cont.) A shear stress between two objects occurs when a force pulls the object along the same plane as the face of the object abutting another object that is being pulled in the opposite direction.

A shear stress within an object will occur when a force parallel to the plane causes one plane of the material to want to slip against another, thus deforming the material.

Mathematical Definition If a fluid is placed between two parallel plates spaced 1.0

cm apart, and a force of 1.0 dyne is applied to each square centimeter of the surface of the upper plate to keep it in motion, the shear stress in the fluid is 1 dyne/cm2 at any point between the two plates.

The formula to calculate average shear stress is:

Ʈ=F/A

where:

Ʈ= the shear stress;

F= the force applied;

A= the cross-sectional area of material with area parallel to the applied force vector.

Sign Convention

Calculating Shear StressMeasure the area, say value A, of the material over which the force is applied. The area of a simple rectangular or square-shaped cross section is obtained by multiplying the length by the height. The area of a circular cross section is calculated by the equation A= pi*r^2. The area of a circle is equal to the value of pi (3.14159) multiplied by the squared radius of the circle. Measure the force that is to be applied over the area, say value F. Simple forces of weight can be measured with a scale that displays results in pounds. Substitute the values obtained in the above steps as the following formula: T=F/A; where T = the shear stress, F = the force applied and A = the cross-sectional area over which the force was applied at first. Divide the numerical value for F by the value for A and the resulting number is the calculated shear stress.

Other Forms of Shear Stress

Pure shear stressPure shear stress is related to pure Shear strain, denoted , by the following equation: = γGwhere is G the Shear modulus of the material, given by G Here E is Young’s modulus is and ʋ is Poison’s ratio.

Other Forms of Shear Stress(Cont.) Beam shear stressIt is defined as the internal shear stress of a beam caused by the shear force applied to the beam.

Where:V = Total shear force at the location in question;Q = Statical moment of area; t = Thickness in the material perpendicular to the shear;I = Moment of inertia of the entire cross sectional area.

Other Forms of Shear Stress(Cont.) Resolved shear stress

It is the shear component of an applied tensile (or compressive) stress resolved along a slip plane that is other than perpendicular or parallel to the stress axis.

τ = σ cos Φ cos λ

Critical resolved shear stress

It is the value of resolved shear stress at which yielding begins; it is a property of the material.

τ =σ (cosΦ cosλ)max

Other Forms of Shear Stress(Cont.) Direct shear stress

It is the stress on the mechanical elements of that surface - something like the stress in a bolt that is connecting two pieces of metal. If the bolt cracks straight across, if failed due to the shear.

Impact shear stress

It is the stress when something lands on a surface - something like when a person falls off a bike and skids across the ground. The shear stress tears their skin.

Testing Machine of Shear Stress

Some Shear Testing Machines

Shear Force vs. Shear Stress

Transverse Shear Force: ΣF = 0 (V = RA

in this case)

Transverse Shear Stress: fv = V/A

Mohr Circle

The circle is centered at the average stress value, and has a radius R equal to the maximum shear stress, as shown in the figure. The maximum shear stress is equal to one-half the difference between the two principal stresses,

Mohr Circle(Cont.)Ɵs is an important angle where the maximum shear stress occurs.

The shear stress equals the maximum shear stress when the stress element is rotated 45° away from the principal directions.

The transformation to the maximum shear stress direction can be illustrated as:

Horizontal and Vertical Shear Stress

Horizontal & Vertical Shear Stress

Shear Stress on BeamsLet us begin by examining a beam of rectangular cross section. We can reasonably assume that the shear stresses τ act parallel to the shear force V.

Let us also assume that the distribution of shear stresses is uniform across the width of the beam.

Shear Stress on Beams(Cont.)Shear stresses on one side of an element are

accompanied by shear stresses of equal magnitude acting on perpendicular faces of an element.

Thus, there will be horizontal shear stresses between horizontal layers (fibers) of the beam, as well as, transverse shear stresses on the vertical cross section.

At any point within the beam these complementary shear stresses are equal in magnitude.

Shear Stress on Beams(Cont.)The existence of horizontal shear stresses in a beam can be demonstrated as follows.

A single bar of depth 2h is much stiffer that two separate bars each of depth h.

Shown below is a rectangular beam in pure bending.

Let Q = First moment of area =∫ydA

τ =VQ/Ib

Where: V = transverse shear force

Q = first moment of area (section above area of interest)

I = moment of inertia

b = width of section

For the rectangular section shown above:

τ =V/2I(h² /4) – y1²)

As shown above, shear stresses vary quadratic ally with the distance y1 from the neutral axis. The maximum shear stress occurs at the neutral axis and is zero at both the top and bottom surface of the beam.

Shear Stress on Beams(Cont.)

For a rectangular cross section, the maximum shear stress is obtained as follows:

Q = (bh/2)(h/4) = bh²/8

I = bh²/12

Substituting yields:

Ʈmax = 3V/2A

For a circular cross section:

Ʈmax = 4V/3A

Shear Stress on Beams(Cont.)

Shear Stress in Steel

Problem: WebCrippling

Solution: WebStiffeners

Steel is affected by the compression component of

Shear. In case of tension there is no problem.

Shear Stress in Concrete

Problem: Diagonal Cracking

Solution: Shear Reinforcement

Concrete is affected by the tensile component of

principal shear stress. In case of compression there is no problem.

Shear Stress Distribution

Shear Stress Distribution in a Rectangular Section

Shear Stress Distribution in a Triangular Section

Shear Stress Distribution(Cont.)

Shear Stress Distribution in a Wide Flange Section

Shear Stress Distribution(Cont.)

Shear Stress Distribution in a Circular Section

Shear Stress Distribution in a T Section

Thank you all…