Shear and diagonal tension in beams

SHEAR AND
DIAGONAL TENSION
IN BEAMS

INTRODUCTION
• Beams must also have an adequate safety margin.
• Shear forces

WHAT IS
SHEAR
FAILURE?
More properly
called diagonal
tension.
01
Not fully
understood.
02
If beam is not
properly
designed?
03

DIRECT SHEAR STRESSES IN BEAMS

SHEAR FAILURE (NILSON, 1997)
OVERALL VIEW
DETAIL NEAR RIGHT SUPPORT

FIGURE SHOWS Point Loading
1
Member failed
immediately
2
formation of the critical
crack in the high-shear
region near the right
support.
3

WHEN ARE THE SHEARING EFFECTS SO LARGE
THAT THEY CANNOT BE IGNORED AS A DESIGN
CONSIDERATION?
• The best way to begin answering this question is to try to approximate the shear
stresses on the cross section of the beam.

POINT LOADING AND UNGLUED STACK

POINT LOADING AND
UNGLUED STACK
• Stacked without fastening
• loaded in a direction normal to the surface
of these slabs
• Stack deforms
• Ends gets staggered

POINT LOADING
AND GLUED
STACK
• The fact that this solid beam does
not exhibit this relative movement
of longitudinal elements after the
slabs are glued indicates the
presence of shearing stresses on
longitudinal planes.

THEORETICAL BACKGROUND
• The concept of stresses acting in homogeneous beams are usually covered in
various textbooks of mechanics of materials (strength of materials).
• It can be shown that when the material is elastic, shear stresses can be computed by

THEORETICAL BACKGROUND (CONT..)
• All points along the length of the beam, where the shear and bending moment are
not zero, and at locations other than the extreme fiber or neutral axis, are subject
to both shearing stresses and bending stresses.
• The combination of these stresses produces maximum normal and shearing
stresses in a specific plane inclined with respect to the axis of the beam.

CONT….
• The distributions of the bending and shear stresses acting individually are shown in

PRINCIPAL PLANES
• The combination of bending moment and shearing stresses is of such a nature that
maximum normal and shearing shearing stresses at a point in a beam exist on
planes that are inclined with the axis of the beam.
• These planes are commonly called principal planes, and the stresses that act on
them are referred to as principal stresses.

CONT….
• The principal stresses in a beam subjected to shear and bending may be computed
using the following equation:
• fpr = principal stress
• f = bending stress computed from Eq. 2
• v = shearing stress computed from Eq. 1

ORIENTATION PRINCIPAL PLANES
• The orientation of the principal planes may be calculated using the following
equation:
• Note that at the neutral axis of the beam, the principal stresses will occur at a 45
degree angle.

“DIAGONAL TENSION FAILURE”
25

DIAGONAL
TENSION
FAILURE
In the beams with which we are
concerned,
where the length over which a shear
failure
could occur (the shear span) is in excess
of
approximately three times the effective
depth.
01
The diagonal tension failure would be
the
mode of failure in shear.
02
26

DIAGONAL TENSION FAILURE
• Such a failure is shown in Figures 1 and 8.
• For longer shear spans in plain concrete
beams, cracks due to flexural tensile stresses
would occur long before cracks due to
diagonal tension.
27

• FIGURE 8
28

• FIGURE 1
29

• The principal stress in tension acts at an
approximately 45 plane to the normal at
sections close to support, as seen in Figure 9.
30

• Because of the low tensile strength of
concrete, diagonal cracking develops along
planes perpendicular to the planes of principal
tensile stress; hence the term diagonal
tension cracks.
• To prevent such cracks, special diagonal
tension reinforcement has to be provided
31

DIAGONAL
TENSION
FAILURE
FIGURE 9
32

BEAMS WITHOUT
DIAGONAL
TENSION
REINFORCEMENT
Modes of Failure:
• The slenderness of the
beam or its shear
span/depth determines the
failure of the plain
concrete beam.
• Figure 10 demonstrates
schematically the
failure patters.
33

BEAMS WITHOUT DIAGONAL TENSION
REINFORCEMENT
• Basically, three modes of failure or their
combinations are of interest:
1. Flexure Failure
2. Diagonal Tension Failure
3. Shear Compression Failure
• The more slender the beam, the stronger
the tendency toward flexure behavior.
34

BEAMS WITHOUT
DIAGONAL
TENSION
REINFORCEMENT
FIGURE 10
35

DESIGN PHILOSOPHY
• The ACI provides design guidelines for shear reinforcement based on the vertical
shear force Vu that develops at any given cross section of a member.
• Diagonal tensile force

REQUIREMENTS (SHEAR OR WEB REINFORCEMENT)
• Vertical stirrups
• Inclined or diagonal stirrups
• The main reinforcement bent at ends to act as inclined stirrups

ACI CODE PROVISIONS FOR SHEAR
REINFORCEMENT
• For member that are subject to shear and flexure only, the amount of shear force
that the concrete (unreinforced for shear)can resist is

REINFORCEMENT
• The ACI also permits the use of more refined equation in replacement of Eq. 5, that
is

REINFORCEMENT
• the code requires that a minimum area of shear reinforcement be provided in all
reinforced concrete flexural members
• when Vu > ½ Vc, except as follows: In slabs and footings
• In concrete joist construction as defined in the code.
• In beams with a total depth of less than 10 in., 2 ½ times the flange thickness, or
one-half the width of the web, whichever is greater.

MINIMUM AREA OF SHEAR REINFORCEMENT
• Vu > ½ Vc, the minimum area of shear reinforcement shall be computed from

ACCORDING TO THE ACI CODE
• The design of beams for shear is based on the following relation:

CONT…
• For inclined stirrups
• For = 450, the expression takes the form
• The design for stirrup spacing can be determined from

ACCORDING TO THE ACI CODE,
• the maximum spacing of stirrups is the smallest value of
• If Vs exceeds , the maximum spacing must not exceed d/4 or 12 in.
• It is not usually good practice to space vertical stirrups closer than 4 in.

CRITICAL SECTION
• The maximum shear usually occurs in this section near the support. For stirrup
design, the section located a distance dfrom the face of the support is called the
critical section Sections located less than a distance d from the face of the support
may be designed for the same Vu as that of the critical section.

Shear and diagonal tension in beams

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