session 12: members analysis under shear (part i) · shear stresses generate in beams due to...

2015/3/18

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BETON PRATEGANG TKS - 4023

Dr.Eng. Achfas Zacoeb, ST., MT.

Jurusan Teknik Sipil

Fakultas Teknik

Universitas Brawijaya

Session 12:

Members Analysis under Shear (Part I)

The Objectives

To avoid shear failure, the beam should fail in flexure

at its ultimate flexural strength. Hence, each mode of failure is addressed in the design for shear. The design involves not only the design of the stirrups, but also limiting the average shear stress in concrete, providing adequate thickness of the web and adequate development length of the longitudinal bars.

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The Objectives (cont’d)

To provide ultimate resistance for shear (VuR) greater

than the shear demand under ultimate loads (Vu). For simply supported prestressed beams, the maximum shear near the support is given by the beam theory. For continuous prestressed beams, a rigorous analysis can be done by the moment distribution method.

Introduction The analysis of reinforced concrete and prestressed

concrete members for shear is more difficult compared

to the analyses for axial load or flexure.

The analysis for axial load and flexure are based on the

following principles of mechanics.

1) Equilibrium of internal and external forces

2) Compatibility of strains in concrete and steel

3) Constitutive relationships of materials.

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Introduction (cont’d)

The conventional analysis for shear is based on

equilibrium of forces by a simple equation.

The compatibility of strains is not considered. The

constitutive relationships (relating stress and strain) of

the materials, concrete or steel, are not used.

Shear stresses generate in beams due to bending or

twisting.

The two types of shear stress are called flexural shear

stress and torsional shear stress, respectively.

Introduction (cont’d)

To understand flexural shear stress, the behaviour of a simply supported beam under uniformly distributed load,

without prestressing, will be explained first. The lecture

will be in the following sequence:

① Stresses in an uncracked (homogenous) beam.

② Types of cracks generated due to the combination of

flexure and shear.

③ Components of shear resistance and the modes of

failure.

④ Effect of prestressing force.

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Stresses in an uncracked beam Fig. 1 shows the variations of shear and moment along the span of a simply supported beam under a uniformly distributed load. The variations of normal stress and shear stress along the depth of a section of the beam are also shown.

Fig. 1 Variations of forces and stresses in a simply supported beam

Stresses in an uncracked beam (cont’d) Under a general loading, the shear force and the moment vary along the length. The normal stress and the shear stress vary along the length, as well as along the depth. The combination of the normal and shear stresses generate a two-dimensional stress field at a point. At any point in the beam, the state of two-dimensional stresses can be expressed in terms of the principal stresses. The Mohr’s circle of stress is helpful to understand the state of stress.

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Stresses in an uncracked beam (cont’d) Before cracking, the stress carried by steel is negligible. When the principal tensile stress exceeds the cracking stress, the concrete cracks and there is redistribution of stresses between concrete and steel. For a point on the neutral axis (Element 1), the shear stress is maximum and the normal stress is zero. The principal tensile stress (σ1) is inclined at 45º to the neutral axis. Fig. 2 shows the state of in-plane stresses at a point on the neutral axis of a beam.

Stresses in an uncracked beam (cont’d)

Fig. 2 State of stresses at a point on the neutral axis of a beam

The Mohr’s circle is a representation of the state of in-plane stresses on surfaces of various inclinations passing through a point. The horizontal and vertical axes represent the normal and shear stresses, respectively. For a state of pure shear, the centre of the Mohr’s circle coincides with the origin of the axes.

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Since the shear force is maximum near the supports, cracks due to shear occur near the supports. The cracks are formed around the neutral axis and perpendicular to the principal tensile stress (σ1). The cracks are thus inclined at 45° to the axis of the beam. Fig. 3 shows the inclination of the cracks forming at the neutral axis.

Fig. 3 Inclination of crack at the level of neutral axis


For a point near the bottom edge of the beam (Element 2), the normal stress is maximum and the shear stress is close to zero. The principal tensile stress (σ1) is almost parallel to the bottom edge. The angle of inclination of σ1 with respect to the axis of the beam (α) is smaller than 45o. Fig. 4 shows the state of in-plane stresses at a point close to the edge under tension.

Fig. 4 State of stresses at a point close to the edge under tension

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Adjacent to the bottom edge (edge under tension), the tensile stress due to flexure is maximum and the shear stress is zero. The state of stress is nearly uniaxial tensile stress. The principal compressive stress is negligible. Since the moment is maximum at mid span, cracks due to flexure occur near mid span. The cracks are formed at the bottom edge and perpendicular to σ1. Since σ1 is parallel to the edge, the cracks are perpendicular to the edge as shown in Fig. 5.

Fig. 5 Inclination of crack close to the edge under tension


The previous concepts can be used to develop the principal stress trajectories. Fig. 6 shows the trajectories for a simply supported beam under a uniformly distributed load. The crack pattern can be predicted from these trajectories.

Fig. 6 Principle stress trajectories

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Types of Cracks

The types and formation of cracks depends on the span-

to-depth ratio of the beam and loading. These variables

influence the moment and shear along the length of the

beam. For a simply supported beam under uniformly

distributed load, without prestressing, three types of

cracks are identified such as:

1. Flexural cracks

2. Web shear cracks

3. Flexure shear cracks

Types of Cracks (cont’d)

Flexural cracks, these cracks form at the bottom near the midspan and propagate upwards.

Web shear cracks, these cracks form near the neutral axis close to the support and propagate inclined to the beam axis.

Flexure shear cracks, these cracks form at the bottom due to flexure and propagate due to both flexure and shear.

The formation of cracks for a beam with large span-to-depth ratio and uniformly distributed loading is shown in Fig. 7.

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Types of Cracks (cont’d)

Fig. 7 Formation of cracks in a reinforced concrete beam

Components of Shear

Resistance

The components of shear resistance are studied based on the internal forces at a flexure shear crack. The forces are shown in Fig. 8. The magnitude and the relative value of each component change with increasing load.

Fig. 8 Internal forces at a flexure shear crack

The notations are as follows: Vcz = Shear carried by uncracked concrete Va = Shear resistance due to aggregate interlock Vd = Shear resistance due to dowel action Vs = Shear carried by stirrups Vp = Vertical component of prestressing force in inclined tendons

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Modes of Failure

For beams with low span-to-depth ratio or inadequate shear reinforcement, the failure can be due to shear. A failure due to shear is sudden as compared to a failure due to flexure. The following five modes of failure due to shear are identified such as:

1. Diagonal tension failure 2. Shear compression failure 3. Shear tension failure 4. Web crushing failure 5. Arch rib failure

The occurrence of a mode of failure depends on the span-to-depth ratio, loading, cross-section of the beam, amount and anchorage of reinforcement.

Modes of Failure (cont’d)

Diagonal tension failure In this mode, an inclined crack propagates rapidly due to inadequate shear reinforcement.

Fig. 9 Diagonal tension failure

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Shear compression failure There is crushing of the concrete near the compression flange above the tip of the inclined crack.

Fig. 10 Shear compression failure


Shear tension failure Due to inadequate anchorage of the longitudinal bars, the diagonal cracks propagate horizontally along the bars.

Fig. 11 Shear tension failure

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Web crushing failure The concrete in the web crushes due to inadequate web thickness.

Fig. 12 Web crushing failure


Arch rib failure For deep beams, the web may buckle and subsequently crush. There can be anchorage failure or failure of the bearing.

Fig. 13 Arch rib failure

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Effect of Prestressing Force

In presence of prestressing force, the flexural cracking occurs at a higher load. For Type 1 and Type 2 sections, there is no flexural crack under service loads. In presence of prestressing force, the web shear cracks also generate under higher load. With increase in the load beyond the cracking load, the cracks generate in a similar sequence. But, the inclinations of the flexure shear and web shear cracks are reduced depending on the amount of prestressing and the profile of the tendon.


(cont’d)

The effect of prestressing force is explained for a beam with a concentric effective prestressing force (Pe) as shown in Fig. 14.

Fig. 14 A simply supported beam under concentric prestressed and uniformly distributed loads

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(cont’d)

For a point at the neutral axis (Element 1), there is normal stress due to the prestressing force (–fpe). The principal tensile stress (σ1) is inclined to the neutral axis at an angle greater than 45o. With the combination of shear stress, the principal compressive stress (σ2) is inclined to the neutral axis at an angle much smaller than 45o. Fig. 15 shows the state of in-plane stresses at a point on the neutral axis for a prestressed beam .


(cont’d)

Fig. 16 shows the formation of cracks for a prestressed beam with large span-to-depth ratio and uniformly distributed loading is shown. This figure can be compared with that for a reinforced concrete beam.

Fig. 15 State of stresses at a point on the neutral axis for a prestressed beam

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(cont’d)

After cracking, in presence of prestressing force, the length and crack width of a diagonal crack are low. Thus, the aggregate interlock and zone of concrete under compression are larger as compared to a non-prestressed beam under the same load. Hence, the shear strength of concrete (Vc) increases in presence of prestressing force. This is accounted for in the expression of Vc.

Fig. 16 Formation of cracks in a prestressed beam

Design of Stirrups The design is done for the critical section. In general

cases, the face of the support is considered as the

critical section. When the reaction at the support

introduces compression at the end of the beam, the

critical section can be selected at a distance effective

depth from the face of the support. The effective depth is

selected as the greater of dp (depth of CGS from the

extreme compression fiber) or ds (depth of centroid of

non-prestressed steel).

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Design of Stirrups (cont’d) To vary the spacing of stirrups along the span, other

sections may be selected for design. Usually the

following scheme is selected for beams under uniform

load.

1. Close spacing for quarter of the span adjacent to the

supports.

2. Wide spacing for half of the span at the middle.

For large beams, more variation of spacing may be

selected. Fig. 17 shows the typical variation of spacing of

stirrups. The span is represented by L.

Design of Stirrups (cont’d)

Fig. 17 Typical variation of spacing of stirrups

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Limit State of Collapse for Shear

The shear is studied based on the capacity of a section which is the limit state of collapse. The capacity (or ultimate resistance) of a section (VuR) consists of a concrete contribution (Vc) and the stirrup contribution (VS). (1) Vc includes Vcz (contribution from uncracked concrete), Va (aggregate interlock) and Vd (dowel action).

Limit State of Collapse for Shear (cont’d)

The value of Vc depends on whether the section is

cracked due to flexure. Two expressions of Vc are

available, one for cracked section and the other for

uncracked section. Usually, the expression for the

uncracked section will govern near the support. The

expression for the cracked section will govern near the

mid span. Of course, both the expressions need to be

evaluated at a particular section. The lower value

obtained from the two expressions is selected.

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For Uncracked Sections

(2)

where,

Vco = the shear causing web shear cracking at CGC

b = breadth of the section

= bw, breadth of the web for flanged sections

D = total depth of the section (h)

ft = tensile strength of concrete = 0.24√fck

fcp = compressive stress in concrete at CGC due

to the prestress = Pe/A

For Uncracked Sections (cont’d)

Eq. (2) can be derived based on the expression of the

principal tensile stress (σ1) at CGC as shown in Fig. 15.

The principal tensile stress is equated to the direct

tensile strength of concrete (ft).

(3)

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Transposing the terms of Eq.3: (4)

Fig. 17 Cross-section of a beam showing the variables for calculating shear stress in the web

I = gross moment of inertia

Q = At y

At = area of section above CGC

y = vertical distance of centroid

of At from CGC.


The term 0.67bD represents Ib/Q for the section. It is

exact for a rectangular section and conservative for other

sections. Only 80% of the prestressing force is

considered in the term 0.8fcp. For a flanged section,

when the CGC is in the flange, the intersection of web

and flange is considered to be the critical location.


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The expression of Vc0 is modified by substituting 0.8fcp

with 0.8 X (the stress in concrete at the level of the

intersection of web and flange). In presence of inclined

tendons or vertical prestress, the vertical component of

the prestressing force (Vp) can be added to Vc0.

(5)


For Cracked Sections

(6)

Vcr is the shear corresponding to flexure shear cracking.

The term (1 – 0.55fpe/fpk)cbd is the additional shear that

changes a flexural crack to a flexure shear crack.

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For Cracked Sections

The notations in Eq. 6 are as follows: fpe = effective prestress in the tendon after all losses ≤ 0.6fpk fpk = characteristic strength of prestressing steel c = ultimate shear stress capacity of concrete, obtained b = breadth of the section

= bw , breadth of the web for flanged sections d = distance from the extreme compression fibre to the centroid of the tendons at the section considered M0 = moment initiating a flexural crack Mu = moment due to ultimate loads at the design section Vu = shear due to ultimate loads at the design section.

For Cracked Sections (cont’d)

The expression of M0 is given in Eq. 7,

(7)

where fpt = magnitude of the compressive stress in

concrete at the level of CGS due to prestress only. An

equal amount of tensile stress is required to decompress

the concrete at the level of CGS. The corresponding

moment is fptI/y. In the expression of M0, I = gross

moment of inertia and y = depth of the CGS from CGC.

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For Cracked Sections (cont’d)

The factor 0.8 implies that M0 is estimated to be 80% of

the moment that decompresses the concrete at the level

of CGS. Since the concrete is cracked and the inclination

of tendon is small away from the supports, any vertical

component of the prestressing force is not added to Vcr.

Maximum Permissible Shear Stress To check the crushing of concrete in shear compression failure, the shear stress is limited to a maximum value (c,max) as shown in Eq. 8. The value of c,max depends on the grade of concrete. (8) where, dt = greater of dp or ds dp = depth of CGS from the extreme compression fiber ds = depth of centroid of regular steel Vu = shear force at a section due to ultimate loads.

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Thanks for Your Attention and

Success for Your Study!

session 12: members analysis under shear (part i) · shear stresses generate in beams due to...

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