STATICS AND STRENGTH OF MATERIALS REVIEW: MODULE 2 …gozips.uakron.edu/.../StaticsStrenthsReviewModule2.pdf · STATICS AND STRENGTH OF MATERIALS REVIEW: MODULE 2. tab ... – Shear

STATICS AND STRENGTH OF MATERIALS REVIEW:

MODULE 2

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TYPES OF BEAMS

• If a beam is freely supported at its ends with

either pins or rollers, it is called a simply

supported beam or simple beam.

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TYPES OF BEAMS

• The beam that is fixed at one end and free

at the other is called a cantilever beam.

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TYPES OF BEAMS

• Supports for beams

with overhanging

ends are either pins

or rollers.

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TYPES OF BEAMS

• These beams, are all determinate because the

three unknown reactions for each beam can be

determined by the equations of static equilibrium.

THEY YIELD 3 UNKNOWNS

3 EQUATIONS ARE AVAILABLE

ΣM

ΣFX

ΣFY

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TYPES OF BEAMS

• Examples of statically indeterminate beams.

A fixed beam in which

both ends are fixed.

A propped beam in which

one end is fixed and the

other end supported by

a roller.

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SHEAR FORCE SIGN CONVENTION

• Shown here is the sign

convention for positive internal

shear force.

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SHEAR FORCE SIGN CONVENTION

• Shown here is the sign convention

for negative internal shear force.

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BENDING MOMENTS

• Shown here is the direction

of positive internal bending

moments.

– compression on the top of the beam and tension on the

bottom of the beam.

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BENDING MOMENTS

• The direction of negative

bending moments is

reversed.– Negative bending moments tend to

cause tension on the top of the

beam and compression on the

bottom of the beam.

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SHEAR AND BENDING-MOMENT DIAGRAMS

• Concentrated loads on

the beam:

– Shear diagram consists of

straight horizontal lines

broken only at new load.

– The moment diagram

consists of straight sloping

lines broken at new loads.

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SHEAR AND BENDING-MOMENT DIAGRAMS

• Uniform loads on the

beam:

– the shear diagram consists

of straight sloping

(diagonal) lines.

– The moment diagram

consists of curved lines

(second-degree curves).

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• The maximum/minimum

moments occur at points

where the value of shear is

zero.

RELATIONSHIP BETWEEN SHEAR AND BENDING MOMENT DIAGRAMS

FOR DETAILED PROCEDURES ON DRAWING SHEAR AND BENDING

MOMENT DIAGRAMS: SEE WEBSITE FOR DOWNLOADABLE DOCUMENT

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BEAMS IN BENDING

Bending stress at any distance y is:

Maximum bending stress occurs at

extreme fibers and max. bending stress:

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HORIZONTAL SHEAR STRESS DUE TO BENDING

Looking at the “stacked” planks in (a) it can be seen

that if they are individually place and NOT attached to

each other, slippage will occur between them.

Each plank is bending individually;

C in top fibers

T in bottom fibers

If an adhesive is applied that bonds the planks

together, then the planks will bend as one beam;

Cmax in top fibers of top plank

Tmax in bottom fibers of bottom blank

The adhesive will then be resisting the “shearing

stress” that occurs in those planes due to vertical

loading.

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• “General Shear Formula” Ss = VQ

I b

Where; V is computed vertical shear force at cross section being considered

Q is the statical moment about the neutral axis of the area outside

the horizontal plane being evaluated

I is moment of inertiab is the width of the cross section in the horizontal plane where the

shear stress is being calculated

HORIZONTAL SHEAR STRESS DUE TO BENDING

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VERTICAL SHEAR STRESS

***For equilibrium to exist: Vertical shear is

equal to horizontal at any given point.

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CALCULATING BEAM DEFLECTIONS

• Deflection, moment and shear equations for

various beams are shown in the Beam Deflection

Tables on the class website.

• Many textbooks will also provide deflection

equations in the Appendices.

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DEFLECTION CALCULATION:SUPERPOSITION METHOD

Consider the cantilever beam shown.

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DEFLECTION CALCULATION:SUPERPOSITION METHOD

Deflection, moment, shear can be calculated by the

method of superposition: