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Published byLouisa Henderson Modified over 5 years ago

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**Beams Beams: Comparison with trusses, plates t**

L, W, t: L >> W and L >> t L W Comparison with trusses, plates Examples: 1. simply supported beams 2. cantilever beams

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**Beams - loads and internal loads**

Loads: concentrated loads, distributed loads, couples (moments) Internal loads: shear force and bending moments

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**Shear Forces, Bending Moments - Sign Conventions**

right section left section Shear forces: positive shear: negative shear: Bending moments: positive moment negative moment

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**Shear Forces, Bending Moments - Static Equilibrium Approach**

Procedure: 1. find reactions; 2. cut the beam at a certain cross section, draw F.B.D. of one piece of the beam; 3. set up equations; 4. solve for shear force and bending moment at that cross section; 5. draw shear and bending moment diagrams. Example 1: Find the shear force and bending diagram at any cross section of the beam shown below.

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**Relationship between Loads, Shear Forces, and Bending Diagram**

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**Beam - Normal Strain M M no transverse load Pure bending problem**

no axial load no torque Observations of the deformed beam under pure bending Length of the longitudinal elements Vertical plane remains plane after deformation Beam deforms like an arc

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**Normal Strain - Analysis**

neutral axis (N.A.): radius of curvature: Coordinate system: q longitudinal strain: r y N.A.

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**Beam - Normal Stress Hooke’s Law: y M M M x Maximum stresses:**

Neutral axis:

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**Flexure Formula y Moment balance: M x Comparison:**

Axially loaded members Torsional shafts:

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Moment of Inertia - I Example 2: h w Example 3: h w w 4h w

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**Design of Beams for Bending Stresses**

Design Criteria: 1. 2. cost as low as possible Design Question: Given the loading and material, how to choose the shape and the size of the beam so that the two design criteria are satisfied?

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**Design of Beams for Bending Stresses**

Procedure: Find Mmax Calculate the required section modulus Pick a beam with the least cross-sectional area or weight Check your answer

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**Design of Beams for Bending Stresses**

Example 4: A beam needs to support a uniform loading with density of 200 lb /ft. The allowable stress is 16,000 psi. Select the shape and the size of the beam if the height of the beam has to be 2 in and only rectangular and circular shapes are allowed. 6 ft

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**Shear Stresses inside Beams**

shear force: V V Horizontal shear stresses: y h1 y1 x h2 s1 s2 tH

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**Shear Stresses inside Beams**

Relationship between the horizontal shear stresses and the vertical shear stresses: y h1 y1 x h2 Shear stresses - force balance V: shear force at the transverse cross section Q: first moment of the cross sectional area above the level at which the shear stress is being evaluated w: width of the beam at the point at which the shear stress is being evaluated I: second moment of inertial of the cross section

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**Shear Stresses inside Beams**

Example 5: Find shear stresses at points A, O and B located at cross section a-a. P a A O a B w

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**Shear Stress Formula - Limitations**

- elementary shear stress theory Assumptions: 1. Linearly elastic material, small deformation 2. The edge of the cross section must be parallel to y axis, not applicable for triangular or semi-circular shape 3. Shear stress must be uniform across the width 4. For rectangular shape, w should not be too large

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**Shear Stresses inside Beams**

Example 6: The transverse shear V is 6000 N. Determine the vertical shear stress at the web.

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**Beams - Examples Example 7: For the beam and loading shown, determine**

(1) the largest normal stress (2) the largest shearing stress (3) the shearing stress at point a

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Deflections of Beam Deflection curve of the beam: deflection of the neutral axis of the beam. y P y x x Derivation: Moment-curvature relationship: Curvature of the deflection curve: (1) Small deflection: (2) (3) Equations (1), (2) and (3) are totally equivalent.

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**Deflections by Integration of the Moment Differential Equation**

Example 8 (approach 1):

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**Deflections by Integration of the Load Differential Equation**

Example 8 (approach 2):

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**Method of Superposition**

q Deflection: y P Deflection: y1 Deflection: y2

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**Method of Superposition**

Example 9

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**Statically Indeterminate Beam**

Number of unknown reactions is larger than the number of independent Equilibrium equations. Propped cantilever beam Clamped-clamped beam Continuous beam

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**Statically Indeterminate Beam**

Example 10. Find the reactions of the propped beam shown below.

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