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Engineering, section 1, Fall 2019
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Goodno & Gere - Mechanics of Materials 9/e (Homework)

James Finch

Engineering, section 1, Fall 2019

Instructor: Dr. Friendly

Current Score : 37 / 38

Due : Sunday, January 27, 2030 12:00 EST

Last Saved : n/a Saving... ()

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Question

Points

1	2	3	4	5	6	7	8	9	10	11	12
18/18	1/1	3/3	2/2	1/1	1/1	2/3	1/1	3/3	1/1	2/2	2/2

Total

37/38 (97.4%)

Instructions

Give students a rigorous, complete, and integrated treatment of the mechanics of materials—an essential subject in mechanical, civil, and structural engineering. The enhanced ninth edition of Goodno/Gere's Mechanics of Materials, published by Cengage Learning, examines the analysis and design of structural members subjected to tension, compression, torsion, and bending—laying the foundation for further study. Available via WebAssign is MindTap Reader, Cengage's next-generation eBook, and other digital resources.

Question 1 is an Active Example. Active Examples build a bridge between practice and homework through helpful Examples from the textbook that guide students through the process needed to master a concept and include worked-out solutions.

Question 2 is a Chapter Quiz Question. Chapter Quiz Questions encourage students to test and apply what they have learned in each chapter. These questions can serve as a quick and useful self-test to help confirm understanding of each concept.

In Question 3 the maximum normal strain ε_max is determined for a steel wire bent around a cylindrical drum.

Question 4 calculates the maximum bending stress σ_max of a simply supported wood beam (linearly elastic material).

In Question 5 students select a suitable size for a simply supported wood beam subjected to unsymmetrical point loads.

Question 6 shows how the manner in which the bending stresses vary along the axis of a nonprismatic beam is not the same as for a prismatic beam.

Question 7 determines the normal stress and the shear stress at point C when a simply supported wood beam is subjected to a uniformly distributed load q.

Question 8 asks for the maximum shear stress in a pole consisting of a circular tube loaded by a linearly varying distributed force.

Question 9 determines the maximum permissible load q of a bridge girder constructed of three plates welded to form a cross section.

Question 10 determines the maximum allowable shear force V_max for a prefabricated wood I-beam serving as a floor joist with a cross section.

Question 11 uses weight and wind force to determine the maximum tensile and compressive stresses σ_t and σ_c, respectively, of an aluminum pole (at its base) for a street light.

Question 12 uses the dimensions of a rectangular beam with notches and a hole, along with the bending moment and the maximum allowable bending stress in the material (steel), to calculate the smallest radius R_min that should be used in the notches, and the diameter d_max of the largest hole that should be drilled at the mid-height of the beam. This demo assignment allows many submissions and allows you to try another version of the same question for practice wherever the problem has randomized values.

Assignment Submission

For this assignment, you submit answers by question parts. The number of submissions remaining for each question part only changes if you submit or change the answer.

Assignment Scoring

Your last submission is used for your score.

1. 18/18 points | Previous Answers GGMechMat9 5.AE.007. My Notes

Question Part

Points

Submissions Used

1/1

2/100

Total

18/18

Design of Beams for Bending Stresses

Question ▲
A simple beam AB of span length 21 ft must support a uniform load
q = 2,000 lb/ft
distributed along the beam in the manner shown in the figure below.

Design of a Simple Beam with Partial Uniform Loads

A horizontal beam A B is supported at its left end A by a pin support and at its right end B by a roller support. Two uniformly distributed loads with intensity q = 2,000 lb/ft act downwards on the beam. One load begins at A and extends rightwards 12 ft. The other load begins at B and extends leftwards 6 ft, leaving a 3 ft segment without a load. Reaction force R_A acts upwards at A and reaction force R_B acts upwards at B.

Considering both the uniform load and the weight of the beam, select a structural steel beam of wide-flange shape to support the loads. Use an allowable bending stress of 18,300 psi.
- Solution ▲
  Use a four-step problem-solving approach. Combine steps as needed for an efficient solution.
  - Conceptualize, Categorize ▲
    
    In this example, proceed as follows.
    
    Find the maximum bending moment in the beam due to the uniform load.
    
    Knowing the maximum moment, find the required section modulus.
    
    Select a trial wide-flange beam from this table or this table and obtain the weight of the beam.
    
    With the weight known, calculate a new value of the bending moment and a new value of the section modulus.
    
    Determine whether the selected beam is still satisfactory. If it is not, select a new beam size and repeat the process until you find a satisfactory size of beam.
    
    Maximum Bending Moment
    
    To assist in locating the cross section of the maximum bending moment, construct the shear-force diagram (see figure below) using the methods described in Chapter 4. As part of that process, determine the reactions at the supports, as follows.
    
    A shear diagram depicts V (lb). V starts at 18,860 and goes down and right linearly to zero a horizontal distance x₁ to the right of the beginning. V continues down and right with the same slope to −5,140, extends horizontally rightwards a short distance, then continues down and right linearly to −17,140.
    
    R_A = 18,860 lb R_B = 17,140 lb
    
    Find the distance
    x₁
    from the left-hand support to the cross section of zero shear force from
    
    V = R_A − qx₁ = 0,
    
    which is valid in the range 0 ≤ x ≤ 12 ft. Solve for
    x₁
    to get
    
    x₁ =
    R_A
    q
    =
    18,860 lb
    2,000 lb/ft
    = 9.430 ft,
    
    which is less than 12 ft. Therefore, the calculation is valid. The maximum bending moment occurs at the cross section where the shear force is zero. Therefore, we get the following.
    
    M_max = R_Ax₁ −
    qx₁²
    2
    = 88,920 lb-ft
  - Analyze ▲
    
    Required Section Modulus
    
    Find the required section modulus (based only upon the load q) as obtained from the equation
    
    S =
    M_max
    σ_allow
    ,
    
    as follows.
    
    S =
    M_max
    σ_allow
    =
    (88,920 lb-ft)(12 in/ft)
    18,300 psi
    = 58.3 in³
    
    Trial Beam
    
    Now consider this table and this table and select the lightest wide-flange beam having a section modulus greater than 58.3 in³. The lightest beam that provides this section modulus is W 12 ✕ 50 with
    S = 64.2 in³.
    This beam weighs 50 lb/ft. (Recall that the tables in the above links are abridged, so a lighter beam may actually be available.)
    
    Now recalculate the reactions, the maximum bending moment, and the required section modulus with the beam loaded by both the uniform load q and its own weight. Under these combined loads, the reactions are
    
    R_A = 19,380 lb R_B = 17,670 lb,
    
    and the distance to the cross section of zero shear becomes the following.
    
    x₁ =
    19,380 lb
    2,050 lb/ft
    = 9.454 ft
    
    The maximum bending moment increases to 91,610 lb-ft, and the new required section modulus is as follows.
    
    S =
    M_max
    σ_allow
    =
    (91,610 lb-ft)(12 in/ft)
    18,300 psi
    = 60.1 in³
  - Finalize ▲
    
    Thus, the W 12 ✕ 50 beam with section modulus
    S = 64.2 in³
    is still satisfactory.
    
    Note: If the new required section modulus exceeded that of the W 12 ✕ 50 beam, you would select a new beam with a larger section modulus and repeat the design process.

Additional Question ▲
- Question 1▲
  
  A steel wide flange beam was selected in this design example (W 12 ✕ 50, see figure below).
  
  A cross section of an I beam has a vertical web and horizontal flanges. A horizontal line through the midpoint of the web is axis 1–1. A vertical line through the midpoint of the web is axis 2–2. The distance from the top of the beam to the bottom is 12.19 in.
  
  Now consider three additional designs for the cross section of this steel beam. We have rectangular (see figure below),
  
  The cross section of a beam is a rectangle of height h and width h⁄4.
  
  circular (see figure below),
  
  The cross section of a beam is a circle of diameter d.
  
  and cruciform (see figure below).
  
  The cross section of a beam is a symmetric cruciform. Each face extends outwards a distance b⁄4 and has a width of b⁄2 centered on the midpoint of the respective face. The maximum height and width are both 4 ✕ (b⁄4) = b.
  
  (i)
  
  Find the cross section dimension (in inches) for each (h for rectangle, d for circle, b for cruciform) so that each has the same cross sectional area as the W 12 ✕ 50 beam used in the example
  (A_x = 14.6 in²).
  [Note: each of the four beams has the same area and so also has the same weight per foot,
  w = 50 lb/ft].
  
  Use known area formulas for each cross section shape.
  
  rectangle h =
  7.64
  in. circle d =
  4.31
  in. cruciform b =
  4.41
  in.
  
  (ii)
  
  Compute the section modulus S (in³) about a horizontal centroidal axis for each beam (rectangle
  S_R,
  circle
  S_C,
  cruciform
  S_Cr).
  [Recall that
  S_W = 64.2 in³
  for the W 12 ✕ 50 beam].
  
  Expect the section modulus values for rectangular, circular and cruciform cross sections to be much less than that for the W shape. Recall that the W shape has its material located as far as practical from the neutral axis so is the most efficient design among the four shapes considered here.
  
  S_R =
  18.6
  in³ S_C =
  7.87
  in³ S_{C_r} =
  8.05
  in³
  
  Find ratios
  S_R
  S_W
  ,
  
  S_C
  S_W
  ,
  and
  S_Cr
  S_W
  .
  
  S_R
  S_W
  =
  0.29
  
  S_C
  S_W
  =
  0.123
  
  S_Cr
  S_W
  =
  0.125
  
  Which beams are the most and second most economical? (Recall that the most economical beam is the one having the largest capacity-to-weight ratio.)
  
  most economical
  W 12 ✕ 50
  second most economical
  rectangle
  
  (iii)
  
  What is the maximum moment (M_max, lb-ft) that each beam can carry? Recall that the allowable bending stress for each steel beam is
  σ_a = 18,300 psi.
  (Enter the magnitudes.)
  
  From the example, the W 12 ✕ 50 beam has moment capacity that exceeds 91,000 lb-ft. Expect the capacities of the rectangle, circle and cruciform shape beams to be much less.
  
  W 12 ✕ 50 M_max =
  97900
  lb-ft rectangle M_max =
  28400
  lb-ft circle M_max =
  12000
  lb-ft cruciform M_max =
  12300
  lb-ft
- Question 2▲
  
  Considering both the uniform load q (see the figure below) and the weight of the beam, and also using an allowable bending stress of 18,300 psi, find the minimum dimensions of each of the three alternative steel beams to support the loads (both applied load q and self-weight
  w = γA).
  
  A horizontal beam A B is supported at its left end A by a pin support and at its right end B by a roller support. Two uniformly distributed loads with intensity q = 2,000 lb/ft act downwards on the beam. One load begins at A and extends rightwards 12 ft. The other load begins at B and extends leftwards 6 ft, leaving a 3 ft segment without a load. Reaction force R_A acts upwards at A and reaction force R_B acts upwards at B.
  
  To begin, use
  M_max = 88,920 lb-ft
  (based on distributed load q alone), then include the distributed weight w (lb/ft) of each beam. Recall that the weight density of steel is
  γ = 490 lb/ft³
  (see this table).
  
  Equate the section modulus S formula for each of the rectangular, circular, and cruciform cross sections to the required section modulus, then solve for the minimum required dimension for each shape.
  
  (i)
  
  Find the minimum height h (in inches) for a rectangular cross section (
  h
  4
  by h, see figure below).
  
  The cross section of a beam is a rectangle of height h and width h⁄4.
  
  11.4
  in.
  
  (ii)
  
  Find the minimum diameter d (in inches) for a circular cross section (see figure below).
  
  The cross section of a beam is a circle of diameter d.
  
  8.75
  in.
  
  (iii)
  
  Find the minimum dimension b (in inches) for a cruciform cross section (see figure below).
  
  The cross section of a beam is a symmetric cruciform. Each face extends outwards a distance b⁄4 and has a width of b⁄2 centered on the midpoint of the respective face. The maximum height and width are both 4 ✕ (b⁄4) = b.
  
  8.88
  in.

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2. 1/1 points | Previous Answers GGMechMat9 5.CQ.POST.001. My Notes

The flexure formula shows that normal stresses in a beam have which quality?

They vary linearly over the beam cross section. They are uniform over the entire beam cross section. They vary non-linearly over the beam cross section. none of these choices

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3. 3/3 points | Previous Answers GGMechMat9 5.4.001. My Notes

A steel wire with a diameter of d = 3/13 in. is bent around a cylindrical drum with a radius of R = 37 in. (see figure).

A wire of diameter d bends part way around a cylinder of radius R.

(a)

Determine the maximum normal strain ε_max.

ε_max =

0.00311

(b)

What is the minimum acceptable radius of the drum (in inches) if the maximum normal strain must remain below yield? Assume E = 25,000 ksi and σ_Y = 100 ksi.

28.7

in.

(c)

If R = 37 in., what is the maximum acceptable diameter of the wire (in inches) if the maximum normal strain must remain below yield?

0.297

in.

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4. 2/2 points | Previous Answers GGMechMat9 5.5.004. My Notes

A simply supported wood beam AB with a span length

L = 4 m

carries a uniform load of intensity

q = 5.7 kN/m

(see Figure (a) below).

Two figures depict a simply supported wood beam of length L, each carrying a different load. The beam has a pin support on the left end at point A and a roller support on the right end at point B. The cross section of the beam has width b and height h > b.

Figure (a). The beam carries a uniform load of intensity q.

Figure (b). The beam carries a trapezoidal distributed load where the intensity at the left end is q/2 and the intensity at the right end is q.

(a)

Calculate the maximum bending stress σ_max (in MPa) for Figure (a) due to the load q if the beam has a rectangular cross section with width

b = 130 mm

and height

h = 270 mm.

7.22

MPa

(b)

Repeat part (a) but use the trapezoidal distributed load shown in Figure (b). (Enter the maximum bending stress σ_max in MPa.)

5.43

MPa

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5. 1/1 points | Previous Answers GGMechMat9 5.6.001. My Notes

A simply supported wood beam having a span length L = 12 ft is subjected to unsymmetrical point loads, as shown in the figure below.

A simply supported wood beam carries two point loads separated by 4 ft. Point load P₁ = 2.5 kips acts 4 ft from the left end of the beam and point load P₂ = 3.2 kips acts 4 ft from the right end of the beam.

Select a suitable size for the beam using the table in Appendix G. The allowable bending stress is 1,800 psi and the wood weighs 35 lb/ft³.

4 in. ✕ 10 in. 3 in. ✕ 6 in. 2 in. ✕ 8 in. correct

6 in. ✕ 10 in.

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6. 1/1 points | Previous Answers GGMechMat9 5.7.006. My Notes

The following problem pertains to fully stressed beams of rectangular cross section. Consider only the bending stresses obtained from the flexure formula and disregard the weights of the beams.

A cantilever beam AB with rectangular cross sections of a constant width b and varying height

h_x

is subjected to a uniform load of intensity q (see figure).

A cantilever beam of horizontal length L is shaped like an isosceles triangle with two long sides and one short side. The short, right side of the beam is attached to a vertical wall and has height h_B. Point A is at the left vertex of the beam and point B is at the top right corner of the beam. At a distance x to the right of point A, the beam has height h_x < h_B.

Above the beam, a uniform load labeled q acts vertically downward along the entire length of the beam.

Two rectangular cross sections are shown of the beam. One has height h_x and width b. The other has height h_B and width b.

How should the height

h_x

vary as a function of x (measured from the free end of the beam) in order to have a fully stressed beam? (Express

h_x

in terms of the height

h_B

at the fixed end of the beam. Use the following as necessary: h_B, x, and L.)

h_x =

xhBL

$webMathematica generated answer key$

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7. 2/3 points | Previous Answers GGMechMat9 5.8.003. My Notes

A simply supported wood beam is subjected to uniformly distributed load q. The width of the beam is 6 in. and the height is 8 in.

A 16 ft long, horizontal, rectangular wood beam is supported under its left end by pin support A and under its right end by roller support B. Point C is located 4 ft to the right of its left end and 3 in. below the top of the beam. A uniformly distributed load q = 500 lb/ft acts downward on the top of the beam, along the entire length of the beam.

A rectangular cross-sectional view of the end of the beam shows that the +y-axis points up and the +z-axis points to the left, where the origin O is in the center of the cross-section. It has height 8 in. along the y-direction and width 6 in. along the z-direction.

Determine the normal stress and the shear stress at Point C. (Enter your answers in psi. Assume the +x-axis points to the right from A. Indicate the direction with the signs of your answers.)

normal stress

-563

psi shear stress

-58.6

psi

Show these stresses on a sketch of a stress element at Point C.

The stress element at point C is represented by a square labeled C. Two arrows, collectively labeled σ, point inward toward the square, one toward its right side and one toward its left side. Four arrows, collectively labeled τ, point along the sides of the square as follows.

Along the top side, an arrow points toward the right.

Along the right side, an arrow points upward.

Along the bottom side, an arrow points toward the left.

Along the left side, an arrow points downward.

The stress element at point C is represented by a square labeled C. Two arrows, collectively labeled σ, point outward away from the square, one away from its right side and one away from its left side. Four arrows, collectively labeled τ, point along the sides of the square as follows.

Along the top side, an arrow points toward the right.

Along the right side, an arrow points upward.

Along the bottom side, an arrow points toward the left.

Along the left side, an arrow points downward.

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8. 1/1 points | Previous Answers GGMechMat9 5.9.003. My Notes

A vertical pole consisting of a circular tube of outer diameter 4 in. and inner diameter 3.2 in. is loaded by a linearly varying distributed force with maximum intensity of

q₀.

A tall, vertically-oriented cylinder has a height of 10 ft, inner diameter d₁, and outer diameter d₂. A linearly varying distributed force acts rightward on the left side of the cylinder. The maximum value q₀ = 500 lb/ft acts at the bottom of the cylinder, and q linearly decreases to a value of zero acting at the top of the cylinder.

Find the maximum shear stress in the pole. (Enter your answer in ksi. Indicate the direction with the sign of your answer.)

1.1

ksi

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9. 3/3 points | Previous Answers GGMechMat9 5.10.008. My Notes

A bridge girder AB on a simple span of length

L = 10 m

supports a distributed load of maximum intensity q at mid-span and minimum intensity

q/2

at supports A and B that includes the weight of the girder (see figure).

Two diagrams of a bridge girder are shown.

In the first diagram, the bridge girder and supports are viewed from the side, with the full length L = 10 m of the girder spanning from pin support A on the left to roller support B on the right. A distributed load is represented by many arrows above the girder pointing vertically down toward the girder. The greatest load is at the center of the girder and labeled q. The least load is found at either end of the girder and labeled q⁄2. The load increases linearly from either end of the girder to the center of the girder.

In the second diagram, the cross section of a wide flange beam is shown. The beam has a shape similar to a capital I, with a narrow vertical web and wide horizontal flanges above and below the web. The horizontal width of each flange is 450 mm, and the vertical thickness of each flange is 32 mm. The horizontal width of the web is 16 mm, and the vertical height of the web, from the point where it meets the top face of the bottom flange to the point where it meets the bottom face of the top flange, is 1800 mm.

The girder is constructed of three plates welded to form the cross section shown. Determine the maximum permissible load q (in kN/m) based upon the following.

(a)

solely based on an allowable bending stress

σ_allow = 110 MPa

362

kN/m

(b)

solely based on an allowable shear stress

τ_allow = 50 MPa

346

kN/m

(c)

accounting for both an allowable bending stress σ_allow and an allowable shear stress τ_allow

346

kN/m

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10. 1/1 points | Previous Answers GGMechMat9 5.11.001. My Notes

A prefabricated wood I-beam serving as a floor joist has the cross section shown in the figure.

The cross section of a wood I-beam is composed of a long vertical web capped on its top and bottom with shorter horizontal flanges, which are centered above and below the web. The origin O is in the center of the cross section, where the +y-axis points up and the +z-axis points to the left. The dimensions of each component of the beam are as follows.

The web has height 9 in. and width 0.625 in.

Each flange has height 0.90 in. and width 5 in.

The allowable load in shear for the glued joints between the web and the flanges is 55 lb/in. in the longitudinal direction. Determine the maximum allowable shear force V_max for the beam. (Enter the magnitude in lb.)

640

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11. 2/2 points | Previous Answers GGMechMat9 5.12.004. My Notes

An aluminum pole for a street light weighs 4,800 N and supports an arm that weighs 500 N (see figure).

Two diagrams of an aluminum pole are shown.

In the first diagram, the pole is upright and extends upward from the base on the ground. The weight of the pole is W₁ = 4,800 N. At the top of the pole, an arm extends up and to the left to end at a lamp. The weight of the arm is W₂ = 500 N, and this weight acts at the center of gravity of the arm at a distance of 1.2 m to the left of the pole. A wind force of P₁ = 100 N pushes on the pole to the left at a height of 9 m above the base. The +x-direction points out of the page, the +y-direction points to the right, and the +z-direction points upward.

In the second diagram, a cross section of the pole is seen from above. The cross section is a cylindrical shell with outer diameter 225 mm and thickness 18 mm. The +x-direction points downward and the +y-direction points to the right.

The center of gravity of the arm is 1.2 m from the axis of the pole. A wind force of 100 N also acts in the

(−y)

direction at 9 m above the base. The outside diameter of the pole (at its base) is 225 mm, and its thickness is 18 mm.

Determine the maximum tensile and compressive stresses

σ_t

and

σ_c,

respectively, in the pole (at its base) due to the weights and the wind force. (Enter your answers in kPa. Include the signs of the values in your answers.)

maximum tensile stress σ_t =

2220

kPa maximum compressive stress σ_c =

-3120

kPa

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12. 2/2 points | Previous Answers GGMechMat9 5.13.005. My Notes

A rectangular beam with notches and a hole (see figure) has dimensions

h = 7.7 in.,

h₁ = 7 in.,

and thickness

b = 0.73 in.

(perpendicular to the plane of the figure).

A rectangular beam with vertical height h extends from left to right. Two curved arrows, one to the left of the beam and one to the right, are labeled M and curve up and away, then up and toward the beam. Four notches and a hole of diameter d are bored through the beam and into the page. One notch is U-shaped: the bottom of the notch is similar to the bottom of a U, and this bottom may or may not be at some vertical depth below the top-front edge of the beam. The beam is bored above the bottom of the notch into the page, and the horizontal width of the top of the U-shaped notch is 2R. The notch is some distance to the right of the top-left corner of the front and back faces.

A second notch forms a similar arc and similar bore at an equal distance to the left of the top-right corner.

A third notch is shaped like an inverted U: the top of the notch is similar to the top of an inverted U, and this top may or may not be at some vertical height above the bottom-front edge of the beam. The beam is bored below the top of the notch into the page, and the horizontal width of the bottom of the inverted U is 2R. The notch is some distance to the right of the bottom-left corner of the front and back faces.

A fourth notch forms a similar arc and similar bore at an equal distance to the left of the bottom-right corner.

The bottommost point of each notch along the top edge and the topmost point of the nearest notch along the bottom edge are separated by a vertical distance h₁.

The beam is subjected to a bending moment

M = 125 kip-in.,

and the maximum allowable bending stress in the material (steel) is

σ_max = 42,000 psi.

(a)

What is the smallest radius R_min (in inches) that should be used in the notches? (Use any necessary data found in this figure.)

0.77

in.

(b)

What is the diameter d_max (in inches) of the largest hole that should be drilled at the midheight of the beam?

Assume

that the ratio

5.65

in.

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Goodno & Gere - Mechanics of Materials 9/e (Homework)

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Design of Beams for Bending Stresses