WebAssign is not supported for this browser version. Some features or content might not work. System requirements

WebAssign

Welcome, demo@demo

(sign out)

Saturday, March 29, 2025 04:20 EDT

Home My Assignments Grades Communication Calendar My eBooks

My Cengage Home Notifications Help My Account

Engineering, section 1, Fall 2019
My Assignments

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

James Finch

Engineering, section 1, Fall 2019

Instructor: Dr. Friendly

Current Score : 35 / 37

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

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

Print Assignment

Question

Points

1	2	3	4	5	6	7	8	9	10	11
18/18	3/3	2/2	1/1	0/1	3/3	1/1	2/3	1/1	2/2	2/2

Total

35/37 (94.6%)

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, SI edition, 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.

In Question 2 the maximum normal strain ε_max is determined for a steel wire bent around a cylindrical drum.

Question 3 calculates the maximum bending stress σ_max of a simply supported wood beam (linearly elastic material).

In Question 4 students select a suitable size for a simply supported wood beam subjected to unsymmetrical point loads.

Question 5 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 6 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 7 asks for the maximum shear stress in a pole consisting of a circular tube loaded by a linearly varying distributed force.

Question 8 determines the maximum permissible load q of a bridge girder constructed of three plates welded to form a cross section.

Question 9 determines the maximum allowable shear force V_max for a prefabricated wood I-beam serving as a floor joist with a cross section.

Question 10 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 11 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 GGMechMat9SI 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 7 m must support a uniform load
q = 60 kN/m
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 = 60 kN/m act downwards on the beam. One load begins at A and extends rightwards 4 m. The other load begins at B and extends leftwards 2 m, leaving a 1 m 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 111 MPa.
- 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 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 (kN). V starts at 188.6 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 −51.4, extends horizontally rightwards a short distance, then continues down and right linearly to −171.4.
    
    R_A = 188.6 kN R_B = 171.4 kN
    
    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 ≤ 4 m. Solve for
    x₁
    to get
    
    x₁ =
    R_A
    q
    =
    188.6 kN
    60 kN/m
    = 3.14 m,
    
    which is less than 4 m. 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
    = 296.3 kN · m
  - 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
    =
    (296.3 ✕ 10⁶ N · mm)
    111 MPa
    = 2.670 ✕ 10⁶ mm³
    
    Trial Beam
    
    Now consider this table and select the lightest wide-flange beam having a section modulus greater than 2,670 cm³. The lightest beam that provides this section modulus is HE 450A with
    S = 2,896 cm³.
    This beam weighs 140 kg/m, or 1.373 kN/m. (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 = 193.4 kN R_B = 176.2 kN,
    
    and the distance to the cross section of zero shear becomes the following.
    
    x₁ =
    193.4 kN
    (60 kN/m + 1.373 kN/m)
    = 3.151 m
    
    The maximum bending moment increases to 304.7 kN · m, and the new required section modulus is as follows.
    
    S =
    M_max
    σ_allow
    =
    (304.7 ✕ 10⁶ N · mm)
    111 MPa
    = 2,739 cm³
  - Finalize ▲
    
    Thus, the HE 450A beam with section modulus
    S = 2,896 cm³
    is still satisfactory.
    
    Note: If the new required section modulus exceeded that of the HE 450A 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 (HE 450A, 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 440 mm.
  
  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 centimeters) for each (h for rectangle, d for circle, b for cruciform) so that each has the same cross sectional area as the HE 450A beam used in the example
  (A_x = 178 cm²).
  [Note: each of the four beams has the same area and so also has the same weight per meter,
  m = 140 kg/m or 1.373 kN/m].
  
  Use known area formulas for each cross section shape.
  
  rectangle h =
  26.7
  cm circle d =
  15.1
  cm cruciform b =
  15.4
  cm
  
  (ii)
  
  Compute the section modulus S (cm³) about a horizontal centroidal axis for each beam (rectangle
  S_R,
  circle
  S_C,
  cruciform
  S_Cr).
  [Recall that
  S_W = 2,896 cm³
  for the HE 450A beam].
  
  Expect the section modulus values for rectangular, circular and cruciform cross sections to be much less than that for the HE shape. Recall that a wide flange shape has its material located as far as practical from the neutral axis so it is the most efficient design among the four shapes considered here.
  
  S_R =
  792
  cm³ S_C =
  335
  cm³ S_{C_r} =
  343
  cm³
  
  Find ratios
  S_R
  S_W
  ,
  
  S_C
  S_W
  ,
  and
  S_Cr
  S_W
  .
  
  S_R
  S_W
  =
  0.273
  
  S_C
  S_W
  =
  0.116
  
  S_Cr
  S_W
  =
  0.118
  
  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
  HE 450A
  second most economical
  rectangle
  
  (iii)
  
  What is the maximum moment (M_max, kN · m) that each beam can carry? Recall that the allowable bending stress for each steel beam is
  σ_a = 111 MPa.
  (Enter the magnitudes.)
  
  From the example, the HE 450A beam has moment capacity that exceeds 296 kN · m. Expect the capacities of the rectangle, circle and cruciform shape beams to be much less.
  
  HE 450A M_max =
  321
  kN · m rectangle M_max =
  87.9
  kN · m circle M_max =
  37.2
  kN · m cruciform M_max =
  38
  kN · m
- 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 111 MPa, 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 = 60 kN/m act downwards on the beam. One load begins at A and extends rightwards 4 m. The other load begins at B and extends leftwards 2 m, leaving a 1 m 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 = 296.3 kN · m
  (based on distributed load q alone), then include the distributed weight w (kN/m) of each beam. Recall that the weight density of steel is
  γ = 77.0 kN/m³
  (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 centimeters) 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.
  
  40.9
  cm
  
  (ii)
  
  Find the minimum diameter d (in centimeters) for a circular cross section (see figure below).
  
  The cross section of a beam is a circle of diameter d.
  
  31.2
  cm
  
  (iii)
  
  Find the minimum dimension b (in centimeters) 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.
  
  31.7
  cm

Viewing Saved Work Revert to Last Response

2. 3/3 points | Previous Answers GGMechMat9SI 5.4.001. My Notes

A steel wire with a diameter of d = 2.3 mm is bent around a cylindrical drum with a radius of R = 1.2 m. (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 =

0.000957

(b)

What is the minimum acceptable radius of the drum (in m) if the maximum normal strain must remain below yield? Assume E = 245 GPa and σ_Y = 690 MPa.

0.407

(c)

If R = 1.2 m, what is the maximum acceptable diameter of the wire (in mm) if the maximum normal strain must remain below yield?

6.78

Viewing Saved Work Revert to Last Response

3. 2/2 points | Previous Answers GGMechMat9SI 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 = 4.7 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 = 160 mm

and height

h = 220 mm.

7.28

MPa

(b)

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

5.48

MPa

Viewing Saved Work Revert to Last Response

4. 1/1 points | Previous Answers GGMechMat9SI 5.6.001. My Notes

A simply supported wood beam having a span length L = 3.6 m is subjected to unsymmetrical point loads, as shown in the figure.

A simply supported wood beam carries two point loads separated by 1.2 m. Point load P₁ = 11 kN acts 1.2 m from the left end of the beam and point load P₂ = 14 kN acts 1.2 m 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 12 MPa and the wood weighs 560 kg/m³.

150 mm ✕ 150 mm 50 mm ✕ 200 mm 38 mm ✕ 125 mm correct

100 mm ✕ 300 mm

Viewing Saved Work Revert to Last Response

5. 0/1 points | Previous Answers GGMechMat9SI 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 =

χhBL

$webMathematica generated answer key$

Viewing Saved Work Revert to Last Response

6. 3/3 points | Previous Answers GGMechMat9SI 5.8.003. My Notes

A simply supported wood beam is subjected to uniformly distributed load q. The width of the beam is 150 mm and the height is 200 mm.

A 4 m 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 1 m to the right of its left end and 75 mm below the top of the beam. A uniformly distributed load q = 3.9 kN/m 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 200 mm along the y-direction and width 150 mm along the z-direction.

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

normal stress

-1.46

MPa shear stress

-0.183

MPa

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.

Viewing Saved Work Revert to Last Response

7. 1/1 points | Previous Answers GGMechMat9SI 5.9.003. My Notes

A vertical pole consisting of a circular tube of outer diameter 121 mm and inner diameter 107 mm is loaded by a linearly varying distributed force with maximum intensity of

q₀.

A tall, vertically-oriented cylinder has a height of 3 m, 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 = 6.2 kN/m 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 MPa. Indicate the direction with the sign of your answer.)

7.4

MPa

Viewing Saved Work Revert to Last Response

8. 2/3 points | Previous Answers GGMechMat9SI 5.10.008. My Notes

A bridge girder AB on a simple span of length

L = 32 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 = 32 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

35.3

kN/m

(b)

solely based on an allowable shear stress

τ_allow = 50 MPa

108

kN/m

(c)

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

35.3

kN/m

Viewing Saved Work Revert to Last Response

9. 1/1 points | Previous Answers GGMechMat9SI 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 170 mm and width 15 mm.

Each flange has height 24 mm and width 130 mm.

The allowable load in shear for the glued joints between the web and the flanges is 11 kN/m in the longitudinal direction. Determine the maximum allowable shear force V_max for the beam. (Enter the magnitude in N.)

2370

Viewing Saved Work Revert to Last Response

10. 2/2 points | Previous Answers GGMechMat9SI 5.12.004. My Notes

An aluminum pole for a street light weighs 4,400 N and supports an arm that weighs 590 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 = 4,400 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 = 590 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 = 250 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 250 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 =

4840

kPa maximum compressive stress σ_c =

-5690

kPa

Viewing Saved Work Revert to Last Response

11. 2/2 points | Previous Answers GGMechMat9SI 5.13.005. My Notes

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

h = 132 mm,

h₁ = 120 mm,

and thickness

b = 129 mm

(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 is 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 is 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 = 45 kN · m,

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

σ_max = 290 MPa.

(a)

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

13.2

(b)

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

Assume

that the ratio

96.7

Viewing Saved Work Revert to Last Response

Home My Assignments

WebAssign

Welcome, demo@demo

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

Current Score : 35 / 37

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

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

Instructions

Assignment Submission

Assignment Scoring

Design of Beams for Bending Stresses