1) Several loads are applied to a cantilevered beam as shown. A WT305 X 70 standard steel shape is used for the beam and the dimensions are shown below. a) Plot the shear force and bending moment diagram of the beam using the graphical method. Be sure to track calculations and clearly specify the maximum positive and negative values for shear and moment. [Ans. to Check: Vmax-positive = 40 kN; Mmax-negative = -70 kN-m] b) Locate the neutral axis of the cross-section relative to the bottom of the shape in mm. [Ans. to Check: y = 233 mm from bottom] c) Compute the moment of inertia in mm* about the z-axis. [Ans. to Check: I₂ = 78,338,100 mm²] d) Calculate the bending stress in the top and bottom of the beam at the point of maximum positive moment (in MPa). Be sure to specify which fiber is in tension and which fiber is in compression. Plot the normal stress distribution for this location using the rules we learned in class. [Ans. to Check: Otop=49.15 MPa (C)] e) Calculate the bending stress in the top and bottom of the beam at the point of maximum negative moment (in MPa). Be sure to specify which fiber is in tension and which fiber is in compression. Plot the normal stress distribution for this location using the rules we learned in class. [Ans. to Check: Otop= 68.8 MPa (T)] f) Use your answers from parts (d) and (e) to determine the location, magnitude, and sign of the maximum normal tensile and compressive stresses due to bending (in MPa). Note: Specify where these stresses happen along the beam (e.g. x = 2m) and where in the section (e.g. top or bottom). 50 kN-m 20 kN 2 m 60 kN 287.8 mm B 3 m Beam loading along its length 230 mm 40 kN/m -13.1 mm 22.2 mm m Cross-section of the beam (view looking down the beam from A to D)
1) Several loads are applied to a cantilevered beam as shown. A WT305 X 70 standard steel shape is used for the beam and the dimensions are shown below. a) Plot the shear force and bending moment diagram of the beam using the graphical method. Be sure to track calculations and clearly specify the maximum positive and negative values for shear and moment. [Ans. to Check: Vmax-positive = 40 kN; Mmax-negative = -70 kN-m] b) Locate the neutral axis of the cross-section relative to the bottom of the shape in mm. [Ans. to Check: y = 233 mm from bottom] c) Compute the moment of inertia in mm* about the z-axis. [Ans. to Check: I₂ = 78,338,100 mm²] d) Calculate the bending stress in the top and bottom of the beam at the point of maximum positive moment (in MPa). Be sure to specify which fiber is in tension and which fiber is in compression. Plot the normal stress distribution for this location using the rules we learned in class. [Ans. to Check: Otop=49.15 MPa (C)] e) Calculate the bending stress in the top and bottom of the beam at the point of maximum negative moment (in MPa). Be sure to specify which fiber is in tension and which fiber is in compression. Plot the normal stress distribution for this location using the rules we learned in class. [Ans. to Check: Otop= 68.8 MPa (T)] f) Use your answers from parts (d) and (e) to determine the location, magnitude, and sign of the maximum normal tensile and compressive stresses due to bending (in MPa). Note: Specify where these stresses happen along the beam (e.g. x = 2m) and where in the section (e.g. top or bottom). 50 kN-m 20 kN 2 m 60 kN 287.8 mm B 3 m Beam loading along its length 230 mm 40 kN/m -13.1 mm 22.2 mm m Cross-section of the beam (view looking down the beam from A to D)
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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I have questions a-d answered and I need help on the last 2 parts
![### Cantilevered Beam Analysis
1) **Problem Statement:**
Several loads are applied to a cantilevered beam as shown. A WT305 X 70 standard steel shape is used for the beam, and the dimensions are detailed below.
2) **Tasks:**
a) **Shear Force and Bending Moment Diagrams:**
- Plot these using the graphical method.
- Identify the maximum positive and negative values for shear and moment.
- **Check Values:** \( V_{\text{max-positive}} = 40 \, \text{kN}; \, M_{\text{max-negative}} = -70 \, \text{kN-m} \)
b) **Neutral Axis:**
- Locate the neutral axis of the cross-section relative to the bottom of the shape in mm.
- **Check Value:** \(\bar{y} = 233 \, \text{mm from bottom}\)
c) **Moment of Inertia:**
- Compute in \(\text{mm}^4\) about the z-axis.
- **Check Value:** \( I_z = 78,338,100 \, \text{mm}^4 \)
d) **Bending Stress (Maximum Positive Moment):**
- Calculate at the top and bottom of the beam with maximum positive moment (in MPa).
- Specify which fiber is in tension and which in compression.
- Plot the normal stress distribution.
- **Check Value:** \(\sigma_{\text{top}} = 49.15 \, \text{MPa} \, [C]\)
e) **Bending Stress (Maximum Negative Moment):**
- Calculate at the top and bottom of the beam with maximum negative moment (in MPa).
- Specify which fiber is in tension and which in compression.
- Plot the normal stress distribution.
- **Check Value:** \(\sigma_{\text{top}} = 68.8 \, \text{MPa} \, [T]\)
f) **Maximum Normal Stresses:**
- Use results from (d) and (e) to determine the location, magnitude, and sign of maximum normal tensile and compressive stresses (in MPa).
- Identify where these stresses occur along the beam (e.g., \( x = 2\, \text{m} \)) and in the section (e](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fccae5729-7489-41ed-a32a-24cdd49b7598%2F152bd469-1c2a-4994-a6b7-f10df14fe025%2F0w7xln3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Cantilevered Beam Analysis
1) **Problem Statement:**
Several loads are applied to a cantilevered beam as shown. A WT305 X 70 standard steel shape is used for the beam, and the dimensions are detailed below.
2) **Tasks:**
a) **Shear Force and Bending Moment Diagrams:**
- Plot these using the graphical method.
- Identify the maximum positive and negative values for shear and moment.
- **Check Values:** \( V_{\text{max-positive}} = 40 \, \text{kN}; \, M_{\text{max-negative}} = -70 \, \text{kN-m} \)
b) **Neutral Axis:**
- Locate the neutral axis of the cross-section relative to the bottom of the shape in mm.
- **Check Value:** \(\bar{y} = 233 \, \text{mm from bottom}\)
c) **Moment of Inertia:**
- Compute in \(\text{mm}^4\) about the z-axis.
- **Check Value:** \( I_z = 78,338,100 \, \text{mm}^4 \)
d) **Bending Stress (Maximum Positive Moment):**
- Calculate at the top and bottom of the beam with maximum positive moment (in MPa).
- Specify which fiber is in tension and which in compression.
- Plot the normal stress distribution.
- **Check Value:** \(\sigma_{\text{top}} = 49.15 \, \text{MPa} \, [C]\)
e) **Bending Stress (Maximum Negative Moment):**
- Calculate at the top and bottom of the beam with maximum negative moment (in MPa).
- Specify which fiber is in tension and which in compression.
- Plot the normal stress distribution.
- **Check Value:** \(\sigma_{\text{top}} = 68.8 \, \text{MPa} \, [T]\)
f) **Maximum Normal Stresses:**
- Use results from (d) and (e) to determine the location, magnitude, and sign of maximum normal tensile and compressive stresses (in MPa).
- Identify where these stresses occur along the beam (e.g., \( x = 2\, \text{m} \)) and in the section (e
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