![Page 1: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/1.jpg)
Stresses in Beams
Basic Topics
Eng. Iqbal Marie
Mechanics of Materials, James M Gere, 6th edition, ISBN 0-534-41793-0
![Page 2: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/2.jpg)
5.2 Pure Bending and Nonuniform Bending
Pure bending = No shear,
Simple Beam in pure bending (M = M1).
Cantilever Beam inpure bending (M = M2).
![Page 3: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/3.jpg)
Simple beam with centralregion in pure bending
![Page 4: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/4.jpg)
5.3 Curvature of a Beam
![Page 5: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/5.jpg)
Sign Convention for Curvature
![Page 6: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/6.jpg)
5.4 Longitudinal Strains in beams
•xy plane is a plane of symmetry
•Loading is applied in xy plane
•Beam deflects in xy plane
•Thickness of the beam, h, remains unchanged
•Axis of the beam coincides with the centroidal line of the cross
section (to be proved later). Loading q
![Page 7: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/7.jpg)
1. Cross sections (mn and pq) remain plane
2. Cross sections remain perpendicular to the axis of the beam
3. For positive moments (hence positive curvature), lines on the lower part of thebeam (nq) are elongated; those on the upper part (mp) are shortened
4. Somewhere between top andbottom there is a line whose lengthdoes not change, and is called“Neutral Axis”
![Page 8: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/8.jpg)
Normal Strain Due to Bending
![Page 9: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/9.jpg)
![Page 10: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/10.jpg)
What is the normal strain?
![Page 11: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/11.jpg)
5.5 Normal Stress Due to Bending
Centroid
![Page 12: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/12.jpg)
Flexural Formula
The resultant moment of the normal stress over the cross section must equal to the applied moment M
![Page 13: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/13.jpg)
![Page 14: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/14.jpg)
Maximum Stresses at a Cross Section
![Page 15: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/15.jpg)
![Page 16: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/16.jpg)
![Page 17: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/17.jpg)
![Page 18: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/18.jpg)
![Page 19: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/19.jpg)
![Page 20: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/20.jpg)
![Page 21: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/21.jpg)
Example : 5-3 The beam is constructed of a glued laminated wood . Determine the maximum compressive and tensile stresses in the beam due to bending?
23.59 k 21.41 k
1. Find reactions
2. Draw shear and moment diagrams
3. Find maximum moment
4. Find the centroid location and moment of inertia
5. Calculate the stresses
NA
![Page 22: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/22.jpg)
Prob. 5.16 Max. Tensile and compressive stresses in the beam
3m75mm
5.4 kN1.2m
100mm75mm
25mm
2.16 kN 3.24 kN
3888 N.m
![Page 23: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/23.jpg)
Example 5-4: Determine the maximum tensile and compressive stress in the beam
Cross section
3.6 k10.8k
Max. comp
Max. tension
![Page 24: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/24.jpg)
![Page 25: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/25.jpg)
![Page 26: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/26.jpg)
![Page 27: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/27.jpg)
5.8 Bending Moments and Shear Forces in Beams
![Page 28: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/28.jpg)
![Page 29: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/29.jpg)
![Page 30: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/30.jpg)
![Page 31: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/31.jpg)
Shear stresses in beams
![Page 32: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/32.jpg)
Shear Stresses in Rectangular Beams
b
Parabolic Distribution
![Page 33: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/33.jpg)
Example 5-11 Determine the normal and shear stresses at Point C
Cross section
![Page 34: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/34.jpg)
Example 5-12, what is the Maximum Permissible Load
![Page 35: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/35.jpg)
Allowable P= ?
![Page 36: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/36.jpg)
5.9 Shear Stresses in beams of Circular cross section
The exact distribution of shear stress in a beam of circular cross section is very complicated and only that along the neutral axis can be determined relatively easily.
![Page 37: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/37.jpg)
5.10 Shear Stresses in the Webs of Beams with Flanges
![Page 38: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/38.jpg)
Shear Force in the Web: = (The are of he shear stress diagram x the thickness of the web)
V = [min h1 + 2/3 h1 (max - min )]t
= t h1/3 ( 2 max+ min )
Shear force in the web is 90% -98% of the total shear force V acting on the cross section
Assuming the web carries
all of the shear force…
avg = V / th1
![Page 39: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/39.jpg)
![Page 40: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/40.jpg)
5.11 Built-Up Beams and Shear Flow
![Page 41: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/41.jpg)
![Page 42: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/42.jpg)
Shear Force in Fasteners :
In many applications, beam sections consist of several pieces of material that are attached together in a number ways: bolts, rivets, nails, glue, weld, etc. In such so called built-up sections we are interested in knowing the amount of shear stress and the resulting shear force at the cross section of fasteners or over the glued surface .
![Page 43: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/43.jpg)
![Page 44: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/44.jpg)
![Page 45: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/45.jpg)
![Page 46: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/46.jpg)
![Page 47: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/47.jpg)
![Page 48: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/48.jpg)
Ex. 5-16 The plywood is fastened to the flanges by wood screws having an allowable load in shear of F = 800 N each if the shear force V acting on the cross section = 10.5 kN. Determine the max. permissible longitudinal spacing s of the screws.
Use 45 mm
![Page 49: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/49.jpg)
Find the spacing for each case
![Page 50: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/50.jpg)
I
![Page 51: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/51.jpg)
![Page 52: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/52.jpg)
![Page 53: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/53.jpg)
5.12 : Beams With Axial Loads
These equations assume linearity of structures
Means supperposition is applied
Therefore if different load types are simultaneously applied
Resultant stress distribution found by superposition
![Page 54: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/54.jpg)
![Page 55: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/55.jpg)
![Page 56: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/56.jpg)
![Page 57: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/57.jpg)
![Page 58: Stresses in Beams Basic Topics - JUfiles – ملفات الجامعات الاردنيةjufiles.com/wp-content/uploads/2016/12/stresses-in-beams.pdf · 2016-12-29 · Stresses in](https://reader030.vdocuments.site/reader030/viewer/2022020316/5b3a68a07f8b9a560a8b8ecd/html5/thumbnails/58.jpg)