# EAS 152/3 Strength Of Materials [Kekuatan Bahan]

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2 Figure 1 shows a vertical bar ABC which is pin supported at its upper end A and loaded by a force of 3.5kN at its lower end C. A horizontal rigid bar BDE is pinned to the vertical bar at joint B and supported by a pin joint at D. Pin at B is designed to be in single shear with diameter of 12.5mm. Figure 2 shows the detail of bolt connection at joint D. The beam is loaded by a force 6.5kN at end E. The vertical bar ABC is made of steel portion AB and aluminium portion BC with modulus of elasticity given as follows: for steel E S = 210GPa and for aluminium E A = 72GPa. Sizes of crosssection for portion AB and BC of vertical bar are as follows: portion AB 50mm (width) and 10mm (thickness); portion BC : 40mm (width) and 10mm (thickness). For beam BDE, the size of cross-section is as follows : 50mm (width) and 10mm (thickness). Calculate: i) Normal stress in portion AB and BC of vertical bar ABC [6 Marks] ii) Shear stress in bolt at joint D [3 Marks] iii) Bearing stress between gusset plate and bolt at joint D [3 Marks] iv) Shear stress in pin at joint B [2 Marks] v) Vertical displacement of point E [6 Marks] A 50mm 650mm 500mm 450mm B 50mm D E 500mm 40mm C 6.5kN 3.5kN Figure 1 3/-

3 mm 7.5 mm 7.5 mm gusset plate rigid bar BDE gusset plate 20mm diameter bolt Figure 2 2. a) Figure 2(a) shows the Mohr s circle for the stressed element shown in Figure 2(b). Using the Mohr s circle given, determine the normal and shear stress on the inclined plane m-m indicated in Figure 2(b). Provide the solution to your answer on Attachment I. [4 Marks] τ (MPa) (0,50) (-50,0) σ (MPa) (0,0) (50,0) 50MPa 45 50MPa (0,-50) Figure 2(a) Figure 2(b) 4/-

4 - 4 - b) Figure 3(c) shows the Mohr s circle for the state of stress of the element shown in Figure 3(d). Show the stresses represented by points P(9,24) and S(9,-24) in a properly oriented element. Provide the solution to your answer on Attachment II. [4 Marks] P (9,24) (20,30) (30,0) (9,-24) S 40MPa 30MPa 20MPa Figure 3(c) Figure 3(d) c) Draw the Mohr s circle for the state of stress shown in Figure 4. Using the Mohr s circle obtained, determine: i) The principal stresses and the principal planes. [6 Marks] ii) The maximum and minimum shear stresses, the associated normal stresses and the planes. [6 Marks] 5MPa 12MPa 15MPa Figure 4 5/-

5 - 5 - Attachment I Subject : EAS152 Angka Giliran : For Question 2(a) (This page must be submitted together with your answer scripts) τ (MPa) (0,50) (-50,0) σ (MPa) (0,0) (50,0) 50MPa 45 50MPa (0,-50) 6/-

6 - 6 - Attachment II Subject : EAS152 Angka Giliran : For Question 2(b) (This page must be submitted together with your answer scripts) P (9,24) (20,30) (30,0) (9,-24) S 40MPa 30MPa 20MPa 7/-

7 A moment load of 2400 N.m is applied at the free end of a hollow 1.5 m long cantilever shaft having inner diameter of 40 mm and outer diameter of 60 mm. i) Determine the maximum shearing stress developed in the shaft. [10 Marks] ii) Find the diameter of a solid shaft under the maximum shearing stress condition in (i). [5 Marks] iii) If the shaft is solid having a diameter of 65 mm, calculate the angle of twist under the same loading condition (Use G = 77 GPa). [5 Marks] 4. A beam as shown in Figure 5 is carrying a 20 kn concentrated load at B and three uniformly distributed loads of 15 kn/m, 5 kn/m and 10 kn/m along segment AB, BC and CD, respectively. The beam is pinned at A and supported by a roller at C. a) Derive the shear force and moment function in terms of x for segment AB, BC and CD of the beam. State clearly the origin point and the range of x taken, in order to derive the functions for each segment of the beam. [12 Marks] b) Draw the shear force and bending moment diagram for the beam. [4 Marks] c) If it is given that the allowable bending stress and shear stress of the material used for the beam are σ b,all = 8.0MPa and τ all = 0.8MPa, respectively, propose the height of rectangular beam which is safe to be used with a factor of safety 1.5 so that it can resist the highest bending moment of the beam. Take the width of the beam as 150 mm. [4 Marks] 8/-

8 kn/m 20 kn 5 kn/m 10 kn/m A B C D 3.0 m 1.0 m 2.0 m Figure 5 5. a) Define actual length, effective length and Euler buckling load. [4 Marks] b) Columns can be categorized by their slenderness ratio. Define slenderness ratio and state THREE (3) types of column according to slenderness ratio as well as the range of each category. [4 Marks] c) A rectangular hollow section as shown in Figure 6 is to be used as a column with one end fixed and another end pin jointed. The actual length of the column is 5.5 m. Take modulus of elasticity as 210 GPa. (i) (ii) (iii) (iv) Calculate the radius of gyration for the section. Calculate the slenderness ratio of the column. Determine the Euler buckling load. Determine the maximum load that can be supported by the column with a factor of safety of 2.0. [10 Marks] d) Suggest one way to increase the column load carrying capacity without changing the cross section of the column. No calculation is needed. [2 Marks] 9/-

9 mm 300 mm 130 mm 150 mm Figure 6 6. a) Figure 7 shows a beam carrying a uniformly distributed load of 10 kn/m along segment AB and a concentrated load of 20 kn at B. Compute the deflection of the beam at B. Use the double integration method. Given the modulus of elasticity and second moment of inertia of the beam are 200GPa and x 10 6 mm 4, respectively [10 Marks] 10 kn/m 20 kn A B 4m Figure 7 10/-

10 b) Figure 8 shows a structure consisting of two members, ACD and BC, which is used to support a horizontal load P. Link BC is 4mm thick and 12.5mm wide. Link BC is connected to support B and joint C using pin with diameter 5mm. Given that the allowable bearing stress between the link and pin at B, C is 200N/mm 2 and the allowable maximum tensile stress in link BC is 175N/mm 2, determine the maximum load P max that the structure is able to carry. [10 Marks] 4mm 600mm 12.5mm D P 200mm 5mm pin B C C link BC 800mm member ACD A A Figure 8

12 mm 7.5 mm 7.5 mm gusset plate rigid bar BDE gusset plate 20mm diameter bolt Rajah 2 2. a) Rajah 2(a) menunjukkan bulatan Mohr untuk elemen tertegas dalam Rajah 2(b). Dengan menggunakan bulatan Mohr yang diberi, tentukan tegasan normal dan ricih atas satah condong m-m yang ditunjukkan dalam Rajah 2(b). Berikan penyelesaian anda di atas Lampiran I. [4 Markah] τ (MPa) (0,50) (-50,0) σ (MPa) (0,0) (50,0) 50MPa 45 50MPa (0,-50) Rajah 2(a) Rajah 2(b) 13/-

13 b) Rajah 3(c) menunjukkan bulatan Mohr untuk keadaan tegasan untuk elemen dalam Rajah 3(d). Tunjukkan tegasan yang diwakili oleh titik P(9,24) dan S(9,-24) dalam satu elemen dengan orientasi yang bersesuaian. Berikan penyelesaian anda di atas Lampiran II. [4 Markah] P (9,24) (20,30) (9,-24) S (30,0) 40MPa 30MPa 20MPa Rajah 3(c) Rajah 3(d) c) Lukiskan bulatan Mohr untuk keadaan tegasan yang ditunjukkan dalam Rajah 4. Dengan menggunakan bulatan Mohr yang diperolehi, tentukan: i) Tegasan utama dan satah utama [6 markah] ii) Tegasan ricih maksima dan minima, tegasan normal yang berkaitan dan satah yang berkaitan. [6 markah] 5MPa 12MPa 15MPa Rajah 4 14/-

14 Lampiran I Kursus : EAS152 Angka Giliran : Untuk Soalan (a) (Helaian ini hendaklah disertakan bersama dengan jawapan) τ (MPa) (0,50) (-50,0) σ (MPa) (0,0) (50,0) 50MPa 45 50MPa (0,-50) 15/-

15 Lampiran II Kursus : EAS152 Angka Giliran : Untuk Soalan 2(b) (Helaian ini hendaklah disertakan bersama dengan jawapan) P (9,24) (20,30) (30,0) (9,-24) S 40MPa 30MPa 20MPa 16/-

16 Satu beban momen 2400Nm dikenakan di hujung bebas satu aci julur geronggang sepanjang 1.5m yang mempunyai garis pusat dalam 40 mm dan garis luar 60 mm. i) Tentukan tegasan ricih maksimum yang terhasil dalam aci tersebut. [10 Markah] ii) Cari garis pusat untuk satu aci di bawah nilai tegasan ricih maksimum seperti dalam (a). [5Markah] iii) Sekiranya aci ialah padu dan mempunyai garis pusat 65 mm, kira sudut piuh di bawah keadaan beban yang sama. (Guna G = 77 Gpa). [5Markah] 4. Satu rasuk ditunjukkan dalam Rajah 5 menanggung beban titik 20 kn dan tiga beban teragih seragam 15 kn/m, 5 kn/m and 10 kn/m masing-masing di sepanjang rentang AB, BC dan CD. Rasuk tersebut disokong pin di A dan disokong rola di C. a) Terbitkan fungsi daya ricih dan momen dalam sebutan x bagi bahagian AB, BC dan CD rasuk tersebut. Nyatakan dengan jelas titik asalan dan had nilai x yang digunakan untuk menerbitkan fungsi-fungsi tersebut bagi setiap bahagian rasuk. [12 Markah] b) Lukiskan gambarajah daya ricih dan momen lentur bagi rasuk tersebut. [4 Markah] c) Jika tegasan lentur dan tegasan ricih yang dibenarkan bagi bahan yang digunakan untuk rasuk tersebut masing-masing ialah σ b,all = 8.0MPa and τ all = 0.8MPa, cadangkan ketinggian rasuk segiempat tepat yang selamat digunakan dengan faktor keselamatan 1.5 supaya ia dapat menanggung nilai momen tertinggi rasuk tersebut. Lebar rasuk ialah 150 mm [4 Markah] 17/-

17 kn/m 20 kn 5 kn/m 10 kn/m A B C D 3.0 m 1.0 m 2.0 m Rajah 5 5. a) Berikan takrif bagi panjang sebenar, panjang efektif dan beban lengkok Euler. [4 Markah] b) Tiang boleh dikategorikan berdasarkan nisbah kelangsingan. Berikan takrif nisbah kelangsingan dan nyatakan TIGA (3) jenis tiang berdasarkan nilai nisbah kelangsingan tersebut serta nyatakan julatnya untuk setiap kategori. [4 Markah] c) Satu keratan segiempat bergeronggang tepat seperti ditunjukkan dalam Rajah 6 akan digunakan sebagai tiang dengan satu hujungnya diikat tegar dan hujung yang lain disokong pin. Tinggi sebenar tiang tersebut ialah 5.5m. Ambil modulus keanjalan sebagai 210 GPa. (i) (ii) (iii) (iv) Kira jejari legaran bagi keratan Kira nisbah kelangsingan tiang Tentukan beban lengkok Euler Tentukan beban maksimum yang boleh ditanggung tiang dengan faktor keselamatan 2.0 [10 Markah] d) Berikan cadangan satu cara untuk meningkatkan keupayaan menanggung beban tiang tanpa mengubah keratan rentas tiang tersebut. Pengiraan tidak diperlukan. [2 Markah] 18/-

18 mm 300 mm 130 mm 150 mm Rajah 6 6. a) Rajah 7 menunjukkan satu rasuk yang ditindaki beban teragih seragam iaitu 10 kn/m disepanjang bahagian AB dan beban tumpu 20 kn di B. Kirakan lenturan di B rasuk tersebut. Guna kaedah kamiran berganda. Diberi modulus keanjalan dan momen inersia masing-masing adalah 200 GPa and x 10 6 mm 4. [10 Markah] 10 kn/m 20 kn A B 4m Rajah 7 19/-

19 b) Rajah 8 menunjukkan satu struktur terdiri daripada dua anggota, ACD dan BC, yang digunakan untuk menyokong satu beban ufuk P. Penghubung BC adalah 4mm tebal dan 12.5mm lebar. Penghubung BC disambung kepada penyokong B dan sambungan C dengan pin yang bergarispusat 5mm. Diberikan bahawa tegasan galas dibenarkan antara penghuhung BC dan pin pada B, C adalah 200N/mm 2 manakala tegasan tegangan maksima dibenarkan dalam penghubung BC adalah 175N/mm 2, tentukan beban maksima P max yang boleh ditanggung oleh struktur berkenaan. [10 Markah] 4mm 600mm 12.5mm D P 200mm 5mm pin B C C link BC 800mm member ACD A A Rajah8 ooooooooo...11/-

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